rāga: ānandabhairavi, tāḷa: ādi.
हंसगमनी सरस्वती वजिनीवती
शासिनी विद्यार्थपोषिणी
अंबुजासनवीणावादिनी
सकलवेदशास्त्रधारिणी वैष्णवी
वेङ्कटप्रणवनुतविश्वविराजिनी
haṁsagamanī sarasvatī vājinīvatī
śāsinī vidyārthapoṣiṇī
aṁbujāsanavīṇāvādinī
sakalavedaśāstradhāriṇī vaiṣṇavī
veṅkaṭapraṇavanutaviśvavirājinī
Sarasvatī, much like the grand river was for the Vedic r̥ṣi-s, is a confluence of so many important and weighty facets of what defines the term br̥hant, or lofty. She is vāgdevi, the goddess of Speech (vāk being the most significant conceptualization in the Vedic thought for it is through the power of Speech, that the mantra-s were inspired and the devabhāṣa saṁskr̥ta comes about). She is a healer, the grand patron and embodiment of the śāstra, arts, music, and scholarly thought.
Meanwhile, the rāga ānandabhairavi is the embodiment of what defines a rakti rāga, providing endless rañjana to the listener amplified by the beauty of its anyasvaraprayoga-s utilizing the antara gāndhāra and śuddha dhaivata (which was the more prominent dhaivata of this rāga in the past, but the catuśśruti dhaivata has now taken that role, despite it nominally still being classified as a janya rāga of naṭabhairavi).
Here is a rough translation of the kr̥ti, which is in the saṁbodhana or vocative declension:
O one who rides a swan, Sarasvatī, powerful one!
Teacher, nurterer of students!
O one who sits on a lotus and plays the vīṇā!
You, who bears all the veda-s and śāstra-s, O vaiṣṇavī,
are worshipped by Veṅkaṭapraṇava, O ruler of the universe!
Those who are Vedically inclined may immediately notice the references to the sarasvatī prārthana mantra-s of the r̥gvedin-s and yajurvedin-s hinted at in the epithets of the kr̥ti. For instance, vājinīvatī, which is derived from vājinīvat, meaning one who possesses swift horses, metaphorically means one who is strong and powerful. But this is a direct refernece to the immortal prārthana mantra chanted before any kārya or course of study with a guru:
प्र णो॑ देवि सर॑स्वती॒ वजे॑भिर्वा॒जिनी॑वती।
धी॒नाम॑वि॒त्र्य॑वतु॥
Similarly, the epithet viśvavirājati is a reference to this other famous mantra:
पा॒व॒का नः सर॑स्वती॒ वजे॑भिर्वा॒जिनी॑वती।
य॒ज्ञं व॑ष्टु धि॒याव॑सुः॥
चो॒द॒यि॒त्री सू॒नृता॑नां॒ चेत॑न्ती सुमती॒नाम्।
य॒ज्ञं द॑धे॒ सर॑स्वती॥
म॒हो अर्णः॒ सर॑स्वती॒ प्र चे॑तयति के॒तुना॑।
धि॒यो विश्वा॒ वि रा॑जती॥
As a lifelong learner and student of the veda, śāstra, saṅgīta, and mathematics, the message of the kr̥ti needs little further elaboration, for what is life without learning?