Optimal. Leaf size=68 \[ \frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (c x^n\right )\right )-\frac{b c n x^{n+3} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+3}{2 n},\frac{3 (n+1)}{2 n},c^2 x^{2 n}\right )}{3 (n+3)} \]
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Rubi [A] time = 0.0409463, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4842, 12, 364} \[ \frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (c x^n\right )\right )-\frac{b c n x^{n+3} \, _2F_1\left (\frac{1}{2},\frac{n+3}{2 n};\frac{3 (n+1)}{2 n};c^2 x^{2 n}\right )}{3 (n+3)} \]
Antiderivative was successfully verified.
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Rule 4842
Rule 12
Rule 364
Rubi steps
\begin{align*} \int x^2 \left (a+b \sin ^{-1}\left (c x^n\right )\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (c x^n\right )\right )-\frac{1}{3} b \int \frac{c n x^{2+n}}{\sqrt{1-c^2 x^{2 n}}} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (c x^n\right )\right )-\frac{1}{3} (b c n) \int \frac{x^{2+n}}{\sqrt{1-c^2 x^{2 n}}} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \sin ^{-1}\left (c x^n\right )\right )-\frac{b c n x^{3+n} \, _2F_1\left (\frac{1}{2},\frac{3+n}{2 n};\frac{3 (1+n)}{2 n};c^2 x^{2 n}\right )}{3 (3+n)}\\ \end{align*}
Mathematica [A] time = 0.0576328, size = 75, normalized size = 1.1 \[ -\frac{b c n x^{n+3} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+3}{2 n},\frac{n+3}{2 n}+1,c^2 x^{2 n}\right )}{3 (n+3)}+\frac{a x^3}{3}+\frac{1}{3} b x^3 \sin ^{-1}\left (c x^n\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\arcsin \left ( c{x}^{n} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a x^{3} + \frac{1}{3} \,{\left (x^{3} \arctan \left (c x^{n}, \sqrt{c x^{n} + 1} \sqrt{-c x^{n} + 1}\right ) + 3 \, c n \int \frac{\sqrt{c x^{n} + 1} \sqrt{-c x^{n} + 1} x^{2} x^{n}}{3 \,{\left (c^{2} x^{2 \, n} - 1\right )}}\,{d x}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 19.5958, size = 66, normalized size = 0.97 \begin{align*} \frac{a x^{3}}{3} + \frac{b x^{3} \operatorname{asin}{\left (c x^{n} \right )}}{3} + \frac{i b x^{3} \Gamma \left (\frac{3}{2 n}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - \frac{3}{2 n} \\ 1 - \frac{3}{2 n} \end{matrix}\middle |{\frac{x^{- 2 n}}{c^{2}}} \right )}}{6 \Gamma \left (1 + \frac{3}{2 n}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (c x^{n}\right ) + a\right )} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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