Optimal. Leaf size=64 \[ \frac{1}{3} x^3 \sinh ^{-1}\left (a x^n\right )-\frac{a n x^{n+3} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+3}{2 n},\frac{3 (n+1)}{2 n},-a^2 x^{2 n}\right )}{3 (n+3)} \]
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Rubi [A] time = 0.0352749, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5902, 12, 364} \[ \frac{1}{3} x^3 \sinh ^{-1}\left (a x^n\right )-\frac{a n x^{n+3} \, _2F_1\left (\frac{1}{2},\frac{n+3}{2 n};\frac{3 (n+1)}{2 n};-a^2 x^{2 n}\right )}{3 (n+3)} \]
Antiderivative was successfully verified.
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Rule 5902
Rule 12
Rule 364
Rubi steps
\begin{align*} \int x^2 \sinh ^{-1}\left (a x^n\right ) \, dx &=\frac{1}{3} x^3 \sinh ^{-1}\left (a x^n\right )-\frac{1}{3} \int \frac{a n x^{2+n}}{\sqrt{1+a^2 x^{2 n}}} \, dx\\ &=\frac{1}{3} x^3 \sinh ^{-1}\left (a x^n\right )-\frac{1}{3} (a n) \int \frac{x^{2+n}}{\sqrt{1+a^2 x^{2 n}}} \, dx\\ &=\frac{1}{3} x^3 \sinh ^{-1}\left (a x^n\right )-\frac{a n x^{3+n} \, _2F_1\left (\frac{1}{2},\frac{3+n}{2 n};\frac{3 (1+n)}{2 n};-a^2 x^{2 n}\right )}{3 (3+n)}\\ \end{align*}
Mathematica [A] time = 0.0470922, size = 66, normalized size = 1.03 \[ \frac{1}{3} x^3 \sinh ^{-1}\left (a x^n\right )-\frac{a n x^{n+3} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+3}{2 n},\frac{n+3}{2 n}+1,-a^2 x^{2 n}\right )}{3 (n+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.014, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}{\it Arcsinh} \left ( a{x}^{n} \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}{9} \, n x^{3} + \frac{1}{3} \, x^{3} \log \left (a x^{n} + \sqrt{a^{2} x^{2 \, n} + 1}\right ) - a n \int \frac{x^{2} x^{n}}{3 \,{\left (a^{3} x^{3 \, n} + a x^{n} +{\left (a^{2} x^{2 \, n} + 1\right )}^{\frac{3}{2}}\right )}}\,{d x} + n \int \frac{x^{2}}{3 \,{\left (a^{2} x^{2 \, n} + 1\right )}}\,{d x} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{asinh}{\left (a x^{n} \right )}\, dx \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 x^{2} \operatorname{arsinh}\left (a x^{n}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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