Optimal. Leaf size=64 \[ \frac {1}{3} x^3 \sinh ^{-1}\left (a x^n\right )-\frac {a n x^{n+3} \, _2F_1\left (\frac {1}{2},\frac {n+3}{2 n};\frac {3 (n+1)}{2 n};-a^2 x^{2 n}\right )}{3 (n+3)} \]
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Rubi [A] time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 12
Rule 364
Rule 5902
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*}
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Mathematica [A] time = 0.05, size = 66, normalized size = 1.03 \[ \frac {1}{3} x^3 \sinh ^{-1}\left (a x^n\right )-\frac {a n x^{n+3} \, _2F_1\left (\frac {1}{2},\frac {n+3}{2 n};\frac {n+3}{2 n}+1;-a^2 x^{2 n}\right )}{3 (n+3)} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {arsinh}\left (a x^{n}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int x^{2} \arcsinh \left (a \,x^{n}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{9} \, n x^{3} + \frac {1}{3} \, x^{3} \log \left (a x^{n} + \sqrt {a^{2} x^{2 \, n} + 1}\right ) - a n \int \frac {x^{2} x^{n}}{3 \, {\left (a^{3} x^{3 \, n} + a x^{n} + {\left (a^{2} x^{2 \, n} + 1\right )}^{\frac {3}{2}}\right )}}\,{d x} + n \int \frac {x^{2}}{3 \, {\left (a^{2} x^{2 \, n} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^2\,\mathrm {asinh}\left (a\,x^n\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {asinh}{\left (a x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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