Optimal. Leaf size=37 \[ \frac{x^5}{25}-\frac{2 x^3}{15}-\frac{1}{5} \left (1-x^2\right )^{5/2} \sin ^{-1}(x)+\frac{x}{5} \]
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Rubi [A] time = 0.0373196, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4677, 194} \[ \frac{x^5}{25}-\frac{2 x^3}{15}-\frac{1}{5} \left (1-x^2\right )^{5/2} \sin ^{-1}(x)+\frac{x}{5} \]
Antiderivative was successfully verified.
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Rule 4677
Rule 194
Rubi steps
\begin{align*} \int x \left (1-x^2\right )^{3/2} \sin ^{-1}(x) \, dx &=-\frac{1}{5} \left (1-x^2\right )^{5/2} \sin ^{-1}(x)+\frac{1}{5} \int \left (1-x^2\right )^2 \, dx\\ &=-\frac{1}{5} \left (1-x^2\right )^{5/2} \sin ^{-1}(x)+\frac{1}{5} \int \left (1-2 x^2+x^4\right ) \, dx\\ &=\frac{x}{5}-\frac{2 x^3}{15}+\frac{x^5}{25}-\frac{1}{5} \left (1-x^2\right )^{5/2} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0181415, size = 35, normalized size = 0.95 \[ \frac{1}{5} \left (\frac{x^5}{5}-\frac{2 x^3}{3}-\left (1-x^2\right )^{5/2} \sin ^{-1}(x)+x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 37, normalized size = 1. \begin{align*} -{\frac{\arcsin \left ( x \right ) \left ({x}^{2}-1 \right ) ^{2}}{5}\sqrt{-{x}^{2}+1}}+{\frac{ \left ( 3\,{x}^{4}-10\,{x}^{2}+15 \right ) x}{75}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42637, size = 36, normalized size = 0.97 \begin{align*} \frac{1}{25} \, x^{5} - \frac{1}{5} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} \arcsin \left (x\right ) - \frac{2}{15} \, x^{3} + \frac{1}{5} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.58846, size = 105, normalized size = 2.84 \begin{align*} \frac{1}{25} \, x^{5} - \frac{2}{15} \, x^{3} - \frac{1}{5} \,{\left (x^{4} - 2 \, x^{2} + 1\right )} \sqrt{-x^{2} + 1} \arcsin \left (x\right ) + \frac{1}{5} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.54366, size = 63, normalized size = 1.7 \begin{align*} \frac{x^{5}}{25} - \frac{x^{4} \sqrt{1 - x^{2}} \operatorname{asin}{\left (x \right )}}{5} - \frac{2 x^{3}}{15} + \frac{2 x^{2} \sqrt{1 - x^{2}} \operatorname{asin}{\left (x \right )}}{5} + \frac{x}{5} - \frac{\sqrt{1 - x^{2}} \operatorname{asin}{\left (x \right )}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08351, size = 46, normalized size = 1.24 \begin{align*} \frac{1}{25} \, x^{5} - \frac{1}{5} \,{\left (x^{2} - 1\right )}^{2} \sqrt{-x^{2} + 1} \arcsin \left (x\right ) - \frac{2}{15} \, x^{3} + \frac{1}{5} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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