Optimal. Leaf size=88 \[ \frac{5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{2 a}+\frac{15 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4 a}+x \sin ^{-1}(a x)^{5/2}-\frac{15}{4} x \sqrt{\sin ^{-1}(a x)} \]
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Rubi [A] time = 0.164965, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4619, 4677, 4723, 3305, 3351} \[ \frac{5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{2 a}+\frac{15 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4 a}+x \sin ^{-1}(a x)^{5/2}-\frac{15}{4} x \sqrt{\sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4619
Rule 4677
Rule 4723
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \sin ^{-1}(a x)^{5/2} \, dx &=x \sin ^{-1}(a x)^{5/2}-\frac{1}{2} (5 a) \int \frac{x \sin ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{2 a}+x \sin ^{-1}(a x)^{5/2}-\frac{15}{4} \int \sqrt{\sin ^{-1}(a x)} \, dx\\ &=-\frac{15}{4} x \sqrt{\sin ^{-1}(a x)}+\frac{5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{2 a}+x \sin ^{-1}(a x)^{5/2}+\frac{1}{8} (15 a) \int \frac{x}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx\\ &=-\frac{15}{4} x \sqrt{\sin ^{-1}(a x)}+\frac{5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{2 a}+x \sin ^{-1}(a x)^{5/2}+\frac{15 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a}\\ &=-\frac{15}{4} x \sqrt{\sin ^{-1}(a x)}+\frac{5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{2 a}+x \sin ^{-1}(a x)^{5/2}+\frac{15 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{4 a}\\ &=-\frac{15}{4} x \sqrt{\sin ^{-1}(a x)}+\frac{5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{2 a}+x \sin ^{-1}(a x)^{5/2}+\frac{15 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4 a}\\ \end{align*}
Mathematica [C] time = 0.0354744, size = 68, normalized size = 0.77 \[ \frac{-\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-i \sin ^{-1}(a x)\right )-\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},i \sin ^{-1}(a x)\right )}{2 a \sqrt{\sin ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.037, size = 88, normalized size = 1. \begin{align*}{\frac{\sqrt{2}}{8\,a\sqrt{\pi }} \left ( 4\, \left ( \arcsin \left ( ax \right ) \right ) ^{5/2}\sqrt{2}\sqrt{\pi }xa+10\, \left ( \arcsin \left ( ax \right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }\sqrt{-{a}^{2}{x}^{2}+1}-15\,\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }xa+15\,\pi \,{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.51359, size = 209, normalized size = 2.38 \begin{align*} -\frac{i \, \arcsin \left (a x\right )^{\frac{5}{2}} e^{\left (i \, \arcsin \left (a x\right )\right )}}{2 \, a} + \frac{i \, \arcsin \left (a x\right )^{\frac{5}{2}} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{2 \, a} + \frac{5 \, \arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (i \, \arcsin \left (a x\right )\right )}}{4 \, a} + \frac{5 \, \arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{4 \, a} + \frac{\left (15 i - 15\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{32 \, a} - \frac{\left (15 i + 15\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{32 \, a} + \frac{15 i \, \sqrt{\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{8 \, a} - \frac{15 i \, \sqrt{\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{8 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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