Optimal. Leaf size=157 \[ \frac{3 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{512 a^4}-\frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{64 a^4}+\frac{3 x^3 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}+\frac{9 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{64 a^3}-\frac{3 \sin ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{3/2} \]
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Rubi [A] time = 0.378891, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4629, 4707, 4641, 4635, 4406, 12, 3305, 3351} \[ \frac{3 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{512 a^4}-\frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{64 a^4}+\frac{3 x^3 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}+\frac{9 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{64 a^3}-\frac{3 \sin ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 4629
Rule 4707
Rule 4641
Rule 4635
Rule 4406
Rule 12
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int x^3 \sin ^{-1}(a x)^{3/2} \, dx &=\frac{1}{4} x^4 \sin ^{-1}(a x)^{3/2}-\frac{1}{8} (3 a) \int \frac{x^4 \sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{3 x^3 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{3/2}-\frac{3}{64} \int \frac{x^3}{\sqrt{\sin ^{-1}(a x)}} \, dx-\frac{9 \int \frac{x^2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{32 a}\\ &=\frac{9 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{64 a^3}+\frac{3 x^3 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^3(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^4}-\frac{9 \int \frac{\sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{64 a^3}-\frac{9 \int \frac{x}{\sqrt{\sin ^{-1}(a x)}} \, dx}{128 a^2}\\ &=\frac{9 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{64 a^3}+\frac{3 x^3 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}-\frac{3 \sin ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 \sqrt{x}}-\frac{\sin (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{64 a^4}-\frac{9 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{128 a^4}\\ &=\frac{9 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{64 a^3}+\frac{3 x^3 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}-\frac{3 \sin ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{512 a^4}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{256 a^4}-\frac{9 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{128 a^4}\\ &=\frac{9 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{64 a^3}+\frac{3 x^3 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}-\frac{3 \sin ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{256 a^4}-\frac{3 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{128 a^4}-\frac{9 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{256 a^4}\\ &=\frac{9 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{64 a^3}+\frac{3 x^3 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}-\frac{3 \sin ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{3/2}+\frac{3 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{512 a^4}-\frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{256 a^4}-\frac{9 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{128 a^4}\\ &=\frac{9 x \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{64 a^3}+\frac{3 x^3 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}-\frac{3 \sin ^{-1}(a x)^{3/2}}{32 a^4}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{3/2}+\frac{3 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{512 a^4}-\frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{64 a^4}\\ \end{align*}
Mathematica [C] time = 0.0327046, size = 130, normalized size = 0.83 \[ \frac{8 \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-2 i \sin ^{-1}(a x)\right )+8 \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},2 i \sin ^{-1}(a x)\right )-\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-4 i \sin ^{-1}(a x)\right )-\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},4 i \sin ^{-1}(a x)\right )}{512 a^4 \sqrt{\sin ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.053, size = 121, normalized size = 0.8 \begin{align*} -{\frac{1}{1024\,{a}^{4}} \left ( -3\,\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +128\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\cos \left ( 2\,\arcsin \left ( ax \right ) \right ) -32\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\cos \left ( 4\,\arcsin \left ( ax \right ) \right ) +48\,\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -96\,\arcsin \left ( ax \right ) \sin \left ( 2\,\arcsin \left ( ax \right ) \right ) +12\,\arcsin \left ( ax \right ) \sin \left ( 4\,\arcsin \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arcsin \left ( ax \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{asin}^{\frac{3}{2}}{\left (a x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.39818, size = 304, normalized size = 1.94 \begin{align*} \frac{\arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (4 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} - \frac{\arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{16 \, a^{4}} - \frac{\arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{16 \, a^{4}} + \frac{\arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (-4 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} + \frac{\left (3 i - 3\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (i - 1\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{4096 \, a^{4}} - \frac{\left (3 i + 3\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (i + 1\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{4096 \, a^{4}} - \frac{\left (3 i - 3\right ) \, \sqrt{\pi } \operatorname{erf}\left (\left (i - 1\right ) \, \sqrt{\arcsin \left (a x\right )}\right )}{256 \, a^{4}} + \frac{\left (3 i + 3\right ) \, \sqrt{\pi } \operatorname{erf}\left (-\left (i + 1\right ) \, \sqrt{\arcsin \left (a x\right )}\right )}{256 \, a^{4}} + \frac{3 i \, \sqrt{\arcsin \left (a x\right )} e^{\left (4 i \, \arcsin \left (a x\right )\right )}}{512 \, a^{4}} - \frac{3 i \, \sqrt{\arcsin \left (a x\right )} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} + \frac{3 i \, \sqrt{\arcsin \left (a x\right )} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} - \frac{3 i \, \sqrt{\arcsin \left (a x\right )} e^{\left (-4 i \, \arcsin \left (a x\right )\right )}}{512 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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