Optimal. Leaf size=263 \[ \frac{15 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{32 a^5}-\frac{5 \sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{192 a^5}+\frac{3 \sqrt{\frac{\pi }{10}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{1600 a^5}+\frac{x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}-\frac{x^3 \sqrt{\sin ^{-1}(a x)}}{15 a^2}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x \sqrt{\sin ^{-1}(a x)}}{5 a^4}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{5/2}-\frac{3}{100} x^5 \sqrt{\sin ^{-1}(a x)} \]
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Rubi [A] time = 0.803417, antiderivative size = 298, normalized size of antiderivative = 1.13, number of steps used = 26, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4629, 4707, 4677, 4619, 4723, 3305, 3351, 3312} \[ \frac{15 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{32 a^5}-\frac{\sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{320 a^5}-\frac{\sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{60 a^5}+\frac{3 \sqrt{\frac{\pi }{10}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{1600 a^5}+\frac{x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}-\frac{x^3 \sqrt{\sin ^{-1}(a x)}}{15 a^2}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x \sqrt{\sin ^{-1}(a x)}}{5 a^4}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{5/2}-\frac{3}{100} x^5 \sqrt{\sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4629
Rule 4707
Rule 4677
Rule 4619
Rule 4723
Rule 3305
Rule 3351
Rule 3312
Rubi steps
\begin{align*} \int x^4 \sin ^{-1}(a x)^{5/2} \, dx &=\frac{1}{5} x^5 \sin ^{-1}(a x)^{5/2}-\frac{1}{2} a \int \frac{x^5 \sin ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{5/2}-\frac{3}{20} \int x^4 \sqrt{\sin ^{-1}(a x)} \, dx-\frac{2 \int \frac{x^3 \sin ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx}{5 a}\\ &=-\frac{3}{100} x^5 \sqrt{\sin ^{-1}(a x)}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac{x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{5/2}-\frac{4 \int \frac{x \sin ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx}{15 a^3}-\frac{\int x^2 \sqrt{\sin ^{-1}(a x)} \, dx}{5 a^2}+\frac{1}{200} (3 a) \int \frac{x^5}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx\\ &=-\frac{x^3 \sqrt{\sin ^{-1}(a x)}}{15 a^2}-\frac{3}{100} x^5 \sqrt{\sin ^{-1}(a x)}+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac{x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{5/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin ^5(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{200 a^5}-\frac{2 \int \sqrt{\sin ^{-1}(a x)} \, dx}{5 a^4}+\frac{\int \frac{x^3}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx}{30 a}\\ &=-\frac{2 x \sqrt{\sin ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\sin ^{-1}(a x)}}{15 a^2}-\frac{3}{100} x^5 \sqrt{\sin ^{-1}(a x)}+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac{x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{5/2}+\frac{3 \operatorname{Subst}\left (\int \left (\frac{5 \sin (x)}{8 \sqrt{x}}-\frac{5 \sin (3 x)}{16 \sqrt{x}}+\frac{\sin (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{200 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\sin ^3(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{30 a^5}+\frac{\int \frac{x}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx}{5 a^3}\\ &=-\frac{2 x \sqrt{\sin ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\sin ^{-1}(a x)}}{15 a^2}-\frac{3}{100} x^5 \sqrt{\sin ^{-1}(a x)}+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac{x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{5/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (5 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{3200 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{640 a^5}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{320 a^5}+\frac{\operatorname{Subst}\left (\int \left (\frac{3 \sin (x)}{4 \sqrt{x}}-\frac{\sin (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{30 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{5 a^5}\\ &=-\frac{2 x \sqrt{\sin ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\sin ^{-1}(a x)}}{15 a^2}-\frac{3}{100} x^5 \sqrt{\sin ^{-1}(a x)}+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac{x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{5/2}+\frac{3 \operatorname{Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{1600 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{120 a^5}-\frac{3 \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{320 a^5}+\frac{3 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{160 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{40 a^5}+\frac{2 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{5 a^5}\\ &=-\frac{2 x \sqrt{\sin ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\sin ^{-1}(a x)}}{15 a^2}-\frac{3}{100} x^5 \sqrt{\sin ^{-1}(a x)}+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac{x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{5/2}+\frac{3 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{160 a^5}+\frac{\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{5 a^5}-\frac{\sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{320 a^5}+\frac{3 \sqrt{\frac{\pi }{10}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{1600 a^5}-\frac{\operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{60 a^5}+\frac{\operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{20 a^5}\\ &=-\frac{2 x \sqrt{\sin ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\sin ^{-1}(a x)}}{15 a^2}-\frac{3}{100} x^5 \sqrt{\sin ^{-1}(a x)}+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac{x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{5/2}+\frac{11 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{160 a^5}+\frac{\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{5 a^5}-\frac{\sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{60 a^5}-\frac{\sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{320 a^5}+\frac{3 \sqrt{\frac{\pi }{10}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{1600 a^5}\\ \end{align*}
Mathematica [C] time = 0.0706802, size = 204, normalized size = 0.78 \[ \frac{i \sqrt{\sin ^{-1}(a x)} \left (33750 \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-i \sin ^{-1}(a x)\right )-33750 \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},i \sin ^{-1}(a x)\right )-625 \sqrt{3} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-3 i \sin ^{-1}(a x)\right )+625 \sqrt{3} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},3 i \sin ^{-1}(a x)\right )+27 \sqrt{5} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-5 i \sin ^{-1}(a x)\right )-27 \sqrt{5} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},5 i \sin ^{-1}(a x)\right )\right )}{540000 a^5 \sqrt{\sin ^{-1}(a x)^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.077, size = 233, normalized size = 0.9 \begin{align*} -{\frac{1}{144000\,{a}^{5}} \left ( -18000\,ax \left ( \arcsin \left ( ax \right ) \right ) ^{3}+9000\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}\sin \left ( 3\,\arcsin \left ( ax \right ) \right ) -1800\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}\sin \left ( 5\,\arcsin \left ( ax \right ) \right ) -27\,\sqrt{5}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{5}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +625\,\sqrt{3}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{3}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -45000\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}+7500\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) -900\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\cos \left ( 5\,\arcsin \left ( ax \right ) \right ) -33750\,\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +67500\,ax\arcsin \left ( ax \right ) -3750\,\arcsin \left ( ax \right ) \sin \left ( 3\,\arcsin \left ( ax \right ) \right ) +270\,\arcsin \left ( ax \right ) \sin \left ( 5\,\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(-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 = 1.47479, size = 625, normalized size = 2.38 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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