Optimal. Leaf size=89 \[ -\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a^2}+\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{8 a}-\frac {\sin ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{3/2} \]
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Rubi [A] time = 0.18, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4629, 4707, 4641, 4635, 4406, 12, 3305, 3351} \[ -\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a^2}+\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{8 a}-\frac {\sin ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3305
Rule 3351
Rule 4406
Rule 4629
Rule 4635
Rule 4641
Rule 4707
Rubi steps
\begin {align*} \int x \sin ^{-1}(a x)^{3/2} \, dx &=\frac {1}{2} x^2 \sin ^{-1}(a x)^{3/2}-\frac {1}{4} (3 a) \int \frac {x^2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{8 a}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{3/2}-\frac {3}{16} \int \frac {x}{\sqrt {\sin ^{-1}(a x)}} \, dx-\frac {3 \int \frac {\sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{8 a}\\ &=\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{8 a}-\frac {\sin ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{3/2}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^2}\\ &=\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{8 a}-\frac {\sin ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{3/2}-\frac {3 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^2}\\ &=\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{8 a}-\frac {\sin ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{3/2}-\frac {3 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a^2}\\ &=\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{8 a}-\frac {\sin ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{3/2}-\frac {3 \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{16 a^2}\\ &=\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{8 a}-\frac {\sin ^{-1}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{3/2}-\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a^2}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 71, normalized size = 0.80 \[ \frac {\sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {5}{2},-2 i \sin ^{-1}(a x)\right )+\sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {5}{2},2 i \sin ^{-1}(a x)\right )}{16 \sqrt {2} a^2 \sqrt {\sin ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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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 [C] time = 0.27, size = 107, normalized size = 1.20 \[ -\frac {\arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{8 \, a^{2}} - \frac {\arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{8 \, a^{2}} - \frac {\left (3 i - 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{128 \, a^{2}} + \frac {\left (3 i + 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{128 \, a^{2}} - \frac {3 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{2}} + \frac {3 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 64, normalized size = 0.72 \[ -\frac {8 \arcsin \left (a x \right )^{2} \cos \left (2 \arcsin \left (a x \right )\right )+3 \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-6 \arcsin \left (a x \right ) \sin \left (2 \arcsin \left (a x \right )\right )}{32 a^{2} \sqrt {\arcsin \left (a x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {asin}\left (a\,x\right )}^{3/2} \,d x \]
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 \[ \int x \operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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