Optimal. Leaf size=172 \[ \frac{3 \sqrt{\pi } b^{3/2} \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{32 c^2}-\frac{3 \sqrt{\pi } b^{3/2} \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{32 c^2}+\frac{3 b x \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2} \]
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Rubi [A] time = 0.524971, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.786, Rules used = {4629, 4707, 4641, 4635, 4406, 12, 3306, 3305, 3351, 3304, 3352} \[ \frac{3 \sqrt{\pi } b^{3/2} \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{32 c^2}-\frac{3 \sqrt{\pi } b^{3/2} \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{32 c^2}+\frac{3 b x \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 4629
Rule 4707
Rule 4641
Rule 4635
Rule 4406
Rule 12
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int x \left (a+b \sin ^{-1}(c x)\right )^{3/2} \, dx &=\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{1}{4} (3 b c) \int \frac{x^2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{3 b x \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{8 c}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{1}{16} \left (3 b^2\right ) \int \frac{x}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx-\frac{(3 b) \int \frac{\sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{1-c^2 x^2}} \, dx}{8 c}\\ &=\frac{3 b x \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^2}\\ &=\frac{3 b x \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^2}\\ &=\frac{3 b x \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^2}\\ &=\frac{3 b x \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{\left (3 b^2 \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{2 a}{b}+2 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^2}+\frac{\left (3 b^2 \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{2 a}{b}+2 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^2}\\ &=\frac{3 b x \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{\left (3 b \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{16 c^2}+\frac{\left (3 b \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{16 c^2}\\ &=\frac{3 b x \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}-\frac{3 b^{3/2} \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{32 c^2}+\frac{3 b^{3/2} \sqrt{\pi } C\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{32 c^2}\\ \end{align*}
Mathematica [C] time = 0.0585441, size = 126, normalized size = 0.73 \[ \frac{b^2 e^{-\frac{2 i a}{b}} \left (\sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{5}{2},-\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{\frac{4 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{5}{2},\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{16 \sqrt{2} c^2 \sqrt{a+b \sin ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.092, size = 267, normalized size = 1.6 \begin{align*} -{\frac{1}{32\,{c}^{2}} \left ( 3\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{a+b\arcsin \left ( cx \right ) }\cos \left ( 2\,{\frac{a}{b}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{{b}^{-1}}\sqrt{\pi }b}} \right ){b}^{2}-3\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{a+b\arcsin \left ( cx \right ) }\sin \left ( 2\,{\frac{a}{b}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{{b}^{-1}}\sqrt{\pi }b}} \right ){b}^{2}+8\, \left ( \arcsin \left ( cx \right ) \right ) ^{2}\cos \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ){b}^{2}+16\,\arcsin \left ( cx \right ) \cos \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ) ab-6\,\arcsin \left ( cx \right ) \sin \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ){b}^{2}+8\,\cos \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ){a}^{2}-6\,\sin \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ) ab \right ){\frac{1}{\sqrt{a+b\arcsin \left ( cx \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac{3}{2}} x\,{d x} \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 \left (a + b \operatorname{asin}{\left (c x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 2.22962, size = 757, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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