Optimal. Leaf size=227 \[ -\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{64 a \sqrt{1-a^2 x^2}}-\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a \sqrt{1-a^2 x^2}}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt{1-a^2 x^2}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \]
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Rubi [A] time = 0.282997, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4649, 4647, 4641, 4635, 4406, 12, 3305, 3351, 4723} \[ -\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{64 a \sqrt{1-a^2 x^2}}-\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a \sqrt{1-a^2 x^2}}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt{1-a^2 x^2}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4649
Rule 4647
Rule 4641
Rule 4635
Rule 4406
Rule 12
Rule 3305
Rule 3351
Rule 4723
Rubi steps
\begin{align*} \int \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)} \, dx &=\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{1}{4} (3 c) \int \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \, dx-\frac{\left (a c \sqrt{c-a^2 c x^2}\right ) \int \frac{x \left (1-a^2 x^2\right )}{\sqrt{\sin ^{-1}(a x)}} \, dx}{8 \sqrt{1-a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \int \frac{\sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{8 \sqrt{1-a^2 x^2}}-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a \sqrt{1-a^2 x^2}}-\frac{\left (3 a c \sqrt{c-a^2 c x^2}\right ) \int \frac{x}{\sqrt{\sin ^{-1}(a x)}} \, dx}{16 \sqrt{1-a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt{1-a^2 x^2}}-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \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 )}{8 a \sqrt{1-a^2 x^2}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a \sqrt{1-a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt{1-a^2 x^2}}-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{64 a \sqrt{1-a^2 x^2}}-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a \sqrt{1-a^2 x^2}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a \sqrt{1-a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt{1-a^2 x^2}}-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{32 a \sqrt{1-a^2 x^2}}-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{16 a \sqrt{1-a^2 x^2}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a \sqrt{1-a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt{1-a^2 x^2}}-\frac{c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{64 a \sqrt{1-a^2 x^2}}-\frac{c \sqrt{\pi } \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a \sqrt{1-a^2 x^2}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{16 a \sqrt{1-a^2 x^2}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt{1-a^2 x^2}}-\frac{c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{64 a \sqrt{1-a^2 x^2}}-\frac{c \sqrt{\pi } \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.21993, size = 166, normalized size = 0.73 \[ \frac{c \sqrt{c-a^2 c x^2} \left (8 \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-2 i \sin ^{-1}(a x)\right )+8 \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},2 i \sin ^{-1}(a x)\right )+\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-4 i \sin ^{-1}(a x)\right )+\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},4 i \sin ^{-1}(a x)\right )+32 \sin ^{-1}(a x)^2\right )}{128 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.197, size = 0, normalized size = 0. \begin{align*} \int \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}\sqrt{\arcsin \left ( ax \right ) }\, dx \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \sqrt{\arcsin \left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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