Optimal. Leaf size=237 \[ -\frac{3 \sqrt{\frac{\pi }{2}} c^2 \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a \sqrt{1-a^2 x^2}}-\frac{\sqrt{3 \pi } c^2 \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{3}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a \sqrt{1-a^2 x^2}}-\frac{15 \sqrt{\pi } c^2 \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt{\sin ^{-1}(a x)}} \]
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Rubi [A] time = 0.186019, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4659, 4723, 4406, 3305, 3351} \[ -\frac{3 \sqrt{\frac{\pi }{2}} c^2 \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a \sqrt{1-a^2 x^2}}-\frac{\sqrt{3 \pi } c^2 \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{3}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a \sqrt{1-a^2 x^2}}-\frac{15 \sqrt{\pi } c^2 \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt{\sin ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 4659
Rule 4723
Rule 4406
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{\left (c-a^2 c x^2\right )^{5/2}}{\sin ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{\left (12 a c^2 \sqrt{c-a^2 c x^2}\right ) \int \frac{x \left (1-a^2 x^2\right )^2}{\sqrt{\sin ^{-1}(a x)}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{\left (12 c^2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos ^5(x) \sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a \sqrt{1-a^2 x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{\left (12 c^2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{5 \sin (2 x)}{32 \sqrt{x}}+\frac{\sin (4 x)}{8 \sqrt{x}}+\frac{\sin (6 x)}{32 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a \sqrt{1-a^2 x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{\left (3 c^2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (6 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a \sqrt{1-a^2 x^2}}-\frac{\left (3 c^2 \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 )}{2 a \sqrt{1-a^2 x^2}}-\frac{\left (15 c^2 \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 )}{8 a \sqrt{1-a^2 x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{\left (3 c^2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \sin \left (6 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{4 a \sqrt{1-a^2 x^2}}-\frac{\left (3 c^2 \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 )}{a \sqrt{1-a^2 x^2}}-\frac{\left (15 c^2 \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 )}{4 a \sqrt{1-a^2 x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{3 c^2 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a \sqrt{1-a^2 x^2}}-\frac{c^2 \sqrt{3 \pi } \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{3}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a \sqrt{1-a^2 x^2}}-\frac{15 c^2 \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 = 1.05562, size = 404, normalized size = 1.7 \[ -\frac{c^2 \sqrt{c-a^2 c x^2} e^{-6 i \sin ^{-1}(a x)} \left (\sqrt{2} e^{6 i \sin ^{-1}(a x)} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \sin ^{-1}(a x)\right )+\sqrt{2} e^{6 i \sin ^{-1}(a x)} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \sin ^{-1}(a x)\right )-12 e^{6 i \sin ^{-1}(a x)} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \sin ^{-1}(a x)\right )-12 e^{6 i \sin ^{-1}(a x)} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \sin ^{-1}(a x)\right )-\sqrt{6} e^{6 i \sin ^{-1}(a x)} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-6 i \sin ^{-1}(a x)\right )-\sqrt{6} e^{6 i \sin ^{-1}(a x)} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},6 i \sin ^{-1}(a x)\right )+64 \sqrt{\pi } e^{6 i \sin ^{-1}(a x)} \sqrt{\sin ^{-1}(a x)} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )+6 e^{2 i \sin ^{-1}(a x)}+15 e^{4 i \sin ^{-1}(a x)}+20 e^{6 i \sin ^{-1}(a x)}+15 e^{8 i \sin ^{-1}(a x)}+6 e^{10 i \sin ^{-1}(a x)}+e^{12 i \sin ^{-1}(a x)}+1\right )}{32 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.178, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}} \left ( \arcsin \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}}\, 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 \frac{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}{\arcsin \left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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