Optimal. Leaf size=431 \[ \frac{15 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4096 a \sqrt{1-a^2 x^2}}+\frac{15 \sqrt{\pi } c \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{128 a \sqrt{1-a^2 x^2}}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt{1-a^2 x^2}}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac{5 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt{1-a^2 x^2}}+\frac{45 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt{1-a^2 x^2}}-\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}-\frac{15}{256} c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \]
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Rubi [A] time = 0.577838, antiderivative size = 431, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4649, 4647, 4641, 4629, 4707, 4635, 4406, 12, 3305, 3351, 4677, 4723} \[ \frac{15 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4096 a \sqrt{1-a^2 x^2}}+\frac{15 \sqrt{\pi } c \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{128 a \sqrt{1-a^2 x^2}}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt{1-a^2 x^2}}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac{5 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt{1-a^2 x^2}}+\frac{45 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt{1-a^2 x^2}}-\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}-\frac{15}{256} c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4649
Rule 4647
Rule 4641
Rule 4629
Rule 4707
Rule 4635
Rule 4406
Rule 12
Rule 3305
Rule 3351
Rule 4677
Rule 4723
Rubi steps
\begin{align*} \int \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2} \, dx &=\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac{1}{4} (3 c) \int \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2} \, dx-\frac{\left (5 a c \sqrt{c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^{3/2} \, dx}{8 \sqrt{1-a^2 x^2}}\\ &=\frac{5 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}-\frac{\left (15 c \sqrt{c-a^2 c x^2}\right ) \int \left (1-a^2 x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)} \, dx}{64 \sqrt{1-a^2 x^2}}+\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \int \frac{\sin ^{-1}(a x)^{5/2}}{\sqrt{1-a^2 x^2}} \, dx}{8 \sqrt{1-a^2 x^2}}-\frac{\left (15 a c \sqrt{c-a^2 c x^2}\right ) \int x \sin ^{-1}(a x)^{3/2} \, dx}{16 \sqrt{1-a^2 x^2}}\\ &=-\frac{15}{256} c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt{1-a^2 x^2}}+\frac{5 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt{1-a^2 x^2}}-\frac{\left (45 c \sqrt{c-a^2 c x^2}\right ) \int \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)} \, dx}{256 \sqrt{1-a^2 x^2}}+\frac{\left (15 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}{512 \sqrt{1-a^2 x^2}}+\frac{\left (45 a^2 c \sqrt{c-a^2 c x^2}\right ) \int \frac{x^2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{64 \sqrt{1-a^2 x^2}}\\ &=-\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}-\frac{15}{256} c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt{1-a^2 x^2}}+\frac{5 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt{1-a^2 x^2}}-\frac{\left (45 c \sqrt{c-a^2 c x^2}\right ) \int \frac{\sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{512 \sqrt{1-a^2 x^2}}+\frac{\left (45 c \sqrt{c-a^2 c x^2}\right ) \int \frac{\sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{128 \sqrt{1-a^2 x^2}}+\frac{\left (15 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 )}{512 a \sqrt{1-a^2 x^2}}+\frac{\left (45 a c \sqrt{c-a^2 c x^2}\right ) \int \frac{x}{\sqrt{\sin ^{-1}(a x)}} \, dx}{1024 \sqrt{1-a^2 x^2}}+\frac{\left (45 a c \sqrt{c-a^2 c x^2}\right ) \int \frac{x}{\sqrt{\sin ^{-1}(a x)}} \, dx}{256 \sqrt{1-a^2 x^2}}\\ &=-\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}-\frac{15}{256} c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{45 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt{1-a^2 x^2}}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt{1-a^2 x^2}}+\frac{5 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt{1-a^2 x^2}}+\frac{\left (15 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 )}{512 a \sqrt{1-a^2 x^2}}+\frac{\left (45 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 )}{1024 a \sqrt{1-a^2 x^2}}+\frac{\left (45 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 )}{256 a \sqrt{1-a^2 x^2}}\\ &=-\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}-\frac{15}{256} c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{45 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt{1-a^2 x^2}}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt{1-a^2 x^2}}+\frac{5 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt{1-a^2 x^2}}+\frac{\left (15 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 )}{4096 a \sqrt{1-a^2 x^2}}+\frac{\left (15 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 )}{2048 a \sqrt{1-a^2 x^2}}+\frac{\left (45 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 )}{1024 a \sqrt{1-a^2 x^2}}+\frac{\left (45 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 )}{256 a \sqrt{1-a^2 x^2}}\\ &=-\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}-\frac{15}{256} c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{45 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt{1-a^2 x^2}}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt{1-a^2 x^2}}+\frac{5 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt{1-a^2 x^2}}+\frac{\left (15 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 )}{2048 a \sqrt{1-a^2 x^2}}+\frac{\left (15 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 )}{1024 a \sqrt{1-a^2 x^2}}+\frac{\left (45 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 )}{2048 a \sqrt{1-a^2 x^2}}+\frac{\left (45 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 )}{512 a \sqrt{1-a^2 x^2}}\\ &=-\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}-\frac{15}{256} c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{45 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt{1-a^2 x^2}}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt{1-a^2 x^2}}+\frac{5 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt{1-a^2 x^2}}+\frac{15 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 )}{4096 a \sqrt{1-a^2 x^2}}+\frac{15 c \sqrt{\pi } \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2048 a \sqrt{1-a^2 x^2}}+\frac{\left (45 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 )}{1024 a \sqrt{1-a^2 x^2}}+\frac{\left (45 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 )}{256 a \sqrt{1-a^2 x^2}}\\ &=-\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}-\frac{15}{256} c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}+\frac{45 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt{1-a^2 x^2}}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt{1-a^2 x^2}}+\frac{5 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt{1-a^2 x^2}}+\frac{15 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 )}{4096 a \sqrt{1-a^2 x^2}}+\frac{15 c \sqrt{\pi } \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{128 a \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.317277, size = 180, normalized size = 0.42 \[ \frac{c \sqrt{c-a^2 c x^2} \left (-7 \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-4 i \sin ^{-1}(a x)\right )-7 \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},4 i \sin ^{-1}(a x)\right )+1680 \sqrt{\pi } \sqrt{\sin ^{-1}(a x)} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )+1536 \sin ^{-1}(a x)^4+3584 \sin \left (2 \sin ^{-1}(a x)\right ) \sin ^{-1}(a x)^3-3360 \sin \left (2 \sin ^{-1}(a x)\right ) \sin ^{-1}(a x)+4480 \sin ^{-1}(a x)^2 \cos \left (2 \sin ^{-1}(a x)\right )\right )}{14336 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.18, size = 0, normalized size = 0. \begin{align*} \int \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}} \left ( \arcsin \left ( ax \right ) \right ) ^{{\frac{5}{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{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arcsin \left (a x\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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