Optimal. Leaf size=307 \[ \frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt{1-c^2 x^2}}-\frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt{1-c^2 x^2}}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}-\frac{3 c^4 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{8 x \sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{1-c^2 x^2}} \]
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Rubi [A] time = 0.32404, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {4695, 4693, 30, 4709, 4183, 2279, 2391, 14} \[ \frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt{1-c^2 x^2}}-\frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt{1-c^2 x^2}}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}-\frac{3 c^4 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{8 x \sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 4695
Rule 4693
Rule 30
Rule 4709
Rule 4183
Rule 2279
Rule 2391
Rule 14
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x^5} \, dx &=-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}-\frac{1}{4} \left (3 c^2 d\right ) \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{1-c^2 x^2}{x^4} \, dx}{4 \sqrt{1-c^2 x^2}}\\ &=\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \left (\frac{1}{x^4}-\frac{c^2}{x^2}\right ) \, dx}{4 \sqrt{1-c^2 x^2}}-\frac{\left (3 b c^3 d \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{x^2} \, dx}{8 \sqrt{1-c^2 x^2}}+\frac{\left (3 c^4 d \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{1-c^2 x^2}} \, dx}{8 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c d \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{8 x \sqrt{1-c^2 x^2}}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}+\frac{\left (3 c^4 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c d \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{8 x \sqrt{1-c^2 x^2}}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}-\frac{3 c^4 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 \sqrt{1-c^2 x^2}}-\frac{\left (3 b c^4 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{\left (3 b c^4 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c d \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{8 x \sqrt{1-c^2 x^2}}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}-\frac{3 c^4 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 \sqrt{1-c^2 x^2}}+\frac{\left (3 i b c^4 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt{1-c^2 x^2}}-\frac{\left (3 i b c^4 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c d \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{8 x \sqrt{1-c^2 x^2}}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}-\frac{3 c^4 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 \sqrt{1-c^2 x^2}}+\frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt{1-c^2 x^2}}-\frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 5.753, size = 494, normalized size = 1.61 \[ -\frac{b c^4 d^2 \sqrt{1-c^2 x^2} \left (-4 i \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+4 i \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-4 \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )+4 \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )-2 \tan \left (\frac{1}{2} \sin ^{-1}(c x)\right )-2 \cot \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin ^{-1}(c x) \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right )+\sin ^{-1}(c x) \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )}{8 \sqrt{d-c^2 d x^2}}+\frac{b c^4 d \sqrt{d-c^2 d x^2} \left (-24 i \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+24 i \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-\frac{16 \sin ^4\left (\frac{1}{2} \sin ^{-1}(c x)\right )}{c^3 x^3}-24 \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )+24 \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )+8 \tan \left (\frac{1}{2} \sin ^{-1}(c x)\right )+8 \cot \left (\frac{1}{2} \sin ^{-1}(c x)\right )-c x \csc ^4\left (\frac{1}{2} \sin ^{-1}(c x)\right )-3 \sin ^{-1}(c x) \csc ^4\left (\frac{1}{2} \sin ^{-1}(c x)\right )+6 \sin ^{-1}(c x) \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right )+3 \sin ^{-1}(c x) \sec ^4\left (\frac{1}{2} \sin ^{-1}(c x)\right )-6 \sin ^{-1}(c x) \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )}{192 \sqrt{1-c^2 x^2}}-\frac{3}{8} a c^4 d^{3/2} \log \left (\sqrt{d} \sqrt{d-c^2 d x^2}+d\right )+\frac{3}{8} a c^4 d^{3/2} \log (x)+\frac{a d \left (5 c^2 x^2-2\right ) \sqrt{d-c^2 d x^2}}{8 x^4} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.273, size = 601, normalized size = 2. \begin{align*} -{\frac{a}{4\,d{x}^{4}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{a{c}^{2}}{8\,d{x}^{2}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{a{c}^{4}}{8} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{3\,a{c}^{4}}{8}{d}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{-{c}^{2}d{x}^{2}+d} \right ) } \right ) }+{\frac{3\,a{c}^{4}d}{8}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{5\,bd\arcsin \left ( cx \right ){c}^{4}}{8\,{c}^{2}{x}^{2}-8}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{5\,bd{c}^{3}}{8\,x \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{7\,b{c}^{2}d\arcsin \left ( cx \right ) }{8\,{x}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{bdc}{12\,{x}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{bd\arcsin \left ( cx \right ) }{4\,{x}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+3\,{\frac{b\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}{c}^{4}d\arcsin \left ( cx \right ) \ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }{8\,{c}^{2}{x}^{2}-8}}-3\,{\frac{b\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}{c}^{4}d\arcsin \left ( cx \right ) \ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) }{8\,{c}^{2}{x}^{2}-8}}-{\frac{3\,ib{c}^{4}d}{8\,{c}^{2}{x}^{2}-8}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{3\,ib{c}^{4}d}{8\,{c}^{2}{x}^{2}-8}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \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: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (a c^{2} d x^{2} - a d +{\left (b c^{2} d x^{2} - b d\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{x^{5}}, x\right ) \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 (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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