Optimal. Leaf size=244 \[ -\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{4 i b \sqrt{1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}} \]
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Rubi [A] time = 0.4882, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4739, 4677, 4657, 4181, 2279, 2391} \[ -\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{4 i b \sqrt{1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}} \]
Antiderivative was successfully verified.
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Rule 4739
Rule 4677
Rule 4657
Rule 4181
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d e \sqrt{d+c d x} \sqrt{e-c e x}}\\ &=\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{c d e \sqrt{d+c d x} \sqrt{e-c e x}}\\ &=\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}\\ &=\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{4 i b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{\left (2 b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{\left (2 b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}\\ &=\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{4 i b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{\left (2 i b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{\left (2 i b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}\\ &=\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{4 i b \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^2 d e \sqrt{d+c d x} \sqrt{e-c e x}}\\ \end{align*}
Mathematica [A] time = 1.3472, size = 453, normalized size = 1.86 \[ \frac{-2 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )+2 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )+a^2+2 a b \sqrt{1-c^2 x^2} \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-2 a b \sqrt{1-c^2 x^2} \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+2 a b \sin ^{-1}(c x)+i \pi b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-2 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )-\pi b^2 \sqrt{1-c^2 x^2} \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+2 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-\pi b^2 \sqrt{1-c^2 x^2} \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+\pi b^2 \sqrt{1-c^2 x^2} \log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )+\pi b^2 \sqrt{1-c^2 x^2} \log \left (-\cos \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )+b^2 \sin ^{-1}(c x)^2}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.38, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2} \left ( cdx+d \right ) ^{-{\frac{3}{2}}} \left ( -cex+e \right ) ^{-{\frac{3}{2}}}}\, dx \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*} \sqrt{d} \sqrt{e} \int \frac{{\left (b^{2} x \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + 2 \, a b x \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}{c^{4} d^{2} e^{2} x^{4} - 2 \, c^{2} d^{2} e^{2} x^{2} + d^{2} e^{2}}\,{d x} + \frac{a^{2}}{\sqrt{-c^{2} d e x^{2} + d e} c^{2} d e} \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 (b^{2} x \arcsin \left (c x\right )^{2} + 2 \, a b x \arcsin \left (c x\right ) + a^{2} x\right )} \sqrt{c d x + d} \sqrt{-c e x + e}}{c^{4} d^{2} e^{2} x^{4} - 2 \, c^{2} d^{2} e^{2} x^{2} + d^{2} e^{2}}, 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 (b \arcsin \left (c x\right ) + a\right )}^{2} x}{{\left (c d x + d\right )}^{\frac{3}{2}}{\left (-c e x + e\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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