Optimal. Leaf size=257 \[ -\frac{i b^2 c \sqrt{c d x+d} \sqrt{e-c e x} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{c \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt{1-c^2 x^2}}-\frac{i c \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}+\frac{2 b c \sqrt{c d x+d} \sqrt{e-c e x} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{\sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \]
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Rubi [A] time = 0.594417, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4739, 4693, 4625, 3717, 2190, 2279, 2391, 4641} \[ -\frac{i b^2 c \sqrt{c d x+d} \sqrt{e-c e x} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{c \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt{1-c^2 x^2}}-\frac{i c \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}+\frac{2 b c \sqrt{c d x+d} \sqrt{e-c e x} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{\sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \]
Antiderivative was successfully verified.
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Rule 4739
Rule 4693
Rule 4625
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rule 4641
Rubi steps
\begin{align*} \int \frac{\sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx &=\frac{\left (\sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{\sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac{\left (2 b c \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x} \, dx}{\sqrt{1-c^2 x^2}}-\frac{\left (c^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{\sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{c \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt{1-c^2 x^2}}+\frac{\left (2 b c \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}\\ &=-\frac{\sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{i c \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}-\frac{c \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt{1-c^2 x^2}}-\frac{\left (4 i b c \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}\\ &=-\frac{\sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{i c \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}-\frac{c \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt{1-c^2 x^2}}+\frac{2 b c \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 c \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}\\ &=-\frac{\sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{i c \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}-\frac{c \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt{1-c^2 x^2}}+\frac{2 b c \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{\left (i b^2 c \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}\\ &=-\frac{\sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{i c \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}-\frac{c \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt{1-c^2 x^2}}+\frac{2 b c \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{i b^2 c \sqrt{d+c d x} \sqrt{e-c e x} \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 1.18786, size = 373, normalized size = 1.45 \[ -\frac{b^2 c \sqrt{c d x+d} \sqrt{e-c e x} \sqrt{-d e \left (1-c^2 x^2\right )} \left (3 i \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+\sin ^{-1}(c x) \left (\frac{3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c x}+\left (\sin ^{-1}(c x)+3 i\right ) \sin ^{-1}(c x)-6 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )\right )\right )}{3 \sqrt{1-c^2 x^2} \sqrt{-(c d x+d) (e-c e x)}}-\frac{a^2 \sqrt{d (c x+1)} \sqrt{-e (c x-1)}}{x}+a^2 c \sqrt{d} \sqrt{e} \tan ^{-1}\left (\frac{c x \sqrt{d (c x+1)} \sqrt{-e (c x-1)}}{\sqrt{d} \sqrt{e} (c x-1) (c x+1)}\right )-\frac{a b c \sqrt{c d x+d} \sqrt{e-c e x} \sqrt{-d e \left (1-c^2 x^2\right )} \left (\frac{2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c x}-2 \log (c x)+\sin ^{-1}(c x)^2\right )}{\sqrt{1-c^2 x^2} \sqrt{-(c d x+d) (e-c e x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.431, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}{{x}^{2}}\sqrt{cdx+d}\sqrt{-cex+e}}\, 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: 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 (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )} \sqrt{c d x + d} \sqrt{-c e x + e}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d \left (c x + 1\right )} \sqrt{- e \left (c x - 1\right )} \left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \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{\sqrt{c d x + d} \sqrt{-c e x + e}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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