Optimal. Leaf size=55 \[ \frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{c d x+d} \sqrt{e-c e x}} \]
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Rubi [A] time = 0.232389, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {4673, 4641} \[ \frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{c d x+d} \sqrt{e-c e x}} \]
Antiderivative was successfully verified.
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Rule 4673
Rule 4641
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{d+c d x} \sqrt{e-c e x}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d+c d x} \sqrt{e-c e x}}\\ &=\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d+c d x} \sqrt{e-c e x}}\\ \end{align*}
Mathematica [B] time = 0.647459, size = 159, normalized size = 2.89 \[ \frac{-\frac{3 a^2 \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left (c^2 x^2-1\right )}\right )}{\sqrt{d} \sqrt{e}}+\frac{3 a b \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{\sqrt{c d x+d} \sqrt{e-c e x}}+\frac{b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^3}{\sqrt{c d x+d} \sqrt{e-c e x}}}{3 c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}{\frac{1}{\sqrt{cdx+d}}}{\frac{1}{\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}}{c^{2} d e x^{2} - d e}, 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{\left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}{\sqrt{d \left (c x + 1\right )} \sqrt{- e \left (c x - 1\right )}}\, 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{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt{c d x + d} \sqrt{-c e x + e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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