Optimal. Leaf size=455 \[ \frac{2 b c^2 e x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{b c e x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}-\frac{2 b e x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{e \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt{1-c^2 x^2}}+\frac{e \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{1}{2} e x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 b^2 e \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x}}{27 c}+\frac{b^2 e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}-\frac{1}{4} b^2 e x \sqrt{c d x+d} \sqrt{e-c e x}-\frac{4 b^2 e \sqrt{c d x+d} \sqrt{e-c e x}}{9 c} \]
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Rubi [A] time = 0.591805, antiderivative size = 455, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {4673, 4763, 4647, 4641, 4627, 321, 216, 4677, 4645, 444, 43} \[ \frac{2 b c^2 e x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{b c e x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}-\frac{2 b e x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{e \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt{1-c^2 x^2}}+\frac{e \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{1}{2} e x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 b^2 e \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x}}{27 c}+\frac{b^2 e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}-\frac{1}{4} b^2 e x \sqrt{c d x+d} \sqrt{e-c e x}-\frac{4 b^2 e \sqrt{c d x+d} \sqrt{e-c e x}}{9 c} \]
Antiderivative was successfully verified.
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Rule 4673
Rule 4763
Rule 4647
Rule 4641
Rule 4627
Rule 321
Rule 216
Rule 4677
Rule 4645
Rule 444
Rule 43
Rubi steps
\begin{align*} \int \sqrt{d+c d x} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{\left (\sqrt{d+c d x} \sqrt{e-c e x}\right ) \int (e-c e x) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (\sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \left (e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-c e x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt{1-c^2 x^2}}-\frac{\left (c e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{2} e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{\left (e \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}{2 \sqrt{1-c^2 x^2}}-\frac{\left (2 b e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \sqrt{1-c^2 x^2}}-\frac{\left (b c e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{2 b e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}-\frac{b c e x^2 \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{2 b c^2 e x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+\frac{1}{2} e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{x \left (1-\frac{c^2 x^2}{3}\right )}{\sqrt{1-c^2 x^2}} \, dx}{3 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}\\ &=-\frac{1}{4} b^2 e x \sqrt{d+c d x} \sqrt{e-c e x}-\frac{2 b e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}-\frac{b c e x^2 \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{2 b c^2 e x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+\frac{1}{2} e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt{1-c^2 x^2}}+\frac{\left (b^2 e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{4 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{c^2 x}{3}}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{3 \sqrt{1-c^2 x^2}}\\ &=-\frac{1}{4} b^2 e x \sqrt{d+c d x} \sqrt{e-c e x}+\frac{b^2 e \sqrt{d+c d x} \sqrt{e-c e x} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}-\frac{2 b e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}-\frac{b c e x^2 \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{2 b c^2 e x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+\frac{1}{2} e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c e \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt{1-c^2 x}}+\frac{1}{3} \sqrt{1-c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt{1-c^2 x^2}}\\ &=-\frac{4 b^2 e \sqrt{d+c d x} \sqrt{e-c e x}}{9 c}-\frac{1}{4} b^2 e x \sqrt{d+c d x} \sqrt{e-c e x}-\frac{2 b^2 e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right )}{27 c}+\frac{b^2 e \sqrt{d+c d x} \sqrt{e-c e x} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}-\frac{2 b e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}-\frac{b c e x^2 \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{2 b c^2 e x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+\frac{1}{2} e x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{e \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{e \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 1.80815, size = 440, normalized size = 0.97 \[ \frac{e \sqrt{c d x+d} \sqrt{e-c e x} \left (-3 \left (4 \left (3 a^2 \sqrt{1-c^2 x^2} \left (2 c^2 x^2-3 c x-2\right )-4 a b c x \left (c^2 x^2-3\right )+9 b^2 \sqrt{1-c^2 x^2}\right )+9 b^2 \sin \left (2 \sin ^{-1}(c x)\right )\right )+54 a b \cos \left (2 \sin ^{-1}(c x)\right )-4 b^2 \cos \left (3 \sin ^{-1}(c x)\right )\right )-108 a^2 \sqrt{d} e^{3/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left (c^2 x^2-1\right )}\right )+18 b e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left (6 a+3 b \sqrt{1-c^2 x^2}+3 b \sin \left (2 \sin ^{-1}(c x)\right )+b \cos \left (3 \sin ^{-1}(c x)\right )\right )-6 b e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left (2 \left (12 a c^2 x^2 \sqrt{1-c^2 x^2}-12 a \sqrt{1-c^2 x^2}-9 a \sin \left (2 \sin ^{-1}(c x)\right )+9 b c x+b \sin \left (3 \sin ^{-1}(c x)\right )\right )-9 b \cos \left (2 \sin ^{-1}(c x)\right )\right )+36 b^2 e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{216 c \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.267, size = 0, normalized size = 0. \begin{align*} \int \sqrt{cdx+d} \left ( -cex+e \right ) ^{{\frac{3}{2}}} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{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: 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 (-{\left (a^{2} c e x - a^{2} e +{\left (b^{2} c e x - b^{2} e\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c e x - a b e\right )} \arcsin \left (c x\right )\right )} \sqrt{c d x + d} \sqrt{-c e x + e}, 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 \sqrt{c d x + d}{\left (-c e x + e\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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