Optimal. Leaf size=362 \[ \frac{(c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}+\frac{3 x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac{b \sqrt{1-c^2 x^2} (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}-\frac{3 b c x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac{1}{4} x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{15 b^2 x (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac{9 b^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac{1}{32} b^2 x (c d x+d)^{3/2} (e-c e x)^{3/2} \]
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Rubi [A] time = 0.42222, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.281, Rules used = {4673, 4649, 4647, 4641, 4627, 321, 216, 4677, 195} \[ \frac{(c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}+\frac{3 x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac{b \sqrt{1-c^2 x^2} (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}-\frac{3 b c x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac{1}{4} x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{15 b^2 x (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac{9 b^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac{1}{32} b^2 x (c d x+d)^{3/2} (e-c e x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 4673
Rule 4649
Rule 4647
Rule 4641
Rule 4627
Rule 321
Rule 216
Rule 4677
Rule 195
Rubi steps
\begin{align*} \int (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{\left ((d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\left (1-c^2 x^2\right )^{3/2}}\\ &=\frac{1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{\left (3 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (b c (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 \left (1-c^2 x^2\right )^{3/2}}\\ &=\frac{b (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3 x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac{\left (3 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (b^2 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (3 b c (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac{1}{32} b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac{3 b c x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac{b (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3 x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac{(d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (3 b^2 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \sqrt{1-c^2 x^2} \, dx}{32 \left (1-c^2 x^2\right )^{3/2}}+\frac{\left (3 b^2 c^2 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac{1}{32} b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac{15 b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}-\frac{3 b c x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac{b (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3 x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac{(d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac{\left (3 b^2 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{64 \left (1-c^2 x^2\right )^{3/2}}+\frac{\left (3 b^2 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{16 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac{1}{32} b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac{15 b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac{9 b^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac{3 b c x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac{b (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3 x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac{(d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.63099, size = 373, normalized size = 1.03 \[ \frac{d e \sqrt{c d x+d} \sqrt{e-c e x} \left (-64 a^2 c^3 x^3 \sqrt{1-c^2 x^2}+160 a^2 c x \sqrt{1-c^2 x^2}+64 a b \cos \left (2 \sin ^{-1}(c x)\right )+4 a b \cos \left (4 \sin ^{-1}(c x)\right )-32 b^2 \sin \left (2 \sin ^{-1}(c x)\right )-b^2 \sin \left (4 \sin ^{-1}(c x)\right )\right )-96 a^2 d^{3/2} e^{3/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left (c^2 x^2-1\right )}\right )+8 b d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left (12 a+8 b \sin \left (2 \sin ^{-1}(c x)\right )+b \sin \left (4 \sin ^{-1}(c x)\right )\right )+4 b d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left (4 a \left (8 \sin \left (2 \sin ^{-1}(c x)\right )+\sin \left (4 \sin ^{-1}(c x)\right )\right )+16 b \cos \left (2 \sin ^{-1}(c x)\right )+b \cos \left (4 \sin ^{-1}(c x)\right )\right )+32 b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{256 c \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left ( cdx+d \right ) ^{{\frac{3}{2}}} \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^{2} d e x^{2} - a^{2} d e +{\left (b^{2} c^{2} d e x^{2} - b^{2} d e\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} d e x^{2} - a b d 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{\left (c d x + d\right )}^{\frac{3}{2}}{\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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