Optimal. Leaf size=225 \[ -\frac{2 b c x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+\frac{2 b x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{\left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}+\frac{2 b^2 \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x}}{27 c^2}+\frac{4 b^2 \sqrt{c d x+d} \sqrt{e-c e x}}{9 c^2} \]
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Rubi [A] time = 0.394956, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {4739, 4677, 4645, 444, 43} \[ -\frac{2 b c x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}+\frac{2 b x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{\left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}+\frac{2 b^2 \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x}}{27 c^2}+\frac{4 b^2 \sqrt{c d x+d} \sqrt{e-c e x}}{9 c^2} \]
Antiderivative was successfully verified.
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Rule 4739
Rule 4677
Rule 4645
Rule 444
Rule 43
Rubi steps
\begin{align*} \int x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{\left (\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{\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^2}+\frac{\left (2 b \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 c \sqrt{1-c^2 x^2}}\\ &=\frac{2 b x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{2 b c 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{\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^2}-\frac{\left (2 b^2 \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{2 b x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{2 b c 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{\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^2}-\frac{\left (b^2 \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{2 b x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{2 b c 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{\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^2}-\frac{\left (b^2 \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 \sqrt{d+c d x} \sqrt{e-c e x}}{9 c^2}+\frac{2 b^2 \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right )}{27 c^2}+\frac{2 b x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt{1-c^2 x^2}}-\frac{2 b c 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{\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^2}\\ \end{align*}
Mathematica [A] time = 0.58508, size = 178, normalized size = 0.79 \[ \frac{\sqrt{c d x+d} \sqrt{e-c e x} \left (9 a^2 \left (c^2 x^2-1\right )^2+6 a b c x \sqrt{1-c^2 x^2} \left (c^2 x^2-3\right )+6 b \sin ^{-1}(c x) \left (3 a \left (c^2 x^2-1\right )^2+b c x \sqrt{1-c^2 x^2} \left (c^2 x^2-3\right )\right )-2 b^2 \left (c^4 x^4-8 c^2 x^2+7\right )+9 b^2 \left (c^2 x^2-1\right )^2 \sin ^{-1}(c x)^2\right )}{27 c^2 \left (c^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.361, size = 0, normalized size = 0. \begin{align*} \int x\sqrt{cdx+d}\sqrt{-cex+e} \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 [A] time = 2.44801, size = 435, normalized size = 1.93 \begin{align*} \frac{{\left ({\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} x^{4} - 2 \,{\left (9 \, a^{2} - 8 \, b^{2}\right )} c^{2} x^{2} + 9 \,{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arcsin \left (c x\right )^{2} + 9 \, a^{2} - 14 \, b^{2} + 18 \,{\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \arcsin \left (c x\right ) + 6 \,{\left (a b c^{3} x^{3} - 3 \, a b c x +{\left (b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}\right )} \sqrt{c d x + d} \sqrt{-c e x + e}}{27 \,{\left (c^{4} x^{2} - c^{2}\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 [B] time = 1.59997, size = 610, normalized size = 2.71 \begin{align*} -\frac{\frac{9 \,{\left (c d x + d\right )}^{\frac{3}{2}} \sqrt{-{\left (c d x + d\right )} d e + 2 \, d^{2} e} a^{2}{\left (\frac{{\left (c d x + d\right )} e^{\left (-6\right )}}{d^{6}} - \frac{2 \, e^{\left (-6\right )}}{d^{5}}\right )}{\left | d \right |}}{c d^{3}} + \frac{6 \,{\left (3 \,{\left (c d x + d\right )}^{\frac{3}{2}} \sqrt{-{\left (c d x + d\right )} d e + 2 \, d^{2} e}{\left (\frac{{\left (c d x + d\right )} e^{\left (-6\right )}}{d^{6}} - \frac{2 \, e^{\left (-6\right )}}{d^{5}}\right )} \arcsin \left (c x\right ) - \frac{{\left ({\left (c d x + d\right )}^{3} - 3 \,{\left (c d x + d\right )}^{2} d\right )} e^{\left (-\frac{11}{2}\right )}}{d^{\frac{9}{2}}{\left | d \right |}}\right )} a b{\left | d \right |}}{c d^{3}} + \frac{{\left (9 \,{\left (c d x + d\right )}^{\frac{3}{2}} \sqrt{-{\left (c d x + d\right )} d e + 2 \, d^{2} e}{\left (\frac{{\left (c d x + d\right )} e^{\left (-6\right )}}{d^{6}} - \frac{2 \, e^{\left (-6\right )}}{d^{5}}\right )} \arcsin \left (c x\right )^{2} + \frac{\sqrt{d}{\left (\frac{6 \, \pi e^{\left (-6\right )}}{d^{2}} - \frac{{\left (6 \,{\left (c^{2} x^{2} - 1\right )} c d^{2} x \arcsin \left (-c x\right ) + 24 \, c d^{2} x \arcsin \left (-c x\right ) - 9 \, \sqrt{-c^{2} x^{2} + 1} c d^{2} x + 18 \,{\left (c^{2} x^{2} - 1\right )} d^{2} \arcsin \left (-c x\right ) + 2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} d^{2} + 9 \, d^{2} \arcsin \left (-c x\right ) - 24 \, \sqrt{-c^{2} x^{2} + 1} d^{2} - 9 \,{\left (4 \, c d x \arcsin \left (-c x\right ) - \sqrt{-c^{2} x^{2} + 1} c d x + 2 \,{\left (c^{2} x^{2} - 1\right )} d \arcsin \left (-c x\right ) + d \arcsin \left (-c x\right ) - 4 \, \sqrt{-c^{2} x^{2} + 1} d\right )} d\right )} e^{\left (-6\right )}}{d^{4}}\right )} e^{\frac{1}{2}}}{{\left | d \right |}}\right )} b^{2}{\left | d \right |}}{c d^{3}}}{4320 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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