Optimal. Leaf size=235 \[ -\frac{2 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}-\frac{4}{3} a b^2 e^2 x+\frac{2 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac{b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}+\frac{2 b^3 e^2 \left (1-(c+d x)^2\right )^{3/2}}{27 d}-\frac{14 b^3 e^2 \sqrt{1-(c+d x)^2}}{9 d}-\frac{4 b^3 e^2 (c+d x) \sin ^{-1}(c+d x)}{3 d} \]
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Rubi [A] time = 0.322174, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {4805, 12, 4627, 4707, 4677, 4619, 261, 266, 43} \[ -\frac{2 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}-\frac{4}{3} a b^2 e^2 x+\frac{2 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac{b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}+\frac{2 b^3 e^2 \left (1-(c+d x)^2\right )^{3/2}}{27 d}-\frac{14 b^3 e^2 \sqrt{1-(c+d x)^2}}{9 d}-\frac{4 b^3 e^2 (c+d x) \sin ^{-1}(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 12
Rule 4627
Rule 4707
Rule 4677
Rule 4619
Rule 261
Rule 266
Rule 43
Rubi steps
\begin{align*} \int (c e+d e x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int e^2 x^2 \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{x^3 \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=\frac{b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}-\frac{\left (2 b e^2\right ) \operatorname{Subst}\left (\int \frac{x \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{3 d}-\frac{\left (2 b^2 e^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d}\\ &=-\frac{2 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}+\frac{2 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac{b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}-\frac{\left (4 b^2 e^2\right ) \operatorname{Subst}\left (\int \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d}+\frac{\left (2 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{9 d}\\ &=-\frac{4}{3} a b^2 e^2 x-\frac{2 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}+\frac{2 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac{b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}+\frac{\left (b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x}} \, dx,x,(c+d x)^2\right )}{9 d}-\frac{\left (4 b^3 e^2\right ) \operatorname{Subst}\left (\int \sin ^{-1}(x) \, dx,x,c+d x\right )}{3 d}\\ &=-\frac{4}{3} a b^2 e^2 x-\frac{4 b^3 e^2 (c+d x) \sin ^{-1}(c+d x)}{3 d}-\frac{2 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}+\frac{2 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac{b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}+\frac{\left (b^3 e^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{\sqrt{1-x}}-\sqrt{1-x}\right ) \, dx,x,(c+d x)^2\right )}{9 d}+\frac{\left (4 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{3 d}\\ &=-\frac{4}{3} a b^2 e^2 x-\frac{14 b^3 e^2 \sqrt{1-(c+d x)^2}}{9 d}+\frac{2 b^3 e^2 \left (1-(c+d x)^2\right )^{3/2}}{27 d}-\frac{4 b^3 e^2 (c+d x) \sin ^{-1}(c+d x)}{3 d}-\frac{2 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}+\frac{2 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac{b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3}{3 d}\\ \end{align*}
Mathematica [A] time = 0.342833, size = 199, normalized size = 0.85 \[ \frac{e^2 \left ((c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^3-b \left (\frac{2}{3} b (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )-\sqrt{1-(c+d x)^2} (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2-2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2+4 b \left (a d x+b \sqrt{1-(c+d x)^2}+b (c+d x) \sin ^{-1}(c+d x)\right )+\frac{2}{9} b^2 \left (c^2+2 c d x+d^2 x^2+2\right ) \sqrt{1-(c+d x)^2}\right )\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 280, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({\frac{{e}^{2} \left ( dx+c \right ) ^{3}{a}^{3}}{3}}+{e}^{2}{b}^{3} \left ({\frac{ \left ( dx+c \right ) ^{3} \left ( \arcsin \left ( dx+c \right ) \right ) ^{3}}{3}}+{\frac{ \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( \left ( dx+c \right ) ^{2}+2 \right ) }{3}\sqrt{1- \left ( dx+c \right ) ^{2}}}-{\frac{4}{3}\sqrt{1- \left ( dx+c \right ) ^{2}}}-{\frac{ \left ( 4\,dx+4\,c \right ) \arcsin \left ( dx+c \right ) }{3}}-{\frac{2\, \left ( dx+c \right ) ^{3}\arcsin \left ( dx+c \right ) }{9}}-{\frac{2\, \left ( dx+c \right ) ^{2}+4}{27}\sqrt{1- \left ( dx+c \right ) ^{2}}} \right ) +3\,{e}^{2}a{b}^{2} \left ( 1/3\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ) ^{3}+2/9\,\arcsin \left ( dx+c \right ) \left ( \left ( dx+c \right ) ^{2}+2 \right ) \sqrt{1- \left ( dx+c \right ) ^{2}}-{\frac{2\, \left ( dx+c \right ) ^{3}}{27}}-4/9\,dx-4/9\,c \right ) +3\,{e}^{2}{a}^{2}b \left ( 1/3\, \left ( dx+c \right ) ^{3}\arcsin \left ( dx+c \right ) +1/9\, \left ( dx+c \right ) ^{2}\sqrt{1- \left ( dx+c \right ) ^{2}}+2/9\,\sqrt{1- \left ( dx+c \right ) ^{2}} \right ) \right ) } \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 [B] time = 2.62982, size = 1107, normalized size = 4.71 \begin{align*} \frac{3 \,{\left (3 \, a^{3} - 2 \, a b^{2}\right )} d^{3} e^{2} x^{3} + 9 \,{\left (3 \, a^{3} - 2 \, a b^{2}\right )} c d^{2} e^{2} x^{2} - 9 \,{\left (4 \, a b^{2} -{\left (3 \, a^{3} - 2 \, a b^{2}\right )} c^{2}\right )} d e^{2} x + 9 \,{\left (b^{3} d^{3} e^{2} x^{3} + 3 \, b^{3} c d^{2} e^{2} x^{2} + 3 \, b^{3} c^{2} d e^{2} x + b^{3} c^{3} e^{2}\right )} \arcsin \left (d x + c\right )^{3} + 27 \,{\left (a b^{2} d^{3} e^{2} x^{3} + 3 \, a b^{2} c d^{2} e^{2} x^{2} + 3 \, a b^{2} c^{2} d e^{2} x + a b^{2} c^{3} e^{2}\right )} \arcsin \left (d x + c\right )^{2} + 3 \,{\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} d^{3} e^{2} x^{3} + 3 \,{\left (9 \, a^{2} b - 2 \, b^{3}\right )} c d^{2} e^{2} x^{2} - 3 \,{\left (4 \, b^{3} -{\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{2}\right )} d e^{2} x -{\left (12 \, b^{3} c -{\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{3}\right )} e^{2}\right )} \arcsin \left (d x + c\right ) +{\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} d^{2} e^{2} x^{2} + 2 \,{\left (9 \, a^{2} b - 2 \, b^{3}\right )} c d e^{2} x +{\left (18 \, a^{2} b - 40 \, b^{3} +{\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{2}\right )} e^{2} + 9 \,{\left (b^{3} d^{2} e^{2} x^{2} + 2 \, b^{3} c d e^{2} x +{\left (b^{3} c^{2} + 2 \, b^{3}\right )} e^{2}\right )} \arcsin \left (d x + c\right )^{2} + 18 \,{\left (a b^{2} d^{2} e^{2} x^{2} + 2 \, a b^{2} c d e^{2} x +{\left (a b^{2} c^{2} + 2 \, a b^{2}\right )} e^{2}\right )} \arcsin \left (d x + c\right )\right )} \sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{27 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.35458, size = 1173, normalized size = 4.99 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.33241, size = 655, normalized size = 2.79 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{2} - 1\right )}{\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right )^{3} e^{2}}{3 \, d} + \frac{{\left ({\left (d x + c\right )}^{2} - 1\right )}{\left (d x + c\right )} a b^{2} \arcsin \left (d x + c\right )^{2} e^{2}}{d} + \frac{{\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right )^{3} e^{2}}{3 \, d} - \frac{{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac{3}{2}} b^{3} \arcsin \left (d x + c\right )^{2} e^{2}}{3 \, d} + \frac{{\left (d x + c\right )}^{3} a^{3} e^{2}}{3 \, d} + \frac{{\left ({\left (d x + c\right )}^{2} - 1\right )}{\left (d x + c\right )} a^{2} b \arcsin \left (d x + c\right ) e^{2}}{d} - \frac{2 \,{\left ({\left (d x + c\right )}^{2} - 1\right )}{\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right ) e^{2}}{9 \, d} + \frac{{\left (d x + c\right )} a b^{2} \arcsin \left (d x + c\right )^{2} e^{2}}{d} - \frac{2 \,{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac{3}{2}} a b^{2} \arcsin \left (d x + c\right ) e^{2}}{3 \, d} + \frac{\sqrt{-{\left (d x + c\right )}^{2} + 1} b^{3} \arcsin \left (d x + c\right )^{2} e^{2}}{d} - \frac{2 \,{\left ({\left (d x + c\right )}^{2} - 1\right )}{\left (d x + c\right )} a b^{2} e^{2}}{9 \, d} + \frac{{\left (d x + c\right )} a^{2} b \arcsin \left (d x + c\right ) e^{2}}{d} - \frac{14 \,{\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right ) e^{2}}{9 \, d} - \frac{{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac{3}{2}} a^{2} b e^{2}}{3 \, d} + \frac{2 \,{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac{3}{2}} b^{3} e^{2}}{27 \, d} + \frac{2 \, \sqrt{-{\left (d x + c\right )}^{2} + 1} a b^{2} \arcsin \left (d x + c\right ) e^{2}}{d} - \frac{14 \,{\left (d x + c\right )} a b^{2} e^{2}}{9 \, d} + \frac{\sqrt{-{\left (d x + c\right )}^{2} + 1} a^{2} b e^{2}}{d} - \frac{14 \, \sqrt{-{\left (d x + c\right )}^{2} + 1} b^{3} e^{2}}{9 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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