Optimal. Leaf size=227 \[ \frac{2 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}-\frac{4}{3} a b^2 e^2 x+\frac{2 b e^2 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}-\frac{b e^2 (c+d x)^2 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d}-\frac{2 b^3 e^2 \left ((c+d x)^2+1\right )^{3/2}}{27 d}+\frac{14 b^3 e^2 \sqrt{(c+d x)^2+1}}{9 d}-\frac{4 b^3 e^2 (c+d x) \sinh ^{-1}(c+d x)}{3 d} \]
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Rubi [A] time = 0.299909, antiderivative size = 227, 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 = {5865, 12, 5661, 5758, 5717, 5653, 261, 266, 43} \[ \frac{2 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}-\frac{4}{3} a b^2 e^2 x+\frac{2 b e^2 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}-\frac{b e^2 (c+d x)^2 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d}-\frac{2 b^3 e^2 \left ((c+d x)^2+1\right )^{3/2}}{27 d}+\frac{14 b^3 e^2 \sqrt{(c+d x)^2+1}}{9 d}-\frac{4 b^3 e^2 (c+d x) \sinh ^{-1}(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 12
Rule 5661
Rule 5758
Rule 5717
Rule 5653
Rule 261
Rule 266
Rule 43
Rubi steps
\begin{align*} \int (c e+d e x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int e^2 x^2 \left (a+b \sinh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int x^2 \left (a+b \sinh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{x^3 \left (a+b \sinh ^{-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 \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d}+\frac{\left (2 b e^2\right ) \operatorname{Subst}\left (\int \frac{x \left (a+b \sinh ^{-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 \sinh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d}\\ &=\frac{2 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}+\frac{2 b e^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-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 \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d}-\frac{\left (4 b^2 e^2\right ) \operatorname{Subst}\left (\int \left (a+b \sinh ^{-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 \sinh ^{-1}(c+d x)\right )}{9 d}+\frac{2 b e^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-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 \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-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 \sinh ^{-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) \sinh ^{-1}(c+d x)}{3 d}+\frac{2 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}+\frac{2 b e^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-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 \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-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) \sinh ^{-1}(c+d x)}{3 d}+\frac{2 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}+\frac{2 b e^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-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 \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d}\\ \end{align*}
Mathematica [A] time = 0.311478, size = 258, normalized size = 1.14 \[ \frac{e^2 \left (a \left (3 a^2+2 b^2\right ) (c+d x)^3+\frac{1}{3} b \sqrt{(c+d x)^2+1} \left (-\left (9 a^2+2 b^2\right ) (c+d x)^2+18 a^2+40 b^2\right )-b \sinh ^{-1}(c+d x) \left (-9 a^2 (c+d x)^3+6 a b \sqrt{(c+d x)^2+1} (c+d x)^2-12 a b \sqrt{(c+d x)^2+1}-2 b^2 (c+d x)^3+12 b^2 (c+d x)\right )-12 a b^2 (c+d x)-3 b^2 \sinh ^{-1}(c+d x)^2 \left (-3 a (c+d x)^3+b \sqrt{(c+d x)^2+1} (c+d x)^2-2 b \sqrt{(c+d x)^2+1}\right )+3 b^3 (c+d x)^3 \sinh ^{-1}(c+d x)^3\right )}{9 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 360, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ({\frac{ \left ( dx+c \right ) ^{3}{e}^{2}{a}^{3}}{3}}+{e}^{2}{b}^{3} \left ({\frac{ \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{3} \left ( dx+c \right ) \left ( 1+ \left ( dx+c \right ) ^{2} \right ) }{3}}-{\frac{ \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{3} \left ( dx+c \right ) }{3}}-{\frac{ \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ) ^{2}}{3}\sqrt{1+ \left ( dx+c \right ) ^{2}}}+{\frac{2\, \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{2}}{3}\sqrt{1+ \left ( dx+c \right ) ^{2}}}+{\frac{ \left ( 2\,dx+2\,c \right ) \left ( 1+ \left ( dx+c \right ) ^{2} \right ){\it Arcsinh} \left ( dx+c \right ) }{9}}-{\frac{ \left ( 14\,dx+14\,c \right ){\it Arcsinh} \left ( dx+c \right ) }{9}}-{\frac{2\, \left ( dx+c \right ) ^{2}}{27}\sqrt{1+ \left ( dx+c \right ) ^{2}}}+{\frac{40}{27}\sqrt{1+ \left ( dx+c \right ) ^{2}}} \right ) +3\,{e}^{2}a{b}^{2} \left ( 1/3\, \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ) \left ( 1+ \left ( dx+c \right ) ^{2} \right ) -1/3\, \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ) -2/9\,{\it Arcsinh} \left ( dx+c \right ) \left ( dx+c \right ) ^{2}\sqrt{1+ \left ( dx+c \right ) ^{2}}+4/9\,{\it Arcsinh} \left ( dx+c \right ) \sqrt{1+ \left ( dx+c \right ) ^{2}}+{\frac{ \left ( 2+2\, \left ( dx+c \right ) ^{2} \right ) \left ( dx+c \right ) }{27}}-{\frac{14\,dx}{27}}-{\frac{14\,c}{27}} \right ) +3\,{e}^{2}{a}^{2}b \left ( 1/3\, \left ( dx+c \right ) ^{3}{\it Arcsinh} \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 = 2.87802, size = 1285, normalized size = 5.66 \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 )} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + 9 \,{\left (3 \, a b^{2} d^{3} e^{2} x^{3} + 9 \, a b^{2} c d^{2} e^{2} x^{2} + 9 \, a b^{2} c^{2} d e^{2} x + 3 \, a b^{2} c^{3} e^{2} -{\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 )} \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\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} - 6 \,{\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 )} \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\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}\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 = 7.19988, size = 1173, normalized size = 5.17 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}^{2}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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