Optimal. Leaf size=227 \[ -\frac {4}{3} a b^2 e^2 x+\frac {14 b^3 e^2 \sqrt {1+(c+d x)^2}}{9 d}-\frac {2 b^3 e^2 \left (1+(c+d x)^2\right )^{3/2}}{27 d}-\frac {4 b^3 e^2 (c+d x) \sinh ^{-1}(c+d x)}{3 d}+\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}+\frac {2 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d} \]
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Rubi [A]
time = 0.21, antiderivative size = 227, normalized size of antiderivative = 1.00, 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 = {5859, 12,
5776, 5812, 5798, 5772, 267, 272, 45} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 267
Rule 272
Rule 5772
Rule 5776
Rule 5798
Rule 5812
Rule 5859
Rubi steps
\begin {align*} \int (c e+d e x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 0.20, size = 258, normalized size = 1.14 \begin {gather*} \frac {e^2 \left (-12 a b^2 (c+d x)+a \left (3 a^2+2 b^2\right ) (c+d x)^3+\frac {1}{3} b \sqrt {1+(c+d x)^2} \left (18 a^2+40 b^2-\left (9 a^2+2 b^2\right ) (c+d x)^2\right )-b \left (12 b^2 (c+d x)-9 a^2 (c+d x)^3-2 b^2 (c+d x)^3-12 a b \sqrt {1+(c+d x)^2}+6 a b (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)-3 b^2 \left (-3 a (c+d x)^3-2 b \sqrt {1+(c+d x)^2}+b (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)^2+3 b^3 (c+d x)^3 \sinh ^{-1}(c+d x)^3\right )}{9 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(679\) vs.
\(2(203)=406\).
time = 3.35, size = 680, normalized size = 3.00
method | result | size |
default | \(\frac {e^{2} \left (d x +c \right )^{3} a^{3}}{3 d}+\frac {b^{3} e^{2} \left (9 \arcsinh \left (d x +c \right )^{3} x^{3} d^{3}+27 \arcsinh \left (d x +c \right )^{3} x^{2} c \,d^{2}+27 \arcsinh \left (d x +c \right )^{3} x \,c^{2} d -9 \arcsinh \left (d x +c \right )^{2} \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x^{2} d^{2}+6 \arcsinh \left (d x +c \right ) x^{3} d^{3}+9 \arcsinh \left (d x +c \right )^{3} c^{3}-18 \arcsinh \left (d x +c \right )^{2} \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x c d +18 \arcsinh \left (d x +c \right ) x^{2} c \,d^{2}-9 \arcsinh \left (d x +c \right )^{2} \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, c^{2}+18 \arcsinh \left (d x +c \right ) x \,c^{2} d -2 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x^{2} d^{2}+6 \arcsinh \left (d x +c \right ) c^{3}-4 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x c d +18 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, \arcsinh \left (d x +c \right )^{2}-36 \arcsinh \left (d x +c \right ) x d -2 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, c^{2}-36 \arcsinh \left (d x +c \right ) c +40 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\right )}{27 d}+\frac {a \,b^{2} e^{2} \left (9 \arcsinh \left (d x +c \right )^{2} x^{3} d^{3}+27 \arcsinh \left (d x +c \right )^{2} x^{2} c \,d^{2}+27 \arcsinh \left (d x +c \right )^{2} x \,c^{2} d -6 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x^{2} d^{2}+2 d^{3} x^{3}+9 \arcsinh \left (d x +c \right )^{2} c^{3}-12 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x c d +6 x^{2} c \,d^{2}-6 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, c^{2}+6 x \,c^{2} d +2 c^{3}+12 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}-12 d x -12 c \right )}{9 d}+\frac {3 a^{2} b \,e^{2} \left (\frac {\left (d x +c \right )^{3} \arcsinh \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d}\) | \(680\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1395 vs.
\(2 (195) = 390\).
time = 0.40, size = 1395, normalized size = 6.15 \begin {gather*} \frac {9 \, {\left ({\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \sinh \left (1\right )^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + 3 \, {\left ({\left (3 \, a^{3} + 2 \, a b^{2}\right )} d^{3} x^{3} + 3 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c d^{2} x^{2} - 3 \, {\left (4 \, a b^{2} - {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c^{2}\right )} d x\right )} \cosh \left (1\right )^{2} + 9 \, {\left (3 \, {\left (a b^{2} d^{3} x^{3} + 3 \, a b^{2} c d^{2} x^{2} + 3 \, a b^{2} c^{2} d x + a b^{2} c^{3}\right )} \cosh \left (1\right )^{2} + 6 \, {\left (a b^{2} d^{3} x^{3} + 3 \, a b^{2} c d^{2} x^{2} + 3 \, a b^{2} c^{2} d x + a b^{2} c^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + 3 \, {\left (a b^{2} d^{3} x^{3} + 3 \, a b^{2} c d^{2} x^{2} + 3 \, a b^{2} c^{2} d x + a b^{2} c^{3}\right )} \sinh \left (1\right )^{2} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2} - 2 \, b^{3}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2} - 2 \, b^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2} - 2 \, b^{3}\right )} \sinh \left (1\right )^{2}\right )}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 6 \, {\left ({\left (3 \, a^{3} + 2 \, a b^{2}\right )} d^{3} x^{3} + 3 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c d^{2} x^{2} - 3 \, {\left (4 \, a b^{2} - {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c^{2}\right )} d x\right )} \cosh \left (1\right ) \sinh \left (1\right ) + 3 \, {\left ({\left (3 \, a^{3} + 2 \, a b^{2}\right )} d^{3} x^{3} + 3 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c d^{2} x^{2} - 3 \, {\left (4 \, a b^{2} - {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c^{2}\right )} d x\right )} \sinh \left (1\right )^{2} + 3 \, {\left ({\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{3} x^{3} + 3 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d^{2} x^{2} - 12 \, b^{3} c + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{3} - 3 \, {\left (4 \, b^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} d x\right )} \cosh \left (1\right )^{2} + 2 \, {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{3} x^{3} + 3 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d^{2} x^{2} - 12 \, b^{3} c + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{3} - 3 \, {\left (4 \, b^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} d x\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{3} x^{3} + 3 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d^{2} x^{2} - 12 \, b^{3} c + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{3} - 3 \, {\left (4 \, b^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} d x\right )} \sinh \left (1\right )^{2} - 6 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left (a b^{2} d^{2} x^{2} + 2 \, a b^{2} c d x + a b^{2} c^{2} - 2 \, a b^{2}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (a b^{2} d^{2} x^{2} + 2 \, a b^{2} c d x + a b^{2} c^{2} - 2 \, a b^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (a b^{2} d^{2} x^{2} + 2 \, a b^{2} c d x + a b^{2} c^{2} - 2 \, a b^{2}\right )} \sinh \left (1\right )^{2}\right )}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{2} x^{2} + 2 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d x - 18 \, a^{2} b - 40 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} \cosh \left (1\right )^{2} + 2 \, {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{2} x^{2} + 2 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d x - 18 \, a^{2} b - 40 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{2} x^{2} + 2 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d x - 18 \, a^{2} b - 40 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} \sinh \left (1\right )^{2}\right )}}{27 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1173 vs.
\(2 (211) = 422\).
time = 0.56, size = 1173, normalized size = 5.17 \begin {gather*} \begin {cases} a^{3} c^{2} e^{2} x + a^{3} c d e^{2} x^{2} + \frac {a^{3} d^{2} e^{2} x^{3}}{3} + \frac {a^{2} b c^{3} e^{2} \operatorname {asinh}{\left (c + d x \right )}}{d} + 3 a^{2} b c^{2} e^{2} x \operatorname {asinh}{\left (c + d x \right )} - \frac {a^{2} b c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{3 d} + 3 a^{2} b c d e^{2} x^{2} \operatorname {asinh}{\left (c + d x \right )} - \frac {2 a^{2} b c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{3} + a^{2} b d^{2} e^{2} x^{3} \operatorname {asinh}{\left (c + d x \right )} - \frac {a^{2} b d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{3} + \frac {2 a^{2} b e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{3 d} + \frac {a b^{2} c^{3} e^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} + 3 a b^{2} c^{2} e^{2} x \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {2 a b^{2} c^{2} e^{2} x}{3} - \frac {2 a b^{2} c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{3 d} + 3 a b^{2} c d e^{2} x^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {2 a b^{2} c d e^{2} x^{2}}{3} - \frac {4 a b^{2} c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{3} + a b^{2} d^{2} e^{2} x^{3} \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {2 a b^{2} d^{2} e^{2} x^{3}}{9} - \frac {2 a b^{2} d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{3} - \frac {4 a b^{2} e^{2} x}{3} + \frac {4 a b^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{3 d} + \frac {b^{3} c^{3} e^{2} \operatorname {asinh}^{3}{\left (c + d x \right )}}{3 d} + \frac {2 b^{3} c^{3} e^{2} \operatorname {asinh}{\left (c + d x \right )}}{9 d} + b^{3} c^{2} e^{2} x \operatorname {asinh}^{3}{\left (c + d x \right )} + \frac {2 b^{3} c^{2} e^{2} x \operatorname {asinh}{\left (c + d x \right )}}{3} - \frac {b^{3} c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{3 d} - \frac {2 b^{3} c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{27 d} + b^{3} c d e^{2} x^{2} \operatorname {asinh}^{3}{\left (c + d x \right )} + \frac {2 b^{3} c d e^{2} x^{2} \operatorname {asinh}{\left (c + d x \right )}}{3} - \frac {2 b^{3} c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{3} - \frac {4 b^{3} c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{27} - \frac {4 b^{3} c e^{2} \operatorname {asinh}{\left (c + d x \right )}}{3 d} + \frac {b^{3} d^{2} e^{2} x^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}{3} + \frac {2 b^{3} d^{2} e^{2} x^{3} \operatorname {asinh}{\left (c + d x \right )}}{9} - \frac {b^{3} d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{3} - \frac {2 b^{3} d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{27} - \frac {4 b^{3} e^{2} x \operatorname {asinh}{\left (c + d x \right )}}{3} + \frac {2 b^{3} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{3 d} + \frac {40 b^{3} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{27 d} & \text {for}\: d \neq 0 \\c^{2} e^{2} x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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