Optimal. Leaf size=368 \[ 2 b^2 d^3 x-\frac {4 b^2 d e^2 x}{3 c^2}+\frac {3}{4} b^2 d^2 e x^2-\frac {3 b^2 e^3 x^2}{32 c^2}+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4-\frac {2 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {4 b d e^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac {3 b d^2 e x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}+\frac {3 b e^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^3}-\frac {2 b d e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {b e^3 x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac {d^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}+\frac {3 d^2 e \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{32 c^4}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.49, antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5828, 5838,
5783, 5798, 8, 5812, 30} \begin {gather*} -\frac {3 e^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{32 c^4}-\frac {2 b d^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac {3 b d^2 e x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}+\frac {3 d^2 e \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2}-\frac {2 b d e^2 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {b e^3 x^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac {4 b d e^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}+\frac {3 b e^3 x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^3}-\frac {d^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}-\frac {4 b^2 d e^2 x}{3 c^2}-\frac {3 b^2 e^3 x^2}{32 c^2}+2 b^2 d^3 x+\frac {3}{4} b^2 d^2 e x^2+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 30
Rule 5783
Rule 5798
Rule 5812
Rule 5828
Rule 5838
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}-\frac {(b c) \int \frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{2 e}\\ &=\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}-\frac {(b c) \int \left (\frac {d^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {4 d^3 e x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {6 d^2 e^2 x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {4 d e^3 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {e^4 x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\right ) \, dx}{2 e}\\ &=\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}-\left (2 b c d^3\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\frac {\left (b c d^4\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 e}-\left (3 b c d^2 e\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\left (2 b c d e^2\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{2} \left (b c e^3\right ) \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {2 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac {3 b d^2 e x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}-\frac {2 b d e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {b e^3 x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac {d^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}+\left (2 b^2 d^3\right ) \int 1 \, dx+\frac {1}{2} \left (3 b^2 d^2 e\right ) \int x \, dx+\frac {\left (3 b d^2 e\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 c}+\frac {1}{3} \left (2 b^2 d e^2\right ) \int x^2 \, dx+\frac {\left (4 b d e^2\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{3 c}+\frac {1}{8} \left (b^2 e^3\right ) \int x^3 \, dx+\frac {\left (3 b e^3\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{8 c}\\ &=2 b^2 d^3 x+\frac {3}{4} b^2 d^2 e x^2+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4-\frac {2 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {4 b d e^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac {3 b d^2 e x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}+\frac {3 b e^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^3}-\frac {2 b d e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {b e^3 x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac {d^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}+\frac {3 d^2 e \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}-\frac {\left (4 b^2 d e^2\right ) \int 1 \, dx}{3 c^2}-\frac {\left (3 b e^3\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{16 c^3}-\frac {\left (3 b^2 e^3\right ) \int x \, dx}{16 c^2}\\ &=2 b^2 d^3 x-\frac {4 b^2 d e^2 x}{3 c^2}+\frac {3}{4} b^2 d^2 e x^2-\frac {3 b^2 e^3 x^2}{32 c^2}+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4-\frac {2 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {4 b d e^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac {3 b d^2 e x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}+\frac {3 b e^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^3}-\frac {2 b d e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {b e^3 x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac {d^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}+\frac {3 d^2 e \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{32 c^4}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.28, size = 354, normalized size = 0.96 \begin {gather*} \frac {c \left (72 a^2 c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-6 a b \sqrt {1+c^2 x^2} \left (-e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )+b^2 c x \left (-3 e^2 (128 d+9 e x)+c^2 \left (576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right )\right )\right )-6 b \left (-3 a \left (24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )+b c \sqrt {1+c^2 x^2} \left (-e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )\right ) \sinh ^{-1}(c x)+9 b^2 \left (24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \sinh ^{-1}(c x)^2}{288 c^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{3} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 586, normalized size = 1.59 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} \operatorname {arsinh}\left (c x\right )^{2} e^{3} + b^{2} d x^{3} \operatorname {arsinh}\left (c x\right )^{2} e^{2} + \frac {3}{2} \, b^{2} d^{2} x^{2} \operatorname {arsinh}\left (c x\right )^{2} e + b^{2} d^{3} x \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac {3}{2} \, a^{2} d^{2} x^{2} e + 2 \, b^{2} d^{3} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d^{3} x + \frac {3}{2} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} a b d^{2} e + \frac {3}{4} \, {\left (c^{2} {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{4}}\right )} - 2 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )} \operatorname {arsinh}\left (c x\right )\right )} b^{2} d^{2} e + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d^{3}}{c} + \frac {2}{3} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d e^{2} - \frac {2}{9} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} d e^{2} + \frac {1}{16} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} a b e^{3} + \frac {1}{32} \, {\left ({\left (\frac {x^{4}}{c^{2}} - \frac {3 \, x^{2}}{c^{4}} + \frac {3 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{6}}\right )} c^{2} - 2 \, {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c \operatorname {arsinh}\left (c x\right )\right )} b^{2} e^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1185 vs.
\(2 (321) = 642\).
time = 0.36, size = 1185, normalized size = 3.22 \begin {gather*} \frac {216 \, {\left (2 \, a^{2} + b^{2}\right )} c^{4} d^{2} x^{2} \cosh \left (1\right ) + 288 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{4} d^{3} x + 9 \, {\left ({\left (8 \, a^{2} + b^{2}\right )} c^{4} x^{4} - 3 \, b^{2} c^{2} x^{2}\right )} \cosh \left (1\right )^{3} + 9 \, {\left ({\left (8 \, a^{2} + b^{2}\right )} c^{4} x^{4} - 3 \, b^{2} c^{2} x^{2}\right )} \sinh \left (1\right )^{3} + 32 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} d x^{3} - 12 \, b^{2} c^{2} d x\right )} \cosh \left (1\right )^{2} + 9 \, {\left (32 \, b^{2} c^{4} d x^{3} \cosh \left (1\right )^{2} + 32 \, b^{2} c^{4} d^{3} x + {\left (8 \, b^{2} c^{4} x^{4} - 3 \, b^{2}\right )} \cosh \left (1\right )^{3} + {\left (8 \, b^{2} c^{4} x^{4} - 3 \, b^{2}\right )} \sinh \left (1\right )^{3} + {\left (32 \, b^{2} c^{4} d x^{3} + 3 \, {\left (8 \, b^{2} c^{4} x^{4} - 3 \, b^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 24 \, {\left (2 \, b^{2} c^{4} d^{2} x^{2} + b^{2} c^{2} d^{2}\right )} \cosh \left (1\right ) + {\left (64 \, b^{2} c^{4} d x^{3} \cosh \left (1\right ) + 48 \, b^{2} c^{4} d^{2} x^{2} + 24 \, b^{2} c^{2} d^{2} + 3 \, {\left (8 \, b^{2} c^{4} x^{4} - 3 \, b^{2}\right )} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + {\left (32 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} d x^{3} - 384 \, b^{2} c^{2} d x + 27 \, {\left ({\left (8 \, a^{2} + b^{2}\right )} c^{4} x^{4} - 3 \, b^{2} c^{2} x^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 6 \, {\left (96 \, a b c^{4} d x^{3} \cosh \left (1\right )^{2} + 96 \, a b c^{4} d^{3} x + 3 \, {\left (8 \, a b c^{4} x^{4} - 3 \, a b\right )} \cosh \left (1\right )^{3} + 3 \, {\left (8 \, a b c^{4} x^{4} - 3 \, a b\right )} \sinh \left (1\right )^{3} + 3 \, {\left (32 \, a b c^{4} d x^{3} + 3 \, {\left (8 \, a b c^{4} x^{4} - 3 \, a b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 72 \, {\left (2 \, a b c^{4} d^{2} x^{2} + a b c^{2} d^{2}\right )} \cosh \left (1\right ) + 3 \, {\left (64 \, a b c^{4} d x^{3} \cosh \left (1\right ) + 48 \, a b c^{4} d^{2} x^{2} + 24 \, a b c^{2} d^{2} + 3 \, {\left (8 \, a b c^{4} x^{4} - 3 \, a b\right )} \cosh \left (1\right )^{2}\right )} \sinh \left (1\right ) - {\left (72 \, b^{2} c^{3} d^{2} x \cosh \left (1\right ) + 96 \, b^{2} c^{3} d^{3} + 3 \, {\left (2 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \cosh \left (1\right )^{3} + 3 \, {\left (2 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \sinh \left (1\right )^{3} + 32 \, {\left (b^{2} c^{3} d x^{2} - 2 \, b^{2} c d\right )} \cosh \left (1\right )^{2} + {\left (32 \, b^{2} c^{3} d x^{2} - 64 \, b^{2} c d + 9 \, {\left (2 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + {\left (72 \, b^{2} c^{3} d^{2} x + 9 \, {\left (2 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \cosh \left (1\right )^{2} + 64 \, {\left (b^{2} c^{3} d x^{2} - 2 \, b^{2} c d\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (216 \, {\left (2 \, a^{2} + b^{2}\right )} c^{4} d^{2} x^{2} + 27 \, {\left ({\left (8 \, a^{2} + b^{2}\right )} c^{4} x^{4} - 3 \, b^{2} c^{2} x^{2}\right )} \cosh \left (1\right )^{2} + 64 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} d x^{3} - 12 \, b^{2} c^{2} d x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right ) - 6 \, {\left (72 \, a b c^{3} d^{2} x \cosh \left (1\right ) + 96 \, a b c^{3} d^{3} + 3 \, {\left (2 \, a b c^{3} x^{3} - 3 \, a b c x\right )} \cosh \left (1\right )^{3} + 3 \, {\left (2 \, a b c^{3} x^{3} - 3 \, a b c x\right )} \sinh \left (1\right )^{3} + 32 \, {\left (a b c^{3} d x^{2} - 2 \, a b c d\right )} \cosh \left (1\right )^{2} + {\left (32 \, a b c^{3} d x^{2} - 64 \, a b c d + 9 \, {\left (2 \, a b c^{3} x^{3} - 3 \, a b c x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + {\left (72 \, a b c^{3} d^{2} x + 9 \, {\left (2 \, a b c^{3} x^{3} - 3 \, a b c x\right )} \cosh \left (1\right )^{2} + 64 \, {\left (a b c^{3} d x^{2} - 2 \, a b c d\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}}{288 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 743 vs.
\(2 (364) = 728\).
time = 0.47, size = 743, normalized size = 2.02 \begin {gather*} \begin {cases} a^{2} d^{3} x + \frac {3 a^{2} d^{2} e x^{2}}{2} + a^{2} d e^{2} x^{3} + \frac {a^{2} e^{3} x^{4}}{4} + 2 a b d^{3} x \operatorname {asinh}{\left (c x \right )} + 3 a b d^{2} e x^{2} \operatorname {asinh}{\left (c x \right )} + 2 a b d e^{2} x^{3} \operatorname {asinh}{\left (c x \right )} + \frac {a b e^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {2 a b d^{3} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {3 a b d^{2} e x \sqrt {c^{2} x^{2} + 1}}{2 c} - \frac {2 a b d e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{3 c} - \frac {a b e^{3} x^{3} \sqrt {c^{2} x^{2} + 1}}{8 c} + \frac {3 a b d^{2} e \operatorname {asinh}{\left (c x \right )}}{2 c^{2}} + \frac {4 a b d e^{2} \sqrt {c^{2} x^{2} + 1}}{3 c^{3}} + \frac {3 a b e^{3} x \sqrt {c^{2} x^{2} + 1}}{16 c^{3}} - \frac {3 a b e^{3} \operatorname {asinh}{\left (c x \right )}}{16 c^{4}} + b^{2} d^{3} x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d^{3} x + \frac {3 b^{2} d^{2} e x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {3 b^{2} d^{2} e x^{2}}{4} + b^{2} d e^{2} x^{3} \operatorname {asinh}^{2}{\left (c x \right )} + \frac {2 b^{2} d e^{2} x^{3}}{9} + \frac {b^{2} e^{3} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {b^{2} e^{3} x^{4}}{32} - \frac {2 b^{2} d^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {3 b^{2} d^{2} e x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2 c} - \frac {2 b^{2} d e^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c} - \frac {b^{2} e^{3} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{8 c} + \frac {3 b^{2} d^{2} e \operatorname {asinh}^{2}{\left (c x \right )}}{4 c^{2}} - \frac {4 b^{2} d e^{2} x}{3 c^{2}} - \frac {3 b^{2} e^{3} x^{2}}{32 c^{2}} + \frac {4 b^{2} d e^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c^{3}} + \frac {3 b^{2} e^{3} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{16 c^{3}} - \frac {3 b^{2} e^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________