Optimal. Leaf size=368 \[ -\frac {3 e^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{32 c^4}-\frac {2 b d^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac {3 b d^2 e x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}+\frac {3 d^2 e \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2}-\frac {2 b d e^2 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {b e^3 x^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac {4 b d e^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}+\frac {3 b e^3 x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^3}-\frac {d^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}-\frac {4 b^2 d e^2 x}{3 c^2}-\frac {3 b^2 e^3 x^2}{32 c^2}+2 b^2 d^3 x+\frac {3}{4} b^2 d^2 e x^2+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4 \]
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Rubi [A] time = 0.76, 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 = {5801, 5821, 5675, 5717, 8, 5758, 30} \[ -\frac {3 b d^2 e x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}+\frac {3 d^2 e \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2}-\frac {2 b d^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {4 b d e^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac {2 b d e^2 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {b e^3 x^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac {3 b e^3 x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^3}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{32 c^4}-\frac {d^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}+\frac {(d+e x)^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 e}-\frac {4 b^2 d e^2 x}{3 c^2}-\frac {3 b^2 e^3 x^2}{32 c^2}+\frac {3}{4} b^2 d^2 e x^2+2 b^2 d^3 x+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4 \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5675
Rule 5717
Rule 5758
Rule 5801
Rule 5821
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*}
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Mathematica [A] time = 0.56, size = 354, normalized size = 0.96 \[ \frac {c \left (72 a^2 c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-6 a b \sqrt {c^2 x^2+1} \left (c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )-e^2 (64 d+9 e x)\right )+b^2 c x \left (c^2 \left (576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right )-3 e^2 (128 d+9 e x)\right )\right )-6 b \sinh ^{-1}(c x) \left (b c \sqrt {c^2 x^2+1} \left (c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )-e^2 (64 d+9 e x)\right )-3 a \left (8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+24 c^2 d^2 e-3 e^3\right )\right )+9 b^2 \sinh ^{-1}(c x)^2 \left (8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+24 c^2 d^2 e-3 e^3\right )}{288 c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 475, normalized size = 1.29 \[ \frac {9 \, {\left (8 \, a^{2} + b^{2}\right )} c^{4} e^{3} x^{4} + 32 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} d e^{2} x^{3} + 27 \, {\left (8 \, {\left (2 \, a^{2} + b^{2}\right )} c^{4} d^{2} e - b^{2} c^{2} e^{3}\right )} x^{2} + 9 \, {\left (8 \, b^{2} c^{4} e^{3} x^{4} + 32 \, b^{2} c^{4} d e^{2} x^{3} + 48 \, b^{2} c^{4} d^{2} e x^{2} + 32 \, b^{2} c^{4} d^{3} x + 24 \, b^{2} c^{2} d^{2} e - 3 \, b^{2} e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 96 \, {\left (3 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{4} d^{3} - 4 \, b^{2} c^{2} d e^{2}\right )} x + 6 \, {\left (24 \, a b c^{4} e^{3} x^{4} + 96 \, a b c^{4} d e^{2} x^{3} + 144 \, a b c^{4} d^{2} e x^{2} + 96 \, a b c^{4} d^{3} x + 72 \, a b c^{2} d^{2} e - 9 \, a b e^{3} - {\left (6 \, b^{2} c^{3} e^{3} x^{3} + 32 \, b^{2} c^{3} d e^{2} x^{2} + 96 \, b^{2} c^{3} d^{3} - 64 \, b^{2} c d e^{2} + 9 \, {\left (8 \, b^{2} c^{3} d^{2} e - b^{2} c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (6 \, a b c^{3} e^{3} x^{3} + 32 \, a b c^{3} d e^{2} x^{2} + 96 \, a b c^{3} d^{3} - 64 \, a b c d e^{2} + 9 \, {\left (8 \, a b c^{3} d^{2} e - a b c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} + 1}}{288 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 641, normalized size = 1.74 \[ \frac {\frac {\left (c e x +c d \right )^{4} a^{2}}{4 c^{3} e}+\frac {b^{2} \left (c^{3} d^{3} \left (\arcsinh \left (c x \right )^{2} c x -2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+\frac {3 c^{2} d^{2} e \left (2 \arcsinh \left (c x \right )^{2} c^{2} x^{2}-2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x +\arcsinh \left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4}+\frac {c d \,e^{2} \left (9 \arcsinh \left (c x \right )^{2} c^{3} x^{3}-6 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+27 \arcsinh \left (c x \right )^{2} c x +2 c^{3} x^{3}-42 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+42 c x \right )}{9}+\frac {e^{3} \left (8 \arcsinh \left (c x \right )^{2} c^{4} x^{4}-4 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+16 \arcsinh \left (c x \right )^{2} c^{2} x^{2}+c^{4} x^{4}-10 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x +5 \arcsinh \left (c x \right )^{2}+5 c^{2} x^{2}+4\right )}{32}-3 c d \,e^{2} \left (\arcsinh \left (c x \right )^{2} c x -2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )-\frac {e^{3} \left (2 \arcsinh \left (c x \right )^{2} c^{2} x^{2}-2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x +\arcsinh \left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4}\right )}{c^{3}}+\frac {2 a b \left (\frac {e^{3} \arcsinh \left (c x \right ) c^{4} x^{4}}{4}+e^{2} \arcsinh \left (c x \right ) c^{4} x^{3} d +\frac {3 e \arcsinh \left (c x \right ) c^{4} x^{2} d^{2}}{2}+\arcsinh \left (c x \right ) c^{4} x \,d^{3}+\frac {\arcsinh \left (c x \right ) c^{4} d^{4}}{4 e}-\frac {e^{4} \left (\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{8}+\frac {3 \arcsinh \left (c x \right )}{8}\right )+4 c d \,e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+6 c^{2} d^{2} e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\arcsinh \left (c x \right )}{2}\right )+4 c^{3} d^{3} e \sqrt {c^{2} x^{2}+1}+c^{4} d^{4} \arcsinh \left (c x \right )}{4 e}\right )}{c^{3}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 590, normalized size = 1.60 \[ \frac {1}{4} \, b^{2} e^{3} x^{4} \operatorname {arsinh}\left (c x\right )^{2} + b^{2} d e^{2} x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{4} \, a^{2} e^{3} x^{4} + \frac {3}{2} \, b^{2} d^{2} e x^{2} \operatorname {arsinh}\left (c x\right )^{2} + a^{2} d e^{2} x^{3} + b^{2} d^{3} x \operatorname {arsinh}\left (c x\right )^{2} + \frac {3}{2} \, a^{2} d^{2} e x^{2} + \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}{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} + 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 {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d^{3}}{c} \]
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 \[ \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.94, size = 743, normalized size = 2.02 \[ \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} \]
Verification of antiderivative is not currently implemented for this CAS.
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