Optimal. Leaf size=329 \[ -\frac {2 b d^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac {4 b d e x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}-\frac {2 b e^2 x^4 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}-\frac {16 b e^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^5}+\frac {8 b d e \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}+\frac {8 b e^2 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^3}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {16 b^2 e^2 x}{75 c^4}-\frac {8 b^2 d e x}{9 c^2}-\frac {8 b^2 e^2 x^3}{225 c^2}+2 b^2 d^2 x+\frac {4}{27} b^2 d e x^3+\frac {2}{125} b^2 e^2 x^5 \]
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Rubi [A] time = 0.58, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5706, 5653, 5717, 8, 5661, 5758, 30} \[ -\frac {2 b d^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac {4 b d e x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac {8 b d e \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}-\frac {2 b e^2 x^4 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac {8 b e^2 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^3}-\frac {16 b e^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^5}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {8 b^2 d e x}{9 c^2}-\frac {8 b^2 e^2 x^3}{225 c^2}+\frac {16 b^2 e^2 x}{75 c^4}+2 b^2 d^2 x+\frac {4}{27} b^2 d e x^3+\frac {2}{125} b^2 e^2 x^5 \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5653
Rule 5661
Rule 5706
Rule 5717
Rule 5758
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 \left (a+b \sinh ^{-1}(c x)\right )^2+2 d e x^2 \left (a+b \sinh ^{-1}(c x)\right )^2+e^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+(2 d e) \int x^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+e^2 \int x^4 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx\\ &=d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2-\left (2 b c d^2\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{3} (4 b c d e) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{5} \left (2 b c e^2\right ) \int \frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {2 b d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac {4 b d e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}-\frac {2 b e^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\left (2 b^2 d^2\right ) \int 1 \, dx+\frac {1}{9} \left (4 b^2 d e\right ) \int x^2 \, dx+\frac {(8 b d e) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{9 c}+\frac {1}{25} \left (2 b^2 e^2\right ) \int x^4 \, dx+\frac {\left (8 b e^2\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{25 c}\\ &=2 b^2 d^2 x+\frac {4}{27} b^2 d e x^3+\frac {2}{125} b^2 e^2 x^5-\frac {2 b d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {8 b d e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}-\frac {4 b d e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac {8 b e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^3}-\frac {2 b e^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (8 b^2 d e\right ) \int 1 \, dx}{9 c^2}-\frac {\left (16 b e^2\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{75 c^3}-\frac {\left (8 b^2 e^2\right ) \int x^2 \, dx}{75 c^2}\\ &=2 b^2 d^2 x-\frac {8 b^2 d e x}{9 c^2}+\frac {4}{27} b^2 d e x^3-\frac {8 b^2 e^2 x^3}{225 c^2}+\frac {2}{125} b^2 e^2 x^5-\frac {2 b d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {8 b d e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}-\frac {16 b e^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^5}-\frac {4 b d e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac {8 b e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^3}-\frac {2 b e^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (16 b^2 e^2\right ) \int 1 \, dx}{75 c^4}\\ &=2 b^2 d^2 x-\frac {8 b^2 d e x}{9 c^2}+\frac {16 b^2 e^2 x}{75 c^4}+\frac {4}{27} b^2 d e x^3-\frac {8 b^2 e^2 x^3}{225 c^2}+\frac {2}{125} b^2 e^2 x^5-\frac {2 b d^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {8 b d e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}-\frac {16 b e^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^5}-\frac {4 b d e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac {8 b e^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^3}-\frac {2 b e^2 x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A] time = 0.43, size = 289, normalized size = 0.88 \[ \frac {225 a^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-30 a b \sqrt {c^2 x^2+1} \left (c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )-4 c^2 e \left (25 d+3 e x^2\right )+24 e^2\right )-30 b \sinh ^{-1}(c x) \left (b \sqrt {c^2 x^2+1} \left (c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )-4 c^2 e \left (25 d+3 e x^2\right )+24 e^2\right )-15 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )\right )+225 b^2 c^5 x \sinh ^{-1}(c x)^2 \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+2 b^2 c x \left (c^4 \left (3375 d^2+250 d e x^2+27 e^2 x^4\right )-60 c^2 e \left (25 d+e x^2\right )+360 e^2\right )}{3375 c^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 380, normalized size = 1.16 \[ \frac {27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{5} e^{2} x^{5} + 10 \, {\left (25 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{5} d e - 12 \, b^{2} c^{3} e^{2}\right )} x^{3} + 225 \, {\left (3 \, b^{2} c^{5} e^{2} x^{5} + 10 \, b^{2} c^{5} d e x^{3} + 15 \, b^{2} c^{5} d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 15 \, {\left (225 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{5} d^{2} - 200 \, b^{2} c^{3} d e + 48 \, b^{2} c e^{2}\right )} x + 30 \, {\left (45 \, a b c^{5} e^{2} x^{5} + 150 \, a b c^{5} d e x^{3} + 225 \, a b c^{5} d^{2} x - {\left (9 \, b^{2} c^{4} e^{2} x^{4} + 225 \, b^{2} c^{4} d^{2} - 100 \, b^{2} c^{2} d e + 24 \, b^{2} e^{2} + 2 \, {\left (25 \, b^{2} c^{4} d e - 6 \, b^{2} c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 30 \, {\left (9 \, a b c^{4} e^{2} x^{4} + 225 \, a b c^{4} d^{2} - 100 \, a b c^{2} d e + 24 \, a b e^{2} + 2 \, {\left (25 \, a b c^{4} d e - 6 \, a b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{3375 \, c^{5}} \]
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 [B] time = 0.10, size = 620, normalized size = 1.88 \[ \frac {\frac {a^{2} \left (\frac {1}{5} e^{2} c^{5} x^{5}+\frac {2}{3} c^{5} d e \,x^{3}+x \,c^{5} d^{2}\right )}{c^{4}}+\frac {b^{2} \left (d^{2} c^{4} \left (\arcsinh \left (c x \right )^{2} c x -2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+\frac {2 c^{2} d e \left (9 \arcsinh \left (c x \right )^{2} c^{3} x^{3}-6 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+27 \arcsinh \left (c x \right )^{2} c x +2 c^{3} x^{3}-42 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+42 c x \right )}{27}-2 c^{2} d e \left (\arcsinh \left (c x \right )^{2} c x -2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+\frac {e^{2} \left (675 \arcsinh \left (c x \right )^{2} c^{5} x^{5}-270 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2250 \arcsinh \left (c x \right )^{2} c^{3} x^{3}+54 c^{5} x^{5}-1140 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+3375 \arcsinh \left (c x \right )^{2} c x +380 c^{3} x^{3}-4470 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+4470 c x \right )}{3375}-\frac {2 e^{2} \left (9 \arcsinh \left (c x \right )^{2} c^{3} x^{3}-6 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+27 \arcsinh \left (c x \right )^{2} c x +2 c^{3} x^{3}-42 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+42 c x \right )}{27}+e^{2} \left (\arcsinh \left (c x \right )^{2} c x -2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c^{4}}+\frac {2 a b \left (\frac {\arcsinh \left (c x \right ) e^{2} c^{5} x^{5}}{5}+\frac {2 \arcsinh \left (c x \right ) c^{5} d e \,x^{3}}{3}+\arcsinh \left (c x \right ) c^{5} x \,d^{2}-\frac {e^{2} \left (\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )}{5}-\frac {2 c^{2} d e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-d^{2} c^{4} \sqrt {c^{2} x^{2}+1}\right )}{c^{4}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 429, normalized size = 1.30 \[ \frac {1}{5} \, b^{2} e^{2} x^{5} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{5} \, a^{2} e^{2} x^{5} + \frac {2}{3} \, b^{2} d e x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{3} \, a^{2} d e x^{3} + b^{2} d^{2} x \operatorname {arsinh}\left (c x\right )^{2} + \frac {4}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d e - \frac {4}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} d e + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b e^{2} - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} e^{2} + 2 \, b^{2} d^{2} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \]
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 \[ \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.92, size = 595, normalized size = 1.81 \[ \begin {cases} a^{2} d^{2} x + \frac {2 a^{2} d e x^{3}}{3} + \frac {a^{2} e^{2} x^{5}}{5} + 2 a b d^{2} x \operatorname {asinh}{\left (c x \right )} + \frac {4 a b d e x^{3} \operatorname {asinh}{\left (c x \right )}}{3} + \frac {2 a b e^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {2 a b d^{2} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {4 a b d e x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} - \frac {2 a b e^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{25 c} + \frac {8 a b d e \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} + \frac {8 a b e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{75 c^{3}} - \frac {16 a b e^{2} \sqrt {c^{2} x^{2} + 1}}{75 c^{5}} + b^{2} d^{2} x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d^{2} x + \frac {2 b^{2} d e x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {4 b^{2} d e x^{3}}{27} + \frac {b^{2} e^{2} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {2 b^{2} e^{2} x^{5}}{125} - \frac {2 b^{2} d^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {4 b^{2} d e x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c} - \frac {2 b^{2} e^{2} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{25 c} - \frac {8 b^{2} d e x}{9 c^{2}} - \frac {8 b^{2} e^{2} x^{3}}{225 c^{2}} + \frac {8 b^{2} d e \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c^{3}} + \frac {8 b^{2} e^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{75 c^{3}} + \frac {16 b^{2} e^{2} x}{75 c^{4}} - \frac {16 b^{2} e^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{75 c^{5}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{2} x + \frac {2 d e x^{3}}{3} + \frac {e^{2} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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