Optimal. Leaf size=329 \[ -\frac{2 b d^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac{4 b d e x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac{8 b d e \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}-\frac{2 b e^2 x^4 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac{8 b e^2 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^3}-\frac{16 b e^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{75 c^5}+d^2 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{2}{3} d e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{5} e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{8 b^2 d e x}{9 c^2}-\frac{8 b^2 e^2 x^3}{225 c^2}+\frac{16 b^2 e^2 x}{75 c^4}+2 b^2 d^2 x+\frac{4}{27} b^2 d e x^3+\frac{2}{125} b^2 e^2 x^5 \]
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Rubi [A] time = 0.581781, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, 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 5706
Rule 5653
Rule 5717
Rule 8
Rule 5661
Rule 5758
Rule 30
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*}
Mathematica [A] time = 0.426275, size = 289, normalized size = 0.88 \[ \frac{225 a^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-30 a b \sqrt{c^2 x^2+1} \left (c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )-4 c^2 e \left (25 d+3 e x^2\right )+24 e^2\right )-30 b \sinh ^{-1}(c x) \left (b \sqrt{c^2 x^2+1} \left (c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )-4 c^2 e \left (25 d+3 e x^2\right )+24 e^2\right )-15 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )\right )+2 b^2 c x \left (c^4 \left (3375 d^2+250 d e x^2+27 e^2 x^4\right )-60 c^2 e \left (25 d+e x^2\right )+360 e^2\right )+225 b^2 c^5 x \sinh ^{-1}(c x)^2 \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{3375 c^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.071, size = 620, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19348, size = 579, normalized size = 1.76 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.92329, size = 845, normalized size = 2.57 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.18093, size = 595, normalized size = 1.81 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.55152, size = 656, normalized size = 1.99 \begin{align*} 2 \,{\left (x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{\sqrt{c^{2} x^{2} + 1}}{c}\right )} a b d^{2} +{\left (x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 2 \, c{\left (\frac{x}{c} - \frac{\sqrt{c^{2} x^{2} + 1} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{2}}\right )}\right )} b^{2} d^{2} + a^{2} d^{2} x + \frac{1}{1125} \,{\left (225 \, a^{2} x^{5} + 30 \,{\left (15 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{3 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} - 10 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} + 1}}{c^{5}}\right )} a b +{\left (225 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 2 \, c{\left (\frac{9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{5}} - \frac{15 \,{\left (3 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} - 10 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{6}}\right )}\right )} b^{2}\right )} e^{2} + \frac{2}{27} \,{\left (9 \, a^{2} d x^{3} + 6 \,{\left (3 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{c^{2} x^{2} + 1}}{c^{3}}\right )} a b d +{\left (9 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 2 \, c{\left (\frac{c^{2} x^{3} - 6 \, x}{c^{3}} - \frac{3 \,{\left ({\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{4}}\right )}\right )} b^{2} d\right )} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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