Optimal. Leaf size=124 \[ \frac{1}{7} c^2 d x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{5} d x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac{b d \left (c^2 x^2+1\right )^{7/2}}{49 c^5}+\frac{8 b d \left (c^2 x^2+1\right )^{5/2}}{175 c^5}-\frac{b d \left (c^2 x^2+1\right )^{3/2}}{105 c^5}-\frac{2 b d \sqrt{c^2 x^2+1}}{35 c^5} \]
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Rubi [A] time = 0.118849, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {14, 5730, 12, 446, 77} \[ \frac{1}{7} c^2 d x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{5} d x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac{b d \left (c^2 x^2+1\right )^{7/2}}{49 c^5}+\frac{8 b d \left (c^2 x^2+1\right )^{5/2}}{175 c^5}-\frac{b d \left (c^2 x^2+1\right )^{3/2}}{105 c^5}-\frac{2 b d \sqrt{c^2 x^2+1}}{35 c^5} \]
Antiderivative was successfully verified.
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Rule 14
Rule 5730
Rule 12
Rule 446
Rule 77
Rubi steps
\begin{align*} \int x^4 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{1}{5} d x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} c^2 d x^7 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac{d x^5 \left (7+5 c^2 x^2\right )}{35 \sqrt{1+c^2 x^2}} \, dx\\ &=\frac{1}{5} d x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} c^2 d x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{35} (b c d) \int \frac{x^5 \left (7+5 c^2 x^2\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=\frac{1}{5} d x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} c^2 d x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{70} (b c d) \operatorname{Subst}\left (\int \frac{x^2 \left (7+5 c^2 x\right )}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{5} d x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} c^2 d x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{70} (b c d) \operatorname{Subst}\left (\int \left (\frac{2}{c^4 \sqrt{1+c^2 x}}+\frac{\sqrt{1+c^2 x}}{c^4}-\frac{8 \left (1+c^2 x\right )^{3/2}}{c^4}+\frac{5 \left (1+c^2 x\right )^{5/2}}{c^4}\right ) \, dx,x,x^2\right )\\ &=-\frac{2 b d \sqrt{1+c^2 x^2}}{35 c^5}-\frac{b d \left (1+c^2 x^2\right )^{3/2}}{105 c^5}+\frac{8 b d \left (1+c^2 x^2\right )^{5/2}}{175 c^5}-\frac{b d \left (1+c^2 x^2\right )^{7/2}}{49 c^5}+\frac{1}{5} d x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{7} c^2 d x^7 \left (a+b \sinh ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.0937715, size = 87, normalized size = 0.7 \[ \frac{d \left (105 a x^5 \left (5 c^2 x^2+7\right )-\frac{b \sqrt{c^2 x^2+1} \left (75 c^6 x^6+57 c^4 x^4-76 c^2 x^2+152\right )}{c^5}+105 b x^5 \left (5 c^2 x^2+7\right ) \sinh ^{-1}(c x)\right )}{3675} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 124, normalized size = 1. \begin{align*}{\frac{1}{{c}^{5}} \left ( da \left ({\frac{{c}^{7}{x}^{7}}{7}}+{\frac{{c}^{5}{x}^{5}}{5}} \right ) +db \left ({\frac{{\it Arcsinh} \left ( cx \right ){c}^{7}{x}^{7}}{7}}+{\frac{{\it Arcsinh} \left ( cx \right ){c}^{5}{x}^{5}}{5}}-{\frac{{c}^{6}{x}^{6}}{49}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{19\,{c}^{4}{x}^{4}}{1225}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{76\,{c}^{2}{x}^{2}}{3675}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{152}{3675}\sqrt{{c}^{2}{x}^{2}+1}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20355, size = 248, normalized size = 2. \begin{align*} \frac{1}{7} \, a c^{2} d x^{7} + \frac{1}{5} \, a d x^{5} + \frac{1}{245} \,{\left (35 \, x^{7} \operatorname{arsinh}\left (c x\right ) -{\left (\frac{5 \, \sqrt{c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac{6 \, \sqrt{c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac{16 \, \sqrt{c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{2} d + \frac{1}{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 )} b d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.255, size = 265, normalized size = 2.14 \begin{align*} \frac{525 \, a c^{7} d x^{7} + 735 \, a c^{5} d x^{5} + 105 \,{\left (5 \, b c^{7} d x^{7} + 7 \, b c^{5} d x^{5}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) -{\left (75 \, b c^{6} d x^{6} + 57 \, b c^{4} d x^{4} - 76 \, b c^{2} d x^{2} + 152 \, b d\right )} \sqrt{c^{2} x^{2} + 1}}{3675 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.9766, size = 151, normalized size = 1.22 \begin{align*} \begin{cases} \frac{a c^{2} d x^{7}}{7} + \frac{a d x^{5}}{5} + \frac{b c^{2} d x^{7} \operatorname{asinh}{\left (c x \right )}}{7} - \frac{b c d x^{6} \sqrt{c^{2} x^{2} + 1}}{49} + \frac{b d x^{5} \operatorname{asinh}{\left (c x \right )}}{5} - \frac{19 b d x^{4} \sqrt{c^{2} x^{2} + 1}}{1225 c} + \frac{76 b d x^{2} \sqrt{c^{2} x^{2} + 1}}{3675 c^{3}} - \frac{152 b d \sqrt{c^{2} x^{2} + 1}}{3675 c^{5}} & \text{for}\: c \neq 0 \\\frac{a d x^{5}}{5} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.49798, size = 238, normalized size = 1.92 \begin{align*} \frac{1}{7} \, a c^{2} d x^{7} + \frac{1}{5} \, a d x^{5} + \frac{1}{245} \,{\left (35 \, x^{7} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{5 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{7}{2}} - 21 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} + 35 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 35 \, \sqrt{c^{2} x^{2} + 1}}{c^{7}}\right )} b c^{2} d + \frac{1}{75} \,{\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 )} b d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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