Optimal. Leaf size=179 \[ -\frac{a+b \text{sech}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}+\frac{2 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{3 \sqrt{d} e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt{d+e x^2}} \]
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Rubi [A] time = 0.263708, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {266, 43, 6301, 12, 573, 152, 93, 207} \[ -\frac{a+b \text{sech}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}+\frac{2 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{3 \sqrt{d} e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 6301
Rule 12
Rule 573
Rule 152
Rule 93
Rule 207
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \text{sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \text{sech}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-2 d-3 e x^2}{3 e^2 x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \text{sech}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-2 d-3 e x^2}{x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 e^2}\\ &=\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \text{sech}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{-2 d-3 e x}{x \sqrt{1-c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e^2}\\ &=\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \text{sech}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{d \left (c^2 d+e\right )}{x \sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{3 d e^2 \left (c^2 d+e\right )}\\ &=\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \text{sech}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{3 e^2}\\ &=\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \text{sech}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}-\frac{\left (2 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{-d+x^2} \, dx,x,\frac{\sqrt{d+e x^2}}{\sqrt{1-c^2 x^2}}\right )}{3 e^2}\\ &=\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{a+b \text{sech}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{2 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{3 \sqrt{d} e^2}\\ \end{align*}
Mathematica [A] time = 0.369211, size = 218, normalized size = 1.22 \[ \frac{-a \left (c^2 d+e\right ) \left (2 d+3 e x^2\right )-b \left (c^2 d+e\right ) \text{sech}^{-1}(c x) \left (2 d+3 e x^2\right )+b e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (d+e x^2\right )}{3 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}-\frac{2 b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \sqrt{-d-e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right )}{3 \sqrt{d} e^2 (c x-1) \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.405, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+b{\rm arcsech} \left (cx\right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{3} \, a{\left (\frac{3 \, x^{2}}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} e} + \frac{2 \, d}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} e^{2}}\right )} + b \int \frac{x^{3} \log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.21954, size = 1635, normalized size = 9.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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