Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{x^3},x\right ) \]
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Rubi [A] time = 0.124839, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{x^3} \, dx &=\int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{x^3} \, dx\\ \end{align*}
Mathematica [A] time = 6.56331, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{x^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.937, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arcsech} \left (cx\right )}{{x}^{3}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a e x^{2} + a d +{\left (b e x^{2} + b d\right )} \operatorname{arsech}\left (c x\right )\right )} \sqrt{e x^{2} + d}}{x^{3}}, x\right ) \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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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