Optimal. Leaf size=79 \[ \frac{e^{3/2} \sqrt{d+e x^2}}{6 d^2 x}-\frac{\sqrt{e} \sqrt{d+e x^2}}{12 d x^3}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{4 x^4} \]
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Rubi [A] time = 0.0262587, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {6221, 271, 264} \[ \frac{e^{3/2} \sqrt{d+e x^2}}{6 d^2 x}-\frac{\sqrt{e} \sqrt{d+e x^2}}{12 d x^3}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 6221
Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{x^5} \, dx &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{4 x^4}+\frac{1}{4} \sqrt{e} \int \frac{1}{x^4 \sqrt{d+e x^2}} \, dx\\ &=-\frac{\sqrt{e} \sqrt{d+e x^2}}{12 d x^3}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{4 x^4}-\frac{e^{3/2} \int \frac{1}{x^2 \sqrt{d+e x^2}} \, dx}{6 d}\\ &=-\frac{\sqrt{e} \sqrt{d+e x^2}}{12 d x^3}+\frac{e^{3/2} \sqrt{d+e x^2}}{6 d^2 x}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{4 x^4}\\ \end{align*}
Mathematica [A] time = 0.0404816, size = 63, normalized size = 0.8 \[ \frac{\sqrt{e} x \sqrt{d+e x^2} \left (2 e x^2-d\right )-3 d^2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{12 d^2 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 62, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,{x}^{4}}{\it Artanh} \left ({x\sqrt{e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) }+{\frac{1}{4\,{d}^{2}x}{e}^{{\frac{3}{2}}}\sqrt{e{x}^{2}+d}}-{\frac{1}{12\,{d}^{2}{x}^{3}}\sqrt{e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973572, size = 82, normalized size = 1.04 \begin{align*} \frac{\sqrt{e x^{2} + d} e^{\frac{3}{2}}}{4 \, d^{2} x} - \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}} \sqrt{e}}{12 \, d^{2} x^{3}} - \frac{\operatorname{artanh}\left (\frac{\sqrt{e} x}{\sqrt{e x^{2} + d}}\right )}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24824, size = 162, normalized size = 2.05 \begin{align*} -\frac{3 \, d^{2} \log \left (\frac{2 \, e x^{2} + 2 \, \sqrt{e x^{2} + d} \sqrt{e} x + d}{d}\right ) - 2 \,{\left (2 \, e x^{3} - d x\right )} \sqrt{e x^{2} + d} \sqrt{e}}{24 \, d^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{x^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40209, size = 144, normalized size = 1.82 \begin{align*} \frac{{\left (3 \,{\left (x e^{\frac{1}{2}} - \sqrt{x^{2} e + d}\right )}^{2} d e - d^{2} e\right )} e}{3 \,{\left ({\left (x e^{\frac{1}{2}} - \sqrt{x^{2} e + d}\right )}^{2} - d\right )}^{3} d} - \frac{\log \left (-\frac{\frac{x e^{\frac{1}{2}}}{\sqrt{x^{2} e + d}} + 1}{\frac{x e^{\frac{1}{2}}}{\sqrt{x^{2} e + d}} - 1}\right )}{8 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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