Optimal. Leaf size=38 \[ \frac{4 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-3 d x-\frac{e x^3}{3} \]
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Rubi [A] time = 0.0337036, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1150, 390, 208} \[ \frac{4 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-3 d x-\frac{e x^3}{3} \]
Antiderivative was successfully verified.
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Rule 1150
Rule 390
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^3}{d^2-e^2 x^4} \, dx &=\int \frac{\left (d+e x^2\right )^2}{d-e x^2} \, dx\\ &=\int \left (-3 d-e x^2+\frac{4 d^2}{d-e x^2}\right ) \, dx\\ &=-3 d x-\frac{e x^3}{3}+\left (4 d^2\right ) \int \frac{1}{d-e x^2} \, dx\\ &=-3 d x-\frac{e x^3}{3}+\frac{4 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.0171963, size = 38, normalized size = 1. \[ \frac{4 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-3 d x-\frac{e x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 31, normalized size = 0.8 \begin{align*} -{\frac{e{x}^{3}}{3}}-3\,dx+4\,{\frac{{d}^{2}}{\sqrt{de}}{\it Artanh} \left ({\frac{ex}{\sqrt{de}}} \right ) } \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 = 1.8496, size = 201, normalized size = 5.29 \begin{align*} \left [-\frac{1}{3} \, e x^{3} + 2 \, d \sqrt{\frac{d}{e}} \log \left (\frac{e x^{2} + 2 \, e x \sqrt{\frac{d}{e}} + d}{e x^{2} - d}\right ) - 3 \, d x, -\frac{1}{3} \, e x^{3} - 4 \, d \sqrt{-\frac{d}{e}} \arctan \left (\frac{e x \sqrt{-\frac{d}{e}}}{d}\right ) - 3 \, d x\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.404648, size = 58, normalized size = 1.53 \begin{align*} - 3 d x - \frac{e x^{3}}{3} - 2 \sqrt{\frac{d^{3}}{e}} \log{\left (x - \frac{\sqrt{\frac{d^{3}}{e}}}{d} \right )} + 2 \sqrt{\frac{d^{3}}{e}} \log{\left (x + \frac{\sqrt{\frac{d^{3}}{e}}}{d} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1757, size = 166, normalized size = 4.37 \begin{align*} 2 \,{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} -{\left (d^{2}\right )}^{\frac{1}{4}}{\left | d \right |} e^{\frac{11}{2}}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{{\left (d^{2}\right )}^{\frac{1}{4}}}\right ) e^{\left (-6\right )} +{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{15}{2}} +{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{15}{2}}\right )} e^{\left (-8\right )} \log \left ({\left |{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} + x \right |}\right ) -{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} +{\left (d^{2}\right )}^{\frac{1}{4}}{\left | d \right |} e^{\frac{11}{2}}\right )} e^{\left (-6\right )} \log \left ({\left | -{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} + x \right |}\right ) - \frac{1}{3} \,{\left (x^{3} e^{7} + 9 \, d x e^{6}\right )} e^{\left (-6\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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