Optimal. Leaf size=51 \[ \frac{8 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-7 d^2 x-\frac{4}{3} d e x^3-\frac{1}{5} e^2 x^5 \]
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Rubi [A] time = 0.0410136, antiderivative size = 51, 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} \[ \frac{8 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-7 d^2 x-\frac{4}{3} d e x^3-\frac{1}{5} e^2 x^5 \]
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 )^4}{d^2-e^2 x^4} \, dx &=\int \frac{\left (d+e x^2\right )^3}{d-e x^2} \, dx\\ &=\int \left (-7 d^2-4 d e x^2-e^2 x^4+\frac{8 d^3}{d-e x^2}\right ) \, dx\\ &=-7 d^2 x-\frac{4}{3} d e x^3-\frac{e^2 x^5}{5}+\left (8 d^3\right ) \int \frac{1}{d-e x^2} \, dx\\ &=-7 d^2 x-\frac{4}{3} d e x^3-\frac{e^2 x^5}{5}+\frac{8 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.023367, size = 51, normalized size = 1. \[ \frac{8 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-7 d^2 x-\frac{4}{3} d e x^3-\frac{1}{5} e^2 x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 42, normalized size = 0.8 \begin{align*} -{\frac{{e}^{2}{x}^{5}}{5}}-{\frac{4\,de{x}^{3}}{3}}-7\,{d}^{2}x+8\,{\frac{{d}^{3}}{\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 = 1.84779, size = 255, normalized size = 5. \begin{align*} \left [-\frac{1}{5} \, e^{2} x^{5} - \frac{4}{3} \, d e x^{3} + 4 \, d^{2} \sqrt{\frac{d}{e}} \log \left (\frac{e x^{2} + 2 \, e x \sqrt{\frac{d}{e}} + d}{e x^{2} - d}\right ) - 7 \, d^{2} x, -\frac{1}{5} \, e^{2} x^{5} - \frac{4}{3} \, d e x^{3} - 8 \, d^{2} \sqrt{-\frac{d}{e}} \arctan \left (\frac{e x \sqrt{-\frac{d}{e}}}{d}\right ) - 7 \, d^{2} x\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.475067, size = 75, normalized size = 1.47 \begin{align*} - 7 d^{2} x - \frac{4 d e x^{3}}{3} - \frac{e^{2} x^{5}}{5} - 4 \sqrt{\frac{d^{5}}{e}} \log{\left (x - \frac{\sqrt{\frac{d^{5}}{e}}}{d^{2}} \right )} + 4 \sqrt{\frac{d^{5}}{e}} \log{\left (x + \frac{\sqrt{\frac{d^{5}}{e}}}{d^{2}} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18287, size = 194, normalized size = 3.8 \begin{align*} 4 \,{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d^{2} e^{\frac{11}{2}} -{\left (d^{2}\right )}^{\frac{1}{4}} d{\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 )} + 2 \,{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d^{2} e^{\frac{15}{2}} +{\left (d^{2}\right )}^{\frac{3}{4}} d 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 ) - 2 \,{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d^{2} e^{\frac{11}{2}} +{\left (d^{2}\right )}^{\frac{1}{4}} d{\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}{15} \,{\left (3 \, x^{5} e^{12} + 20 \, d x^{3} e^{11} + 105 \, d^{2} x e^{10}\right )} e^{\left (-10\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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