Optimal. Leaf size=89 \[ \frac{5 x}{16 d^3 \left (d+e x^2\right )}+\frac{x}{8 d^2 \left (d+e x^2\right )^2}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{16 d^{7/2} \sqrt{e}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{7/2} \sqrt{e}} \]
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Rubi [A] time = 0.0827575, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1150, 414, 527, 522, 208, 205} \[ \frac{5 x}{16 d^3 \left (d+e x^2\right )}+\frac{x}{8 d^2 \left (d+e x^2\right )^2}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{16 d^{7/2} \sqrt{e}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{7/2} \sqrt{e}} \]
Antiderivative was successfully verified.
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Rule 1150
Rule 414
Rule 527
Rule 522
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )} \, dx &=\int \frac{1}{\left (d-e x^2\right ) \left (d+e x^2\right )^3} \, dx\\ &=\frac{x}{8 d^2 \left (d+e x^2\right )^2}-\frac{\int \frac{-7 d e+3 e^2 x^2}{\left (d-e x^2\right ) \left (d+e x^2\right )^2} \, dx}{8 d^2 e}\\ &=\frac{x}{8 d^2 \left (d+e x^2\right )^2}+\frac{5 x}{16 d^3 \left (d+e x^2\right )}+\frac{\int \frac{18 d^2 e^2-10 d e^3 x^2}{\left (d-e x^2\right ) \left (d+e x^2\right )} \, dx}{32 d^4 e^2}\\ &=\frac{x}{8 d^2 \left (d+e x^2\right )^2}+\frac{5 x}{16 d^3 \left (d+e x^2\right )}+\frac{\int \frac{1}{d-e x^2} \, dx}{8 d^3}+\frac{7 \int \frac{1}{d+e x^2} \, dx}{16 d^3}\\ &=\frac{x}{8 d^2 \left (d+e x^2\right )^2}+\frac{5 x}{16 d^3 \left (d+e x^2\right )}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{16 d^{7/2} \sqrt{e}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{7/2} \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.0584231, size = 76, normalized size = 0.85 \[ \frac{\frac{\sqrt{d} x \left (7 d+5 e x^2\right )}{\left (d+e x^2\right )^2}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}}{16 d^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 73, normalized size = 0.8 \begin{align*}{\frac{5\,e{x}^{3}}{16\,{d}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{7\,x}{16\,{d}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{7}{16\,{d}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{1}{8\,{d}^{3}}{\it Artanh} \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \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 [B] time = 1.82766, size = 608, normalized size = 6.83 \begin{align*} \left [\frac{5 \, d e^{2} x^{3} + 7 \, d^{2} e x + 7 \,{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) +{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt{d e} \log \left (\frac{e x^{2} + 2 \, \sqrt{d e} x + d}{e x^{2} - d}\right )}{16 \,{\left (d^{4} e^{3} x^{4} + 2 \, d^{5} e^{2} x^{2} + d^{6} e\right )}}, \frac{10 \, d e^{2} x^{3} + 14 \, d^{2} e x - 4 \,{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt{-d e} \arctan \left (\frac{\sqrt{-d e} x}{d}\right ) - 7 \,{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt{-d e} \log \left (\frac{e x^{2} - 2 \, \sqrt{-d e} x - d}{e x^{2} + d}\right )}{32 \,{\left (d^{4} e^{3} x^{4} + 2 \, d^{5} e^{2} x^{2} + d^{6} e\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.90657, size = 255, normalized size = 2.87 \begin{align*} - \frac{\sqrt{\frac{1}{d^{7} e}} \log{\left (- \frac{20 d^{11} e \left (\frac{1}{d^{7} e}\right )^{\frac{3}{2}}}{371} - \frac{351 d^{4} \sqrt{\frac{1}{d^{7} e}}}{371} + x \right )}}{16} + \frac{\sqrt{\frac{1}{d^{7} e}} \log{\left (\frac{20 d^{11} e \left (\frac{1}{d^{7} e}\right )^{\frac{3}{2}}}{371} + \frac{351 d^{4} \sqrt{\frac{1}{d^{7} e}}}{371} + x \right )}}{16} - \frac{7 \sqrt{- \frac{1}{d^{7} e}} \log{\left (- \frac{245 d^{11} e \left (- \frac{1}{d^{7} e}\right )^{\frac{3}{2}}}{106} - \frac{351 d^{4} \sqrt{- \frac{1}{d^{7} e}}}{106} + x \right )}}{32} + \frac{7 \sqrt{- \frac{1}{d^{7} e}} \log{\left (\frac{245 d^{11} e \left (- \frac{1}{d^{7} e}\right )^{\frac{3}{2}}}{106} + \frac{351 d^{4} \sqrt{- \frac{1}{d^{7} e}}}{106} + x \right )}}{32} + \frac{7 d x + 5 e x^{3}}{16 d^{5} + 32 d^{4} e x^{2} + 16 d^{3} e^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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