Optimal. Leaf size=223 \[ -\frac{x \left (-35 a^2 e^4-6 a c d^2 e^2+93 c^2 d^4\right )}{128 d^4 e^4 \left (d+e x^2\right )}+\frac{x \left (35 a^2+\frac{6 a c d^2}{e^2}+\frac{163 c^2 d^4}{e^4}\right )}{192 d^3 \left (d+e x^2\right )^2}+\frac{x \left (7 a^2-\frac{18 a c d^2}{e^2}-\frac{25 c^2 d^4}{e^4}\right )}{48 d^2 \left (d+e x^2\right )^3}+\frac{\left (35 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{128 d^{9/2} e^{9/2}}+\frac{x \left (a e^2+c d^2\right )^2}{8 d e^4 \left (d+e x^2\right )^4} \]
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Rubi [A] time = 0.338634, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {1158, 1814, 1157, 385, 205} \[ -\frac{x \left (-35 a^2 e^4-6 a c d^2 e^2+93 c^2 d^4\right )}{128 d^4 e^4 \left (d+e x^2\right )}+\frac{x \left (35 a^2+\frac{6 a c d^2}{e^2}+\frac{163 c^2 d^4}{e^4}\right )}{192 d^3 \left (d+e x^2\right )^2}+\frac{x \left (7 a^2-\frac{18 a c d^2}{e^2}-\frac{25 c^2 d^4}{e^4}\right )}{48 d^2 \left (d+e x^2\right )^3}+\frac{\left (35 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{128 d^{9/2} e^{9/2}}+\frac{x \left (a e^2+c d^2\right )^2}{8 d e^4 \left (d+e x^2\right )^4} \]
Antiderivative was successfully verified.
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Rule 1158
Rule 1814
Rule 1157
Rule 385
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+c x^4\right )^2}{\left (d+e x^2\right )^5} \, dx &=\frac{\left (c d^2+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}-\frac{\int \frac{-7 a^2+\frac{c^2 d^4}{e^4}+\frac{2 a c d^2}{e^2}-\frac{8 c d \left (c d^2+2 a e^2\right ) x^2}{e^3}+\frac{8 c^2 d^2 x^4}{e^2}-\frac{8 c^2 d x^6}{e}}{\left (d+e x^2\right )^4} \, dx}{8 d}\\ &=\frac{\left (c d^2+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}+\frac{\left (7 a^2-\frac{25 c^2 d^4}{e^4}-\frac{18 a c d^2}{e^2}\right ) x}{48 d^2 \left (d+e x^2\right )^3}+\frac{\int \frac{35 a^2+\frac{19 c^2 d^4}{e^4}+\frac{6 a c d^2}{e^2}-\frac{96 c^2 d^3 x^2}{e^3}+\frac{48 c^2 d^2 x^4}{e^2}}{\left (d+e x^2\right )^3} \, dx}{48 d^2}\\ &=\frac{\left (c d^2+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}+\frac{\left (7 a^2-\frac{25 c^2 d^4}{e^4}-\frac{18 a c d^2}{e^2}\right ) x}{48 d^2 \left (d+e x^2\right )^3}+\frac{\left (35 a^2+\frac{163 c^2 d^4}{e^4}+\frac{6 a c d^2}{e^2}\right ) x}{192 d^3 \left (d+e x^2\right )^2}-\frac{\int \frac{-3 \left (35 a^2-\frac{29 c^2 d^4}{e^4}+\frac{6 a c d^2}{e^2}\right )-\frac{192 c^2 d^3 x^2}{e^3}}{\left (d+e x^2\right )^2} \, dx}{192 d^3}\\ &=\frac{\left (c d^2+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}+\frac{\left (7 a^2-\frac{25 c^2 d^4}{e^4}-\frac{18 a c d^2}{e^2}\right ) x}{48 d^2 \left (d+e x^2\right )^3}+\frac{\left (35 a^2+\frac{163 c^2 d^4}{e^4}+\frac{6 a c d^2}{e^2}\right ) x}{192 d^3 \left (d+e x^2\right )^2}-\frac{\left (93 c^2 d^4-6 a c d^2 e^2-35 a^2 e^4\right ) x}{128 d^4 e^4 \left (d+e x^2\right )}+\frac{\left (35 c^2 d^4+6 a c d^2 e^2+35 a^2 e^4\right ) \int \frac{1}{d+e x^2} \, dx}{128 d^4 e^4}\\ &=\frac{\left (c d^2+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}+\frac{\left (7 a^2-\frac{25 c^2 d^4}{e^4}-\frac{18 a c d^2}{e^2}\right ) x}{48 d^2 \left (d+e x^2\right )^3}+\frac{\left (35 a^2+\frac{163 c^2 d^4}{e^4}+\frac{6 a c d^2}{e^2}\right ) x}{192 d^3 \left (d+e x^2\right )^2}-\frac{\left (93 c^2 d^4-6 a c d^2 e^2-35 a^2 e^4\right ) x}{128 d^4 e^4 \left (d+e x^2\right )}+\frac{\left (35 c^2 d^4+6 a c d^2 e^2+35 a^2 e^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{128 d^{9/2} e^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.190922, size = 200, normalized size = 0.9 \[ \frac{\frac{\sqrt{d} \sqrt{e} x \left (a^2 e^4 \left (511 d^2 e x^2+279 d^3+385 d e^2 x^4+105 e^3 x^6\right )-6 a c d^2 e^2 \left (11 d^2 e x^2+3 d^3-11 d e^2 x^4-3 e^3 x^6\right )-c^2 d^4 \left (385 d^2 e x^2+105 d^3+511 d e^2 x^4+279 e^3 x^6\right )\right )}{\left (d+e x^2\right )^4}+3 \left (35 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{384 d^{9/2} e^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 231, normalized size = 1. \begin{align*}{\frac{1}{ \left ( e{x}^{2}+d \right ) ^{4}} \left ({\frac{ \left ( 35\,{a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}-93\,{c}^{2}{d}^{4} \right ){x}^{7}}{128\,{d}^{4}e}}+{\frac{ \left ( 385\,{a}^{2}{e}^{4}+66\,ac{d}^{2}{e}^{2}-511\,{c}^{2}{d}^{4} \right ){x}^{5}}{384\,{d}^{3}{e}^{2}}}+{\frac{ \left ( 511\,{a}^{2}{e}^{4}-66\,ac{d}^{2}{e}^{2}-385\,{c}^{2}{d}^{4} \right ){x}^{3}}{384\,{d}^{2}{e}^{3}}}+{\frac{ \left ( 93\,{a}^{2}{e}^{4}-6\,ac{d}^{2}{e}^{2}-35\,{c}^{2}{d}^{4} \right ) x}{128\,d{e}^{4}}} \right ) }+{\frac{35\,{a}^{2}}{128\,{d}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,ac}{64\,{d}^{2}{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{35\,{c}^{2}}{128\,{e}^{4}}\arctan \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 [A] time = 1.93905, size = 1697, normalized size = 7.61 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.93223, size = 335, normalized size = 1.5 \begin{align*} - \frac{\sqrt{- \frac{1}{d^{9} e^{9}}} \left (35 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log{\left (- d^{5} e^{4} \sqrt{- \frac{1}{d^{9} e^{9}}} + x \right )}}{256} + \frac{\sqrt{- \frac{1}{d^{9} e^{9}}} \left (35 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log{\left (d^{5} e^{4} \sqrt{- \frac{1}{d^{9} e^{9}}} + x \right )}}{256} + \frac{x^{7} \left (105 a^{2} e^{7} + 18 a c d^{2} e^{5} - 279 c^{2} d^{4} e^{3}\right ) + x^{5} \left (385 a^{2} d e^{6} + 66 a c d^{3} e^{4} - 511 c^{2} d^{5} e^{2}\right ) + x^{3} \left (511 a^{2} d^{2} e^{5} - 66 a c d^{4} e^{3} - 385 c^{2} d^{6} e\right ) + x \left (279 a^{2} d^{3} e^{4} - 18 a c d^{5} e^{2} - 105 c^{2} d^{7}\right )}{384 d^{8} e^{4} + 1536 d^{7} e^{5} x^{2} + 2304 d^{6} e^{6} x^{4} + 1536 d^{5} e^{7} x^{6} + 384 d^{4} e^{8} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12563, size = 267, normalized size = 1.2 \begin{align*} \frac{{\left (35 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 35 \, a^{2} e^{4}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{9}{2}\right )}}{128 \, d^{\frac{9}{2}}} - \frac{{\left (279 \, c^{2} d^{4} x^{7} e^{3} + 511 \, c^{2} d^{5} x^{5} e^{2} - 18 \, a c d^{2} x^{7} e^{5} + 385 \, c^{2} d^{6} x^{3} e - 66 \, a c d^{3} x^{5} e^{4} + 105 \, c^{2} d^{7} x - 105 \, a^{2} x^{7} e^{7} + 66 \, a c d^{4} x^{3} e^{3} - 385 \, a^{2} d x^{5} e^{6} + 18 \, a c d^{5} x e^{2} - 511 \, a^{2} d^{2} x^{3} e^{5} - 279 \, a^{2} d^{3} x e^{4}\right )} e^{\left (-4\right )}}{384 \,{\left (x^{2} e + d\right )}^{4} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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