Optimal. Leaf size=155 \[ \frac{x \left (3 a^2-\frac{10 a c d^2}{e^2}-\frac{13 c^2 d^4}{e^4}\right )}{8 d^2 \left (d+e x^2\right )}+\frac{\left (3 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} e^{9/2}}+\frac{x \left (a e^2+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}-\frac{3 c^2 d x}{e^4}+\frac{c^2 x^3}{3 e^3} \]
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Rubi [A] time = 0.252468, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {1158, 1814, 1153, 205} \[ \frac{x \left (3 a^2-\frac{10 a c d^2}{e^2}-\frac{13 c^2 d^4}{e^4}\right )}{8 d^2 \left (d+e x^2\right )}+\frac{\left (3 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} e^{9/2}}+\frac{x \left (a e^2+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}-\frac{3 c^2 d x}{e^4}+\frac{c^2 x^3}{3 e^3} \]
Antiderivative was successfully verified.
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Rule 1158
Rule 1814
Rule 1153
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx &=\frac{\left (c d^2+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}-\frac{\int \frac{-3 a^2+\frac{c^2 d^4}{e^4}+\frac{2 a c d^2}{e^2}-\frac{4 c d \left (c d^2+2 a e^2\right ) x^2}{e^3}+\frac{4 c^2 d^2 x^4}{e^2}-\frac{4 c^2 d x^6}{e}}{\left (d+e x^2\right )^2} \, dx}{4 d}\\ &=\frac{\left (c d^2+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}+\frac{\left (3 a^2-\frac{13 c^2 d^4}{e^4}-\frac{10 a c d^2}{e^2}\right ) x}{8 d^2 \left (d+e x^2\right )}+\frac{\int \frac{3 a^2+\frac{11 c^2 d^4}{e^4}+\frac{6 a c d^2}{e^2}-\frac{16 c^2 d^3 x^2}{e^3}+\frac{8 c^2 d^2 x^4}{e^2}}{d+e x^2} \, dx}{8 d^2}\\ &=\frac{\left (c d^2+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}+\frac{\left (3 a^2-\frac{13 c^2 d^4}{e^4}-\frac{10 a c d^2}{e^2}\right ) x}{8 d^2 \left (d+e x^2\right )}+\frac{\int \left (-\frac{24 c^2 d^3}{e^4}+\frac{8 c^2 d^2 x^2}{e^3}+\frac{35 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4}{e^4 \left (d+e x^2\right )}\right ) \, dx}{8 d^2}\\ &=-\frac{3 c^2 d x}{e^4}+\frac{c^2 x^3}{3 e^3}+\frac{\left (c d^2+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}+\frac{\left (3 a^2-\frac{13 c^2 d^4}{e^4}-\frac{10 a c d^2}{e^2}\right ) x}{8 d^2 \left (d+e x^2\right )}+\frac{\left (35 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \int \frac{1}{d+e x^2} \, dx}{8 d^2 e^4}\\ &=-\frac{3 c^2 d x}{e^4}+\frac{c^2 x^3}{3 e^3}+\frac{\left (c d^2+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}+\frac{\left (3 a^2-\frac{13 c^2 d^4}{e^4}-\frac{10 a c d^2}{e^2}\right ) x}{8 d^2 \left (d+e x^2\right )}+\frac{\left (35 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} e^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.111127, size = 154, normalized size = 0.99 \[ \frac{x \left (3 a^2 e^4 \left (5 d+3 e x^2\right )-6 a c d^2 e^2 \left (3 d+5 e x^2\right )-c^2 d^2 \left (175 d^2 e x^2+105 d^3+56 d e^2 x^4-8 e^3 x^6\right )\right )}{24 d^2 e^4 \left (d+e x^2\right )^2}+\frac{\left (3 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} e^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 211, normalized size = 1.4 \begin{align*}{\frac{{c}^{2}{x}^{3}}{3\,{e}^{3}}}-3\,{\frac{{c}^{2}dx}{{e}^{4}}}+{\frac{3\,{a}^{2}e{x}^{3}}{8\, \left ( e{x}^{2}+d \right ) ^{2}{d}^{2}}}-{\frac{5\,a{x}^{3}c}{4\,e \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{13\,{d}^{2}{x}^{3}{c}^{2}}{8\,{e}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{5\,{a}^{2}x}{8\, \left ( e{x}^{2}+d \right ) ^{2}d}}-{\frac{3\,adxc}{4\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{11\,{d}^{3}x{c}^{2}}{8\,{e}^{4} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{3\,{a}^{2}}{8\,{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,ac}{4\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{35\,{c}^{2}{d}^{2}}{8\,{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 = 1.9364, size = 1071, normalized size = 6.91 \begin{align*} \left [\frac{16 \, c^{2} d^{3} e^{4} x^{7} - 112 \, c^{2} d^{4} e^{3} x^{5} - 2 \,{\left (175 \, c^{2} d^{5} e^{2} + 30 \, a c d^{3} e^{4} - 9 \, a^{2} d e^{6}\right )} x^{3} - 3 \,{\left (35 \, c^{2} d^{6} + 6 \, a c d^{4} e^{2} + 3 \, a^{2} d^{2} e^{4} +{\left (35 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + 3 \, a^{2} e^{6}\right )} x^{4} + 2 \,{\left (35 \, c^{2} d^{5} e + 6 \, a c d^{3} e^{3} + 3 \, a^{2} d e^{5}\right )} x^{2}\right )} \sqrt{-d e} \log \left (\frac{e x^{2} - 2 \, \sqrt{-d e} x - d}{e x^{2} + d}\right ) - 6 \,{\left (35 \, c^{2} d^{6} e + 6 \, a c d^{4} e^{3} - 5 \, a^{2} d^{2} e^{5}\right )} x}{48 \,{\left (d^{3} e^{7} x^{4} + 2 \, d^{4} e^{6} x^{2} + d^{5} e^{5}\right )}}, \frac{8 \, c^{2} d^{3} e^{4} x^{7} - 56 \, c^{2} d^{4} e^{3} x^{5} -{\left (175 \, c^{2} d^{5} e^{2} + 30 \, a c d^{3} e^{4} - 9 \, a^{2} d e^{6}\right )} x^{3} + 3 \,{\left (35 \, c^{2} d^{6} + 6 \, a c d^{4} e^{2} + 3 \, a^{2} d^{2} e^{4} +{\left (35 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + 3 \, a^{2} e^{6}\right )} x^{4} + 2 \,{\left (35 \, c^{2} d^{5} e + 6 \, a c d^{3} e^{3} + 3 \, a^{2} d e^{5}\right )} x^{2}\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) - 3 \,{\left (35 \, c^{2} d^{6} e + 6 \, a c d^{4} e^{3} - 5 \, a^{2} d^{2} e^{5}\right )} x}{24 \,{\left (d^{3} e^{7} x^{4} + 2 \, d^{4} e^{6} x^{2} + d^{5} e^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.85728, size = 257, normalized size = 1.66 \begin{align*} - \frac{3 c^{2} d x}{e^{4}} + \frac{c^{2} x^{3}}{3 e^{3}} - \frac{\sqrt{- \frac{1}{d^{5} e^{9}}} \left (3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log{\left (- d^{3} e^{4} \sqrt{- \frac{1}{d^{5} e^{9}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{d^{5} e^{9}}} \left (3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log{\left (d^{3} e^{4} \sqrt{- \frac{1}{d^{5} e^{9}}} + x \right )}}{16} + \frac{x^{3} \left (3 a^{2} e^{5} - 10 a c d^{2} e^{3} - 13 c^{2} d^{4} e\right ) + x \left (5 a^{2} d e^{4} - 6 a c d^{3} e^{2} - 11 c^{2} d^{5}\right )}{8 d^{4} e^{4} + 16 d^{3} e^{5} x^{2} + 8 d^{2} e^{6} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19466, size = 196, normalized size = 1.26 \begin{align*} \frac{1}{3} \,{\left (c^{2} x^{3} e^{6} - 9 \, c^{2} d x e^{5}\right )} e^{\left (-9\right )} + \frac{{\left (35 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{9}{2}\right )}}{8 \, d^{\frac{5}{2}}} - \frac{{\left (13 \, c^{2} d^{4} x^{3} e + 11 \, c^{2} d^{5} x + 10 \, a c d^{2} x^{3} e^{3} + 6 \, a c d^{3} x e^{2} - 3 \, a^{2} x^{3} e^{5} - 5 \, a^{2} d x e^{4}\right )} e^{\left (-4\right )}}{8 \,{\left (x^{2} e + d\right )}^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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