Optimal. Leaf size=184 \[ \frac{x \left (5 a^2+\frac{2 a c d^2}{e^2}+\frac{29 c^2 d^4}{e^4}\right )}{16 d^3 \left (d+e x^2\right )}+\frac{x \left (5 a^2-\frac{14 a c d^2}{e^2}-\frac{19 c^2 d^4}{e^4}\right )}{24 d^2 \left (d+e x^2\right )^2}-\frac{\left (-5 a^2 e^4-2 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{16 d^{7/2} e^{9/2}}+\frac{x \left (a e^2+c d^2\right )^2}{6 d e^4 \left (d+e x^2\right )^3}+\frac{c^2 x}{e^4} \]
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Rubi [A] time = 0.296625, antiderivative size = 184, 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, 388, 205} \[ \frac{x \left (5 a^2+\frac{2 a c d^2}{e^2}+\frac{29 c^2 d^4}{e^4}\right )}{16 d^3 \left (d+e x^2\right )}+\frac{x \left (5 a^2-\frac{14 a c d^2}{e^2}-\frac{19 c^2 d^4}{e^4}\right )}{24 d^2 \left (d+e x^2\right )^2}-\frac{\left (-5 a^2 e^4-2 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{16 d^{7/2} e^{9/2}}+\frac{x \left (a e^2+c d^2\right )^2}{6 d e^4 \left (d+e x^2\right )^3}+\frac{c^2 x}{e^4} \]
Antiderivative was successfully verified.
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Rule 1158
Rule 1814
Rule 1157
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+c x^4\right )^2}{\left (d+e x^2\right )^4} \, dx &=\frac{\left (c d^2+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}-\frac{\int \frac{-5 a^2+\frac{c^2 d^4}{e^4}+\frac{2 a c d^2}{e^2}-\frac{6 c d \left (c d^2+2 a e^2\right ) x^2}{e^3}+\frac{6 c^2 d^2 x^4}{e^2}-\frac{6 c^2 d x^6}{e}}{\left (d+e x^2\right )^3} \, dx}{6 d}\\ &=\frac{\left (c d^2+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}+\frac{\left (5 a^2-\frac{19 c^2 d^4}{e^4}-\frac{14 a c d^2}{e^2}\right ) x}{24 d^2 \left (d+e x^2\right )^2}+\frac{\int \frac{3 \left (5 a^2+\frac{5 c^2 d^4}{e^4}+\frac{2 a c d^2}{e^2}\right )-\frac{48 c^2 d^3 x^2}{e^3}+\frac{24 c^2 d^2 x^4}{e^2}}{\left (d+e x^2\right )^2} \, dx}{24 d^2}\\ &=\frac{\left (c d^2+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}+\frac{\left (5 a^2-\frac{19 c^2 d^4}{e^4}-\frac{14 a c d^2}{e^2}\right ) x}{24 d^2 \left (d+e x^2\right )^2}+\frac{\left (5 a^2+\frac{29 c^2 d^4}{e^4}+\frac{2 a c d^2}{e^2}\right ) x}{16 d^3 \left (d+e x^2\right )}-\frac{\int \frac{-3 \left (5 a^2-\frac{19 c^2 d^4}{e^4}+\frac{2 a c d^2}{e^2}\right )-\frac{48 c^2 d^3 x^2}{e^3}}{d+e x^2} \, dx}{48 d^3}\\ &=\frac{c^2 x}{e^4}+\frac{\left (c d^2+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}+\frac{\left (5 a^2-\frac{19 c^2 d^4}{e^4}-\frac{14 a c d^2}{e^2}\right ) x}{24 d^2 \left (d+e x^2\right )^2}+\frac{\left (5 a^2+\frac{29 c^2 d^4}{e^4}+\frac{2 a c d^2}{e^2}\right ) x}{16 d^3 \left (d+e x^2\right )}-\frac{\left (35 c^2 d^4-2 a c d^2 e^2-5 a^2 e^4\right ) \int \frac{1}{d+e x^2} \, dx}{16 d^3 e^4}\\ &=\frac{c^2 x}{e^4}+\frac{\left (c d^2+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}+\frac{\left (5 a^2-\frac{19 c^2 d^4}{e^4}-\frac{14 a c d^2}{e^2}\right ) x}{24 d^2 \left (d+e x^2\right )^2}+\frac{\left (5 a^2+\frac{29 c^2 d^4}{e^4}+\frac{2 a c d^2}{e^2}\right ) x}{16 d^3 \left (d+e x^2\right )}-\frac{\left (35 c^2 d^4-2 a c d^2 e^2-5 a^2 e^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{16 d^{7/2} e^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.14253, size = 174, normalized size = 0.95 \[ \frac{x \left (a^2 e^4 \left (33 d^2+40 d e x^2+15 e^2 x^4\right )-2 a c d^2 e^2 \left (3 d^2+8 d e x^2-3 e^2 x^4\right )+c^2 d^3 \left (280 d^2 e x^2+105 d^3+231 d e^2 x^4+48 e^3 x^6\right )\right )}{48 d^3 e^4 \left (d+e x^2\right )^3}-\frac{\left (-5 a^2 e^4-2 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{16 d^{7/2} e^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 262, normalized size = 1.4 \begin{align*}{\frac{{c}^{2}x}{{e}^{4}}}+{\frac{5\,{e}^{2}{x}^{5}{a}^{2}}{16\, \left ( e{x}^{2}+d \right ) ^{3}{d}^{3}}}+{\frac{ac{x}^{5}}{8\, \left ( e{x}^{2}+d \right ) ^{3}d}}+{\frac{29\,d{x}^{5}{c}^{2}}{16\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{5\,{a}^{2}e{x}^{3}}{6\, \left ( e{x}^{2}+d \right ) ^{3}{d}^{2}}}-{\frac{a{x}^{3}c}{3\,e \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{17\,{d}^{2}{x}^{3}{c}^{2}}{6\,{e}^{3} \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{11\,{a}^{2}x}{16\, \left ( e{x}^{2}+d \right ) ^{3}d}}-{\frac{adxc}{8\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{19\,{d}^{3}x{c}^{2}}{16\,{e}^{4} \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{5\,{a}^{2}}{16\,{d}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{ac}{8\,d{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{35\,{c}^{2}d}{16\,{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 = 2.00367, size = 1366, normalized size = 7.42 \begin{align*} \left [\frac{96 \, c^{2} d^{4} e^{4} x^{7} + 6 \,{\left (77 \, c^{2} d^{5} e^{3} + 2 \, a c d^{3} e^{5} + 5 \, a^{2} d e^{7}\right )} x^{5} + 16 \,{\left (35 \, c^{2} d^{6} e^{2} - 2 \, a c d^{4} e^{4} + 5 \, a^{2} d^{2} e^{6}\right )} x^{3} + 3 \,{\left (35 \, c^{2} d^{7} - 2 \, a c d^{5} e^{2} - 5 \, a^{2} d^{3} e^{4} +{\left (35 \, c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} - 5 \, a^{2} e^{7}\right )} x^{6} + 3 \,{\left (35 \, c^{2} d^{5} e^{2} - 2 \, a c d^{3} e^{4} - 5 \, a^{2} d e^{6}\right )} x^{4} + 3 \,{\left (35 \, c^{2} d^{6} e - 2 \, a c d^{4} e^{3} - 5 \, a^{2} d^{2} 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^{7} e - 2 \, a c d^{5} e^{3} + 11 \, a^{2} d^{3} e^{5}\right )} x}{96 \,{\left (d^{4} e^{8} x^{6} + 3 \, d^{5} e^{7} x^{4} + 3 \, d^{6} e^{6} x^{2} + d^{7} e^{5}\right )}}, \frac{48 \, c^{2} d^{4} e^{4} x^{7} + 3 \,{\left (77 \, c^{2} d^{5} e^{3} + 2 \, a c d^{3} e^{5} + 5 \, a^{2} d e^{7}\right )} x^{5} + 8 \,{\left (35 \, c^{2} d^{6} e^{2} - 2 \, a c d^{4} e^{4} + 5 \, a^{2} d^{2} e^{6}\right )} x^{3} - 3 \,{\left (35 \, c^{2} d^{7} - 2 \, a c d^{5} e^{2} - 5 \, a^{2} d^{3} e^{4} +{\left (35 \, c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} - 5 \, a^{2} e^{7}\right )} x^{6} + 3 \,{\left (35 \, c^{2} d^{5} e^{2} - 2 \, a c d^{3} e^{4} - 5 \, a^{2} d e^{6}\right )} x^{4} + 3 \,{\left (35 \, c^{2} d^{6} e - 2 \, a c d^{4} e^{3} - 5 \, a^{2} d^{2} e^{5}\right )} x^{2}\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) + 3 \,{\left (35 \, c^{2} d^{7} e - 2 \, a c d^{5} e^{3} + 11 \, a^{2} d^{3} e^{5}\right )} x}{48 \,{\left (d^{4} e^{8} x^{6} + 3 \, d^{5} e^{7} x^{4} + 3 \, d^{6} e^{6} x^{2} + d^{7} e^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.05803, size = 292, normalized size = 1.59 \begin{align*} \frac{c^{2} x}{e^{4}} - \frac{\sqrt{- \frac{1}{d^{7} e^{9}}} \left (5 a^{2} e^{4} + 2 a c d^{2} e^{2} - 35 c^{2} d^{4}\right ) \log{\left (- d^{4} e^{4} \sqrt{- \frac{1}{d^{7} e^{9}}} + x \right )}}{32} + \frac{\sqrt{- \frac{1}{d^{7} e^{9}}} \left (5 a^{2} e^{4} + 2 a c d^{2} e^{2} - 35 c^{2} d^{4}\right ) \log{\left (d^{4} e^{4} \sqrt{- \frac{1}{d^{7} e^{9}}} + x \right )}}{32} + \frac{x^{5} \left (15 a^{2} e^{6} + 6 a c d^{2} e^{4} + 87 c^{2} d^{4} e^{2}\right ) + x^{3} \left (40 a^{2} d e^{5} - 16 a c d^{3} e^{3} + 136 c^{2} d^{5} e\right ) + x \left (33 a^{2} d^{2} e^{4} - 6 a c d^{4} e^{2} + 57 c^{2} d^{6}\right )}{48 d^{6} e^{4} + 144 d^{5} e^{5} x^{2} + 144 d^{4} e^{6} x^{4} + 48 d^{3} e^{7} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15817, size = 225, normalized size = 1.22 \begin{align*} c^{2} x e^{\left (-4\right )} - \frac{{\left (35 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - 5 \, a^{2} e^{4}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{9}{2}\right )}}{16 \, d^{\frac{7}{2}}} + \frac{{\left (87 \, c^{2} d^{4} x^{5} e^{2} + 136 \, c^{2} d^{5} x^{3} e + 6 \, a c d^{2} x^{5} e^{4} + 57 \, c^{2} d^{6} x - 16 \, a c d^{3} x^{3} e^{3} + 15 \, a^{2} x^{5} e^{6} - 6 \, a c d^{4} x e^{2} + 40 \, a^{2} d x^{3} e^{5} + 33 \, a^{2} d^{2} x e^{4}\right )} e^{\left (-4\right )}}{48 \,{\left (x^{2} e + d\right )}^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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