Optimal. Leaf size=142 \[ \frac{x \left (a e^2-b d e+c d^2\right )}{4 d^3 \left (d+e x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (35 a e^2-15 b d e+3 c d^2\right )}{8 d^{9/2} \sqrt{e}}+\frac{x \left (3 c d^2-e (7 b d-11 a e)\right )}{8 d^4 \left (d+e x^2\right )}-\frac{b d-3 a e}{d^4 x}-\frac{a}{3 d^3 x^3} \]
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Rubi [A] time = 0.217412, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {1259, 1261, 205} \[ \frac{x \left (a e^2-b d e+c d^2\right )}{4 d^3 \left (d+e x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (35 a e^2-15 b d e+3 c d^2\right )}{8 d^{9/2} \sqrt{e}}+\frac{x \left (3 c d^2-e (7 b d-11 a e)\right )}{8 d^4 \left (d+e x^2\right )}-\frac{b d-3 a e}{d^4 x}-\frac{a}{3 d^3 x^3} \]
Antiderivative was successfully verified.
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Rule 1259
Rule 1261
Rule 205
Rubi steps
\begin{align*} \int \frac{a+b x^2+c x^4}{x^4 \left (d+e x^2\right )^3} \, dx &=\frac{\left (c d^2-b d e+a e^2\right ) x}{4 d^3 \left (d+e x^2\right )^2}+\frac{\int \frac{4 a d^2 e^2+4 d e^2 (b d-a e) x^2+3 e^2 \left (c d^2-b d e+a e^2\right ) x^4}{x^4 \left (d+e x^2\right )^2} \, dx}{4 d^3 e^2}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) x}{4 d^3 \left (d+e x^2\right )^2}+\frac{\left (3 c d^2-e (7 b d-11 a e)\right ) x}{8 d^4 \left (d+e x^2\right )}+\frac{\int \frac{8 a d^4 e^4+8 d^3 e^4 (b d-2 a e) x^2+d^2 e^4 \left (3 c d^2-e (7 b d-11 a e)\right ) x^4}{x^4 \left (d+e x^2\right )} \, dx}{8 d^6 e^4}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) x}{4 d^3 \left (d+e x^2\right )^2}+\frac{\left (3 c d^2-e (7 b d-11 a e)\right ) x}{8 d^4 \left (d+e x^2\right )}+\frac{\int \left (\frac{8 a d^3 e^4}{x^4}+\frac{8 d^2 e^4 (b d-3 a e)}{x^2}+\frac{d^2 e^4 \left (3 c d^2-15 b d e+35 a e^2\right )}{d+e x^2}\right ) \, dx}{8 d^6 e^4}\\ &=-\frac{a}{3 d^3 x^3}-\frac{b d-3 a e}{d^4 x}+\frac{\left (c d^2-b d e+a e^2\right ) x}{4 d^3 \left (d+e x^2\right )^2}+\frac{\left (3 c d^2-e (7 b d-11 a e)\right ) x}{8 d^4 \left (d+e x^2\right )}+\frac{\left (3 c d^2-15 b d e+35 a e^2\right ) \int \frac{1}{d+e x^2} \, dx}{8 d^4}\\ &=-\frac{a}{3 d^3 x^3}-\frac{b d-3 a e}{d^4 x}+\frac{\left (c d^2-b d e+a e^2\right ) x}{4 d^3 \left (d+e x^2\right )^2}+\frac{\left (3 c d^2-e (7 b d-11 a e)\right ) x}{8 d^4 \left (d+e x^2\right )}+\frac{\left (3 c d^2-15 b d e+35 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{9/2} \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.0858429, size = 141, normalized size = 0.99 \[ \frac{x \left (11 a e^2-7 b d e+3 c d^2\right )}{8 d^4 \left (d+e x^2\right )}+\frac{x \left (a e^2-b d e+c d^2\right )}{4 d^3 \left (d+e x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (35 a e^2-15 b d e+3 c d^2\right )}{8 d^{9/2} \sqrt{e}}+\frac{3 a e-b d}{d^4 x}-\frac{a}{3 d^3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 207, normalized size = 1.5 \begin{align*} -{\frac{a}{3\,{d}^{3}{x}^{3}}}+3\,{\frac{ae}{{d}^{4}x}}-{\frac{b}{{d}^{3}x}}+{\frac{11\,{x}^{3}a{e}^{3}}{8\,{d}^{4} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{7\,{x}^{3}b{e}^{2}}{8\,{d}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{3\,{x}^{3}ce}{8\,{d}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{13\,a{e}^{2}x}{8\,{d}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{9\,bex}{8\,{d}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{5\,cx}{8\,d \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{35\,a{e}^{2}}{8\,{d}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{15\,be}{8\,{d}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,c}{8\,{d}^{2}}\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 = 1.91339, size = 1031, normalized size = 7.26 \begin{align*} \left [\frac{6 \,{\left (3 \, c d^{3} e^{2} - 15 \, b d^{2} e^{3} + 35 \, a d e^{4}\right )} x^{6} - 16 \, a d^{4} e + 10 \,{\left (3 \, c d^{4} e - 15 \, b d^{3} e^{2} + 35 \, a d^{2} e^{3}\right )} x^{4} - 16 \,{\left (3 \, b d^{4} e - 7 \, a d^{3} e^{2}\right )} x^{2} - 3 \,{\left ({\left (3 \, c d^{2} e^{2} - 15 \, b d e^{3} + 35 \, a e^{4}\right )} x^{7} + 2 \,{\left (3 \, c d^{3} e - 15 \, b d^{2} e^{2} + 35 \, a d e^{3}\right )} x^{5} +{\left (3 \, c d^{4} - 15 \, b d^{3} e + 35 \, a d^{2} e^{2}\right )} x^{3}\right )} \sqrt{-d e} \log \left (\frac{e x^{2} - 2 \, \sqrt{-d e} x - d}{e x^{2} + d}\right )}{48 \,{\left (d^{5} e^{3} x^{7} + 2 \, d^{6} e^{2} x^{5} + d^{7} e x^{3}\right )}}, \frac{3 \,{\left (3 \, c d^{3} e^{2} - 15 \, b d^{2} e^{3} + 35 \, a d e^{4}\right )} x^{6} - 8 \, a d^{4} e + 5 \,{\left (3 \, c d^{4} e - 15 \, b d^{3} e^{2} + 35 \, a d^{2} e^{3}\right )} x^{4} - 8 \,{\left (3 \, b d^{4} e - 7 \, a d^{3} e^{2}\right )} x^{2} + 3 \,{\left ({\left (3 \, c d^{2} e^{2} - 15 \, b d e^{3} + 35 \, a e^{4}\right )} x^{7} + 2 \,{\left (3 \, c d^{3} e - 15 \, b d^{2} e^{2} + 35 \, a d e^{3}\right )} x^{5} +{\left (3 \, c d^{4} - 15 \, b d^{3} e + 35 \, a d^{2} e^{2}\right )} x^{3}\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right )}{24 \,{\left (d^{5} e^{3} x^{7} + 2 \, d^{6} e^{2} x^{5} + d^{7} e x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.40683, size = 214, normalized size = 1.51 \begin{align*} - \frac{\sqrt{- \frac{1}{d^{9} e}} \left (35 a e^{2} - 15 b d e + 3 c d^{2}\right ) \log{\left (- d^{5} \sqrt{- \frac{1}{d^{9} e}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{d^{9} e}} \left (35 a e^{2} - 15 b d e + 3 c d^{2}\right ) \log{\left (d^{5} \sqrt{- \frac{1}{d^{9} e}} + x \right )}}{16} + \frac{- 8 a d^{3} + x^{6} \left (105 a e^{3} - 45 b d e^{2} + 9 c d^{2} e\right ) + x^{4} \left (175 a d e^{2} - 75 b d^{2} e + 15 c d^{3}\right ) + x^{2} \left (56 a d^{2} e - 24 b d^{3}\right )}{24 d^{6} x^{3} + 48 d^{5} e x^{5} + 24 d^{4} e^{2} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11525, size = 173, normalized size = 1.22 \begin{align*} \frac{{\left (3 \, c d^{2} - 15 \, b d e + 35 \, a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{1}{2}\right )}}{8 \, d^{\frac{9}{2}}} + \frac{3 \, c d^{2} x^{3} e - 7 \, b d x^{3} e^{2} + 5 \, c d^{3} x + 11 \, a x^{3} e^{3} - 9 \, b d^{2} x e + 13 \, a d x e^{2}}{8 \,{\left (x^{2} e + d\right )}^{2} d^{4}} - \frac{3 \, b d x^{2} - 9 \, a x^{2} e + a d}{3 \, d^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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