Optimal. Leaf size=89 \[ -\frac{x \left (a e^2-b d e+c d^2\right )}{2 d^2 e \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e (b d-3 a e)+c d^2\right )}{2 d^{5/2} e^{3/2}}-\frac{a}{d^2 x} \]
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Rubi [A] time = 0.118452, antiderivative size = 86, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {1259, 453, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e (b d-3 a e)+c d^2\right )}{2 d^{5/2} e^{3/2}}-\frac{x \left (\frac{c}{e}-\frac{b d-a e}{d^2}\right )}{2 \left (d+e x^2\right )}-\frac{a}{d^2 x} \]
Antiderivative was successfully verified.
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Rule 1259
Rule 453
Rule 205
Rubi steps
\begin{align*} \int \frac{a+b x^2+c x^4}{x^2 \left (d+e x^2\right )^2} \, dx &=-\frac{\left (\frac{c}{e}-\frac{b d-a e}{d^2}\right ) x}{2 \left (d+e x^2\right )}-\frac{\int \frac{-2 a d e^2-e \left (c d^2+e (b d-a e)\right ) x^2}{x^2 \left (d+e x^2\right )} \, dx}{2 d^2 e^2}\\ &=-\frac{a}{d^2 x}-\frac{\left (\frac{c}{e}-\frac{b d-a e}{d^2}\right ) x}{2 \left (d+e x^2\right )}+\frac{1}{2} \left (\frac{c}{e}+\frac{b d-3 a e}{d^2}\right ) \int \frac{1}{d+e x^2} \, dx\\ &=-\frac{a}{d^2 x}-\frac{\left (\frac{c}{e}-\frac{b d-a e}{d^2}\right ) x}{2 \left (d+e x^2\right )}+\frac{\left (c d^2+e (b d-3 a e)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2} e^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.061662, size = 89, normalized size = 1. \[ -\frac{x \left (a e^2-b d e+c d^2\right )}{2 d^2 e \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (-3 a e^2+b d e+c d^2\right )}{2 d^{5/2} e^{3/2}}-\frac{a}{d^2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 121, normalized size = 1.4 \begin{align*} -{\frac{a}{{d}^{2}x}}-{\frac{exa}{2\,{d}^{2} \left ( e{x}^{2}+d \right ) }}+{\frac{xb}{2\,d \left ( e{x}^{2}+d \right ) }}-{\frac{xc}{2\,e \left ( e{x}^{2}+d \right ) }}-{\frac{3\,ae}{2\,{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{b}{2\,d}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{c}{2\,e}\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 = 1.89304, size = 555, normalized size = 6.24 \begin{align*} \left [-\frac{4 \, a d^{2} e^{2} + 2 \,{\left (c d^{3} e - b d^{2} e^{2} + 3 \, a d e^{3}\right )} x^{2} -{\left ({\left (c d^{2} e + b d e^{2} - 3 \, a e^{3}\right )} x^{3} +{\left (c d^{3} + b d^{2} e - 3 \, a d e^{2}\right )} x\right )} \sqrt{-d e} \log \left (\frac{e x^{2} + 2 \, \sqrt{-d e} x - d}{e x^{2} + d}\right )}{4 \,{\left (d^{3} e^{3} x^{3} + d^{4} e^{2} x\right )}}, -\frac{2 \, a d^{2} e^{2} +{\left (c d^{3} e - b d^{2} e^{2} + 3 \, a d e^{3}\right )} x^{2} -{\left ({\left (c d^{2} e + b d e^{2} - 3 \, a e^{3}\right )} x^{3} +{\left (c d^{3} + b d^{2} e - 3 \, a d e^{2}\right )} x\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right )}{2 \,{\left (d^{3} e^{3} x^{3} + d^{4} e^{2} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.33698, size = 155, normalized size = 1.74 \begin{align*} \frac{\sqrt{- \frac{1}{d^{5} e^{3}}} \left (3 a e^{2} - b d e - c d^{2}\right ) \log{\left (- d^{3} e \sqrt{- \frac{1}{d^{5} e^{3}}} + x \right )}}{4} - \frac{\sqrt{- \frac{1}{d^{5} e^{3}}} \left (3 a e^{2} - b d e - c d^{2}\right ) \log{\left (d^{3} e \sqrt{- \frac{1}{d^{5} e^{3}}} + x \right )}}{4} - \frac{2 a d e + x^{2} \left (3 a e^{2} - b d e + c d^{2}\right )}{2 d^{3} e x + 2 d^{2} e^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09488, size = 112, normalized size = 1.26 \begin{align*} \frac{{\left (c d^{2} + b d e - 3 \, a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{3}{2}\right )}}{2 \, d^{\frac{5}{2}}} - \frac{{\left (c d^{2} x^{2} - b d x^{2} e + 3 \, a x^{2} e^{2} + 2 \, a d e\right )} e^{\left (-1\right )}}{2 \,{\left (x^{3} e + d x\right )} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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