Optimal. Leaf size=83 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 c d^2-e (a e+b d)\right )}{2 d^{3/2} e^{5/2}}+\frac{x \left (a+\frac{d (c d-b e)}{e^2}\right )}{2 d \left (d+e x^2\right )}+\frac{c x}{e^2} \]
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Rubi [A] time = 0.0929843, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1157, 388, 205} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 c d^2-e (a e+b d)\right )}{2 d^{3/2} e^{5/2}}+\frac{x \left (a+\frac{d (c d-b e)}{e^2}\right )}{2 d \left (d+e x^2\right )}+\frac{c x}{e^2} \]
Antiderivative was successfully verified.
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Rule 1157
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{a+b x^2+c x^4}{\left (d+e x^2\right )^2} \, dx &=\frac{\left (a+\frac{d (c d-b e)}{e^2}\right ) x}{2 d \left (d+e x^2\right )}-\frac{\int \frac{\frac{c d^2-e (b d+a e)}{e^2}-\frac{2 c d x^2}{e}}{d+e x^2} \, dx}{2 d}\\ &=\frac{c x}{e^2}+\frac{\left (a+\frac{d (c d-b e)}{e^2}\right ) x}{2 d \left (d+e x^2\right )}-\frac{\left (3 c d^2-e (b d+a e)\right ) \int \frac{1}{d+e x^2} \, dx}{2 d e^2}\\ &=\frac{c x}{e^2}+\frac{\left (a+\frac{d (c d-b e)}{e^2}\right ) x}{2 d \left (d+e x^2\right )}-\frac{\left (3 c d^2-e (b d+a e)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} e^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0544369, size = 88, normalized size = 1.06 \[ \frac{x \left (a e^2-b d e+c d^2\right )}{2 d e^2 \left (d+e x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (-a e^2-b d e+3 c d^2\right )}{2 d^{3/2} e^{5/2}}+\frac{c x}{e^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 118, normalized size = 1.4 \begin{align*}{\frac{cx}{{e}^{2}}}+{\frac{xa}{2\,d \left ( e{x}^{2}+d \right ) }}-{\frac{xb}{2\,e \left ( e{x}^{2}+d \right ) }}+{\frac{dxc}{2\,{e}^{2} \left ( e{x}^{2}+d \right ) }}+{\frac{a}{2\,d}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{b}{2\,e}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{3\,cd}{2\,{e}^{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 = 1.84387, size = 541, normalized size = 6.52 \begin{align*} \left [\frac{4 \, c d^{2} e^{2} x^{3} +{\left (3 \, c d^{3} - b d^{2} e - a d e^{2} +{\left (3 \, c d^{2} e - b d e^{2} - a e^{3}\right )} x^{2}\right )} \sqrt{-d e} \log \left (\frac{e x^{2} - 2 \, \sqrt{-d e} x - d}{e x^{2} + d}\right ) + 2 \,{\left (3 \, c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} x}{4 \,{\left (d^{2} e^{4} x^{2} + d^{3} e^{3}\right )}}, \frac{2 \, c d^{2} e^{2} x^{3} -{\left (3 \, c d^{3} - b d^{2} e - a d e^{2} +{\left (3 \, c d^{2} e - b d e^{2} - a e^{3}\right )} x^{2}\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) +{\left (3 \, c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} x}{2 \,{\left (d^{2} e^{4} x^{2} + d^{3} e^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.0066, size = 153, normalized size = 1.84 \begin{align*} \frac{c x}{e^{2}} + \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{2 d^{2} e^{2} + 2 d e^{3} x^{2}} - \frac{\sqrt{- \frac{1}{d^{3} e^{5}}} \left (a e^{2} + b d e - 3 c d^{2}\right ) \log{\left (- d^{2} e^{2} \sqrt{- \frac{1}{d^{3} e^{5}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{d^{3} e^{5}}} \left (a e^{2} + b d e - 3 c d^{2}\right ) \log{\left (d^{2} e^{2} \sqrt{- \frac{1}{d^{3} e^{5}}} + x \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12464, size = 101, normalized size = 1.22 \begin{align*} c x e^{\left (-2\right )} - \frac{{\left (3 \, c d^{2} - b d e - a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{5}{2}\right )}}{2 \, d^{\frac{3}{2}}} + \frac{{\left (c d^{2} x - b d x e + a x e^{2}\right )} e^{\left (-2\right )}}{2 \,{\left (x^{2} e + d\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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