Optimal. Leaf size=64 \[ \frac{x}{c}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{3/2} \sqrt{e} \sqrt{c d-b e}} \]
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Rubi [A] time = 0.078468, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1149, 388, 208} \[ \frac{x}{c}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{3/2} \sqrt{e} \sqrt{c d-b e}} \]
Antiderivative was successfully verified.
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Rule 1149
Rule 388
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^2}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx &=\int \frac{d+e x^2}{\frac{-c d^2+b d e}{d}+c e x^2} \, dx\\ &=\frac{x}{c}-\frac{\left (-c d e+\frac{e \left (-c d^2+b d e\right )}{d}\right ) \int \frac{1}{\frac{-c d^2+b d e}{d}+c e x^2} \, dx}{c e}\\ &=\frac{x}{c}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{3/2} \sqrt{e} \sqrt{c d-b e}}\\ \end{align*}
Mathematica [A] time = 0.0551712, size = 63, normalized size = 0.98 \[ \frac{x}{c}-\frac{(b e-2 c d) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right )}{c^{3/2} \sqrt{e} \sqrt{b e-c d}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 79, normalized size = 1.2 \begin{align*}{\frac{x}{c}}-{\frac{be}{c}\arctan \left ({cex{\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}}+2\,{\frac{d}{\sqrt{ \left ( be-cd \right ) ce}}\arctan \left ({\frac{cex}{\sqrt{ \left ( be-cd \right ) ce}}} \right ) } \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 = 2.00839, size = 424, normalized size = 6.62 \begin{align*} \left [-\frac{\sqrt{c^{2} d e - b c e^{2}}{\left (2 \, c d - b e\right )} \log \left (\frac{c e x^{2} + c d - b e + 2 \, \sqrt{c^{2} d e - b c e^{2}} x}{c e x^{2} - c d + b e}\right ) - 2 \,{\left (c^{2} d e - b c e^{2}\right )} x}{2 \,{\left (c^{3} d e - b c^{2} e^{2}\right )}}, -\frac{\sqrt{-c^{2} d e + b c e^{2}}{\left (2 \, c d - b e\right )} \arctan \left (-\frac{\sqrt{-c^{2} d e + b c e^{2}} x}{c d - b e}\right ) -{\left (c^{2} d e - b c e^{2}\right )} x}{c^{3} d e - b c^{2} e^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.568459, size = 212, normalized size = 3.31 \begin{align*} \frac{\sqrt{- \frac{1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right ) \log{\left (x + \frac{- b c e \sqrt{- \frac{1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right ) + c^{2} d \sqrt{- \frac{1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right )}{b e - 2 c d} \right )}}{2} - \frac{\sqrt{- \frac{1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right ) \log{\left (x + \frac{b c e \sqrt{- \frac{1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right ) - c^{2} d \sqrt{- \frac{1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right )}{b e - 2 c d} \right )}}{2} + \frac{x}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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