Optimal. Leaf size=49 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{\sqrt{c} \sqrt{e} \sqrt{c d-b e}} \]
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Rubi [A] time = 0.0288674, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {1149, 208} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{\sqrt{c} \sqrt{e} \sqrt{c d-b e}} \]
Antiderivative was successfully verified.
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Rule 1149
Rule 208
Rubi steps
\begin{align*} \int \frac{d+e x^2}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx &=\int \frac{1}{\frac{-c d^2+b d e}{d}+c e x^2} \, dx\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{\sqrt{c} \sqrt{e} \sqrt{c d-b e}}\\ \end{align*}
Mathematica [A] time = 0.0125817, size = 48, normalized size = 0.98 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right )}{\sqrt{c} \sqrt{e} \sqrt{b e-c d}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 33, normalized size = 0.7 \begin{align*}{\arctan \left ({cex{\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \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 = 1.60306, size = 279, normalized size = 5.69 \begin{align*} \left [\frac{\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 \, \sqrt{c^{2} d e - b c e^{2}}}, -\frac{\sqrt{-c^{2} d e + b c e^{2}} \arctan \left (-\frac{\sqrt{-c^{2} d e + b c e^{2}} x}{c d - b e}\right )}{c^{2} d e - b c e^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.218389, size = 124, normalized size = 2.53 \begin{align*} - \frac{\sqrt{- \frac{1}{c e \left (b e - c d\right )}} \log{\left (- b e \sqrt{- \frac{1}{c e \left (b e - c d\right )}} + c d \sqrt{- \frac{1}{c e \left (b e - c d\right )}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{c e \left (b e - c d\right )}} \log{\left (b e \sqrt{- \frac{1}{c e \left (b e - c d\right )}} - c d \sqrt{- \frac{1}{c e \left (b e - c d\right )}} + x \right )}}{2} \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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