Optimal. Leaf size=55 \[ \frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{5/2}}-\frac{c d x}{e^2}+\frac{c x^3}{3 e} \]
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Rubi [A] time = 0.0347125, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1154, 205} \[ \frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{5/2}}-\frac{c d x}{e^2}+\frac{c x^3}{3 e} \]
Antiderivative was successfully verified.
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Rule 1154
Rule 205
Rubi steps
\begin{align*} \int \frac{a+c x^4}{d+e x^2} \, dx &=\int \left (-\frac{c d}{e^2}+\frac{c x^2}{e}+\frac{c d^2+a e^2}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=-\frac{c d x}{e^2}+\frac{c x^3}{3 e}+\left (a+\frac{c d^2}{e^2}\right ) \int \frac{1}{d+e x^2} \, dx\\ &=-\frac{c d x}{e^2}+\frac{c x^3}{3 e}+\frac{\left (c d^2+a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0360523, size = 55, normalized size = 1. \[ \frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{5/2}}-\frac{c d x}{e^2}+\frac{c x^3}{3 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 57, normalized size = 1. \begin{align*}{\frac{c{x}^{3}}{3\,e}}-{\frac{cdx}{{e}^{2}}}+{a\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{c{d}^{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 = 1.93919, size = 293, normalized size = 5.33 \begin{align*} \left [\frac{2 \, c d e^{2} x^{3} - 6 \, c d^{2} e x - 3 \,{\left (c d^{2} + a e^{2}\right )} \sqrt{-d e} \log \left (\frac{e x^{2} - 2 \, \sqrt{-d e} x - d}{e x^{2} + d}\right )}{6 \, d e^{3}}, \frac{c d e^{2} x^{3} - 3 \, c d^{2} e x + 3 \,{\left (c d^{2} + a e^{2}\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right )}{3 \, d e^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.427377, size = 104, normalized size = 1.89 \begin{align*} - \frac{c d x}{e^{2}} + \frac{c x^{3}}{3 e} - \frac{\sqrt{- \frac{1}{d e^{5}}} \left (a e^{2} + c d^{2}\right ) \log{\left (- d e^{2} \sqrt{- \frac{1}{d e^{5}}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{d e^{5}}} \left (a e^{2} + c d^{2}\right ) \log{\left (d e^{2} \sqrt{- \frac{1}{d e^{5}}} + x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12393, size = 59, normalized size = 1.07 \begin{align*} \frac{{\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{5}{2}\right )}}{\sqrt{d}} + \frac{1}{3} \,{\left (c x^{3} e^{2} - 3 \, c d x e\right )} e^{\left (-3\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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