Optimal. Leaf size=66 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right )}{\sqrt{d} e^{5/2}}-\frac{x (c d-b e)}{e^2}+\frac{c x^3}{3 e} \]
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Rubi [A] time = 0.0446231, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1153, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right )}{\sqrt{d} e^{5/2}}-\frac{x (c d-b e)}{e^2}+\frac{c x^3}{3 e} \]
Antiderivative was successfully verified.
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Rule 1153
Rule 205
Rubi steps
\begin{align*} \int \frac{a+b x^2+c x^4}{d+e x^2} \, dx &=\int \left (-\frac{c d-b e}{e^2}+\frac{c x^2}{e}+\frac{c d^2-b d e+a e^2}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=-\frac{(c d-b e) x}{e^2}+\frac{c x^3}{3 e}+\frac{\left (c d^2-b d e+a e^2\right ) \int \frac{1}{d+e x^2} \, dx}{e^2}\\ &=-\frac{(c d-b e) x}{e^2}+\frac{c x^3}{3 e}+\frac{\left (c d^2-b d e+a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0512229, size = 65, normalized size = 0.98 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right )}{\sqrt{d} e^{5/2}}+\frac{x (b e-c d)}{e^2}+\frac{c x^3}{3 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 84, normalized size = 1.3 \begin{align*}{\frac{c{x}^{3}}{3\,e}}+{\frac{bx}{e}}-{\frac{cdx}{{e}^{2}}}+{a\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{bd}{e}\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 = 1.63718, size = 347, normalized size = 5.26 \begin{align*} \left [\frac{2 \, c d e^{2} x^{3} - 3 \,{\left (c d^{2} - b d e + 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 \,{\left (c d^{2} e - b d e^{2}\right )} x}{6 \, d e^{3}}, \frac{c d e^{2} x^{3} + 3 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) - 3 \,{\left (c d^{2} e - b d e^{2}\right )} x}{3 \, d e^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.640437, size = 117, normalized size = 1.77 \begin{align*} \frac{c x^{3}}{3 e} - \frac{\sqrt{- \frac{1}{d e^{5}}} \left (a e^{2} - b d e + 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} - b d e + c d^{2}\right ) \log{\left (d e^{2} \sqrt{- \frac{1}{d e^{5}}} + x \right )}}{2} + \frac{x \left (b e - c d\right )}{e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26151, size = 76, normalized size = 1.15 \begin{align*} \frac{{\left (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 )}}{\sqrt{d}} + \frac{1}{3} \,{\left (c x^{3} e^{2} - 3 \, c d x e + 3 \, b x e^{2}\right )} e^{\left (-3\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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