Optimal. Leaf size=49 \[ -\frac{d \log \left (a+c x^2\right )}{2 a}+\frac{e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}+\frac{d \log (x)}{a} \]
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Rubi [A] time = 0.0419312, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {801, 635, 205, 260} \[ -\frac{d \log \left (a+c x^2\right )}{2 a}+\frac{e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}+\frac{d \log (x)}{a} \]
Antiderivative was successfully verified.
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Rule 801
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{d+e x}{x \left (a+c x^2\right )} \, dx &=\int \left (\frac{d}{a x}+\frac{a e-c d x}{a \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{d \log (x)}{a}+\frac{\int \frac{a e-c d x}{a+c x^2} \, dx}{a}\\ &=\frac{d \log (x)}{a}-\frac{(c d) \int \frac{x}{a+c x^2} \, dx}{a}+e \int \frac{1}{a+c x^2} \, dx\\ &=\frac{e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}+\frac{d \log (x)}{a}-\frac{d \log \left (a+c x^2\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0206574, size = 49, normalized size = 1. \[ -\frac{d \log \left (a+c x^2\right )}{2 a}+\frac{e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}+\frac{d \log (x)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 39, normalized size = 0.8 \begin{align*}{\frac{d\ln \left ( x \right ) }{a}}-{\frac{d\ln \left ( c{x}^{2}+a \right ) }{2\,a}}+{e\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \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 = 1.55701, size = 269, normalized size = 5.49 \begin{align*} \left [-\frac{c d \log \left (c x^{2} + a\right ) - 2 \, c d \log \left (x\right ) + \sqrt{-a c} e \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right )}{2 \, a c}, -\frac{c d \log \left (c x^{2} + a\right ) - 2 \, c d \log \left (x\right ) - 2 \, \sqrt{a c} e \arctan \left (\frac{\sqrt{a c} x}{a}\right )}{2 \, a c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.3596, size = 321, normalized size = 6.55 \begin{align*} \left (- \frac{d}{2 a} - \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right ) \log{\left (x + \frac{- 12 a^{2} c d \left (- \frac{d}{2 a} - \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right )^{2} + 2 a^{2} e^{2} \left (- \frac{d}{2 a} - \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right ) + 6 a c d^{2} \left (- \frac{d}{2 a} - \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right ) - 2 a d e^{2} + 6 c d^{3}}{a e^{3} + 9 c d^{2} e} \right )} + \left (- \frac{d}{2 a} + \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right ) \log{\left (x + \frac{- 12 a^{2} c d \left (- \frac{d}{2 a} + \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right )^{2} + 2 a^{2} e^{2} \left (- \frac{d}{2 a} + \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right ) + 6 a c d^{2} \left (- \frac{d}{2 a} + \frac{e \sqrt{- a^{3} c}}{2 a^{2} c}\right ) - 2 a d e^{2} + 6 c d^{3}}{a e^{3} + 9 c d^{2} e} \right )} + \frac{d \log{\left (x \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11762, size = 54, normalized size = 1.1 \begin{align*} \frac{\arctan \left (\frac{c x}{\sqrt{a c}}\right ) e}{\sqrt{a c}} - \frac{d \log \left (c x^{2} + a\right )}{2 \, a} + \frac{d \log \left ({\left | x \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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