Optimal. Leaf size=49 \[ -\frac{\sqrt{a} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{3/2}}+\frac{e \log \left (a+c x^2\right )}{2 c}+\frac{d x}{c} \]
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Rubi [A] time = 0.0300692, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {1394, 774, 635, 205, 260} \[ -\frac{\sqrt{a} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{3/2}}+\frac{e \log \left (a+c x^2\right )}{2 c}+\frac{d x}{c} \]
Antiderivative was successfully verified.
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Rule 1394
Rule 774
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{d+\frac{e}{x}}{c+\frac{a}{x^2}} \, dx &=\int \frac{x (e+d x)}{a+c x^2} \, dx\\ &=\frac{d x}{c}+\frac{\int \frac{-a d+c e x}{a+c x^2} \, dx}{c}\\ &=\frac{d x}{c}-\frac{(a d) \int \frac{1}{a+c x^2} \, dx}{c}+e \int \frac{x}{a+c x^2} \, dx\\ &=\frac{d x}{c}-\frac{\sqrt{a} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{3/2}}+\frac{e \log \left (a+c x^2\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.0233652, size = 49, normalized size = 1. \[ -\frac{\sqrt{a} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{3/2}}+\frac{e \log \left (a+c x^2\right )}{2 c}+\frac{d x}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 43, normalized size = 0.9 \begin{align*}{\frac{dx}{c}}+{\frac{e\ln \left ( c{x}^{2}+a \right ) }{2\,c}}-{\frac{ad}{c}\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 = 1.27228, size = 238, normalized size = 4.86 \begin{align*} \left [\frac{d \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{a}{c}} - a}{c x^{2} + a}\right ) + 2 \, d x + e \log \left (c x^{2} + a\right )}{2 \, c}, -\frac{2 \, d \sqrt{\frac{a}{c}} \arctan \left (\frac{c x \sqrt{\frac{a}{c}}}{a}\right ) - 2 \, d x - e \log \left (c x^{2} + a\right )}{2 \, c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.438742, size = 112, normalized size = 2.29 \begin{align*} \left (\frac{e}{2 c} - \frac{d \sqrt{- a c^{3}}}{2 c^{3}}\right ) \log{\left (x + \frac{- 2 c \left (\frac{e}{2 c} - \frac{d \sqrt{- a c^{3}}}{2 c^{3}}\right ) + e}{d} \right )} + \left (\frac{e}{2 c} + \frac{d \sqrt{- a c^{3}}}{2 c^{3}}\right ) \log{\left (x + \frac{- 2 c \left (\frac{e}{2 c} + \frac{d \sqrt{- a c^{3}}}{2 c^{3}}\right ) + e}{d} \right )} + \frac{d x}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09732, size = 58, normalized size = 1.18 \begin{align*} -\frac{a d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c} + \frac{d x}{c} + \frac{e \log \left (c x^{2} + a\right )}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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