Optimal. Leaf size=43 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}-\frac{e \log \left (a-c x^2\right )}{2 c} \]
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Rubi [A] time = 0.0151123, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {635, 208, 260} \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}-\frac{e \log \left (a-c x^2\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 635
Rule 208
Rule 260
Rubi steps
\begin{align*} \int \frac{d+e x}{a-c x^2} \, dx &=d \int \frac{1}{a-c x^2} \, dx+e \int \frac{x}{a-c x^2} \, dx\\ &=\frac{d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}-\frac{e \log \left (a-c x^2\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.014611, size = 43, normalized size = 1. \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}-\frac{e \log \left (a-c x^2\right )}{2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 34, normalized size = 0.8 \begin{align*} -{\frac{e\ln \left ( c{x}^{2}-a \right ) }{2\,c}}+{d{\it Artanh} \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 = 1.58712, size = 228, normalized size = 5.3 \begin{align*} \left [-\frac{a e \log \left (c x^{2} - a\right ) - \sqrt{a c} d \log \left (\frac{c x^{2} + 2 \, \sqrt{a c} x + a}{c x^{2} - a}\right )}{2 \, a c}, -\frac{a e \log \left (c x^{2} - a\right ) + 2 \, \sqrt{-a c} d \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 = 0.313423, size = 119, normalized size = 2.77 \begin{align*} - \left (\frac{e}{2 c} - \frac{d \sqrt{a c^{3}}}{2 a c^{2}}\right ) \log{\left (x + \frac{- 2 a c \left (\frac{e}{2 c} - \frac{d \sqrt{a c^{3}}}{2 a c^{2}}\right ) + a e}{c d} \right )} - \left (\frac{e}{2 c} + \frac{d \sqrt{a c^{3}}}{2 a c^{2}}\right ) \log{\left (x + \frac{- 2 a c \left (\frac{e}{2 c} + \frac{d \sqrt{a c^{3}}}{2 a c^{2}}\right ) + a e}{c d} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13916, size = 50, normalized size = 1.16 \begin{align*} -\frac{d \arctan \left (\frac{c x}{\sqrt{-a c}}\right )}{\sqrt{-a 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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