Optimal. Leaf size=55 \[ \frac{a^2 e+c^2 d x}{2 a^2 c^2 \left (a^2-c^2 x^2\right )}+\frac{d \tanh ^{-1}\left (\frac{c x}{a}\right )}{2 a^3 c} \]
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Rubi [A] time = 0.0162884, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {639, 208} \[ \frac{a^2 e+c^2 d x}{2 a^2 c^2 \left (a^2-c^2 x^2\right )}+\frac{d \tanh ^{-1}\left (\frac{c x}{a}\right )}{2 a^3 c} \]
Antiderivative was successfully verified.
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Rule 639
Rule 208
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (a^2-c^2 x^2\right )^2} \, dx &=\frac{a^2 e+c^2 d x}{2 a^2 c^2 \left (a^2-c^2 x^2\right )}+\frac{d \int \frac{1}{a^2-c^2 x^2} \, dx}{2 a^2}\\ &=\frac{a^2 e+c^2 d x}{2 a^2 c^2 \left (a^2-c^2 x^2\right )}+\frac{d \tanh ^{-1}\left (\frac{c x}{a}\right )}{2 a^3 c}\\ \end{align*}
Mathematica [A] time = 0.0132764, size = 58, normalized size = 1.05 \[ \frac{a^2 (-e)-c^2 d x}{2 a^2 c^2 \left (c^2 x^2-a^2\right )}+\frac{d \tanh ^{-1}\left (\frac{c x}{a}\right )}{2 a^3 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 102, normalized size = 1.9 \begin{align*}{\frac{d\ln \left ( cx+a \right ) }{4\,c{a}^{3}}}+{\frac{e}{4\,{c}^{2}a \left ( cx+a \right ) }}-{\frac{d}{4\,c{a}^{2} \left ( cx+a \right ) }}-{\frac{d\ln \left ( cx-a \right ) }{4\,c{a}^{3}}}-{\frac{e}{4\,{c}^{2}a \left ( cx-a \right ) }}-{\frac{d}{4\,c{a}^{2} \left ( cx-a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05251, size = 92, normalized size = 1.67 \begin{align*} -\frac{c^{2} d x + a^{2} e}{2 \,{\left (a^{2} c^{4} x^{2} - a^{4} c^{2}\right )}} + \frac{d \log \left (c x + a\right )}{4 \, a^{3} c} - \frac{d \log \left (c x - a\right )}{4 \, a^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51561, size = 173, normalized size = 3.15 \begin{align*} -\frac{2 \, a c^{2} d x + 2 \, a^{3} e -{\left (c^{3} d x^{2} - a^{2} c d\right )} \log \left (c x + a\right ) +{\left (c^{3} d x^{2} - a^{2} c d\right )} \log \left (c x - a\right )}{4 \,{\left (a^{3} c^{4} x^{2} - a^{5} c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.71487, size = 54, normalized size = 0.98 \begin{align*} - \frac{a^{2} e + c^{2} d x}{- 2 a^{4} c^{2} + 2 a^{2} c^{4} x^{2}} + \frac{d \left (- \frac{\log{\left (- \frac{a}{c} + x \right )}}{4} + \frac{\log{\left (\frac{a}{c} + x \right )}}{4}\right )}{a^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15322, size = 96, normalized size = 1.75 \begin{align*} \frac{d \log \left ({\left | c x + a \right |}\right )}{4 \, a^{3} c} - \frac{d \log \left ({\left | c x - a \right |}\right )}{4 \, a^{3} c} - \frac{c^{2} d x + a^{2} e}{2 \,{\left (c^{2} x^{2} - a^{2}\right )} a^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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