Optimal. Leaf size=77 \[ \frac{x (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac{(2 a e+c d) \log (a-c x)}{4 a c^4}-\frac{(c d-2 a e) \log (a+c x)}{4 a c^4} \]
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Rubi [A] time = 0.0488583, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {819, 633, 31} \[ \frac{x (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac{(2 a e+c d) \log (a-c x)}{4 a c^4}-\frac{(c d-2 a e) \log (a+c x)}{4 a c^4} \]
Antiderivative was successfully verified.
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Rule 819
Rule 633
Rule 31
Rubi steps
\begin{align*} \int \frac{x^2 (d+e x)}{\left (a^2-c^2 x^2\right )^2} \, dx &=\frac{x (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac{\int \frac{a^2 d+2 a^2 e x}{a^2-c^2 x^2} \, dx}{2 a^2 c^2}\\ &=\frac{x (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac{(c d-2 a e) \int \frac{1}{-a c-c^2 x} \, dx}{4 a c^2}-\frac{(c d+2 a e) \int \frac{1}{a c-c^2 x} \, dx}{4 a c^2}\\ &=\frac{x (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac{(c d+2 a e) \log (a-c x)}{4 a c^4}-\frac{(c d-2 a e) \log (a+c x)}{4 a c^4}\\ \end{align*}
Mathematica [A] time = 0.036751, size = 64, normalized size = 0.83 \[ \frac{\frac{a^2 e+c^2 d x}{a^2-c^2 x^2}+e \log \left (a^2-c^2 x^2\right )-\frac{c d \tanh ^{-1}\left (\frac{c x}{a}\right )}{a}}{2 c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 118, normalized size = 1.5 \begin{align*}{\frac{ae}{4\,{c}^{4} \left ( cx+a \right ) }}-{\frac{d}{4\,{c}^{3} \left ( cx+a \right ) }}+{\frac{\ln \left ( cx+a \right ) e}{2\,{c}^{4}}}-{\frac{\ln \left ( cx+a \right ) d}{4\,a{c}^{3}}}-{\frac{ae}{4\,{c}^{4} \left ( cx-a \right ) }}-{\frac{d}{4\,{c}^{3} \left ( cx-a \right ) }}+{\frac{\ln \left ( cx-a \right ) e}{2\,{c}^{4}}}+{\frac{\ln \left ( cx-a \right ) d}{4\,a{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07014, size = 107, normalized size = 1.39 \begin{align*} -\frac{c^{2} d x + a^{2} e}{2 \,{\left (c^{6} x^{2} - a^{2} c^{4}\right )}} - \frac{{\left (c d - 2 \, a e\right )} \log \left (c x + a\right )}{4 \, a c^{4}} + \frac{{\left (c d + 2 \, a e\right )} \log \left (c x - a\right )}{4 \, a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59396, size = 235, normalized size = 3.05 \begin{align*} -\frac{2 \, a c^{2} d x + 2 \, a^{3} e -{\left (a^{2} c d - 2 \, a^{3} e -{\left (c^{3} d - 2 \, a c^{2} e\right )} x^{2}\right )} \log \left (c x + a\right ) +{\left (a^{2} c d + 2 \, a^{3} e -{\left (c^{3} d + 2 \, a c^{2} e\right )} x^{2}\right )} \log \left (c x - a\right )}{4 \,{\left (a c^{6} x^{2} - a^{3} c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.762366, size = 109, normalized size = 1.42 \begin{align*} - \frac{a^{2} e + c^{2} d x}{- 2 a^{2} c^{4} + 2 c^{6} x^{2}} + \frac{\left (2 a e - c d\right ) \log{\left (x + \frac{2 a^{2} e - a \left (2 a e - c d\right )}{c^{2} d} \right )}}{4 a c^{4}} + \frac{\left (2 a e + c d\right ) \log{\left (x + \frac{2 a^{2} e - a \left (2 a e + c d\right )}{c^{2} d} \right )}}{4 a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12963, size = 115, normalized size = 1.49 \begin{align*} -\frac{d x + \frac{a^{2} e}{c^{2}}}{2 \,{\left (c x + a\right )}{\left (c x - a\right )} c^{2}} - \frac{{\left (c d - 2 \, a e\right )} \log \left ({\left | c x + a \right |}\right )}{4 \, a c^{4}} + \frac{{\left (c d + 2 \, a e\right )} \log \left ({\left | c x - a \right |}\right )}{4 \, a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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