Optimal. Leaf size=63 \[ -\frac{a (a e+c d) \log (a-c x)}{2 c^4}+\frac{a (c d-a e) \log (a+c x)}{2 c^4}-\frac{d x}{c^2}-\frac{e x^2}{2 c^2} \]
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Rubi [A] time = 0.0559836, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {801, 633, 31} \[ -\frac{a (a e+c d) \log (a-c x)}{2 c^4}+\frac{a (c d-a e) \log (a+c x)}{2 c^4}-\frac{d x}{c^2}-\frac{e x^2}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 801
Rule 633
Rule 31
Rubi steps
\begin{align*} \int \frac{x^2 (d+e x)}{a^2-c^2 x^2} \, dx &=\int \left (-\frac{d}{c^2}-\frac{e x}{c^2}+\frac{a^2 d+a^2 e x}{c^2 \left (a^2-c^2 x^2\right )}\right ) \, dx\\ &=-\frac{d x}{c^2}-\frac{e x^2}{2 c^2}+\frac{\int \frac{a^2 d+a^2 e x}{a^2-c^2 x^2} \, dx}{c^2}\\ &=-\frac{d x}{c^2}-\frac{e x^2}{2 c^2}-\frac{(a (c d-a e)) \int \frac{1}{-a c-c^2 x} \, dx}{2 c^2}+\frac{(a (c d+a e)) \int \frac{1}{a c-c^2 x} \, dx}{2 c^2}\\ &=-\frac{d x}{c^2}-\frac{e x^2}{2 c^2}-\frac{a (c d+a e) \log (a-c x)}{2 c^4}+\frac{a (c d-a e) \log (a+c x)}{2 c^4}\\ \end{align*}
Mathematica [A] time = 0.0123887, size = 56, normalized size = 0.89 \[ -\frac{a^2 e \log \left (a^2-c^2 x^2\right )}{2 c^4}+\frac{a d \tanh ^{-1}\left (\frac{c x}{a}\right )}{c^3}-\frac{d x}{c^2}-\frac{e x^2}{2 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 78, normalized size = 1.2 \begin{align*} -{\frac{e{x}^{2}}{2\,{c}^{2}}}-{\frac{dx}{{c}^{2}}}-{\frac{{a}^{2}\ln \left ( cx+a \right ) e}{2\,{c}^{4}}}+{\frac{a\ln \left ( cx+a \right ) d}{2\,{c}^{3}}}-{\frac{{a}^{2}\ln \left ( cx-a \right ) e}{2\,{c}^{4}}}-{\frac{a\ln \left ( cx-a \right ) d}{2\,{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04531, size = 82, normalized size = 1.3 \begin{align*} -\frac{e x^{2} + 2 \, d x}{2 \, c^{2}} + \frac{{\left (a c d - a^{2} e\right )} \log \left (c x + a\right )}{2 \, c^{4}} - \frac{{\left (a c d + a^{2} e\right )} \log \left (c x - a\right )}{2 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62095, size = 130, normalized size = 2.06 \begin{align*} -\frac{c^{2} e x^{2} + 2 \, c^{2} d x -{\left (a c d - a^{2} e\right )} \log \left (c x + a\right ) +{\left (a c d + a^{2} e\right )} \log \left (c x - a\right )}{2 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.593743, size = 88, normalized size = 1.4 \begin{align*} - \frac{a \left (a e - c d\right ) \log{\left (x + \frac{a^{2} e - a \left (a e - c d\right )}{c^{2} d} \right )}}{2 c^{4}} - \frac{a \left (a e + c d\right ) \log{\left (x + \frac{a^{2} e - a \left (a e + c d\right )}{c^{2} d} \right )}}{2 c^{4}} - \frac{d x}{c^{2}} - \frac{e x^{2}}{2 c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16154, size = 97, normalized size = 1.54 \begin{align*} \frac{{\left (a c d - a^{2} e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, c^{4}} - \frac{{\left (a c d + a^{2} e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, c^{4}} - \frac{c^{2} x^{2} e + 2 \, c^{2} d x}{2 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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