Optimal. Leaf size=50 \[ -\frac{(a e+c d) \log (a-c x)}{2 c^3}-\frac{(c d-a e) \log (a+c x)}{2 c^3}-\frac{e x}{c^2} \]
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Rubi [A] time = 0.0384204, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {774, 633, 31} \[ -\frac{(a e+c d) \log (a-c x)}{2 c^3}-\frac{(c d-a e) \log (a+c x)}{2 c^3}-\frac{e x}{c^2} \]
Antiderivative was successfully verified.
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Rule 774
Rule 633
Rule 31
Rubi steps
\begin{align*} \int \frac{x (d+e x)}{a^2-c^2 x^2} \, dx &=-\frac{e x}{c^2}-\frac{\int \frac{-a^2 e-c^2 d x}{a^2-c^2 x^2} \, dx}{c^2}\\ &=-\frac{e x}{c^2}+\frac{(c d-a e) \int \frac{1}{-a c-c^2 x} \, dx}{2 c}+\frac{(c d+a e) \int \frac{1}{a c-c^2 x} \, dx}{2 c}\\ &=-\frac{e x}{c^2}-\frac{(c d+a e) \log (a-c x)}{2 c^3}-\frac{(c d-a e) \log (a+c x)}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.0092586, size = 42, normalized size = 0.84 \[ -\frac{d \log \left (a^2-c^2 x^2\right )}{2 c^2}+\frac{a e \tanh ^{-1}\left (\frac{c x}{a}\right )}{c^3}-\frac{e x}{c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 63, normalized size = 1.3 \begin{align*} -{\frac{ex}{{c}^{2}}}+{\frac{\ln \left ( cx+a \right ) ae}{2\,{c}^{3}}}-{\frac{\ln \left ( cx+a \right ) d}{2\,{c}^{2}}}-{\frac{\ln \left ( cx-a \right ) ae}{2\,{c}^{3}}}-{\frac{\ln \left ( cx-a \right ) d}{2\,{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07019, size = 63, normalized size = 1.26 \begin{align*} -\frac{e x}{c^{2}} - \frac{{\left (c d - a e\right )} \log \left (c x + a\right )}{2 \, c^{3}} - \frac{{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5489, size = 100, normalized size = 2. \begin{align*} -\frac{2 \, c e x +{\left (c d - a e\right )} \log \left (c x + a\right ) +{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.633854, size = 60, normalized size = 1.2 \begin{align*} - \frac{e x}{c^{2}} + \frac{\left (a e - c d\right ) \log{\left (x + \frac{d + \frac{a e - c d}{c}}{e} \right )}}{2 c^{3}} - \frac{\left (a e + c d\right ) \log{\left (x + \frac{d - \frac{a e + c d}{c}}{e} \right )}}{2 c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14182, size = 70, normalized size = 1.4 \begin{align*} -\frac{x e}{c^{2}} - \frac{{\left (c d - a e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, c^{3}} - \frac{{\left (c d + a e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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