Optimal. Leaf size=75 \[ \frac{c^2 d \log (x)}{a^4}-\frac{c (a e+c d) \log (a-c x)}{2 a^4}-\frac{c (c d-a e) \log (a+c x)}{2 a^4}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x} \]
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Rubi [A] time = 0.0661303, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {801} \[ \frac{c^2 d \log (x)}{a^4}-\frac{c (a e+c d) \log (a-c x)}{2 a^4}-\frac{c (c d-a e) \log (a+c x)}{2 a^4}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x} \]
Antiderivative was successfully verified.
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Rule 801
Rubi steps
\begin{align*} \int \frac{d+e x}{x^3 \left (a^2-c^2 x^2\right )} \, dx &=\int \left (\frac{d}{a^2 x^3}+\frac{e}{a^2 x^2}+\frac{c^2 d}{a^4 x}+\frac{c^2 (c d+a e)}{2 a^4 (a-c x)}+\frac{c^2 (-c d+a e)}{2 a^4 (a+c x)}\right ) \, dx\\ &=-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x}+\frac{c^2 d \log (x)}{a^4}-\frac{c (c d+a e) \log (a-c x)}{2 a^4}-\frac{c (c d-a e) \log (a+c x)}{2 a^4}\\ \end{align*}
Mathematica [A] time = 0.0173343, size = 68, normalized size = 0.91 \[ -\frac{c^2 d \log \left (a^2-c^2 x^2\right )}{2 a^4}+\frac{c^2 d \log (x)}{a^4}+\frac{c e \tanh ^{-1}\left (\frac{c x}{a}\right )}{a^3}-\frac{d}{2 a^2 x^2}-\frac{e}{a^2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 90, normalized size = 1.2 \begin{align*} -{\frac{d}{2\,{a}^{2}{x}^{2}}}-{\frac{e}{{a}^{2}x}}+{\frac{{c}^{2}d\ln \left ( x \right ) }{{a}^{4}}}+{\frac{c\ln \left ( cx+a \right ) e}{2\,{a}^{3}}}-{\frac{{c}^{2}\ln \left ( cx+a \right ) d}{2\,{a}^{4}}}-{\frac{c\ln \left ( cx-a \right ) e}{2\,{a}^{3}}}-{\frac{{c}^{2}\ln \left ( cx-a \right ) d}{2\,{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03212, size = 95, normalized size = 1.27 \begin{align*} \frac{c^{2} d \log \left (x\right )}{a^{4}} - \frac{{\left (c^{2} d - a c e\right )} \log \left (c x + a\right )}{2 \, a^{4}} - \frac{{\left (c^{2} d + a c e\right )} \log \left (c x - a\right )}{2 \, a^{4}} - \frac{2 \, e x + d}{2 \, a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6253, size = 170, normalized size = 2.27 \begin{align*} \frac{2 \, c^{2} d x^{2} \log \left (x\right ) - 2 \, a^{2} e x -{\left (c^{2} d - a c e\right )} x^{2} \log \left (c x + a\right ) -{\left (c^{2} d + a c e\right )} x^{2} \log \left (c x - a\right ) - a^{2} d}{2 \, a^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.76543, size = 236, normalized size = 3.15 \begin{align*} - \frac{d + 2 e x}{2 a^{2} x^{2}} + \frac{c^{2} d \log{\left (x \right )}}{a^{4}} + \frac{c \left (a e - c d\right ) \log{\left (x + \frac{- 2 a^{2} c^{2} d e^{2} + a^{2} c e^{2} \left (a e - c d\right ) - 6 c^{4} d^{3} - 3 c^{3} d^{2} \left (a e - c d\right ) + 3 c^{2} d \left (a e - c d\right )^{2}}{a^{2} c^{2} e^{3} - 9 c^{4} d^{2} e} \right )}}{2 a^{4}} - \frac{c \left (a e + c d\right ) \log{\left (x + \frac{- 2 a^{2} c^{2} d e^{2} - a^{2} c e^{2} \left (a e + c d\right ) - 6 c^{4} d^{3} + 3 c^{3} d^{2} \left (a e + c d\right ) + 3 c^{2} d \left (a e + c d\right )^{2}}{a^{2} c^{2} e^{3} - 9 c^{4} d^{2} e} \right )}}{2 a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.157, size = 126, normalized size = 1.68 \begin{align*} \frac{c^{2} d \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (c^{3} d - a c^{2} e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, a^{4} c} - \frac{{\left (c^{3} d + a c^{2} e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, a^{4} c} - \frac{2 \, a^{2} x e + a^{2} d}{2 \, a^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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