Optimal. Leaf size=86 \[ \frac{(e f-d g)^2 \log (d+e x)}{4 d^2 e^3}-\frac{(e f-3 d g) (d g+e f) \log (d-e x)}{4 d^2 e^3}+\frac{(d g+e f)^2}{2 d e^3 (d-e x)} \]
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Rubi [A] time = 0.0815934, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {799, 88} \[ \frac{(e f-d g)^2 \log (d+e x)}{4 d^2 e^3}-\frac{(e f-3 d g) (d g+e f) \log (d-e x)}{4 d^2 e^3}+\frac{(d g+e f)^2}{2 d e^3 (d-e x)} \]
Antiderivative was successfully verified.
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Rule 799
Rule 88
Rubi steps
\begin{align*} \int \frac{(d+e x) (f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac{(f+g x)^2}{(d-e x)^2 (d+e x)} \, dx\\ &=\int \left (\frac{(e f+d g)^2}{2 d e^2 (d-e x)^2}+\frac{(e f-3 d g) (e f+d g)}{4 d^2 e^2 (d-e x)}+\frac{(-e f+d g)^2}{4 d^2 e^2 (d+e x)}\right ) \, dx\\ &=\frac{(e f+d g)^2}{2 d e^3 (d-e x)}-\frac{(e f-3 d g) (e f+d g) \log (d-e x)}{4 d^2 e^3}+\frac{(e f-d g)^2 \log (d+e x)}{4 d^2 e^3}\\ \end{align*}
Mathematica [A] time = 0.0483859, size = 91, normalized size = 1.06 \[ \frac{(d-e x) \left (3 d^2 g^2+2 d e f g-e^2 f^2\right ) \log (d-e x)+(d-e x) (e f-d g)^2 \log (d+e x)+2 d (d g+e f)^2}{4 d^2 e^3 (d-e x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.09, size = 156, normalized size = 1.8 \begin{align*} -{\frac{{g}^{2}d}{2\,{e}^{3} \left ( ex-d \right ) }}-{\frac{fg}{{e}^{2} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{2\,de \left ( ex-d \right ) }}+{\frac{3\,\ln \left ( ex-d \right ){g}^{2}}{4\,{e}^{3}}}+{\frac{\ln \left ( ex-d \right ) fg}{2\,d{e}^{2}}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{4\,{d}^{2}e}}+{\frac{\ln \left ( ex+d \right ){g}^{2}}{4\,{e}^{3}}}-{\frac{\ln \left ( ex+d \right ) fg}{2\,d{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){f}^{2}}{4\,{d}^{2}e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975871, size = 154, normalized size = 1.79 \begin{align*} -\frac{e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}}{2 \,{\left (d e^{4} x - d^{2} e^{3}\right )}} + \frac{{\left (e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{4 \, d^{2} e^{3}} - \frac{{\left (e^{2} f^{2} - 2 \, d e f g - 3 \, d^{2} g^{2}\right )} \log \left (e x - d\right )}{4 \, d^{2} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71683, size = 343, normalized size = 3.99 \begin{align*} -\frac{2 \, d e^{2} f^{2} + 4 \, d^{2} e f g + 2 \, d^{3} g^{2} +{\left (d e^{2} f^{2} - 2 \, d^{2} e f g + d^{3} g^{2} -{\left (e^{3} f^{2} - 2 \, d e^{2} f g + d^{2} e g^{2}\right )} x\right )} \log \left (e x + d\right ) -{\left (d e^{2} f^{2} - 2 \, d^{2} e f g - 3 \, d^{3} g^{2} -{\left (e^{3} f^{2} - 2 \, d e^{2} f g - 3 \, d^{2} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{4 \,{\left (d^{2} e^{4} x - d^{3} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.14259, size = 180, normalized size = 2.09 \begin{align*} - \frac{d^{2} g^{2} + 2 d e f g + e^{2} f^{2}}{- 2 d^{2} e^{3} + 2 d e^{4} x} + \frac{\left (d g - e f\right )^{2} \log{\left (x + \frac{2 d^{3} g^{2} - d \left (d g - e f\right )^{2}}{d^{2} e g^{2} + 2 d e^{2} f g - e^{3} f^{2}} \right )}}{4 d^{2} e^{3}} + \frac{\left (d g + e f\right ) \left (3 d g - e f\right ) \log{\left (x + \frac{2 d^{3} g^{2} - d \left (d g + e f\right ) \left (3 d g - e f\right )}{d^{2} e g^{2} + 2 d e^{2} f g - e^{3} f^{2}} \right )}}{4 d^{2} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14363, size = 215, normalized size = 2.5 \begin{align*} \frac{1}{2} \, g^{2} e^{\left (-3\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac{{\left (d^{2} g^{2} + 2 \, d f g e - f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{4 \, d{\left | d \right |}} - \frac{{\left ({\left (d^{2} g^{2} + 2 \, d f g e + f^{2} e^{2}\right )} x +{\left (d^{3} g^{2} e + 2 \, d^{2} f g e^{2} + d f^{2} e^{3}\right )} e^{\left (-2\right )}\right )} e^{\left (-2\right )}}{2 \,{\left (x^{2} e^{2} - d^{2}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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