Optimal. Leaf size=50 \[ \frac{(d g+e f)^2}{e^3 (d-e x)}+\frac{2 g (d g+e f) \log (d-e x)}{e^3}+\frac{g^2 x}{e^2} \]
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Rubi [A] time = 0.0570058, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {848, 43} \[ \frac{(d g+e f)^2}{e^3 (d-e x)}+\frac{2 g (d g+e f) \log (d-e x)}{e^3}+\frac{g^2 x}{e^2} \]
Antiderivative was successfully verified.
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Rule 848
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^2 (f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac{(f+g x)^2}{(d-e x)^2} \, dx\\ &=\int \left (\frac{g^2}{e^2}+\frac{(e f+d g)^2}{e^2 (-d+e x)^2}+\frac{2 g (e f+d g)}{e^2 (-d+e x)}\right ) \, dx\\ &=\frac{g^2 x}{e^2}+\frac{(e f+d g)^2}{e^3 (d-e x)}+\frac{2 g (e f+d g) \log (d-e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.04354, size = 46, normalized size = 0.92 \[ \frac{\frac{(d g+e f)^2}{d-e x}+2 g (d g+e f) \log (d-e x)+e g^2 x}{e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 96, normalized size = 1.9 \begin{align*}{\frac{{g}^{2}x}{{e}^{2}}}+2\,{\frac{d\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}+2\,{\frac{\ln \left ( ex-d \right ) fg}{{e}^{2}}}-{\frac{{d}^{2}{g}^{2}}{{e}^{3} \left ( ex-d \right ) }}-2\,{\frac{dfg}{{e}^{2} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{e \left ( ex-d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984881, size = 93, normalized size = 1.86 \begin{align*} \frac{g^{2} x}{e^{2}} - \frac{e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}}{e^{4} x - d e^{3}} + \frac{2 \,{\left (e f g + d g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70168, size = 184, normalized size = 3.68 \begin{align*} \frac{e^{2} g^{2} x^{2} - d e g^{2} x - e^{2} f^{2} - 2 \, d e f g - d^{2} g^{2} - 2 \,{\left (d e f g + d^{2} g^{2} -{\left (e^{2} f g + d e g^{2}\right )} x\right )} \log \left (e x - d\right )}{e^{4} x - d e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.545168, size = 60, normalized size = 1.2 \begin{align*} - \frac{d^{2} g^{2} + 2 d e f g + e^{2} f^{2}}{- d e^{3} + e^{4} x} + \frac{g^{2} x}{e^{2}} + \frac{2 g \left (d g + e f\right ) \log{\left (- d + e x \right )}}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19154, size = 216, normalized size = 4.32 \begin{align*} g^{2} x e^{\left (-2\right )} +{\left (d g^{2} e + f g e^{2}\right )} e^{\left (-4\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac{{\left (d^{2} g^{2} e^{2} + d f g e^{3}\right )} e^{\left (-5\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 )}{{\left | d \right |}} - \frac{{\left (d^{3} g^{2} e + 2 \, d^{2} f g e^{2} + d f^{2} e^{3} +{\left (d^{2} g^{2} e^{2} + 2 \, d f g e^{3} + f^{2} e^{4}\right )} x\right )} e^{\left (-4\right )}}{x^{2} e^{2} - d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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