Optimal. Leaf size=107 \[ \frac{x \left (8 d^2 g^2+8 d e f g+e^2 f^2\right )}{e^2}+\frac{4 d^2 (d g+e f)^2}{e^3 (d-e x)}+\frac{4 d (d g+e f) (3 d g+e f) \log (d-e x)}{e^3}+\frac{g x^2 (2 d g+e f)}{e}+\frac{g^2 x^3}{3} \]
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Rubi [A] time = 0.136072, antiderivative size = 107, 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, 88} \[ \frac{x \left (8 d^2 g^2+8 d e f g+e^2 f^2\right )}{e^2}+\frac{4 d^2 (d g+e f)^2}{e^3 (d-e x)}+\frac{4 d (d g+e f) (3 d g+e f) \log (d-e x)}{e^3}+\frac{g x^2 (2 d g+e f)}{e}+\frac{g^2 x^3}{3} \]
Antiderivative was successfully verified.
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Rule 848
Rule 88
Rubi steps
\begin{align*} \int \frac{(d+e x)^4 (f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^2 (f+g x)^2}{(d-e x)^2} \, dx\\ &=\int \left (\frac{e^2 f^2+8 d e f g+8 d^2 g^2}{e^2}+\frac{2 g (e f+2 d g) x}{e}+g^2 x^2+\frac{4 d (-e f-3 d g) (e f+d g)}{e^2 (d-e x)}+\frac{4 d^2 (e f+d g)^2}{e^2 (-d+e x)^2}\right ) \, dx\\ &=\frac{\left (e^2 f^2+8 d e f g+8 d^2 g^2\right ) x}{e^2}+\frac{g (e f+2 d g) x^2}{e}+\frac{g^2 x^3}{3}+\frac{4 d^2 (e f+d g)^2}{e^3 (d-e x)}+\frac{4 d (e f+d g) (e f+3 d g) \log (d-e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.0884237, size = 115, normalized size = 1.07 \[ \frac{x \left (8 d^2 g^2+8 d e f g+e^2 f^2\right )}{e^2}+\frac{4 d \left (3 d^2 g^2+4 d e f g+e^2 f^2\right ) \log (d-e x)}{e^3}-\frac{4 d^2 (d g+e f)^2}{e^3 (e x-d)}+\frac{g x^2 (2 d g+e f)}{e}+\frac{g^2 x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 167, normalized size = 1.6 \begin{align*}{\frac{{g}^{2}{x}^{3}}{3}}+2\,{\frac{d{x}^{2}{g}^{2}}{e}}+{x}^{2}fg+8\,{\frac{{d}^{2}{g}^{2}x}{{e}^{2}}}+8\,{\frac{dfgx}{e}}+x{f}^{2}+12\,{\frac{{d}^{3}\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}+16\,{\frac{{d}^{2}\ln \left ( ex-d \right ) fg}{{e}^{2}}}+4\,{\frac{d\ln \left ( ex-d \right ){f}^{2}}{e}}-4\,{\frac{{d}^{4}{g}^{2}}{{e}^{3} \left ( ex-d \right ) }}-8\,{\frac{{d}^{3}fg}{{e}^{2} \left ( ex-d \right ) }}-4\,{\frac{{d}^{2}{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 = 0.97571, size = 190, normalized size = 1.78 \begin{align*} -\frac{4 \,{\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2}\right )}}{e^{4} x - d e^{3}} + \frac{e^{2} g^{2} x^{3} + 3 \,{\left (e^{2} f g + 2 \, d e g^{2}\right )} x^{2} + 3 \,{\left (e^{2} f^{2} + 8 \, d e f g + 8 \, d^{2} g^{2}\right )} x}{3 \, e^{2}} + \frac{4 \,{\left (d e^{2} f^{2} + 4 \, d^{2} e f g + 3 \, d^{3} 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 = 1.75058, size = 421, normalized size = 3.93 \begin{align*} \frac{e^{4} g^{2} x^{4} - 12 \, d^{2} e^{2} f^{2} - 24 \, d^{3} e f g - 12 \, d^{4} g^{2} +{\left (3 \, e^{4} f g + 5 \, d e^{3} g^{2}\right )} x^{3} + 3 \,{\left (e^{4} f^{2} + 7 \, d e^{3} f g + 6 \, d^{2} e^{2} g^{2}\right )} x^{2} - 3 \,{\left (d e^{3} f^{2} + 8 \, d^{2} e^{2} f g + 8 \, d^{3} e g^{2}\right )} x - 12 \,{\left (d^{2} e^{2} f^{2} + 4 \, d^{3} e f g + 3 \, d^{4} g^{2} -{\left (d e^{3} f^{2} + 4 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{3 \,{\left (e^{4} x - d e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.829104, size = 122, normalized size = 1.14 \begin{align*} \frac{4 d \left (d g + e f\right ) \left (3 d g + e f\right ) \log{\left (- d + e x \right )}}{e^{3}} + \frac{g^{2} x^{3}}{3} - \frac{4 d^{4} g^{2} + 8 d^{3} e f g + 4 d^{2} e^{2} f^{2}}{- d e^{3} + e^{4} x} + \frac{x^{2} \left (2 d g^{2} + e f g\right )}{e} + \frac{x \left (8 d^{2} g^{2} + 8 d e f g + e^{2} f^{2}\right )}{e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18845, size = 338, normalized size = 3.16 \begin{align*} 2 \,{\left (3 \, d^{3} g^{2} e^{3} + 4 \, d^{2} f g e^{4} + d f^{2} e^{5}\right )} e^{\left (-6\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac{1}{3} \,{\left (g^{2} x^{3} e^{12} + 6 \, d g^{2} x^{2} e^{11} + 24 \, d^{2} g^{2} x e^{10} + 3 \, f g x^{2} e^{12} + 24 \, d f g x e^{11} + 3 \, f^{2} x e^{12}\right )} e^{\left (-12\right )} + \frac{2 \,{\left (3 \, d^{4} g^{2} e^{4} + 4 \, d^{3} f g e^{5} + d^{2} f^{2} e^{6}\right )} e^{\left (-7\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{4 \,{\left (d^{5} g^{2} e^{3} + 2 \, d^{4} f g e^{4} + d^{3} f^{2} e^{5} +{\left (d^{4} g^{2} e^{4} + 2 \, d^{3} f g e^{5} + d^{2} f^{2} e^{6}\right )} x\right )} e^{\left (-6\right )}}{x^{2} e^{2} - d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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