Optimal. Leaf size=61 \[ -\frac{2 g (d g+e f)}{e^3 (d-e x)}+\frac{(d g+e f)^2}{2 e^3 (d-e x)^2}-\frac{g^2 \log (d-e x)}{e^3} \]
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Rubi [A] time = 0.0597248, antiderivative size = 61, 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} \[ -\frac{2 g (d g+e f)}{e^3 (d-e x)}+\frac{(d g+e f)^2}{2 e^3 (d-e x)^2}-\frac{g^2 \log (d-e x)}{e^3} \]
Antiderivative was successfully verified.
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Rule 848
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx &=\int \frac{(f+g x)^2}{(d-e x)^3} \, dx\\ &=\int \left (\frac{(e f+d g)^2}{e^2 (d-e x)^3}-\frac{2 g (e f+d g)}{e^2 (d-e x)^2}+\frac{g^2}{e^2 (d-e x)}\right ) \, dx\\ &=\frac{(e f+d g)^2}{2 e^3 (d-e x)^2}-\frac{2 g (e f+d g)}{e^3 (d-e x)}-\frac{g^2 \log (d-e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.025836, size = 49, normalized size = 0.8 \[ \frac{\frac{(d g+e f) (e (f+4 g x)-3 d g)}{(d-e x)^2}-2 g^2 \log (d-e x)}{2 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 105, normalized size = 1.7 \begin{align*} -{\frac{\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}+{\frac{{d}^{2}{g}^{2}}{2\,{e}^{3} \left ( ex-d \right ) ^{2}}}+{\frac{dfg}{{e}^{2} \left ( ex-d \right ) ^{2}}}+{\frac{{f}^{2}}{2\,e \left ( ex-d \right ) ^{2}}}+2\,{\frac{{g}^{2}d}{{e}^{3} \left ( ex-d \right ) }}+2\,{\frac{fg}{{e}^{2} \left ( ex-d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00053, size = 109, normalized size = 1.79 \begin{align*} \frac{e^{2} f^{2} - 2 \, d e f g - 3 \, d^{2} g^{2} + 4 \,{\left (e^{2} f g + d e g^{2}\right )} x}{2 \,{\left (e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}\right )}} - \frac{g^{2} \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 = 1.72297, size = 205, normalized size = 3.36 \begin{align*} \frac{e^{2} f^{2} - 2 \, d e f g - 3 \, d^{2} g^{2} + 4 \,{\left (e^{2} f g + d e g^{2}\right )} x - 2 \,{\left (e^{2} g^{2} x^{2} - 2 \, d e g^{2} x + d^{2} g^{2}\right )} \log \left (e x - d\right )}{2 \,{\left (e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.708844, size = 80, normalized size = 1.31 \begin{align*} \frac{- 3 d^{2} g^{2} - 2 d e f g + e^{2} f^{2} + x \left (4 d e g^{2} + 4 e^{2} f g\right )}{2 d^{2} e^{3} - 4 d e^{4} x + 2 e^{5} x^{2}} - \frac{g^{2} \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 = 1.17624, size = 263, normalized size = 4.31 \begin{align*} -\frac{d g^{2} 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 )}{2 \,{\left | d \right |}} - \frac{1}{2} \, g^{2} e^{\left (-3\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac{{\left (4 \,{\left (d^{2} g^{2} e^{4} + d f g e^{5}\right )} x^{3} +{\left (5 \, d^{3} g^{2} e^{3} + 6 \, d^{2} f g e^{4} + d f^{2} e^{5}\right )} x^{2} - 2 \,{\left (d^{4} g^{2} e^{2} - d^{2} f^{2} e^{4}\right )} x -{\left (3 \, d^{5} g^{2} e^{3} + 2 \, d^{4} f g e^{4} - d^{3} f^{2} e^{5}\right )} e^{\left (-2\right )}\right )} e^{\left (-4\right )}}{2 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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