Optimal. Leaf size=122 \[ \frac{e^2 f^2-d^2 g^2}{4 d^3 e^3 (d-e x)}-\frac{(e f-d g)^2}{8 d^3 e^3 (d+e x)}+\frac{(d g+e f)^2}{8 d^2 e^3 (d-e x)^2}+\frac{(d g+3 e f) (e f-d g) \tanh ^{-1}\left (\frac{e x}{d}\right )}{8 d^4 e^3} \]
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Rubi [A] time = 0.11539, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {799, 88, 208} \[ \frac{e^2 f^2-d^2 g^2}{4 d^3 e^3 (d-e x)}-\frac{(e f-d g)^2}{8 d^3 e^3 (d+e x)}+\frac{(d g+e f)^2}{8 d^2 e^3 (d-e x)^2}+\frac{(d g+3 e f) (e f-d g) \tanh ^{-1}\left (\frac{e x}{d}\right )}{8 d^4 e^3} \]
Antiderivative was successfully verified.
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Rule 799
Rule 88
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x) (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx &=\int \frac{(f+g x)^2}{(d-e x)^3 (d+e x)^2} \, dx\\ &=\int \left (\frac{(e f+d g)^2}{4 d^2 e^2 (d-e x)^3}+\frac{e^2 f^2-d^2 g^2}{4 d^3 e^2 (d-e x)^2}+\frac{(-e f+d g)^2}{8 d^3 e^2 (d+e x)^2}+\frac{(e f-d g) (3 e f+d g)}{8 d^3 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=\frac{(e f+d g)^2}{8 d^2 e^3 (d-e x)^2}+\frac{e^2 f^2-d^2 g^2}{4 d^3 e^3 (d-e x)}-\frac{(e f-d g)^2}{8 d^3 e^3 (d+e x)}+\frac{((e f-d g) (3 e f+d g)) \int \frac{1}{d^2-e^2 x^2} \, dx}{8 d^3 e^2}\\ &=\frac{(e f+d g)^2}{8 d^2 e^3 (d-e x)^2}+\frac{e^2 f^2-d^2 g^2}{4 d^3 e^3 (d-e x)}-\frac{(e f-d g)^2}{8 d^3 e^3 (d+e x)}+\frac{(e f-d g) (3 e f+d g) \tanh ^{-1}\left (\frac{e x}{d}\right )}{8 d^4 e^3}\\ \end{align*}
Mathematica [A] time = 0.104088, size = 140, normalized size = 1.15 \[ \frac{\frac{4 d e^2 f^2-4 d^3 g^2}{d-e x}+\left (d^2 g^2+2 d e f g-3 e^2 f^2\right ) \log (d-e x)+\left (-d^2 g^2-2 d e f g+3 e^2 f^2\right ) \log (d+e x)+\frac{2 d^2 (d g+e f)^2}{(d-e x)^2}-\frac{2 d (e f-d g)^2}{d+e x}}{16 d^4 e^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.055, size = 257, normalized size = 2.1 \begin{align*}{\frac{{g}^{2}}{4\,d{e}^{3} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{4\,e{d}^{3} \left ( ex-d \right ) }}+{\frac{{g}^{2}}{8\,{e}^{3} \left ( ex-d \right ) ^{2}}}+{\frac{fg}{4\,d{e}^{2} \left ( ex-d \right ) ^{2}}}+{\frac{{f}^{2}}{8\,e{d}^{2} \left ( ex-d \right ) ^{2}}}+{\frac{\ln \left ( ex-d \right ){g}^{2}}{16\,{e}^{3}{d}^{2}}}+{\frac{\ln \left ( ex-d \right ) fg}{8\,{e}^{2}{d}^{3}}}-{\frac{3\,\ln \left ( ex-d \right ){f}^{2}}{16\,e{d}^{4}}}-{\frac{\ln \left ( ex+d \right ){g}^{2}}{16\,{e}^{3}{d}^{2}}}-{\frac{\ln \left ( ex+d \right ) fg}{8\,{e}^{2}{d}^{3}}}+{\frac{3\,\ln \left ( ex+d \right ){f}^{2}}{16\,e{d}^{4}}}-{\frac{{g}^{2}}{8\,d{e}^{3} \left ( ex+d \right ) }}+{\frac{fg}{4\,{d}^{2}{e}^{2} \left ( ex+d \right ) }}-{\frac{{f}^{2}}{8\,e{d}^{3} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03317, size = 285, normalized size = 2.34 \begin{align*} \frac{2 \, d^{2} e^{2} f^{2} + 4 \, d^{3} e f g - 2 \, d^{4} g^{2} -{\left (3 \, e^{4} f^{2} - 2 \, d e^{3} f g - d^{2} e^{2} g^{2}\right )} x^{2} +{\left (3 \, d e^{3} f^{2} - 2 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} x}{8 \,{\left (d^{3} e^{6} x^{3} - d^{4} e^{5} x^{2} - d^{5} e^{4} x + d^{6} e^{3}\right )}} + \frac{{\left (3 \, e^{2} f^{2} - 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x + d\right )}{16 \, d^{4} e^{3}} - \frac{{\left (3 \, e^{2} f^{2} - 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x - d\right )}{16 \, d^{4} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71767, size = 810, normalized size = 6.64 \begin{align*} \frac{4 \, d^{3} e^{2} f^{2} + 8 \, d^{4} e f g - 4 \, d^{5} g^{2} - 2 \,{\left (3 \, d e^{4} f^{2} - 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} + 2 \,{\left (3 \, d^{2} e^{3} f^{2} - 2 \, d^{3} e^{2} f g + 3 \, d^{4} e g^{2}\right )} x +{\left (3 \, d^{3} e^{2} f^{2} - 2 \, d^{4} e f g - d^{5} g^{2} +{\left (3 \, e^{5} f^{2} - 2 \, d e^{4} f g - d^{2} e^{3} g^{2}\right )} x^{3} -{\left (3 \, d e^{4} f^{2} - 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} -{\left (3 \, d^{2} e^{3} f^{2} - 2 \, d^{3} e^{2} f g - d^{4} e g^{2}\right )} x\right )} \log \left (e x + d\right ) -{\left (3 \, d^{3} e^{2} f^{2} - 2 \, d^{4} e f g - d^{5} g^{2} +{\left (3 \, e^{5} f^{2} - 2 \, d e^{4} f g - d^{2} e^{3} g^{2}\right )} x^{3} -{\left (3 \, d e^{4} f^{2} - 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} -{\left (3 \, d^{2} e^{3} f^{2} - 2 \, d^{3} e^{2} f g - d^{4} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{16 \,{\left (d^{4} e^{6} x^{3} - d^{5} e^{5} x^{2} - d^{6} e^{4} x + d^{7} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.45898, size = 277, normalized size = 2.27 \begin{align*} \frac{- 2 d^{4} g^{2} + 4 d^{3} e f g + 2 d^{2} e^{2} f^{2} + x^{2} \left (d^{2} e^{2} g^{2} + 2 d e^{3} f g - 3 e^{4} f^{2}\right ) + x \left (3 d^{3} e g^{2} - 2 d^{2} e^{2} f g + 3 d e^{3} f^{2}\right )}{8 d^{6} e^{3} - 8 d^{5} e^{4} x - 8 d^{4} e^{5} x^{2} + 8 d^{3} e^{6} x^{3}} + \frac{\left (d g - e f\right ) \left (d g + 3 e f\right ) \log{\left (- \frac{d \left (d g - e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 2 d e f g - 3 e^{2} f^{2}\right )} + x \right )}}{16 d^{4} e^{3}} - \frac{\left (d g - e f\right ) \left (d g + 3 e f\right ) \log{\left (\frac{d \left (d g - e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 2 d e f g - 3 e^{2} f^{2}\right )} + x \right )}}{16 d^{4} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14288, size = 258, normalized size = 2.11 \begin{align*} \frac{{\left (d^{2} g^{2} + 2 \, d f g e - 3 \, 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 )}{16 \, d^{3}{\left | d \right |}} + \frac{{\left (d^{2} g^{2} x^{3} e^{4} + 4 \, d^{3} g^{2} x^{2} e^{3} + d^{4} g^{2} x e^{2} - 2 \, d^{5} g^{2} e + 2 \, d f g x^{3} e^{5} + 2 \, d^{3} f g x e^{3} + 4 \, d^{4} f g e^{2} - 3 \, f^{2} x^{3} e^{6} + 5 \, d^{2} f^{2} x e^{4} + 2 \, d^{3} f^{2} e^{3}\right )} e^{\left (-4\right )}}{8 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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