Optimal. Leaf size=87 \[ -\frac{(3 d g+e f) (e f-d g)}{4 d^2 e^3 (d+e x)}+\frac{(d g+e f)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{4 d^3 e^3}-\frac{(e f-d g)^2}{4 d e^3 (d+e x)^2} \]
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Rubi [A] time = 0.0966412, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {848, 88, 208} \[ -\frac{(3 d g+e f) (e f-d g)}{4 d^2 e^3 (d+e x)}+\frac{(d g+e f)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{4 d^3 e^3}-\frac{(e f-d g)^2}{4 d e^3 (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 848
Rule 88
Rule 208
Rubi steps
\begin{align*} \int \frac{(f+g x)^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )} \, dx &=\int \frac{(f+g x)^2}{(d-e x) (d+e x)^3} \, dx\\ &=\int \left (\frac{(-e f+d g)^2}{2 d e^2 (d+e x)^3}+\frac{(e f-d g) (e f+3 d g)}{4 d^2 e^2 (d+e x)^2}+\frac{(e f+d g)^2}{4 d^2 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=-\frac{(e f-d g)^2}{4 d e^3 (d+e x)^2}-\frac{(e f-d g) (e f+3 d g)}{4 d^2 e^3 (d+e x)}+\frac{(e f+d g)^2 \int \frac{1}{d^2-e^2 x^2} \, dx}{4 d^2 e^2}\\ &=-\frac{(e f-d g)^2}{4 d e^3 (d+e x)^2}-\frac{(e f-d g) (e f+3 d g)}{4 d^2 e^3 (d+e x)}+\frac{(e f+d g)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{4 d^3 e^3}\\ \end{align*}
Mathematica [A] time = 0.07658, size = 87, normalized size = 1. \[ \frac{\frac{2 d (d g-e f) \left (2 d^2 g+d e (2 f+3 g x)+e^2 f x\right )}{(d+e x)^2}+(d g+e f)^2 (-\log (d-e x))+(d g+e f)^2 \log (d+e x)}{8 d^3 e^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.055, size = 206, normalized size = 2.4 \begin{align*} -{\frac{\ln \left ( ex-d \right ){g}^{2}}{8\,d{e}^{3}}}-{\frac{\ln \left ( ex-d \right ) fg}{4\,{d}^{2}{e}^{2}}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{8\,{d}^{3}e}}+{\frac{3\,{g}^{2}}{4\,{e}^{3} \left ( ex+d \right ) }}-{\frac{fg}{2\,d{e}^{2} \left ( ex+d \right ) }}-{\frac{{f}^{2}}{4\,{d}^{2}e \left ( ex+d \right ) }}-{\frac{{g}^{2}d}{4\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{fg}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{{f}^{2}}{4\,de \left ( ex+d \right ) ^{2}}}+{\frac{\ln \left ( ex+d \right ){g}^{2}}{8\,d{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) fg}{4\,{d}^{2}{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){f}^{2}}{8\,{d}^{3}e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.972392, size = 201, normalized size = 2.31 \begin{align*} -\frac{2 \, d e^{2} f^{2} - 2 \, d^{3} g^{2} +{\left (e^{3} f^{2} + 2 \, d e^{2} f g - 3 \, d^{2} e g^{2}\right )} x}{4 \,{\left (d^{2} e^{5} x^{2} + 2 \, d^{3} e^{4} x + d^{4} 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 )}{8 \, d^{3} e^{3}} - \frac{{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{8 \, d^{3} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.82193, size = 548, normalized size = 6.3 \begin{align*} -\frac{4 \, d^{2} e^{2} f^{2} - 4 \, d^{4} g^{2} + 2 \,{\left (d e^{3} f^{2} + 2 \, d^{2} e^{2} f g - 3 \, d^{3} e g^{2}\right )} x -{\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2} +{\left (e^{4} f^{2} + 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{2} + 2 \,{\left (d e^{3} f^{2} + 2 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x\right )} \log \left (e x + d\right ) +{\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2} +{\left (e^{4} f^{2} + 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{2} + 2 \,{\left (d e^{3} f^{2} + 2 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{8 \,{\left (d^{3} e^{5} x^{2} + 2 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.11523, size = 185, normalized size = 2.13 \begin{align*} \frac{2 d^{3} g^{2} - 2 d e^{2} f^{2} + x \left (3 d^{2} e g^{2} - 2 d e^{2} f g - e^{3} f^{2}\right )}{4 d^{4} e^{3} + 8 d^{3} e^{4} x + 4 d^{2} e^{5} x^{2}} - \frac{\left (d g + e f\right )^{2} \log{\left (- \frac{d \left (d g + e f\right )^{2}}{e \left (d^{2} g^{2} + 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{8 d^{3} e^{3}} + \frac{\left (d g + e f\right )^{2} \log{\left (\frac{d \left (d g + e f\right )^{2}}{e \left (d^{2} g^{2} + 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{8 d^{3} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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