Optimal. Leaf size=50 \[ -\frac{g x (d g+e f)}{e^2}-\frac{(d g+e f)^2 \log (d-e x)}{e^3}-\frac{(f+g x)^2}{2 e} \]
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Rubi [A] time = 0.0340709, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {799, 43} \[ -\frac{g x (d g+e f)}{e^2}-\frac{(d g+e f)^2 \log (d-e x)}{e^3}-\frac{(f+g x)^2}{2 e} \]
Antiderivative was successfully verified.
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Rule 799
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x) (f+g x)^2}{d^2-e^2 x^2} \, dx &=\int \frac{(f+g x)^2}{d-e x} \, dx\\ &=\int \left (-\frac{g (e f+d g)}{e^2}+\frac{(e f+d g)^2}{e^2 (d-e x)}-\frac{g (f+g x)}{e}\right ) \, dx\\ &=-\frac{g (e f+d g) x}{e^2}-\frac{(f+g x)^2}{2 e}-\frac{(e f+d g)^2 \log (d-e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.0192171, size = 43, normalized size = 0.86 \[ -\frac{e g x (2 d g+4 e f+e g x)+2 (d g+e f)^2 \log (d-e x)}{2 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 82, normalized size = 1.6 \begin{align*} -{\frac{{g}^{2}{x}^{2}}{2\,e}}-{\frac{{g}^{2}dx}{{e}^{2}}}-2\,{\frac{fgx}{e}}-{\frac{\ln \left ( ex-d \right ){d}^{2}{g}^{2}}{{e}^{3}}}-2\,{\frac{\ln \left ( ex-d \right ) dfg}{{e}^{2}}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.947378, size = 85, normalized size = 1.7 \begin{align*} -\frac{e g^{2} x^{2} + 2 \,{\left (2 \, e f g + d g^{2}\right )} x}{2 \, e^{2}} - \frac{{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} 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 = 1.78179, size = 136, normalized size = 2.72 \begin{align*} -\frac{e^{2} g^{2} x^{2} + 2 \,{\left (2 \, e^{2} f g + d e g^{2}\right )} x + 2 \,{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{2 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.402451, size = 46, normalized size = 0.92 \begin{align*} - \frac{g^{2} x^{2}}{2 e} - \frac{x \left (d g^{2} + 2 e f g\right )}{e^{2}} - \frac{\left (d g + e f\right )^{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 = 1.19032, size = 181, normalized size = 3.62 \begin{align*} -\frac{1}{2} \,{\left (d^{2} g^{2} e + 2 \, d f g e^{2} + f^{2} e^{3}\right )} e^{\left (-4\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac{1}{2} \,{\left (g^{2} x^{2} e^{3} + 2 \, d g^{2} x e^{2} + 4 \, f g x e^{3}\right )} e^{\left (-4\right )} - \frac{{\left (d^{3} g^{2} + 2 \, d^{2} f g e + d 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 )}{2 \,{\left | d \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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