Optimal. Leaf size=83 \[ \frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^2 (e f-d g)}-\frac{\log (f+g x) \left (a g^2-b f g+c f^2\right )}{g^2 (e f-d g)}+\frac{c x}{e g} \]
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Rubi [A] time = 0.100025, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {893} \[ \frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^2 (e f-d g)}-\frac{\log (f+g x) \left (a g^2-b f g+c f^2\right )}{g^2 (e f-d g)}+\frac{c x}{e g} \]
Antiderivative was successfully verified.
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Rule 893
Rubi steps
\begin{align*} \int \frac{a+b x+c x^2}{(d+e x) (f+g x)} \, dx &=\int \left (\frac{c}{e g}+\frac{c d^2-b d e+a e^2}{e (e f-d g) (d+e x)}+\frac{c f^2-b f g+a g^2}{g (-e f+d g) (f+g x)}\right ) \, dx\\ &=\frac{c x}{e g}+\frac{\left (c d^2-b d e+a e^2\right ) \log (d+e x)}{e^2 (e f-d g)}-\frac{\left (c f^2-b f g+a g^2\right ) \log (f+g x)}{g^2 (e f-d g)}\\ \end{align*}
Mathematica [A] time = 0.0538467, size = 85, normalized size = 1.02 \[ -\frac{\log (d+e x) \left (-a e^2+b d e-c d^2\right )}{e^2 (e f-d g)}-\frac{\log (f+g x) \left (a g^2-b f g+c f^2\right )}{g^2 (e f-d g)}+\frac{c x}{e g} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 142, normalized size = 1.7 \begin{align*}{\frac{cx}{eg}}-{\frac{\ln \left ( ex+d \right ) a}{dg-ef}}+{\frac{\ln \left ( ex+d \right ) bd}{ \left ( dg-ef \right ) e}}-{\frac{\ln \left ( ex+d \right ) c{d}^{2}}{ \left ( dg-ef \right ){e}^{2}}}+{\frac{\ln \left ( gx+f \right ) a}{dg-ef}}-{\frac{\ln \left ( gx+f \right ) bf}{ \left ( dg-ef \right ) g}}+{\frac{\ln \left ( gx+f \right ) c{f}^{2}}{{g}^{2} \left ( dg-ef \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01263, size = 117, normalized size = 1.41 \begin{align*} \frac{{\left (c d^{2} - b d e + a e^{2}\right )} \log \left (e x + d\right )}{e^{3} f - d e^{2} g} - \frac{{\left (c f^{2} - b f g + a g^{2}\right )} \log \left (g x + f\right )}{e f g^{2} - d g^{3}} + \frac{c x}{e g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30229, size = 198, normalized size = 2.39 \begin{align*} \frac{{\left (c d^{2} - b d e + a e^{2}\right )} g^{2} \log \left (e x + d\right ) +{\left (c e^{2} f g - c d e g^{2}\right )} x -{\left (c e^{2} f^{2} - b e^{2} f g + a e^{2} g^{2}\right )} \log \left (g x + f\right )}{e^{3} f g^{2} - d e^{2} g^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 11.337, size = 420, normalized size = 5.06 \begin{align*} \frac{c x}{e g} + \frac{\left (a g^{2} - b f g + c f^{2}\right ) \log{\left (x + \frac{a d e g^{2} + a e^{2} f g - 2 b d e f g + c d^{2} f g + c d e f^{2} - \frac{d^{2} e g \left (a g^{2} - b f g + c f^{2}\right )}{d g - e f} + \frac{2 d e^{2} f \left (a g^{2} - b f g + c f^{2}\right )}{d g - e f} - \frac{e^{3} f^{2} \left (a g^{2} - b f g + c f^{2}\right )}{g \left (d g - e f\right )}}{2 a e^{2} g^{2} - b d e g^{2} - b e^{2} f g + c d^{2} g^{2} + c e^{2} f^{2}} \right )}}{g^{2} \left (d g - e f\right )} - \frac{\left (a e^{2} - b d e + c d^{2}\right ) \log{\left (x + \frac{a d e g^{2} + a e^{2} f g - 2 b d e f g + c d^{2} f g + c d e f^{2} + \frac{d^{2} g^{3} \left (a e^{2} - b d e + c d^{2}\right )}{e \left (d g - e f\right )} - \frac{2 d f g^{2} \left (a e^{2} - b d e + c d^{2}\right )}{d g - e f} + \frac{e f^{2} g \left (a e^{2} - b d e + c d^{2}\right )}{d g - e f}}{2 a e^{2} g^{2} - b d e g^{2} - b e^{2} f g + c d^{2} g^{2} + c e^{2} f^{2}} \right )}}{e^{2} \left (d g - e f\right )} \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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