Optimal. Leaf size=84 \[ \frac{(b e-a f) (d e-c f) \log (e+f x)}{f^2 (f g-e h)}-\frac{(b g-a h) (d g-c h) \log (g+h x)}{h^2 (f g-e h)}+\frac{b d x}{f h} \]
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Rubi [A] time = 0.0860184, antiderivative size = 84, 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 = {142} \[ \frac{(b e-a f) (d e-c f) \log (e+f x)}{f^2 (f g-e h)}-\frac{(b g-a h) (d g-c h) \log (g+h x)}{h^2 (f g-e h)}+\frac{b d x}{f h} \]
Antiderivative was successfully verified.
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Rule 142
Rubi steps
\begin{align*} \int \frac{(a+b x) (c+d x)}{(e+f x) (g+h x)} \, dx &=\int \left (\frac{b d}{f h}+\frac{(-b e+a f) (-d e+c f)}{f (f g-e h) (e+f x)}+\frac{(-b g+a h) (-d g+c h)}{h (-f g+e h) (g+h x)}\right ) \, dx\\ &=\frac{b d x}{f h}+\frac{(b e-a f) (d e-c f) \log (e+f x)}{f^2 (f g-e h)}-\frac{(b g-a h) (d g-c h) \log (g+h x)}{h^2 (f g-e h)}\\ \end{align*}
Mathematica [A] time = 0.0706683, size = 85, normalized size = 1.01 \[ \frac{f (b d h x (f g-e h)-f (b g-a h) (d g-c h) \log (g+h x))+h^2 (b e-a f) (d e-c f) \log (e+f x)}{f^2 h^2 (f g-e h)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 196, normalized size = 2.3 \begin{align*}{\frac{bdx}{fh}}-{\frac{\ln \left ( fx+e \right ) ac}{eh-fg}}+{\frac{\ln \left ( fx+e \right ) ade}{f \left ( eh-fg \right ) }}+{\frac{\ln \left ( fx+e \right ) bce}{f \left ( eh-fg \right ) }}-{\frac{\ln \left ( fx+e \right ) bd{e}^{2}}{{f}^{2} \left ( eh-fg \right ) }}+{\frac{\ln \left ( hx+g \right ) ac}{eh-fg}}-{\frac{\ln \left ( hx+g \right ) adg}{h \left ( eh-fg \right ) }}-{\frac{\ln \left ( hx+g \right ) bcg}{h \left ( eh-fg \right ) }}+{\frac{\ln \left ( hx+g \right ) bd{g}^{2}}{{h}^{2} \left ( eh-fg \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.37506, size = 140, normalized size = 1.67 \begin{align*} \frac{b d x}{f h} + \frac{{\left (b d e^{2} + a c f^{2} -{\left (b c + a d\right )} e f\right )} \log \left (f x + e\right )}{f^{3} g - e f^{2} h} - \frac{{\left (b d g^{2} + a c h^{2} -{\left (b c + a d\right )} g h\right )} \log \left (h x + g\right )}{f g h^{2} - e h^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48352, size = 242, normalized size = 2.88 \begin{align*} \frac{{\left (b d e^{2} + a c f^{2} -{\left (b c + a d\right )} e f\right )} h^{2} \log \left (f x + e\right ) +{\left (b d f^{2} g h - b d e f h^{2}\right )} x -{\left (b d f^{2} g^{2} + a c f^{2} h^{2} -{\left (b c + a d\right )} f^{2} g h\right )} \log \left (h x + g\right )}{f^{3} g h^{2} - e f^{2} h^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 10.4987, size = 507, normalized size = 6.04 \begin{align*} \frac{b d x}{f h} + \frac{\left (a h - b g\right ) \left (c h - d g\right ) \log{\left (x + \frac{a c e f h^{2} + a c f^{2} g h - 2 a d e f g h - 2 b c e f g h + b d e^{2} g h + b d e f g^{2} - \frac{e^{2} f h \left (a h - b g\right ) \left (c h - d g\right )}{e h - f g} + \frac{2 e f^{2} g \left (a h - b g\right ) \left (c h - d g\right )}{e h - f g} - \frac{f^{3} g^{2} \left (a h - b g\right ) \left (c h - d g\right )}{h \left (e h - f g\right )}}{2 a c f^{2} h^{2} - a d e f h^{2} - a d f^{2} g h - b c e f h^{2} - b c f^{2} g h + b d e^{2} h^{2} + b d f^{2} g^{2}} \right )}}{h^{2} \left (e h - f g\right )} - \frac{\left (a f - b e\right ) \left (c f - d e\right ) \log{\left (x + \frac{a c e f h^{2} + a c f^{2} g h - 2 a d e f g h - 2 b c e f g h + b d e^{2} g h + b d e f g^{2} + \frac{e^{2} h^{3} \left (a f - b e\right ) \left (c f - d e\right )}{f \left (e h - f g\right )} - \frac{2 e g h^{2} \left (a f - b e\right ) \left (c f - d e\right )}{e h - f g} + \frac{f g^{2} h \left (a f - b e\right ) \left (c f - d e\right )}{e h - f g}}{2 a c f^{2} h^{2} - a d e f h^{2} - a d f^{2} g h - b c e f h^{2} - b c f^{2} g h + b d e^{2} h^{2} + b d f^{2} g^{2}} \right )}}{f^{2} \left (e h - f g\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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