Integrand size = 25, antiderivative size = 84 \[ \int \frac {(a+b x) (c+d x)}{(e+f x) (g+h x)} \, dx=\frac {b d x}{f h}+\frac {(b e-a f) (d e-c f) \log (e+f x)}{f^2 (f g-e h)}-\frac {(b g-a h) (d g-c h) \log (g+h x)}{h^2 (f g-e h)} \]
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Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {147} \[ \int \frac {(a+b x) (c+d x)}{(e+f x) (g+h x)} \, dx=\frac {(b e-a f) (d e-c f) \log (e+f x)}{f^2 (f g-e h)}-\frac {(b g-a h) (d g-c h) \log (g+h x)}{h^2 (f g-e h)}+\frac {b d x}{f h} \]
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Rule 147
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b x) (c+d x)}{(e+f x) (g+h x)} \, dx=\frac {(b e-a f) (d e-c f) h^2 \log (e+f x)+f (b d h (f g-e h) x-f (b g-a h) (d g-c h) \log (g+h x))}{f^2 h^2 (f g-e h)} \]
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Time = 1.57 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.21
method | result | size |
default | \(\frac {b d x}{f h}+\frac {\left (a c \,h^{2}-a d g h -b c g h +b d \,g^{2}\right ) \ln \left (h x +g \right )}{h^{2} \left (e h -f g \right )}+\frac {\left (-a c \,f^{2}+a d e f +b c e f -b d \,e^{2}\right ) \ln \left (f x +e \right )}{f^{2} \left (e h -f g \right )}\) | \(102\) |
norman | \(\frac {b d x}{f h}+\frac {\left (a c \,h^{2}-a d g h -b c g h +b d \,g^{2}\right ) \ln \left (h x +g \right )}{h^{2} \left (e h -f g \right )}-\frac {\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) \ln \left (f x +e \right )}{\left (e h -f g \right ) f^{2}}\) | \(103\) |
parallelrisch | \(-\frac {\ln \left (f x +e \right ) a c \,f^{2} h^{2}-\ln \left (f x +e \right ) a d e f \,h^{2}-\ln \left (f x +e \right ) b c e f \,h^{2}+\ln \left (f x +e \right ) b d \,e^{2} h^{2}-\ln \left (h x +g \right ) a c \,f^{2} h^{2}+\ln \left (h x +g \right ) a d \,f^{2} g h +\ln \left (h x +g \right ) b c \,f^{2} g h -\ln \left (h x +g \right ) b d \,f^{2} g^{2}-x b d e f \,h^{2}+x b d \,f^{2} g h}{f^{2} h^{2} \left (e h -f g \right )}\) | \(159\) |
risch | \(\frac {b d x}{f h}-\frac {\ln \left (f x +e \right ) a c}{e h -f g}+\frac {\ln \left (f x +e \right ) a d e}{\left (e h -f g \right ) f}+\frac {\ln \left (f x +e \right ) b c e}{\left (e h -f g \right ) f}-\frac {\ln \left (f x +e \right ) b d \,e^{2}}{\left (e h -f g \right ) f^{2}}+\frac {\ln \left (-h x -g \right ) a c}{e h -f g}-\frac {\ln \left (-h x -g \right ) a d g}{h \left (e h -f g \right )}-\frac {\ln \left (-h x -g \right ) b c g}{h \left (e h -f g \right )}+\frac {\ln \left (-h x -g \right ) b d \,g^{2}}{h^{2} \left (e h -f g \right )}\) | \(208\) |
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Time = 0.26 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b x) (c+d x)}{(e+f x) (g+h x)} \, dx=\frac {{\left (b d e^{2} + a c f^{2} - {\left (b c + a d\right )} e f\right )} h^{2} \log \left (f x + e\right ) + {\left (b d f^{2} g h - b d e f h^{2}\right )} x - {\left (b d f^{2} g^{2} + a c f^{2} h^{2} - {\left (b c + a d\right )} f^{2} g h\right )} \log \left (h x + g\right )}{f^{3} g h^{2} - e f^{2} h^{3}} \]
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Timed out. \[ \int \frac {(a+b x) (c+d x)}{(e+f x) (g+h x)} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b x) (c+d x)}{(e+f x) (g+h x)} \, dx=\frac {b d x}{f h} + \frac {{\left (b d e^{2} + a c f^{2} - {\left (b c + a d\right )} e f\right )} \log \left (f x + e\right )}{f^{3} g - e f^{2} h} - \frac {{\left (b d g^{2} + a c h^{2} - {\left (b c + a d\right )} g h\right )} \log \left (h x + g\right )}{f g h^{2} - e h^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.29 \[ \int \frac {(a+b x) (c+d x)}{(e+f x) (g+h x)} \, dx=\frac {b d x}{f h} + \frac {{\left (b d e^{2} - b c e f - a d e f + a c f^{2}\right )} \log \left ({\left | f x + e \right |}\right )}{f^{3} g - e f^{2} h} - \frac {{\left (b d g^{2} - b c g h - a d g h + a c h^{2}\right )} \log \left ({\left | h x + g \right |}\right )}{f g h^{2} - e h^{3}} \]
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Time = 3.47 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.25 \[ \int \frac {(a+b x) (c+d x)}{(e+f x) (g+h x)} \, dx=\frac {\ln \left (e+f\,x\right )\,\left (a\,c\,f^2-f\,\left (a\,d\,e+b\,c\,e\right )+b\,d\,e^2\right )}{f^3\,g-e\,f^2\,h}+\frac {\ln \left (g+h\,x\right )\,\left (a\,c\,h^2-h\,\left (a\,d\,g+b\,c\,g\right )+b\,d\,g^2\right )}{e\,h^3-f\,g\,h^2}+\frac {b\,d\,x}{f\,h} \]
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