Optimal. Leaf size=108 \[ -\frac {(b c-a d) \log (c+d x)}{(d e-c f) (d g-c h)}+\frac {(b e-a f) \log (e+f x)}{(d e-c f) (f g-e h)}-\frac {(b g-a h) \log (g+h x)}{(d g-c h) (f g-e h)} \]
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Rubi [A] time = 0.11, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {148} \[ -\frac {(b c-a d) \log (c+d x)}{(d e-c f) (d g-c h)}+\frac {(b e-a f) \log (e+f x)}{(d e-c f) (f g-e h)}-\frac {(b g-a h) \log (g+h x)}{(d g-c h) (f g-e h)} \]
Antiderivative was successfully verified.
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Rule 148
Rubi steps
\begin {align*} \int \frac {a+b x}{(c+d x) (e+f x) (g+h x)} \, dx &=\int \left (\frac {d (-b c+a d)}{(d e-c f) (d g-c h) (c+d x)}+\frac {f (-b e+a f)}{(d e-c f) (-f g+e h) (e+f x)}+\frac {h (-b g+a h)}{(d g-c h) (f g-e h) (g+h x)}\right ) \, dx\\ &=-\frac {(b c-a d) \log (c+d x)}{(d e-c f) (d g-c h)}+\frac {(b e-a f) \log (e+f x)}{(d e-c f) (f g-e h)}-\frac {(b g-a h) \log (g+h x)}{(d g-c h) (f g-e h)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 102, normalized size = 0.94 \[ \frac {(b c-a d) \log (c+d x) (f g-e h)-(b e-a f) (d g-c h) \log (e+f x)+(b g-a h) (d e-c f) \log (g+h x)}{(d e-c f) (d g-c h) (e h-f g)} \]
Antiderivative was successfully verified.
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fricas [A] time = 152.52, size = 160, normalized size = 1.48 \[ -\frac {{\left ({\left (b c - a d\right )} f g - {\left (b c - a d\right )} e h\right )} \log \left (d x + c\right ) - {\left ({\left (b d e - a d f\right )} g - {\left (b c e - a c f\right )} h\right )} \log \left (f x + e\right ) + {\left ({\left (b d e - b c f\right )} g - {\left (a d e - a c f\right )} h\right )} \log \left (h x + g\right )}{{\left (d^{2} e f - c d f^{2}\right )} g^{2} - {\left (d^{2} e^{2} - c^{2} f^{2}\right )} g h + {\left (c d e^{2} - c^{2} e f\right )} h^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 162, normalized size = 1.50 \[ \frac {{\left (b c d - a d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{c d^{2} f g - c^{2} d f h - d^{3} g e + c d^{2} h e} + \frac {{\left (a f^{2} - b f e\right )} \log \left ({\left | f x + e \right |}\right )}{c f^{3} g - d f^{2} g e - c f^{2} h e + d f h e^{2}} - \frac {{\left (b g h - a h^{2}\right )} \log \left ({\left | h x + g \right |}\right )}{d f g^{2} h - c f g h^{2} - d g h^{2} e + c h^{3} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 179, normalized size = 1.66 \[ \frac {a d \ln \left (d x +c \right )}{\left (c f -d e \right ) \left (c h -d g \right )}-\frac {a f \ln \left (f x +e \right )}{\left (c f -d e \right ) \left (e h -f g \right )}+\frac {a h \ln \left (h x +g \right )}{\left (c h -d g \right ) \left (e h -f g \right )}-\frac {b c \ln \left (d x +c \right )}{\left (c f -d e \right ) \left (c h -d g \right )}+\frac {b e \ln \left (f x +e \right )}{\left (c f -d e \right ) \left (e h -f g \right )}-\frac {b g \ln \left (h x +g \right )}{\left (c h -d g \right ) \left (e h -f g \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 134, normalized size = 1.24 \[ -\frac {{\left (b c - a d\right )} \log \left (d x + c\right )}{{\left (d^{2} e - c d f\right )} g - {\left (c d e - c^{2} f\right )} h} + \frac {{\left (b e - a f\right )} \log \left (f x + e\right )}{{\left (d e f - c f^{2}\right )} g - {\left (d e^{2} - c e f\right )} h} - \frac {{\left (b g - a h\right )} \log \left (h x + g\right )}{d f g^{2} + c e h^{2} - {\left (d e + c f\right )} g h} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.17, size = 127, normalized size = 1.18 \[ \frac {\ln \left (e+f\,x\right )\,\left (a\,f-b\,e\right )}{c\,f^2\,g+d\,e^2\,h-c\,e\,f\,h-d\,e\,f\,g}+\frac {\ln \left (g+h\,x\right )\,\left (a\,h-b\,g\right )}{c\,e\,h^2+d\,f\,g^2-c\,f\,g\,h-d\,e\,g\,h}+\frac {\ln \left (c+d\,x\right )\,\left (a\,d-b\,c\right )}{d^2\,e\,g+c^2\,f\,h-c\,d\,e\,h-c\,d\,f\,g} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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