Optimal. Leaf size=163 \[ \frac {b^2 \log (a+b x)}{(b c-a d) (b e-a f) (b g-a h)}-\frac {d^2 \log (c+d x)}{(b c-a d) (d e-c f) (d g-c h)}+\frac {f^2 \log (e+f x)}{(b e-a f) (d e-c f) (f g-e h)}-\frac {h^2 \log (g+h x)}{(b g-a h) (d g-c h) (f g-e h)} \]
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Rubi [A] time = 0.21, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {180} \[ \frac {b^2 \log (a+b x)}{(b c-a d) (b e-a f) (b g-a h)}-\frac {d^2 \log (c+d x)}{(b c-a d) (d e-c f) (d g-c h)}+\frac {f^2 \log (e+f x)}{(b e-a f) (d e-c f) (f g-e h)}-\frac {h^2 \log (g+h x)}{(b g-a h) (d g-c h) (f g-e h)} \]
Antiderivative was successfully verified.
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Rule 180
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) (c+d x) (e+f x) (g+h x)} \, dx &=\int \left (\frac {b^3}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}-\frac {d^3}{(b c-a d) (-d e+c f) (-d g+c h) (c+d x)}-\frac {f^3}{(b e-a f) (d e-c f) (-f g+e h) (e+f x)}-\frac {h^3}{(b g-a h) (d g-c h) (f g-e h) (g+h x)}\right ) \, dx\\ &=\frac {b^2 \log (a+b x)}{(b c-a d) (b e-a f) (b g-a h)}-\frac {d^2 \log (c+d x)}{(b c-a d) (d e-c f) (d g-c h)}+\frac {f^2 \log (e+f x)}{(b e-a f) (d e-c f) (f g-e h)}-\frac {h^2 \log (g+h x)}{(b g-a h) (d g-c h) (f g-e h)}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 164, normalized size = 1.01 \[ \frac {b^2 \log (a+b x)}{(b c-a d) (b e-a f) (b g-a h)}-\frac {d^2 \log (c+d x)}{(b c-a d) (c f-d e) (c h-d g)}-\frac {f^2 \log (e+f x)}{(b e-a f) (d e-c f) (e h-f g)}-\frac {h^2 \log (g+h x)}{(b g-a h) (d g-c h) (f g-e h)} \]
Antiderivative was successfully verified.
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fricas [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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giac [B] time = 1.19, size = 363, normalized size = 2.23 \[ -\frac {b^{3} \log \left ({\left | b x + a \right |}\right )}{a b^{3} c f g - a^{2} b^{2} d f g - a^{2} b^{2} c f h + a^{3} b d f h - b^{4} c g e + a b^{3} d g e + a b^{3} c h e - a^{2} b^{2} d h e} + \frac {d^{3} \log \left ({\left | d x + c \right |}\right )}{b c^{2} d^{2} f g - a c d^{3} f g - b c^{3} d f h + a c^{2} d^{2} f h - b c d^{3} g e + a d^{4} g e + b c^{2} d^{2} h e - a c d^{3} h e} + \frac {f^{3} \log \left ({\left | f x + e \right |}\right )}{a c f^{4} g - b c f^{3} g e - a d f^{3} g e - a c f^{3} h e + b d f^{2} g e^{2} + b c f^{2} h e^{2} + a d f^{2} h e^{2} - b d f h e^{3}} - \frac {h^{3} \log \left ({\left | h x + g \right |}\right )}{b d f g^{3} h - b c f g^{2} h^{2} - a d f g^{2} h^{2} + a c f g h^{3} - b d g^{2} h^{2} e + b c g h^{3} e + a d g h^{3} e - a c h^{4} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 164, normalized size = 1.01 \[ -\frac {b^{2} \ln \left (b x +a \right )}{\left (a d -b c \right ) \left (a f -b e \right ) \left (a h -b g \right )}+\frac {d^{2} \ln \left (d x +c \right )}{\left (a d -b c \right ) \left (c f -d e \right ) \left (c h -d g \right )}-\frac {f^{2} \ln \left (f x +e \right )}{\left (c f -d e \right ) \left (e h -f g \right ) \left (a f -b e \right )}+\frac {h^{2} \ln \left (h x +g \right )}{\left (c h -d g \right ) \left (a h -b g \right ) \left (e h -f g \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 310, normalized size = 1.90 \[ \frac {b^{2} \log \left (b x + a\right )}{{\left ({\left (b^{3} c - a b^{2} d\right )} e - {\left (a b^{2} c - a^{2} b d\right )} f\right )} g - {\left ({\left (a b^{2} c - a^{2} b d\right )} e - {\left (a^{2} b c - a^{3} d\right )} f\right )} h} - \frac {d^{2} \log \left (d x + c\right )}{{\left ({\left (b c d^{2} - a d^{3}\right )} e - {\left (b c^{2} d - a c d^{2}\right )} f\right )} g - {\left ({\left (b c^{2} d - a c d^{2}\right )} e - {\left (b c^{3} - a c^{2} d\right )} f\right )} h} + \frac {f^{2} \log \left (f x + e\right )}{{\left (b d e^{2} f + a c f^{3} - {\left (b c + a d\right )} e f^{2}\right )} g - {\left (b d e^{3} + a c e f^{2} - {\left (b c + a d\right )} e^{2} f\right )} h} - \frac {h^{2} \log \left (h x + g\right )}{b d f g^{3} - a c e h^{3} - {\left (b d e + {\left (b c + a d\right )} f\right )} g^{2} h + {\left (a c f + {\left (b c + a d\right )} e\right )} g h^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.62, size = 317, normalized size = 1.94 \[ \frac {b^2\,\ln \left (a+b\,x\right )}{b^3\,c\,e\,g-a^3\,d\,f\,h-a\,b^2\,c\,e\,h-a\,b^2\,c\,f\,g-a\,b^2\,d\,e\,g+a^2\,b\,c\,f\,h+a^2\,b\,d\,e\,h+a^2\,b\,d\,f\,g}+\frac {d^2\,\ln \left (c+d\,x\right )}{a\,d^3\,e\,g-b\,c^3\,f\,h-a\,c\,d^2\,e\,h-a\,c\,d^2\,f\,g-b\,c\,d^2\,e\,g+a\,c^2\,d\,f\,h+b\,c^2\,d\,e\,h+b\,c^2\,d\,f\,g}+\frac {f^2\,\ln \left (e+f\,x\right )}{a\,c\,f^3\,g-b\,d\,e^3\,h-a\,c\,e\,f^2\,h-a\,d\,e\,f^2\,g-b\,c\,e\,f^2\,g+a\,d\,e^2\,f\,h+b\,c\,e^2\,f\,h+b\,d\,e^2\,f\,g}+\frac {h^2\,\ln \left (g+h\,x\right )}{a\,c\,e\,h^3-b\,d\,f\,g^3-a\,c\,f\,g\,h^2-a\,d\,e\,g\,h^2-b\,c\,e\,g\,h^2+a\,d\,f\,g^2\,h+b\,c\,f\,g^2\,h+b\,d\,e\,g^2\,h} \]
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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