Optimal. Leaf size=86 \[ \frac {b \log (a+b x)}{(b c-a d) (b e-a f)}-\frac {d \log (c+d x)}{(b c-a d) (d e-c f)}+\frac {f \log (e+f x)}{(b e-a f) (d e-c f)} \]
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Rubi [A]
time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {84}
\begin {gather*} \frac {b \log (a+b x)}{(b c-a d) (b e-a f)}-\frac {d \log (c+d x)}{(b c-a d) (d e-c f)}+\frac {f \log (e+f x)}{(b e-a f) (d e-c f)} \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) (c+d x) (e+f x)} \, dx &=\int \left (\frac {b^2}{(b c-a d) (b e-a f) (a+b x)}+\frac {d^2}{(b c-a d) (-d e+c f) (c+d x)}+\frac {f^2}{(b e-a f) (d e-c f) (e+f x)}\right ) \, dx\\ &=\frac {b \log (a+b x)}{(b c-a d) (b e-a f)}-\frac {d \log (c+d x)}{(b c-a d) (d e-c f)}+\frac {f \log (e+f x)}{(b e-a f) (d e-c f)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 80, normalized size = 0.93 \begin {gather*} \frac {b (-d e+c f) \log (a+b x)+d (b e-a f) \log (c+d x)+(-b c+a d) f \log (e+f x)}{(b c-a d) (b e-a f) (-d e+c f)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 87, normalized size = 1.01
method | result | size |
default | \(\frac {b \ln \left (b x +a \right )}{\left (a d -b c \right ) \left (a f -b e \right )}-\frac {d \ln \left (d x +c \right )}{\left (a d -b c \right ) \left (c f -d e \right )}+\frac {f \ln \left (f x +e \right )}{\left (a f -b e \right ) \left (c f -d e \right )}\) | \(87\) |
norman | \(\frac {f \ln \left (f x +e \right )}{a c \,f^{2}-a d e f -b c e f +b d \,e^{2}}+\frac {b \ln \left (b x +a \right )}{\left (a d -b c \right ) \left (a f -b e \right )}-\frac {d \ln \left (d x +c \right )}{\left (a d -b c \right ) \left (c f -d e \right )}\) | \(94\) |
risch | \(\frac {b \ln \left (b x +a \right )}{a^{2} d f -a b c f -a b d e +b^{2} c e}-\frac {d \ln \left (d x +c \right )}{a c d f -a \,d^{2} e -b \,c^{2} f +b c d e}+\frac {f \ln \left (-f x -e \right )}{a c \,f^{2}-a d e f -b c e f +b d \,e^{2}}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 117, normalized size = 1.36 \begin {gather*} \frac {b \log \left (b x + a\right )}{b^{2} c e - a b d e - {\left (a b c - a^{2} d\right )} f} - \frac {d \log \left (d x + c\right )}{b c d e - a d^{2} e - {\left (b c^{2} - a c d\right )} f} + \frac {f \log \left (f x + e\right )}{a c f^{2} + b d e^{2} - {\left (b c e + a d e\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 6.24, size = 115, normalized size = 1.34 \begin {gather*} \frac {{\left (b c - a d\right )} f \log \left (f x + e\right ) - {\left (b c f - b d e\right )} \log \left (b x + a\right ) + {\left (a d f - b d e\right )} \log \left (d x + c\right )}{{\left (a b c^{2} - a^{2} c d\right )} f^{2} - {\left (b^{2} c^{2} - a^{2} d^{2}\right )} f e + {\left (b^{2} c d - a b d^{2}\right )} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.63, size = 137, normalized size = 1.59 \begin {gather*} -\frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{a b^{2} c f - a^{2} b d f - b^{3} c e + a b^{2} d e} + \frac {d^{2} \log \left ({\left | d x + c \right |}\right )}{b c^{2} d f - a c d^{2} f - b c d^{2} e + a d^{3} e} + \frac {f^{2} \log \left ({\left | f x + e \right |}\right )}{a c f^{3} - b c f^{2} e - a d f^{2} e + b d f e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.38, size = 106, normalized size = 1.23 \begin {gather*} \frac {b\,\ln \left (a+b\,x\right )}{b^2\,c\,e+a^2\,d\,f-a\,b\,c\,f-a\,b\,d\,e}+\frac {d\,\ln \left (c+d\,x\right )}{a\,d^2\,e+b\,c^2\,f-a\,c\,d\,f-b\,c\,d\,e}+\frac {f\,\ln \left (e+f\,x\right )}{a\,c\,f^2+b\,d\,e^2-a\,d\,e\,f-b\,c\,e\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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