Integrand size = 46, antiderivative size = 86 \[ \int \frac {1}{a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\frac {b \log (a+b x)}{(b c-a d) (b e-a f)}-\frac {d \log (c+d x)}{(b c-a d) (d e-c f)}+\frac {f \log (e+f x)}{(b e-a f) (d e-c f)} \]
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Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {2083} \[ \int \frac {1}{a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\frac {b \log (a+b x)}{(b c-a d) (b e-a f)}-\frac {d \log (c+d x)}{(b c-a d) (d e-c f)}+\frac {f \log (e+f x)}{(b e-a f) (d e-c f)} \]
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Rule 2083
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.93 \[ \int \frac {1}{a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\frac {b (-d e+c f) \log (a+b x)+d (b e-a f) \log (c+d x)+(-b c+a d) f \log (e+f x)}{(b c-a d) (b e-a f) (-d e+c f)} \]
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Time = 0.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.01
method | result | size |
default | \(\frac {f \ln \left (f x +e \right )}{\left (c f -e d \right ) \left (a f -b e \right )}-\frac {d \ln \left (d x +c \right )}{\left (c f -e d \right ) \left (d a -b c \right )}+\frac {b \ln \left (b x +a \right )}{\left (a f -b e \right ) \left (d a -b c \right )}\) | \(87\) |
norman | \(\frac {f \ln \left (f x +e \right )}{a c \,f^{2}-a d e f -b c e f +d \,e^{2} b}+\frac {b \ln \left (b x +a \right )}{\left (a f -b e \right ) \left (d a -b c \right )}-\frac {d \ln \left (d x +c \right )}{\left (c f -e d \right ) \left (d a -b c \right )}\) | \(94\) |
parallelrisch | \(\frac {\ln \left (b x +a \right ) b c f -\ln \left (b x +a \right ) b d e -\ln \left (d x +c \right ) a d f +\ln \left (d x +c \right ) b d e +\ln \left (f x +e \right ) a d f -\ln \left (f x +e \right ) b c f}{\left (a c \,f^{2}-a d e f -b c e f +d \,e^{2} b \right ) \left (d a -b c \right )}\) | \(103\) |
risch | \(-\frac {d \ln \left (d x +c \right )}{a c d f -a \,d^{2} e -b \,c^{2} f +b c d e}+\frac {f \ln \left (-f x -e \right )}{a c \,f^{2}-a d e f -b c e f +d \,e^{2} b}+\frac {b \ln \left (b x +a \right )}{a^{2} d f -a b c f -a b d e +b^{2} c e}\) | \(111\) |
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Time = 2.91 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.30 \[ \int \frac {1}{a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\frac {{\left (b c - a d\right )} f \log \left (f x + e\right ) + {\left (b d e - b c f\right )} \log \left (b x + a\right ) - {\left (b d e - a d f\right )} \log \left (d x + c\right )}{{\left (b^{2} c d - a b d^{2}\right )} e^{2} - {\left (b^{2} c^{2} - a^{2} d^{2}\right )} e f + {\left (a b c^{2} - a^{2} c d\right )} f^{2}} \]
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Timed out. \[ \int \frac {1}{a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.30 \[ \int \frac {1}{a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\frac {b \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} e - {\left (a b c - a^{2} d\right )} f} - \frac {d \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} e - {\left (b c^{2} - a c d\right )} f} + \frac {f \log \left (f x + e\right )}{b d e^{2} + a c f^{2} - {\left (b c + a d\right )} e f} \]
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Time = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.52 \[ \int \frac {1}{a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3} c e - a b^{2} d e - a b^{2} c f + a^{2} b d f} - \frac {d^{2} \log \left ({\left | d x + c \right |}\right )}{b c d^{2} e - a d^{3} e - b c^{2} d f + a c d^{2} f} + \frac {f^{2} \log \left ({\left | f x + e \right |}\right )}{b d e^{2} f - b c e f^{2} - a d e f^{2} + a c f^{3}} \]
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Time = 9.71 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.23 \[ \int \frac {1}{a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\frac {b\,\ln \left (a+b\,x\right )}{b^2\,c\,e+a^2\,d\,f-a\,b\,c\,f-a\,b\,d\,e}+\frac {d\,\ln \left (c+d\,x\right )}{a\,d^2\,e+b\,c^2\,f-a\,c\,d\,f-b\,c\,d\,e}+\frac {f\,\ln \left (e+f\,x\right )}{a\,c\,f^2+b\,d\,e^2-a\,d\,e\,f-b\,c\,e\,f} \]
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