Integrand size = 32, antiderivative size = 102 \[ \int \frac {(a+b \log (c (d+e x))) (f+g \log (c (d+e x)))}{(d+e x)^2} \, dx=-\frac {b g}{e (d+e x)}-\frac {g (a+b+b \log (c (d+e x)))}{e (d+e x)}-\frac {b (f+g \log (c (d+e x)))}{e (d+e x)}-\frac {(a+b \log (c (d+e x))) (f+g \log (c (d+e x)))}{e (d+e x)} \]
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Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2416, 12, 2341, 2413} \[ \int \frac {(a+b \log (c (d+e x))) (f+g \log (c (d+e x)))}{(d+e x)^2} \, dx=-\frac {(a+b \log (c (d+e x))) (g \log (c (d+e x))+f)}{e (d+e x)}-\frac {g (a+b \log (c (d+e x))+b)}{e (d+e x)}-\frac {b (g \log (c (d+e x))+f)}{e (d+e x)}-\frac {b g}{e (d+e x)} \]
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Rule 12
Rule 2341
Rule 2413
Rule 2416
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {c^2 (a+b \log (x)) (f+g \log (x))}{x^2} \, dx,x,c (d+e x)\right )}{c e} \\ & = \frac {c \text {Subst}\left (\int \frac {(a+b \log (x)) (f+g \log (x))}{x^2} \, dx,x,c (d+e x)\right )}{e} \\ & = -\frac {b (f+g \log (c (d+e x)))}{e (d+e x)}-\frac {(a+b \log (c (d+e x))) (f+g \log (c (d+e x)))}{e (d+e x)}-\frac {(c g) \text {Subst}\left (\int \frac {-a \left (1+\frac {b}{a}\right )-b \log (x)}{x^2} \, dx,x,c (d+e x)\right )}{e} \\ & = -\frac {b g}{e (d+e x)}-\frac {g (a+b+b \log (c (d+e x)))}{e (d+e x)}-\frac {b (f+g \log (c (d+e x)))}{e (d+e x)}-\frac {(a+b \log (c (d+e x))) (f+g \log (c (d+e x)))}{e (d+e x)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.57 \[ \int \frac {(a+b \log (c (d+e x))) (f+g \log (c (d+e x)))}{(d+e x)^2} \, dx=-\frac {a (f+g)+b (f+2 g)+(a g+b (f+2 g)) \log (c (d+e x))+b g \log ^2(c (d+e x))}{e (d+e x)} \]
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Time = 0.57 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.69
method | result | size |
norman | \(\frac {-\frac {a f +a g +b f +2 b g}{e}-\frac {\left (a g +b f +2 b g \right ) \ln \left (c \left (e x +d \right )\right )}{e}-\frac {b g \ln \left (c \left (e x +d \right )\right )^{2}}{e}}{e x +d}\) | \(70\) |
parallelrisch | \(\frac {-\ln \left (c \left (e x +d \right )\right )^{2} b \,e^{2} g -\ln \left (c \left (e x +d \right )\right ) a \,e^{2} g -\ln \left (c \left (e x +d \right )\right ) b \,e^{2} f -2 \ln \left (c \left (e x +d \right )\right ) b \,e^{2} g -a \,e^{2} f -a \,e^{2} g -b \,e^{2} f -2 b \,e^{2} g}{\left (e x +d \right ) e^{3}}\) | \(103\) |
risch | \(-\frac {b g \ln \left (c \left (e x +d \right )\right )^{2}}{e \left (e x +d \right )}-\frac {\left (a g +b f +2 b g \right ) \ln \left (c \left (e x +d \right )\right )}{e \left (e x +d \right )}-\frac {a f}{e \left (e x +d \right )}-\frac {a g}{e \left (e x +d \right )}-\frac {b f}{e \left (e x +d \right )}-\frac {2 b g}{e \left (e x +d \right )}\) | \(113\) |
parts | \(-\frac {a f}{e \left (e x +d \right )}+\frac {\left (a g +b f \right ) c \left (-\frac {\ln \left (c e x +c d \right )}{c e x +c d}-\frac {1}{c e x +c d}\right )}{e}+\frac {b g c \left (-\frac {\ln \left (c e x +c d \right )^{2}}{c e x +c d}-\frac {2 \ln \left (c e x +c d \right )}{c e x +c d}-\frac {2}{c e x +c d}\right )}{e}\) | \(126\) |
derivativedivides | \(\frac {-\frac {c^{2} a f}{c e x +c d}+c^{2} a g \left (-\frac {\ln \left (c e x +c d \right )}{c e x +c d}-\frac {1}{c e x +c d}\right )+c^{2} b f \left (-\frac {\ln \left (c e x +c d \right )}{c e x +c d}-\frac {1}{c e x +c d}\right )+c^{2} b g \left (-\frac {\ln \left (c e x +c d \right )^{2}}{c e x +c d}-\frac {2 \ln \left (c e x +c d \right )}{c e x +c d}-\frac {2}{c e x +c d}\right )}{c e}\) | \(169\) |
default | \(\frac {-\frac {c^{2} a f}{c e x +c d}+c^{2} a g \left (-\frac {\ln \left (c e x +c d \right )}{c e x +c d}-\frac {1}{c e x +c d}\right )+c^{2} b f \left (-\frac {\ln \left (c e x +c d \right )}{c e x +c d}-\frac {1}{c e x +c d}\right )+c^{2} b g \left (-\frac {\ln \left (c e x +c d \right )^{2}}{c e x +c d}-\frac {2 \ln \left (c e x +c d \right )}{c e x +c d}-\frac {2}{c e x +c d}\right )}{c e}\) | \(169\) |
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Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.60 \[ \int \frac {(a+b \log (c (d+e x))) (f+g \log (c (d+e x)))}{(d+e x)^2} \, dx=-\frac {b g \log \left (c e x + c d\right )^{2} + {\left (a + b\right )} f + {\left (a + 2 \, b\right )} g + {\left (b f + {\left (a + 2 \, b\right )} g\right )} \log \left (c e x + c d\right )}{e^{2} x + d e} \]
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Time = 0.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b \log (c (d+e x))) (f+g \log (c (d+e x)))}{(d+e x)^2} \, dx=- \frac {b g \log {\left (c \left (d + e x\right ) \right )}^{2}}{d e + e^{2} x} + \frac {\left (- a g - b f - 2 b g\right ) \log {\left (c \left (d + e x\right ) \right )}}{d e + e^{2} x} - \frac {a f + a g + b f + 2 b g}{d e + e^{2} x} \]
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Time = 0.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.56 \[ \int \frac {(a+b \log (c (d+e x))) (f+g \log (c (d+e x)))}{(d+e x)^2} \, dx=-b {\left (\frac {c e}{c e^{3} x + c d e^{2}} + \frac {\log \left (c e x + c d\right )}{e^{2} x + d e}\right )} f - a {\left (\frac {c e}{c e^{3} x + c d e^{2}} + \frac {\log \left (c e x + c d\right )}{e^{2} x + d e}\right )} g - \frac {a f}{e^{2} x + d e} - \frac {{\left (c^{2} \log \left (c e x + c d\right )^{2} + 2 \, c^{2} \log \left (c e x + c d\right ) + 2 \, c^{2}\right )} b g}{{\left (c e x + c d\right )} c e} \]
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Time = 0.33 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b \log (c (d+e x))) (f+g \log (c (d+e x)))}{(d+e x)^2} \, dx=-\frac {{\left (\frac {b c g \log \left ({\left (e x + d\right )} c\right )^{2}}{{\left (e x + d\right )} e^{2}} + \frac {{\left (b c^{2} f + a c^{2} g + 2 \, b c^{2} g\right )} \log \left ({\left (e x + d\right )} c\right )}{{\left (e x + d\right )} c e^{2}} + \frac {a c^{2} f + b c^{2} f + a c^{2} g + 2 \, b c^{2} g}{{\left (e x + d\right )} c e^{2}}\right )} e}{c} \]
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Time = 1.59 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b \log (c (d+e x))) (f+g \log (c (d+e x)))}{(d+e x)^2} \, dx=-\frac {d\,\left (b\,g\,{\ln \left (c\,d+c\,e\,x\right )}^2+a\,g\,\ln \left (c\,d+c\,e\,x\right )+b\,f\,\ln \left (c\,d+c\,e\,x\right )+2\,b\,g\,\ln \left (c\,d+c\,e\,x\right )\right )-e\,\left (a\,f\,x+a\,g\,x+b\,f\,x+2\,b\,g\,x\right )}{d^2\,e+x\,d\,e^2} \]
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