Integrand size = 28, antiderivative size = 92 \[ \int \frac {\log (c (d+e x)) (a+b \log (c (d+e x)))}{(d+e x)^2} \, dx=-\frac {b}{e (d+e x)}-\frac {b \log (c (d+e x))}{e (d+e x)}-\frac {\log (c (d+e x)) (a+b \log (c (d+e x)))}{e (d+e x)}-\frac {a+b+b \log (c (d+e x))}{e (d+e x)} \]
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Time = 0.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2416, 12, 2341, 2413} \[ \int \frac {\log (c (d+e x)) (a+b \log (c (d+e x)))}{(d+e x)^2} \, dx=-\frac {\log (c (d+e x)) (a+b \log (c (d+e x)))}{e (d+e x)}-\frac {a+b \log (c (d+e x))+b}{e (d+e x)}-\frac {b \log (c (d+e x))}{e (d+e x)}-\frac {b}{e (d+e x)} \]
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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 \log (x) (a+b \log (x))}{x^2} \, dx,x,c (d+e x)\right )}{c e} \\ & = \frac {c \text {Subst}\left (\int \frac {\log (x) (a+b \log (x))}{x^2} \, dx,x,c (d+e x)\right )}{e} \\ & = -\frac {b \log (c (d+e x))}{e (d+e x)}-\frac {\log (c (d+e x)) (a+b \log (c (d+e x)))}{e (d+e x)}-\frac {c \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}{e (d+e x)}-\frac {b \log (c (d+e x))}{e (d+e x)}-\frac {\log (c (d+e x)) (a+b \log (c (d+e x)))}{e (d+e x)}-\frac {a+b+b \log (c (d+e x))}{e (d+e x)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.47 \[ \int \frac {\log (c (d+e x)) (a+b \log (c (d+e x)))}{(d+e x)^2} \, dx=-\frac {a+2 b+(a+2 b) \log (c (d+e x))+b \log ^2(c (d+e x))}{e (d+e x)} \]
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Time = 0.58 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.59
method | result | size |
norman | \(\frac {-\frac {a +2 b}{e}-\frac {b \ln \left (c \left (e x +d \right )\right )^{2}}{e}-\frac {\left (a +2 b \right ) \ln \left (c \left (e x +d \right )\right )}{e}}{e x +d}\) | \(54\) |
parallelrisch | \(\frac {-\ln \left (c \left (e x +d \right )\right )^{2} b \,e^{2}-\ln \left (c \left (e x +d \right )\right ) a \,e^{2}-2 \ln \left (c \left (e x +d \right )\right ) b \,e^{2}-a \,e^{2}-2 e^{2} b}{\left (e x +d \right ) e^{3}}\) | \(69\) |
risch | \(-\frac {b \ln \left (c \left (e x +d \right )\right )^{2}}{e \left (e x +d \right )}-\frac {\left (a +2 b \right ) \ln \left (c \left (e x +d \right )\right )}{e \left (e x +d \right )}-\frac {a}{e \left (e x +d \right )}-\frac {2 b}{e \left (e x +d \right )}\) | \(76\) |
parts | \(\frac {a c \left (-\frac {\ln \left (c e x +c d \right )}{c e x +c d}-\frac {1}{c e x +c d}\right )}{e}+\frac {b c \left (-\frac {\ln \left (c e x +c d \right )^{2}}{c e x +c d}-\frac {2 \ln \left (c e x +c d \right )}{c e x +c d}-\frac {2}{c e x +c d}\right )}{e}\) | \(105\) |
derivativedivides | \(\frac {c^{2} a \left (-\frac {\ln \left (c e x +c d \right )}{c e x +c d}-\frac {1}{c e x +c d}\right )+c^{2} b \left (-\frac {\ln \left (c e x +c d \right )^{2}}{c e x +c d}-\frac {2 \ln \left (c e x +c d \right )}{c e x +c d}-\frac {2}{c e x +c d}\right )}{c e}\) | \(110\) |
default | \(\frac {c^{2} a \left (-\frac {\ln \left (c e x +c d \right )}{c e x +c d}-\frac {1}{c e x +c d}\right )+c^{2} b \left (-\frac {\ln \left (c e x +c d \right )^{2}}{c e x +c d}-\frac {2 \ln \left (c e x +c d \right )}{c e x +c d}-\frac {2}{c e x +c d}\right )}{c e}\) | \(110\) |
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Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.50 \[ \int \frac {\log (c (d+e x)) (a+b \log (c (d+e x)))}{(d+e x)^2} \, dx=-\frac {b \log \left (c e x + c d\right )^{2} + {\left (a + 2 \, b\right )} \log \left (c e x + c d\right ) + a + 2 \, b}{e^{2} x + d e} \]
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Time = 0.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.61 \[ \int \frac {\log (c (d+e x)) (a+b \log (c (d+e x)))}{(d+e x)^2} \, dx=- \frac {b \log {\left (c \left (d + e x\right ) \right )}^{2}}{d e + e^{2} x} + \frac {\left (- a - 2 b\right ) \log {\left (c \left (d + e x\right ) \right )}}{d e + e^{2} x} - \frac {a + 2 b}{d e + e^{2} x} \]
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Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.08 \[ \int \frac {\log (c (d+e x)) (a+b \log (c (d+e x)))}{(d+e x)^2} \, dx=-{\left (b {\left (\frac {c e}{c e^{3} x + c d e^{2}} + \frac {\log \left (c e x + c d\right )}{e^{2} x + d e}\right )} + \frac {a}{e^{2} x + d e}\right )} \log \left ({\left (e x + d\right )} c\right ) - \frac {{\left (b {\left (\log \left (c\right ) + 2\right )} + b \log \left (e x + d\right ) + a\right )} e}{e^{3} x + d e^{2}} \]
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Time = 0.32 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.98 \[ \int \frac {\log (c (d+e x)) (a+b \log (c (d+e x)))}{(d+e x)^2} \, dx=-\frac {e {\left (\frac {b c \log \left ({\left (e x + d\right )} c\right )^{2}}{{\left (e x + d\right )} e^{2}} + \frac {{\left (a c^{2} + 2 \, b c^{2}\right )} \log \left ({\left (e x + d\right )} c\right )}{{\left (e x + d\right )} c e^{2}} + \frac {a c^{2} + 2 \, b c^{2}}{{\left (e x + d\right )} c e^{2}}\right )}}{c} \]
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Time = 1.58 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.68 \[ \int \frac {\log (c (d+e x)) (a+b \log (c (d+e x)))}{(d+e x)^2} \, dx=-\frac {d\,\left (b\,{\ln \left (c\,\left (d+e\,x\right )\right )}^2+a\,\ln \left (c\,\left (d+e\,x\right )\right )+2\,b\,\ln \left (c\,\left (d+e\,x\right )\right )\right )-e\,\left (a\,x+2\,b\,x\right )}{d\,e\,\left (d+e\,x\right )} \]
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