Integrand size = 29, antiderivative size = 110 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right ) \, dx=-((b f+a g) n x)+2 b g n^2 x-\frac {2 b g n (d+e x) \log \left (c (d+e x)^n\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {d \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b e g} \]
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Time = 0.12 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2478, 2458, 2388, 2338, 2332} \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {d \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b e g}+x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )-n x (a g+b f)-\frac {2 b g n (d+e x) \log \left (c (d+e x)^n\right )}{e}+2 b g n^2 x \]
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Rule 2332
Rule 2338
Rule 2388
Rule 2458
Rule 2478
Rubi steps \begin{align*} \text {integral}& = x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-(e n) \int \frac {x \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx \\ & = x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-n \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right ) \left (b f+a g+2 b g \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right ) \\ & = x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac {n \text {Subst}\left (\int \left (b f+a g+2 b g \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}+\frac {(d n) \text {Subst}\left (\int \frac {b f+a g+2 b g \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{e} \\ & = -((b f+a g) n x)+x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {d \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b e g}-\frac {(2 b g n) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e} \\ & = -((b f+a g) n x)+2 b g n^2 x-\frac {2 b g n (d+e x) \log \left (c (d+e x)^n\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {d \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b e g} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.69 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {e (a (f-g n)+b n (-f+2 g n)) x+(a g+b (f-2 g n)) (d+e x) \log \left (c (d+e x)^n\right )+b g (d+e x) \log ^2\left (c (d+e x)^n\right )}{e} \]
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Time = 0.40 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05
method | result | size |
norman | \(\left (2 b g \,n^{2}-a g n -b f n +a f \right ) x +\left (-2 b g n +a g +b f \right ) x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )+b g x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}+\frac {n \left (-2 d g b n +a d g +b d f \right ) \ln \left (e x +d \right )}{e}+\frac {d g b \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}}{e}\) | \(116\) |
parts | \(x a f +\left (a g +b f \right ) \left (x \ln \left (c \left (e x +d \right )^{n}\right )-e n \left (\frac {x}{e}-\frac {d \ln \left (e x +d \right )}{e^{2}}\right )\right )+b g x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}+\frac {d g b \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}}{e}+2 b g \,n^{2} x -\frac {2 n^{2} d g b \ln \left (e x +d \right )}{e}-2 b g n x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )\) | \(131\) |
default | \(x a f +x a g \ln \left (c \left (e x +d \right )^{n}\right )-a g n x +\frac {a g n d \ln \left (e x +d \right )}{e}+x b \ln \left (c \left (e x +d \right )^{n}\right ) f -b f n x +\frac {b f n d \ln \left (e x +d \right )}{e}+b g x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}+\frac {d g b \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )^{2}}{e}+2 b g \,n^{2} x -\frac {2 n^{2} d g b \ln \left (e x +d \right )}{e}-2 b g n x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )\) | \(156\) |
parallelrisch | \(\frac {x \ln \left (c \left (e x +d \right )^{n}\right )^{2} b e g n -2 x \ln \left (c \left (e x +d \right )^{n}\right ) b e g \,n^{2}+2 x b e g \,n^{3}+x \ln \left (c \left (e x +d \right )^{n}\right ) a e g n +x \ln \left (c \left (e x +d \right )^{n}\right ) b e f n -x a e g \,n^{2}-x b e f \,n^{2}+\ln \left (c \left (e x +d \right )^{n}\right )^{2} b d g n -2 \ln \left (c \left (e x +d \right )^{n}\right ) b d g \,n^{2}-2 b d g \,n^{3}+x a e f n +\ln \left (c \left (e x +d \right )^{n}\right ) a d g n +\ln \left (c \left (e x +d \right )^{n}\right ) b d f n +a d g \,n^{2}+b d f \,n^{2}-a d f n}{e n}\) | \(204\) |
risch | \(\text {Expression too large to display}\) | \(1245\) |
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Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.42 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {b e g x \log \left (c\right )^{2} + {\left (b e g n^{2} x + b d g n^{2}\right )} \log \left (e x + d\right )^{2} - {\left (2 \, b e g n - b e f - a e g\right )} x \log \left (c\right ) + {\left (2 \, b e g n^{2} + a e f - {\left (b e f + a e g\right )} n\right )} x - {\left (2 \, b d g n^{2} - {\left (b d f + a d g\right )} n + {\left (2 \, b e g n^{2} - {\left (b e f + a e g\right )} n\right )} x - 2 \, {\left (b e g n x + b d g n\right )} \log \left (c\right )\right )} \log \left (e x + d\right )}{e} \]
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Time = 0.30 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.72 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right ) \, dx=\begin {cases} \frac {a d g \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + a f x - a g n x + a g x \log {\left (c \left (d + e x\right )^{n} \right )} + \frac {b d f \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {2 b d g n \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {b d g \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} - b f n x + b f x \log {\left (c \left (d + e x\right )^{n} \right )} + 2 b g n^{2} x - 2 b g n x \log {\left (c \left (d + e x\right )^{n} \right )} + b g x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} & \text {for}\: e \neq 0 \\x \left (a + b \log {\left (c d^{n} \right )}\right ) \left (f + g \log {\left (c d^{n} \right )}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.50 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right ) \, dx=-b e f n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} - a e g n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} + b g x \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + b f x \log \left ({\left (e x + d\right )}^{n} c\right ) + a g x \log \left ({\left (e x + d\right )}^{n} c\right ) - {\left (2 \, e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac {{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n^{2}}{e}\right )} b g + a f x \]
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Time = 0.34 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.90 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {{\left (e x + d\right )} b g n^{2} \log \left (e x + d\right )^{2}}{e} - \frac {2 \, {\left (e x + d\right )} b g n^{2} \log \left (e x + d\right )}{e} + \frac {2 \, {\left (e x + d\right )} b g n \log \left (e x + d\right ) \log \left (c\right )}{e} + \frac {2 \, {\left (e x + d\right )} b g n^{2}}{e} + \frac {{\left (e x + d\right )} b f n \log \left (e x + d\right )}{e} + \frac {{\left (e x + d\right )} a g n \log \left (e x + d\right )}{e} - \frac {2 \, {\left (e x + d\right )} b g n \log \left (c\right )}{e} + \frac {{\left (e x + d\right )} b g \log \left (c\right )^{2}}{e} - \frac {{\left (e x + d\right )} b f n}{e} - \frac {{\left (e x + d\right )} a g n}{e} + \frac {{\left (e x + d\right )} b f \log \left (c\right )}{e} + \frac {{\left (e x + d\right )} a g \log \left (c\right )}{e} + \frac {{\left (e x + d\right )} a f}{e} \]
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Time = 1.34 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.93 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right ) \, dx={\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (b\,g\,x+\frac {b\,d\,g}{e}\right )+x\,\left (a\,f-a\,g\,n-b\,f\,n+2\,b\,g\,n^2\right )+x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (a\,g+b\,f-2\,b\,g\,n\right )+\frac {\ln \left (d+e\,x\right )\,\left (a\,d\,g\,n-2\,b\,d\,g\,n^2+b\,d\,f\,n\right )}{e} \]
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