Integrand size = 24, antiderivative size = 317 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^4} \, dx=-\frac {b^2 e^2 n^2}{3 g (e f-d g)^2 (f+g x)}-\frac {b^2 e^3 n^2 \log (d+e x)}{3 g (e f-d g)^3}+\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) (f+g x)^2}-\frac {2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g)^3 (f+g x)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^3}+\frac {b^2 e^3 n^2 \log (f+g x)}{g (e f-d g)^3}-\frac {2 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1+\frac {e f-d g}{g (d+e x)}\right )}{3 g (e f-d g)^3}+\frac {2 b^2 e^3 n^2 \operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right )}{3 g (e f-d g)^3} \]
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Time = 0.37 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2445, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^4} \, dx=-\frac {2 b e^3 n \log \left (\frac {e f-d g}{g (d+e x)}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^3}-\frac {2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (f+g x) (e f-d g)^3}+\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (f+g x)^2 (e f-d g)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^3}+\frac {2 b^2 e^3 n^2 \operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right )}{3 g (e f-d g)^3}-\frac {b^2 e^3 n^2 \log (d+e x)}{3 g (e f-d g)^3}+\frac {b^2 e^3 n^2 \log (f+g x)}{g (e f-d g)^3}-\frac {b^2 e^2 n^2}{3 g (f+g x) (e f-d g)^2} \]
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Rule 31
Rule 46
Rule 2351
Rule 2356
Rule 2379
Rule 2389
Rule 2438
Rule 2445
Rule 2458
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^3}+\frac {(2 b e n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^3} \, dx}{3 g} \\ & = -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^3}+\frac {(2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^3} \, dx,x,d+e x\right )}{3 g} \\ & = -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^3}-\frac {(2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^3} \, dx,x,d+e x\right )}{3 (e f-d g)}+\frac {(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^2} \, dx,x,d+e x\right )}{3 g (e f-d g)} \\ & = \frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) (f+g x)^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^3}-\frac {(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^2} \, dx,x,d+e x\right )}{3 (e f-d g)^2}+\frac {\left (2 b e^2 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )} \, dx,x,d+e x\right )}{3 g (e f-d g)^2}-\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^2} \, dx,x,d+e x\right )}{3 g (e f-d g)} \\ & = \frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) (f+g x)^2}-\frac {2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g)^3 (f+g x)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^3}-\frac {2 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1+\frac {e f-d g}{g (d+e x)}\right )}{3 g (e f-d g)^3}+\frac {\left (2 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {1}{\frac {e f-d g}{e}+\frac {g x}{e}} \, dx,x,d+e x\right )}{3 (e f-d g)^3}+\frac {\left (2 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {e f-d g}{g x}\right )}{x} \, dx,x,d+e x\right )}{3 g (e f-d g)^3}-\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \left (\frac {e^2}{(e f-d g)^2 x}-\frac {e^2 g}{(e f-d g) (e f-d g+g x)^2}-\frac {e^2 g}{(e f-d g)^2 (e f-d g+g x)}\right ) \, dx,x,d+e x\right )}{3 g (e f-d g)} \\ & = -\frac {b^2 e^2 n^2}{3 g (e f-d g)^2 (f+g x)}-\frac {b^2 e^3 n^2 \log (d+e x)}{3 g (e f-d g)^3}+\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) (f+g x)^2}-\frac {2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g)^3 (f+g x)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^3}+\frac {b^2 e^3 n^2 \log (f+g x)}{g (e f-d g)^3}-\frac {2 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1+\frac {e f-d g}{g (d+e x)}\right )}{3 g (e f-d g)^3}+\frac {2 b^2 e^3 n^2 \text {Li}_2\left (-\frac {e f-d g}{g (d+e x)}\right )}{3 g (e f-d g)^3} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^4} \, dx=\frac {-\left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac {e (f+g x) \left (b (e f-d g)^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )+2 b e (e f-d g) n (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )+e^2 (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b^2 e^2 n^2 (f+g x)^2 (\log (d+e x)-\log (f+g x))-b^2 e n^2 (f+g x) (e f-d g+e (f+g x) \log (d+e x)-e (f+g x) \log (f+g x))-2 b e^2 n (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )-2 b^2 e^2 n^2 (f+g x)^2 \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )}{(e f-d g)^3}}{3 g (f+g x)^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.19 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.38
method | result | size |
risch | \(-\frac {b^{2} \ln \left (\left (e x +d \right )^{n}\right )^{2}}{3 \left (g x +f \right )^{3} g}-\frac {2 b^{2} n \,e^{3} \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (e x +d \right )}{3 g \left (d g -e f \right )^{3}}-\frac {b^{2} n e \ln \left (\left (e x +d \right )^{n}\right )}{3 g \left (d g -e f \right ) \left (g x +f \right )^{2}}+\frac {2 b^{2} n \,e^{3} \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{3 g \left (d g -e f \right )^{3}}+\frac {2 b^{2} n \,e^{2} \ln \left (\left (e x +d \right )^{n}\right )}{3 g \left (d g -e f \right )^{2} \left (g x +f \right )}+\frac {b^{2} n^{2} e^{3} \ln \left (e x +d \right )}{g \left (d g -e f \right )^{3}}-\frac {b^{2} n^{2} e^{2}}{3 g \left (d g -e f \right )^{2} \left (g x +f \right )}-\frac {b^{2} n^{2} e^{3} \ln \left (g x +f \right )}{g \left (d g -e f \right )^{3}}+\frac {b^{2} n^{2} e^{3} \ln \left (e x +d \right )^{2}}{3 g \left (d g -e f \right )^{3}}-\frac {2 b^{2} n^{2} e^{3} \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{3 g \left (d g -e f \right )^{3}}-\frac {2 b^{2} n^{2} e^{3} \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{3 g \left (d g -e f \right )^{3}}+\left (-i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )+i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b +i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b -i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b +2 b \ln \left (c \right )+2 a \right ) b \left (-\frac {\ln \left (\left (e x +d \right )^{n}\right )}{3 \left (g x +f \right )^{3} g}+\frac {n e \left (-\frac {e^{2} \ln \left (e x +d \right )}{\left (d g -e f \right )^{3}}-\frac {1}{2 \left (d g -e f \right ) \left (g x +f \right )^{2}}+\frac {e^{2} \ln \left (g x +f \right )}{\left (d g -e f \right )^{3}}+\frac {e}{\left (d g -e f \right )^{2} \left (g x +f \right )}\right )}{3 g}\right )-\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )+i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b +i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b -i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b +2 b \ln \left (c \right )+2 a \right )}^{2}}{12 \left (g x +f \right )^{3} g}\) | \(755\) |
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\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x + f\right )}^{4}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^4} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2}}{\left (f + g x\right )^{4}}\, dx \]
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\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x + f\right )}^{4}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x + f\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{{\left (f+g\,x\right )}^4} \,d x \]
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