Integrand size = 25, antiderivative size = 202 \[ \int \frac {(f+g x) \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=-\frac {b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )}{d^2 e (d+e x)}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (e f-d g)}-\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 (e f-d g) (d+e x)^2}+\frac {b^2 (e f-d g) n^2 \log (d+e x)}{d^2 e^2}-\frac {b (e f+d g) n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}-\frac {b^2 (e f+d g) n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2 e^2} \]
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Time = 0.22 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2398, 2404, 2338, 2351, 31, 2354, 2438} \[ \int \frac {(f+g x) \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=-\frac {b n (d g+e f) \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e^2}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (e f-d g)}-\frac {b n x (e f-d g) \left (a+b \log \left (c x^n\right )\right )}{d^2 e (d+e x)}-\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 (d+e x)^2 (e f-d g)}-\frac {b^2 n^2 (d g+e f) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2 e^2}+\frac {b^2 n^2 (e f-d g) \log (d+e x)}{d^2 e^2} \]
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Rule 31
Rule 2338
Rule 2351
Rule 2354
Rule 2398
Rule 2404
Rule 2438
Rubi steps \begin{align*} \text {integral}& = -\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 (e f-d g) (d+e x)^2}+\frac {(b n) \int \frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )}{x (d+e x)^2} \, dx}{e f-d g} \\ & = -\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 (e f-d g) (d+e x)^2}+\frac {(b n) \int \left (\frac {f^2 \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac {(-e f+d g)^2 \left (a+b \log \left (c x^n\right )\right )}{d e (d+e x)^2}+\frac {\left (-e^2 f^2+d^2 g^2\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e (d+e x)}\right ) \, dx}{e f-d g} \\ & = -\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 (e f-d g) (d+e x)^2}+\frac {\left (b f^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d^2 (e f-d g)}-\frac {(b (e f-d g) n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d e}-\frac {(b (e f+d g) n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^2 e} \\ & = -\frac {b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )}{d^2 e (d+e x)}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (e f-d g)}-\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 (e f-d g) (d+e x)^2}-\frac {b (e f+d g) n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}+\frac {\left (b^2 (e f-d g) n^2\right ) \int \frac {1}{d+e x} \, dx}{d^2 e}+\frac {\left (b^2 (e f+d g) n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^2 e^2} \\ & = -\frac {b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )}{d^2 e (d+e x)}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (e f-d g)}-\frac {(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 (e f-d g) (d+e x)^2}+\frac {b^2 (e f-d g) n^2 \log (d+e x)}{d^2 e^2}-\frac {b (e f+d g) n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2 e^2}-\frac {b^2 (e f+d g) n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2 e^2} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.21 \[ \int \frac {(f+g x) \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\frac {-\frac {(e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}-\frac {2 g \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {2 g \left (\left (a+b \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )-2 b n \log \left (1+\frac {e x}{d}\right )\right )-2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )}{d}+\frac {(e f-d g) \left (2 b d n \left (a+b \log \left (c x^n\right )\right )+(d+e x) \left (a+b \log \left (c x^n\right )\right )^2-2 b^2 n^2 (d+e x) (\log (x)-\log (d+e x))-2 b n (d+e x) \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-2 b^2 n^2 (d+e x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )}{d^2 (d+e x)}}{2 e^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.77 (sec) , antiderivative size = 740, normalized size of antiderivative = 3.66
method | result | size |
risch | \(-\frac {b^{2} \ln \left (x^{n}\right )^{2} g}{e^{2} \left (e x +d \right )}+\frac {b^{2} \ln \left (x^{n}\right )^{2} d g}{2 e^{2} \left (e x +d \right )^{2}}-\frac {b^{2} \ln \left (x^{n}\right )^{2} f}{2 e \left (e x +d \right )^{2}}-\frac {b^{2} n \ln \left (x^{n}\right ) g}{e^{2} \left (e x +d \right )}+\frac {b^{2} n \ln \left (x^{n}\right ) f}{e d \left (e x +d \right )}-\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right ) g}{e^{2} d}-\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right ) f}{e \,d^{2}}+\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right ) g}{e^{2} d}+\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right ) f}{e \,d^{2}}-\frac {b^{2} n^{2} \ln \left (x \right )^{2} g}{2 e^{2} d}-\frac {b^{2} n^{2} \ln \left (x \right )^{2} f}{2 e \,d^{2}}-\frac {b^{2} n^{2} \ln \left (e x +d \right ) g}{e^{2} d}+\frac {b^{2} n^{2} \ln \left (e x +d \right ) f}{e \,d^{2}}+\frac {b^{2} n^{2} \ln \left (x \right ) g}{e^{2} d}-\frac {b^{2} n^{2} \ln \left (x \right ) f}{e \,d^{2}}+\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) g}{e^{2} d}+\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) f}{e \,d^{2}}+\frac {b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right ) g}{e^{2} d}+\frac {b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right ) f}{e \,d^{2}}+\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (-\frac {\ln \left (x^{n}\right ) g}{e^{2} \left (e x +d \right )}+\frac {\ln \left (x^{n}\right ) d g}{2 e^{2} \left (e x +d \right )^{2}}-\frac {\ln \left (x^{n}\right ) f}{2 e \left (e x +d \right )^{2}}-\frac {n \left (\frac {\left (d g +e f \right ) \ln \left (e x +d \right )}{d^{2}}+\frac {d g -e f}{d \left (e x +d \right )}+\frac {\left (-d g -e f \right ) \ln \left (x \right )}{d^{2}}\right )}{2 e^{2}}\right )+\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2} \left (-\frac {g}{e^{2} \left (e x +d \right )}-\frac {-d g +e f}{2 e^{2} \left (e x +d \right )^{2}}\right )}{4}\) | \(740\) |
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\[ \int \frac {(f+g x) \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {(f+g x) \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2} \left (f + g x\right )}{\left (d + e x\right )^{3}}\, dx \]
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\[ \int \frac {(f+g x) \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {(f+g x) \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(f+g x) \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]
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