Integrand size = 25, antiderivative size = 186 \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=\frac {a x}{g^2}-\frac {b n x}{g^2}+\frac {b e f^2 n \log (d+e x)}{g^3 (e f-d g)}+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3 (f+g x)}-\frac {b e f^2 n \log (f+g x)}{g^3 (e f-d g)}-\frac {2 f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^3}-\frac {2 b f n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^3} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {45, 2463, 2436, 2332, 2442, 36, 31, 2441, 2440, 2438} \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3 (f+g x)}-\frac {2 f \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac {a x}{g^2}+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {b e f^2 n \log (d+e x)}{g^3 (e f-d g)}-\frac {b e f^2 n \log (f+g x)}{g^3 (e f-d g)}-\frac {2 b f n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^3}-\frac {b n x}{g^2} \]
[In]
[Out]
Rule 31
Rule 36
Rule 45
Rule 2332
Rule 2436
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \log \left (c (d+e x)^n\right )}{g^2}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)^2}-\frac {2 f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)}\right ) \, dx \\ & = \frac {\int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g^2}-\frac {(2 f) \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{g^2}+\frac {f^2 \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^2} \, dx}{g^2} \\ & = \frac {a x}{g^2}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3 (f+g x)}-\frac {2 f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^3}+\frac {b \int \log \left (c (d+e x)^n\right ) \, dx}{g^2}+\frac {(2 b e f n) \int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g^3}+\frac {\left (b e f^2 n\right ) \int \frac {1}{(d+e x) (f+g x)} \, dx}{g^3} \\ & = \frac {a x}{g^2}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3 (f+g x)}-\frac {2 f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^3}+\frac {b \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g^2}+\frac {(2 b f n) \text {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g^3}+\frac {\left (b e^2 f^2 n\right ) \int \frac {1}{d+e x} \, dx}{g^3 (e f-d g)}-\frac {\left (b e f^2 n\right ) \int \frac {1}{f+g x} \, dx}{g^2 (e f-d g)} \\ & = \frac {a x}{g^2}-\frac {b n x}{g^2}+\frac {b e f^2 n \log (d+e x)}{g^3 (e f-d g)}+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3 (f+g x)}-\frac {b e f^2 n \log (f+g x)}{g^3 (e f-d g)}-\frac {2 f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^3}-\frac {2 b f n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^3} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.82 \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=\frac {a g x-b g n x+\frac {b g (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x}+\frac {b e f^2 n (\log (d+e x)-\log (f+g x))}{e f-d g}-2 f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )-2 b f n \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )}{g^3} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.84 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.34
method | result | size |
risch | \(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) x}{g^{2}}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) f^{2}}{g^{3} \left (g x +f \right )}-\frac {2 b \ln \left (\left (e x +d \right )^{n}\right ) f \ln \left (g x +f \right )}{g^{3}}-\frac {b n x}{g^{2}}-\frac {b f n}{g^{3}}+\frac {b e n \,f^{2} \ln \left (g x +f \right )}{g^{3} \left (d g -e f \right )}+\frac {b n \ln \left (\left (g x +f \right ) e +d g -e f \right ) d^{2}}{e g \left (d g -e f \right )}-\frac {b n \ln \left (\left (g x +f \right ) e +d g -e f \right ) d f}{g^{2} \left (d g -e f \right )}-\frac {b e n \ln \left (\left (g x +f \right ) e +d g -e f \right ) f^{2}}{g^{3} \left (d g -e f \right )}+\frac {2 b n f \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{g^{3}}+\frac {2 b n f \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{g^{3}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x}{g^{2}}-\frac {f^{2}}{g^{3} \left (g x +f \right )}-\frac {2 f \ln \left (g x +f \right )}{g^{3}}\right )\) | \(435\) |
[In]
[Out]
\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{2}}{{\left (g x + f\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=\int \frac {x^{2} \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )}{\left (f + g x\right )^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{2}}{{\left (g x + f\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{2}}{{\left (g x + f\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{{\left (f+g\,x\right )}^2} \,d x \]
[In]
[Out]