Integrand size = 25, antiderivative size = 181 \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=-\frac {a f x}{g^2}+\frac {b f n x}{g^2}+\frac {b d n x}{2 e g}-\frac {b n x^2}{4 g}-\frac {b d^2 n \log (d+e x)}{2 e^2 g}-\frac {b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^3}+\frac {b f^2 n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^3} \]
-a*f*x/g^2+b*f*n*x/g^2+1/2*b*d*n*x/e/g-1/4*b*n*x^2/g-1/2*b*d^2*n*ln(e*x+d) /e^2/g-b*f*(e*x+d)*ln(c*(e*x+d)^n)/e/g^2+1/2*x^2*(a+b*ln(c*(e*x+d)^n))/g+f ^2*(a+b*ln(c*(e*x+d)^n))*ln(e*(g*x+f)/(-d*g+e*f))/g^3+b*f^2*n*polylog(2,-g *(e*x+d)/(-d*g+e*f))/g^3
Time = 0.09 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.94 \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=-\frac {a f x}{g^2}+\frac {b f n x}{g^2}+\frac {b n \left (\frac {2 d x}{e}-x^2-\frac {2 d^2 \log (d+e x)}{e^2}\right )}{4 g}-\frac {b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^3}+\frac {b f^2 n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^3} \]
-((a*f*x)/g^2) + (b*f*n*x)/g^2 + (b*n*((2*d*x)/e - x^2 - (2*d^2*Log[d + e* x])/e^2))/(4*g) - (b*f*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) + (x^2*(a + b *Log[c*(d + e*x)^n]))/(2*g) + (f^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g^3 + (b*f^2*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g)) ])/g^3
Time = 0.40 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2863, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {f^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}-\frac {a f x}{g^2}-\frac {b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {b d^2 n \log (d+e x)}{2 e^2 g}+\frac {b f^2 n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^3}+\frac {b d n x}{2 e g}+\frac {b f n x}{g^2}-\frac {b n x^2}{4 g}\) |
-((a*f*x)/g^2) + (b*f*n*x)/g^2 + (b*d*n*x)/(2*e*g) - (b*n*x^2)/(4*g) - (b* d^2*n*Log[d + e*x])/(2*e^2*g) - (b*f*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) + (x^2*(a + b*Log[c*(d + e*x)^n]))/(2*g) + (f^2*(a + b*Log[c*(d + e*x)^n] )*Log[(e*(f + g*x))/(e*f - d*g)])/g^3 + (b*f^2*n*PolyLog[2, -((g*(d + e*x) )/(e*f - d*g))])/g^3
3.3.43.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) ^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c , d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.80 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.14
| method | result | size |
| risch | \(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) x^{2}}{2 g}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) f x}{g^{2}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) f^{2} \ln \left (g x +f \right )}{g^{3}}-\frac {b n \,f^{2} \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{g^{3}}-\frac {b n \,f^{2} \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{g^{3}}-\frac {b n \,x^{2}}{4 g}+\frac {b f n x}{g^{2}}+\frac {5 b \,f^{2} n}{4 g^{3}}+\frac {b d n x}{2 e g}+\frac {b d f n}{2 e \,g^{2}}-\frac {b n \,d^{2} \ln \left (\left (g x +f \right ) e +d g -e f \right )}{2 e^{2} g}-\frac {b n d \ln \left (\left (g x +f \right ) e +d g -e f \right ) f}{e \,g^{2}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\frac {1}{2} g \,x^{2}-f x}{g^{2}}+\frac {f^{2} \ln \left (g x +f \right )}{g^{3}}\right )\) | \(388\) |
1/2*b*ln((e*x+d)^n)/g*x^2-b*ln((e*x+d)^n)/g^2*f*x+b*ln((e*x+d)^n)*f^2/g^3* ln(g*x+f)-b*n/g^3*f^2*dilog(((g*x+f)*e+d*g-e*f)/(d*g-e*f))-b*n/g^3*f^2*ln( g*x+f)*ln(((g*x+f)*e+d*g-e*f)/(d*g-e*f))-1/4*b*n*x^2/g+b*f*n*x/g^2+5/4*b*f ^2*n/g^3+1/2*b*d*n*x/e/g+1/2*b*d*f*n/e/g^2-1/2*b/e^2*n/g*d^2*ln((g*x+f)*e+ d*g-e*f)-b/e*n/g^2*d*ln((g*x+f)*e+d*g-e*f)*f+(-1/2*I*b*Pi*csgn(I*c)*csgn(I *(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2 +1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/2*I*b*Pi*csgn(I*c*(e *x+d)^n)^3+b*ln(c)+a)*(1/g^2*(1/2*g*x^2-f*x)+f^2/g^3*ln(g*x+f))
\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{2}}{g x + f} \,d x } \]
\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\int \frac {x^{2} \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )}{f + g x}\, dx \]
\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{2}}{g x + f} \,d x } \]
1/2*a*(2*f^2*log(g*x + f)/g^3 + (g*x^2 - 2*f*x)/g^2) + b*integrate((x^2*lo g((e*x + d)^n) + x^2*log(c))/(g*x + f), x)
\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{2}}{g x + f} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{f+g\,x} \,d x \]