Integrand size = 25, antiderivative size = 250 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3 (f+g x)} \, dx=-\frac {b e n}{2 d f x}-\frac {b e^2 n \log (x)}{2 d^2 f}-\frac {b e g n \log (x)}{d f^2}+\frac {b e^2 n \log (d+e x)}{2 d^2 f}+\frac {b e g n \log (d+e x)}{d f^2}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 f x^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 x}+\frac {g^2 \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}-\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{f^3}-\frac {b g^2 n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{f^3}+\frac {b g^2 n \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{f^3} \]
-1/2*b*e*n/d/f/x-1/2*b*e^2*n*ln(x)/d^2/f-b*e*g*n*ln(x)/d/f^2+1/2*b*e^2*n*l n(e*x+d)/d^2/f+b*e*g*n*ln(e*x+d)/d/f^2+1/2*(-a-b*ln(c*(e*x+d)^n))/f/x^2+g* (a+b*ln(c*(e*x+d)^n))/f^2/x+g^2*ln(-e*x/d)*(a+b*ln(c*(e*x+d)^n))/f^3-g^2*( a+b*ln(c*(e*x+d)^n))*ln(e*(g*x+f)/(-d*g+e*f))/f^3-b*g^2*n*polylog(2,-g*(e* x+d)/(-d*g+e*f))/f^3+b*g^2*n*polylog(2,1+e*x/d)/f^3
Time = 0.14 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3 (f+g x)} \, dx=-\frac {\frac {2 b e f g n (\log (x)-\log (d+e x))}{d}+\frac {b e f^2 n (d+e x \log (x)-e x \log (d+e x))}{d^2 x}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^2}-\frac {2 f g \left (a+b \log \left (c (d+e x)^n\right )\right )}{x}-2 g^2 \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+2 g^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )+2 b g^2 n \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-2 b g^2 n \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{2 f^3} \]
-1/2*((2*b*e*f*g*n*(Log[x] - Log[d + e*x]))/d + (b*e*f^2*n*(d + e*x*Log[x] - e*x*Log[d + e*x]))/(d^2*x) + (f^2*(a + b*Log[c*(d + e*x)^n]))/x^2 - (2* f*g*(a + b*Log[c*(d + e*x)^n]))/x - 2*g^2*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]) + 2*g^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g )] + 2*b*g^2*n*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 2*b*g^2*n*PolyLo g[2, 1 + (e*x)/d])/f^3
Time = 0.49 (sec) , antiderivative size = 250, 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.
| \(\displaystyle \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3 (f+g x)} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (-\frac {g^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 (f+g x)}+\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3 x}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 x^2}+\frac {a+b \log \left (c (d+e x)^n\right )}{f x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {g^2 \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}-\frac {g^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 x}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 f x^2}-\frac {b e^2 n \log (x)}{2 d^2 f}+\frac {b e^2 n \log (d+e x)}{2 d^2 f}-\frac {b g^2 n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{f^3}+\frac {b g^2 n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{f^3}-\frac {b e g n \log (x)}{d f^2}+\frac {b e g n \log (d+e x)}{d f^2}-\frac {b e n}{2 d f x}\) |
-1/2*(b*e*n)/(d*f*x) - (b*e^2*n*Log[x])/(2*d^2*f) - (b*e*g*n*Log[x])/(d*f^ 2) + (b*e^2*n*Log[d + e*x])/(2*d^2*f) + (b*e*g*n*Log[d + e*x])/(d*f^2) - ( a + b*Log[c*(d + e*x)^n])/(2*f*x^2) + (g*(a + b*Log[c*(d + e*x)^n]))/(f^2* x) + (g^2*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/f^3 - (g^2*(a + b*Lo g[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/f^3 - (b*g^2*n*PolyLog[2 , -((g*(d + e*x))/(e*f - d*g))])/f^3 + (b*g^2*n*PolyLog[2, 1 + (e*x)/d])/f ^3
3.3.48.3.1 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 = 0.96 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.72
| method | result | size |
| risch | \(-\frac {b \ln \left (\left (e x +d \right )^{n}\right )}{2 f \,x^{2}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) g^{2} \ln \left (x \right )}{f^{3}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) g}{f^{2} x}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) g^{2} \ln \left (g x +f \right )}{f^{3}}+\frac {b e g n \ln \left (e x +d \right )}{d \,f^{2}}+\frac {b \,e^{2} n \ln \left (e x +d \right )}{2 d^{2} f}-\frac {b e g n \ln \left (x \right )}{d \,f^{2}}-\frac {b \,e^{2} n \ln \left (x \right )}{2 d^{2} f}-\frac {b e n}{2 d f x}-\frac {b n \,g^{2} \operatorname {dilog}\left (\frac {e x +d}{d}\right )}{f^{3}}-\frac {b n \,g^{2} \ln \left (x \right ) \ln \left (\frac {e x +d}{d}\right )}{f^{3}}+\frac {b n \,g^{2} \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{f^{3}}+\frac {b n \,g^{2} \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{f^{3}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {1}{2 f \,x^{2}}+\frac {g^{2} \ln \left (x \right )}{f^{3}}+\frac {g}{f^{2} x}-\frac {g^{2} \ln \left (g x +f \right )}{f^{3}}\right )\) | \(429\) |
-1/2*b*ln((e*x+d)^n)/f/x^2+b*ln((e*x+d)^n)/f^3*g^2*ln(x)+b*ln((e*x+d)^n)/f ^2*g/x-b*ln((e*x+d)^n)/f^3*g^2*ln(g*x+f)+b*e*g*n*ln(e*x+d)/d/f^2+1/2*b*e^2 *n*ln(e*x+d)/d^2/f-b*e*g*n*ln(x)/d/f^2-1/2*b*e^2*n*ln(x)/d^2/f-1/2*b*e*n/d /f/x-b*n/f^3*g^2*dilog((e*x+d)/d)-b*n/f^3*g^2*ln(x)*ln((e*x+d)/d)+b*n/f^3* g^2*dilog(((g*x+f)*e+d*g-e*f)/(d*g-e*f))+b*n/f^3*g^2*ln(g*x+f)*ln(((g*x+f) *e+d*g-e*f)/(d*g-e*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)* (-1/2/f/x^2+1/f^3*g^2*ln(x)+1/f^2*g/x-1/f^3*g^2*ln(g*x+f))
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3 (f+g x)} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )} x^{3}} \,d x } \]
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3 (f+g x)} \, dx=\int \frac {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}{x^{3} \left (f + g x\right )}\, dx \]
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3 (f+g x)} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )} x^{3}} \,d x } \]
-1/2*a*(2*g^2*log(g*x + f)/f^3 - 2*g^2*log(x)/f^3 - (2*g*x - f)/(f^2*x^2)) + b*integrate((log((e*x + d)^n) + log(c))/(g*x^4 + f*x^3), x)
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3 (f+g x)} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3 (f+g x)} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{x^3\,\left (f+g\,x\right )} \,d x \]