Integrand size = 29, antiderivative size = 402 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)^3} \, dx=-\frac {b e n}{2 (e h-d i) (g h-f i) (h+i x)}-\frac {b e g n \log (d+e x)}{(e h-d i) (g h-f i)^2}-\frac {b e^2 n \log (d+e x)}{2 (e h-d i)^2 (g h-f i)}+\frac {a+b \log \left (c (d+e x)^n\right )}{2 (g h-f i) (h+i x)^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2 (h+i x)}+\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{(g h-f i)^3}+\frac {b e g n \log (h+i x)}{(e h-d i) (g h-f i)^2}+\frac {b e^2 n \log (h+i x)}{2 (e h-d i)^2 (g h-f i)}-\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (h+i x)}{e h-d i}\right )}{(g h-f i)^3}+\frac {b g^2 n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{(g h-f i)^3}-\frac {b g^2 n \operatorname {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right )}{(g h-f i)^3} \]
-1/2*b*e*n/(-d*i+e*h)/(-f*i+g*h)/(i*x+h)-b*e*g*n*ln(e*x+d)/(-d*i+e*h)/(-f* i+g*h)^2-1/2*b*e^2*n*ln(e*x+d)/(-d*i+e*h)^2/(-f*i+g*h)+1/2*(a+b*ln(c*(e*x+ d)^n))/(-f*i+g*h)/(i*x+h)^2+g*(a+b*ln(c*(e*x+d)^n))/(-f*i+g*h)^2/(i*x+h)+g ^2*(a+b*ln(c*(e*x+d)^n))*ln(e*(g*x+f)/(-d*g+e*f))/(-f*i+g*h)^3+b*e*g*n*ln( i*x+h)/(-d*i+e*h)/(-f*i+g*h)^2+1/2*b*e^2*n*ln(i*x+h)/(-d*i+e*h)^2/(-f*i+g* h)-g^2*(a+b*ln(c*(e*x+d)^n))*ln(e*(i*x+h)/(-d*i+e*h))/(-f*i+g*h)^3+b*g^2*n *polylog(2,-g*(e*x+d)/(-d*g+e*f))/(-f*i+g*h)^3-b*g^2*n*polylog(2,-i*(e*x+d )/(-d*i+e*h))/(-f*i+g*h)^3
Time = 0.20 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.77 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)^3} \, dx=\frac {\frac {(g h-f i)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(h+i x)^2}+\frac {2 g (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )}{h+i x}+2 g^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )-\frac {2 b e g (g h-f i) n (\log (d+e x)-\log (h+i x))}{e h-d i}-\frac {b e (g h-f i)^2 n (e h-d i+e (h+i x) \log (d+e x)-e (h+i x) \log (h+i x))}{(e h-d i)^2 (h+i x)}-2 g^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (h+i x)}{e h-d i}\right )+2 b g^2 n \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-2 b g^2 n \operatorname {PolyLog}\left (2,\frac {i (d+e x)}{-e h+d i}\right )}{2 (g h-f i)^3} \]
(((g*h - f*i)^2*(a + b*Log[c*(d + e*x)^n]))/(h + i*x)^2 + (2*g*(g*h - f*i) *(a + b*Log[c*(d + e*x)^n]))/(h + i*x) + 2*g^2*(a + b*Log[c*(d + e*x)^n])* Log[(e*(f + g*x))/(e*f - d*g)] - (2*b*e*g*(g*h - f*i)*n*(Log[d + e*x] - Lo g[h + i*x]))/(e*h - d*i) - (b*e*(g*h - f*i)^2*n*(e*h - d*i + e*(h + i*x)*L og[d + e*x] - e*(h + i*x)*Log[h + i*x]))/((e*h - d*i)^2*(h + i*x)) - 2*g^2 *(a + b*Log[c*(d + e*x)^n])*Log[(e*(h + i*x))/(e*h - d*i)] + 2*b*g^2*n*Pol yLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 2*b*g^2*n*PolyLog[2, (i*(d + e*x)) /(-(e*h) + d*i)])/(2*(g*h - f*i)^3)
Time = 0.67 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2865, 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.
| \(\displaystyle \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)^3} \, dx\) |
| \(\Big \downarrow \) 2865 |
| \(\displaystyle \int \left (\frac {g^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x) (g h-f i)^3}-\frac {g^2 i \left (a+b \log \left (c (d+e x)^n\right )\right )}{(h+i x) (g h-f i)^3}-\frac {g i \left (a+b \log \left (c (d+e x)^n\right )\right )}{(h+i x)^2 (g h-f i)^2}-\frac {i \left (a+b \log \left (c (d+e x)^n\right )\right )}{(h+i x)^3 (g h-f i)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {g^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^3}-\frac {g^2 \log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^3}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(h+i x) (g h-f i)^2}+\frac {a+b \log \left (c (d+e x)^n\right )}{2 (h+i x)^2 (g h-f i)}-\frac {b e^2 n \log (d+e x)}{2 (e h-d i)^2 (g h-f i)}+\frac {b e^2 n \log (h+i x)}{2 (e h-d i)^2 (g h-f i)}+\frac {b g^2 n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{(g h-f i)^3}-\frac {b g^2 n \operatorname {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right )}{(g h-f i)^3}-\frac {b e n}{2 (h+i x) (e h-d i) (g h-f i)}-\frac {b e g n \log (d+e x)}{(e h-d i) (g h-f i)^2}+\frac {b e g n \log (h+i x)}{(e h-d i) (g h-f i)^2}\) |
-1/2*(b*e*n)/((e*h - d*i)*(g*h - f*i)*(h + i*x)) - (b*e*g*n*Log[d + e*x])/ ((e*h - d*i)*(g*h - f*i)^2) - (b*e^2*n*Log[d + e*x])/(2*(e*h - d*i)^2*(g*h - f*i)) + (a + b*Log[c*(d + e*x)^n])/(2*(g*h - f*i)*(h + i*x)^2) + (g*(a + b*Log[c*(d + e*x)^n]))/((g*h - f*i)^2*(h + i*x)) + (g^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/(g*h - f*i)^3 + (b*e*g*n*Log[h + i*x])/((e*h - d*i)*(g*h - f*i)^2) + (b*e^2*n*Log[h + i*x])/(2*(e*h - d*i )^2*(g*h - f*i)) - (g^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(h + i*x))/(e*h - d*i)])/(g*h - f*i)^3 + (b*g^2*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))] )/(g*h - f*i)^3 - (b*g^2*n*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)^3
3.3.23.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy mbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[ RFx, x] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.17 (sec) , antiderivative size = 771, normalized size of antiderivative = 1.92
| method | result | size |
| risch | \(-\frac {b \ln \left (\left (e x +d \right )^{n}\right )}{2 \left (f i -g h \right ) \left (i x +h \right )^{2}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) g^{2} \ln \left (i x +h \right )}{\left (f i -g h \right )^{3}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) g}{\left (f i -g h \right )^{2} \left (i x +h \right )}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) g^{2} \ln \left (g x +f \right )}{\left (f i -g h \right )^{3}}+\frac {b e n \ln \left (e x +d \right ) d g i}{\left (f i -g h \right )^{2} \left (d i -e h \right )^{2}}+\frac {b \,e^{2} n \ln \left (e x +d \right ) f i}{2 \left (f i -g h \right )^{2} \left (d i -e h \right )^{2}}-\frac {3 b \,e^{2} n \ln \left (e x +d \right ) g h}{2 \left (f i -g h \right )^{2} \left (d i -e h \right )^{2}}-\frac {b e n \ln \left (i x +h \right ) d g i}{\left (f i -g h \right )^{2} \left (d i -e h \right )^{2}}-\frac {b \,e^{2} n \ln \left (i x +h \right ) f i}{2 \left (f i -g h \right )^{2} \left (d i -e h \right )^{2}}+\frac {3 b \,e^{2} n \ln \left (i x +h \right ) g h}{2 \left (f i -g h \right )^{2} \left (d i -e h \right )^{2}}-\frac {b e n f i}{2 \left (f i -g h \right )^{2} \left (d i -e h \right ) \left (i x +h \right )}+\frac {b e n g h}{2 \left (f i -g h \right )^{2} \left (d i -e h \right ) \left (i x +h \right )}+\frac {b n \,g^{2} \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{\left (f i -g h \right )^{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 )}{\left (f i -g h \right )^{3}}-\frac {b n \,g^{2} \operatorname {dilog}\left (\frac {\left (i x +h \right ) e +d i -e h}{d i -e h}\right )}{\left (f i -g h \right )^{3}}-\frac {b n \,g^{2} \ln \left (i x +h \right ) \ln \left (\frac {\left (i x +h \right ) e +d i -e h}{d i -e h}\right )}{\left (f i -g h \right )^{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 \left (f i -g h \right ) \left (i x +h \right )^{2}}+\frac {g^{2} \ln \left (i x +h \right )}{\left (f i -g h \right )^{3}}+\frac {g}{\left (f i -g h \right )^{2} \left (i x +h \right )}-\frac {g^{2} \ln \left (g x +f \right )}{\left (f i -g h \right )^{3}}\right )\) | \(771\) |
-1/2*b*ln((e*x+d)^n)/(f*i-g*h)/(i*x+h)^2+b*ln((e*x+d)^n)*g^2/(f*i-g*h)^3*l n(i*x+h)+b*ln((e*x+d)^n)*g/(f*i-g*h)^2/(i*x+h)-b*ln((e*x+d)^n)*g^2/(f*i-g* h)^3*ln(g*x+f)+b*e*n/(f*i-g*h)^2/(d*i-e*h)^2*ln(e*x+d)*d*g*i+1/2*b*e^2*n/( f*i-g*h)^2/(d*i-e*h)^2*ln(e*x+d)*f*i-3/2*b*e^2*n/(f*i-g*h)^2/(d*i-e*h)^2*l n(e*x+d)*g*h-b*e*n/(f*i-g*h)^2/(d*i-e*h)^2*ln(i*x+h)*d*g*i-1/2*b*e^2*n/(f* i-g*h)^2/(d*i-e*h)^2*ln(i*x+h)*f*i+3/2*b*e^2*n/(f*i-g*h)^2/(d*i-e*h)^2*ln( i*x+h)*g*h-1/2*b*e*n/(f*i-g*h)^2/(d*i-e*h)/(i*x+h)*f*i+1/2*b*e*n/(f*i-g*h) ^2/(d*i-e*h)/(i*x+h)*g*h+b*n*g^2/(f*i-g*h)^3*dilog(((g*x+f)*e+d*g-e*f)/(d* g-e*f))+b*n*g^2/(f*i-g*h)^3*ln(g*x+f)*ln(((g*x+f)*e+d*g-e*f)/(d*g-e*f))-b* n*g^2/(f*i-g*h)^3*dilog(((i*x+h)*e+d*i-e*h)/(d*i-e*h))-b*n*g^2/(f*i-g*h)^3 *ln(i*x+h)*ln(((i*x+h)*e+d*i-e*h)/(d*i-e*h))+(-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*i-g*h)/(i*x+h)^2+g^2/(f*i-g*h)^3*ln(i*x+h)+ g/(f*i-g*h)^2/(i*x+h)-g^2/(f*i-g*h)^3*ln(g*x+f))
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)^3} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )} {\left (i x + h\right )}^{3}} \,d x } \]
integral((b*log((e*x + d)^n*c) + a)/(g*i^3*x^4 + f*h^3 + (3*g*h*i^2 + f*i^ 3)*x^3 + 3*(g*h^2*i + f*h*i^2)*x^2 + (g*h^3 + 3*f*h^2*i)*x), x)
Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)^3} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)^3} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )} {\left (i x + h\right )}^{3}} \,d x } \]
1/2*(2*g^2*log(g*x + f)/(g^3*h^3 - 3*f*g^2*h^2*i + 3*f^2*g*h*i^2 - f^3*i^3 ) - 2*g^2*log(i*x + h)/(g^3*h^3 - 3*f*g^2*h^2*i + 3*f^2*g*h*i^2 - f^3*i^3) + (2*g*i*x + 3*g*h - f*i)/(g^2*h^4 - 2*f*g*h^3*i + f^2*h^2*i^2 + (g^2*h^2 *i^2 - 2*f*g*h*i^3 + f^2*i^4)*x^2 + 2*(g^2*h^3*i - 2*f*g*h^2*i^2 + f^2*h*i ^3)*x))*a + b*integrate((log((e*x + d)^n) + log(c))/(g*i^3*x^4 + f*h^3 + ( 3*g*h*i^2 + f*i^3)*x^3 + 3*(g*h^2*i + f*h*i^2)*x^2 + (g*h^3 + 3*f*h^2*i)*x ), x)
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)^3} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )} {\left (i x + h\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)^3} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{\left (f+g\,x\right )\,{\left (h+i\,x\right )}^3} \,d x \]