Integrand size = 32, antiderivative size = 433 \[ \int (g x)^{-1+3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\frac {2 b k n (g x)^{3 m}}{27 g m^2}+\frac {4 b e^2 k n x^{-2 m} (g x)^{3 m}}{9 f^2 g m^2}-\frac {5 b e k n x^{-m} (g x)^{3 m}}{36 f g m^2}-\frac {k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac {e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac {e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (e+f x^m\right )}{9 f^3 g m^2}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 f^3 g m^2}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}-\frac {b n (g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}+\frac {(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \operatorname {PolyLog}\left (2,1+\frac {f x^m}{e}\right )}{3 f^3 g m^2} \]
2/27*b*k*n*(g*x)^(3*m)/g/m^2+4/9*b*e^2*k*n*(g*x)^(3*m)/f^2/g/m^2/(x^(2*m)) -5/36*b*e*k*n*(g*x)^(3*m)/f/g/m^2/(x^m)-1/9*k*(g*x)^(3*m)*(a+b*ln(c*x^n))/ g/m-1/3*e^2*k*(g*x)^(3*m)*(a+b*ln(c*x^n))/f^2/g/m/(x^(2*m))+1/6*e*k*(g*x)^ (3*m)*(a+b*ln(c*x^n))/f/g/m/(x^m)-1/9*b*e^3*k*n*(g*x)^(3*m)*ln(e+f*x^m)/f^ 3/g/m^2/(x^(3*m))-1/3*b*e^3*k*n*(g*x)^(3*m)*ln(-f*x^m/e)*ln(e+f*x^m)/f^3/g /m^2/(x^(3*m))+1/3*e^3*k*(g*x)^(3*m)*(a+b*ln(c*x^n))*ln(e+f*x^m)/f^3/g/m/( x^(3*m))-1/9*b*n*(g*x)^(3*m)*ln(d*(e+f*x^m)^k)/g/m^2+1/3*(g*x)^(3*m)*(a+b* ln(c*x^n))*ln(d*(e+f*x^m)^k)/g/m-1/3*b*e^3*k*n*(g*x)^(3*m)*polylog(2,1+f*x ^m/e)/f^3/g/m^2/(x^(3*m))
Time = 0.40 (sec) , antiderivative size = 410, normalized size of antiderivative = 0.95 \[ \int (g x)^{-1+3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\frac {x^{-3 m} (g x)^{3 m} \left (-36 a e^2 f k m x^m+48 b e^2 f k n x^m+18 a e f^2 k m x^{2 m}-15 b e f^2 k n x^{2 m}-12 a f^3 k m x^{3 m}+8 b f^3 k n x^{3 m}-36 b e^3 k m^2 n \log ^2(x)-36 b e^2 f k m x^m \log \left (c x^n\right )+18 b e f^2 k m x^{2 m} \log \left (c x^n\right )-12 b f^3 k m x^{3 m} \log \left (c x^n\right )+36 a e^3 k m \log \left (e-e x^m\right )-12 b e^3 k n \log \left (e-e x^m\right )+36 b e^3 k m \log \left (c x^n\right ) \log \left (e-e x^m\right )-36 b e^3 k n \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )+12 e^3 k m \log (x) \left (3 a m-b n+3 b m \log \left (c x^n\right )-3 b n \log \left (e-e x^m\right )+3 b n \log \left (e+f x^m\right )\right )+36 a f^3 m x^{3 m} \log \left (d \left (e+f x^m\right )^k\right )-12 b f^3 n x^{3 m} \log \left (d \left (e+f x^m\right )^k\right )+36 b f^3 m x^{3 m} \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-36 b e^3 k n \operatorname {PolyLog}\left (2,1+\frac {f x^m}{e}\right )\right )}{108 f^3 g m^2} \]
((g*x)^(3*m)*(-36*a*e^2*f*k*m*x^m + 48*b*e^2*f*k*n*x^m + 18*a*e*f^2*k*m*x^ (2*m) - 15*b*e*f^2*k*n*x^(2*m) - 12*a*f^3*k*m*x^(3*m) + 8*b*f^3*k*n*x^(3*m ) - 36*b*e^3*k*m^2*n*Log[x]^2 - 36*b*e^2*f*k*m*x^m*Log[c*x^n] + 18*b*e*f^2 *k*m*x^(2*m)*Log[c*x^n] - 12*b*f^3*k*m*x^(3*m)*Log[c*x^n] + 36*a*e^3*k*m*L og[e - e*x^m] - 12*b*e^3*k*n*Log[e - e*x^m] + 36*b*e^3*k*m*Log[c*x^n]*Log[ e - e*x^m] - 36*b*e^3*k*n*Log[-((f*x^m)/e)]*Log[e + f*x^m] + 12*e^3*k*m*Lo g[x]*(3*a*m - b*n + 3*b*m*Log[c*x^n] - 3*b*n*Log[e - e*x^m] + 3*b*n*Log[e + f*x^m]) + 36*a*f^3*m*x^(3*m)*Log[d*(e + f*x^m)^k] - 12*b*f^3*n*x^(3*m)*L og[d*(e + f*x^m)^k] + 36*b*f^3*m*x^(3*m)*Log[c*x^n]*Log[d*(e + f*x^m)^k] - 36*b*e^3*k*n*PolyLog[2, 1 + (f*x^m)/e]))/(108*f^3*g*m^2*x^(3*m))
Time = 0.79 (sec) , antiderivative size = 424, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2823, 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.
| \(\displaystyle \int (g x)^{3 m-1} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx\) |
| \(\Big \downarrow \) 2823 |
| \(\displaystyle -b n \int \left (\frac {e^3 k (g x)^{3 m} \log \left (f x^m+e\right ) x^{-3 m-1}}{3 f^3 g m}-\frac {e^2 k (g x)^{3 m} x^{-2 m-1}}{3 f^2 g m}+\frac {e k (g x)^{3 m} x^{-m-1}}{6 f g m}-\frac {k (g x)^{3 m}}{9 g m x}+\frac {(g x)^{3 m} \log \left (d \left (f x^m+e\right )^k\right )}{3 g m x}\right )dx+\frac {(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3 g m}-\frac {e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac {e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}-\frac {k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3 g m}-\frac {e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac {e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}-\frac {k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-b n \left (\frac {(g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \operatorname {PolyLog}\left (2,\frac {f x^m}{e}+1\right )}{3 f^3 g m^2}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \log \left (e+f x^m\right )}{9 f^3 g m^2}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 f^3 g m^2}-\frac {4 e^2 k x^{-2 m} (g x)^{3 m}}{9 f^2 g m^2}+\frac {5 e k x^{-m} (g x)^{3 m}}{36 f g m^2}-\frac {2 k (g x)^{3 m}}{27 g m^2}\right )\) |
-1/9*(k*(g*x)^(3*m)*(a + b*Log[c*x^n]))/(g*m) - (e^2*k*(g*x)^(3*m)*(a + b* Log[c*x^n]))/(3*f^2*g*m*x^(2*m)) + (e*k*(g*x)^(3*m)*(a + b*Log[c*x^n]))/(6 *f*g*m*x^m) + (e^3*k*(g*x)^(3*m)*(a + b*Log[c*x^n])*Log[e + f*x^m])/(3*f^3 *g*m*x^(3*m)) + ((g*x)^(3*m)*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/(3*g *m) - b*n*((-2*k*(g*x)^(3*m))/(27*g*m^2) - (4*e^2*k*(g*x)^(3*m))/(9*f^2*g* m^2*x^(2*m)) + (5*e*k*(g*x)^(3*m))/(36*f*g*m^2*x^m) + (e^3*k*(g*x)^(3*m)*L og[e + f*x^m])/(9*f^3*g*m^2*x^(3*m)) + (e^3*k*(g*x)^(3*m)*Log[-((f*x^m)/e) ]*Log[e + f*x^m])/(3*f^3*g*m^2*x^(3*m)) + ((g*x)^(3*m)*Log[d*(e + f*x^m)^k ])/(9*g*m^2) + (e^3*k*(g*x)^(3*m)*PolyLog[2, 1 + (f*x^m)/e])/(3*f^3*g*m^2* x^(3*m)))
3.2.50.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* (e + f*x^m)^r], x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[1/x u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] && RationalQ[q])) && NeQ[q, -1]
\[\int \left (g x \right )^{-1+3 m} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \,x^{m}\right )^{k}\right )d x\]
Time = 0.30 (sec) , antiderivative size = 368, normalized size of antiderivative = 0.85 \[ \int (g x)^{-1+3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\frac {36 \, b e^{3} g^{3 \, m - 1} k m n \log \left (x\right ) \log \left (\frac {f x^{m} + e}{e}\right ) + 36 \, b e^{3} g^{3 \, m - 1} k n {\rm Li}_2\left (-\frac {f x^{m} + e}{e} + 1\right ) - 4 \, {\left (3 \, b f^{3} k m \log \left (c\right ) + 3 \, a f^{3} k m - 2 \, b f^{3} k n - 3 \, {\left (3 \, b f^{3} m \log \left (c\right ) + 3 \, a f^{3} m - b f^{3} n\right )} \log \left (d\right ) + 3 \, {\left (b f^{3} k m n - 3 \, b f^{3} m n \log \left (d\right )\right )} \log \left (x\right )\right )} g^{3 \, m - 1} x^{3 \, m} + 3 \, {\left (6 \, b e f^{2} k m n \log \left (x\right ) + 6 \, b e f^{2} k m \log \left (c\right ) + 6 \, a e f^{2} k m - 5 \, b e f^{2} k n\right )} g^{3 \, m - 1} x^{2 \, m} - 12 \, {\left (3 \, b e^{2} f k m n \log \left (x\right ) + 3 \, b e^{2} f k m \log \left (c\right ) + 3 \, a e^{2} f k m - 4 \, b e^{2} f k n\right )} g^{3 \, m - 1} x^{m} + 12 \, {\left ({\left (3 \, b f^{3} k m n \log \left (x\right ) + 3 \, b f^{3} k m \log \left (c\right ) + 3 \, a f^{3} k m - b f^{3} k n\right )} g^{3 \, m - 1} x^{3 \, m} + {\left (3 \, b e^{3} k m \log \left (c\right ) + 3 \, a e^{3} k m - b e^{3} k n\right )} g^{3 \, m - 1}\right )} \log \left (f x^{m} + e\right )}{108 \, f^{3} m^{2}} \]
1/108*(36*b*e^3*g^(3*m - 1)*k*m*n*log(x)*log((f*x^m + e)/e) + 36*b*e^3*g^( 3*m - 1)*k*n*dilog(-(f*x^m + e)/e + 1) - 4*(3*b*f^3*k*m*log(c) + 3*a*f^3*k *m - 2*b*f^3*k*n - 3*(3*b*f^3*m*log(c) + 3*a*f^3*m - b*f^3*n)*log(d) + 3*( b*f^3*k*m*n - 3*b*f^3*m*n*log(d))*log(x))*g^(3*m - 1)*x^(3*m) + 3*(6*b*e*f ^2*k*m*n*log(x) + 6*b*e*f^2*k*m*log(c) + 6*a*e*f^2*k*m - 5*b*e*f^2*k*n)*g^ (3*m - 1)*x^(2*m) - 12*(3*b*e^2*f*k*m*n*log(x) + 3*b*e^2*f*k*m*log(c) + 3* a*e^2*f*k*m - 4*b*e^2*f*k*n)*g^(3*m - 1)*x^m + 12*((3*b*f^3*k*m*n*log(x) + 3*b*f^3*k*m*log(c) + 3*a*f^3*k*m - b*f^3*k*n)*g^(3*m - 1)*x^(3*m) + (3*b* e^3*k*m*log(c) + 3*a*e^3*k*m - b*e^3*k*n)*g^(3*m - 1))*log(f*x^m + e))/(f^ 3*m^2)
Timed out. \[ \int (g x)^{-1+3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\text {Timed out} \]
\[ \int (g x)^{-1+3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \left (g x\right )^{3 \, m - 1} \log \left ({\left (f x^{m} + e\right )}^{k} d\right ) \,d x } \]
1/9*(3*b*g^(3*m)*m*x^(3*m)*log(x^n) + (3*a*g^(3*m)*m + (3*g^(3*m)*m*log(c) - g^(3*m)*n)*b)*x^(3*m))*log((f*x^m + e)^k)/(g*m^2) + integrate(-1/9*((3* (f*g^(3*m)*k*m - 3*f*g^(3*m)*m*log(d))*a - (f*g^(3*m)*k*n - 3*(f*g^(3*m)*k *m - 3*f*g^(3*m)*m*log(d))*log(c))*b)*x^(4*m) - 9*(b*e*g^(3*m)*m*log(c)*lo g(d) + a*e*g^(3*m)*m*log(d))*x^(3*m) - 3*(3*b*e*g^(3*m)*m*x^(3*m)*log(d) - (f*g^(3*m)*k*m - 3*f*g^(3*m)*m*log(d))*b*x^(4*m))*log(x^n))/(f*g*m*x*x^m + e*g*m*x), x)
\[ \int (g x)^{-1+3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \left (g x\right )^{3 \, m - 1} \log \left ({\left (f x^{m} + e\right )}^{k} d\right ) \,d x } \]
Timed out. \[ \int (g x)^{-1+3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\int \ln \left (d\,{\left (e+f\,x^m\right )}^k\right )\,{\left (g\,x\right )}^{3\,m-1}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]