Integrand size = 32, antiderivative size = 484 \[ \int (g x)^{-1-3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=-\frac {5 b f k n x^m (g x)^{-3 m}}{36 e g m^2}+\frac {4 b f^2 k n x^{2 m} (g x)^{-3 m}}{9 e^2 g m^2}+\frac {b f^3 k n x^{3 m} (g x)^{-3 m} \log (x)}{9 e^3 g m}-\frac {b f^3 k n x^{3 m} (g x)^{-3 m} \log ^2(x)}{6 e^3 g}-\frac {f k x^m (g x)^{-3 m} \left (a+b \log \left (c x^n\right )\right )}{6 e g m}+\frac {f^2 k x^{2 m} (g x)^{-3 m} \left (a+b \log \left (c x^n\right )\right )}{3 e^2 g m}+\frac {f^3 k x^{3 m} (g x)^{-3 m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3 g}-\frac {b f^3 k n x^{3 m} (g x)^{-3 m} \log \left (e+f x^m\right )}{9 e^3 g m^2}+\frac {b f^3 k n x^{3 m} (g x)^{-3 m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 e^3 g m^2}-\frac {f^3 k x^{3 m} (g x)^{-3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 e^3 g m}-\frac {b n (g x)^{-3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}-\frac {(g x)^{-3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}+\frac {b f^3 k n x^{3 m} (g x)^{-3 m} \operatorname {PolyLog}\left (2,1+\frac {f x^m}{e}\right )}{3 e^3 g m^2} \]
-5/36*b*f*k*n*x^m/e/g/m^2/((g*x)^(3*m))+4/9*b*f^2*k*n*x^(2*m)/e^2/g/m^2/(( g*x)^(3*m))+1/9*b*f^3*k*n*x^(3*m)*ln(x)/e^3/g/m/((g*x)^(3*m))-1/6*b*f^3*k* n*x^(3*m)*ln(x)^2/e^3/g/((g*x)^(3*m))-1/6*f*k*x^m*(a+b*ln(c*x^n))/e/g/m/(( g*x)^(3*m))+1/3*f^2*k*x^(2*m)*(a+b*ln(c*x^n))/e^2/g/m/((g*x)^(3*m))+1/3*f^ 3*k*x^(3*m)*ln(x)*(a+b*ln(c*x^n))/e^3/g/((g*x)^(3*m))-1/9*b*f^3*k*n*x^(3*m )*ln(e+f*x^m)/e^3/g/m^2/((g*x)^(3*m))+1/3*b*f^3*k*n*x^(3*m)*ln(-f*x^m/e)*l n(e+f*x^m)/e^3/g/m^2/((g*x)^(3*m))-1/3*f^3*k*x^(3*m)*(a+b*ln(c*x^n))*ln(e+ f*x^m)/e^3/g/m/((g*x)^(3*m))-1/9*b*n*ln(d*(e+f*x^m)^k)/g/m^2/((g*x)^(3*m)) -1/3*(a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k)/g/m/((g*x)^(3*m))+1/3*b*f^3*k*n*x^( 3*m)*polylog(2,1+f*x^m/e)/e^3/g/m^2/((g*x)^(3*m))
Time = 0.42 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.74 \[ \int (g x)^{-1-3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\frac {(g x)^{-3 m} \left (-6 a e^2 f k m x^m-5 b e^2 f k n x^m+12 a e f^2 k m x^{2 m}+16 b e f^2 k n x^{2 m}-6 b f^3 k m^2 n x^{3 m} \log ^2(x)-6 b e^2 f k m x^m \log \left (c x^n\right )+12 b e f^2 k m x^{2 m} \log \left (c x^n\right )-12 a f^3 k m x^{3 m} \log \left (f-f x^{-m}\right )-4 b f^3 k n x^{3 m} \log \left (f-f x^{-m}\right )-12 b f^3 k m x^{3 m} \log \left (c x^n\right ) \log \left (f-f x^{-m}\right )-12 a e^3 m \log \left (d \left (e+f x^m\right )^k\right )-4 b e^3 n \log \left (d \left (e+f x^m\right )^k\right )-12 b e^3 m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+4 f^3 k m x^{3 m} \log (x) \left (3 a m+b n+3 b m \log \left (c x^n\right )+3 b n \log \left (f-f x^{-m}\right )-3 b n \log \left (1+\frac {f x^m}{e}\right )\right )-12 b f^3 k n x^{3 m} \operatorname {PolyLog}\left (2,-\frac {f x^m}{e}\right )\right )}{36 e^3 g m^2} \]
(-6*a*e^2*f*k*m*x^m - 5*b*e^2*f*k*n*x^m + 12*a*e*f^2*k*m*x^(2*m) + 16*b*e* f^2*k*n*x^(2*m) - 6*b*f^3*k*m^2*n*x^(3*m)*Log[x]^2 - 6*b*e^2*f*k*m*x^m*Log [c*x^n] + 12*b*e*f^2*k*m*x^(2*m)*Log[c*x^n] - 12*a*f^3*k*m*x^(3*m)*Log[f - f/x^m] - 4*b*f^3*k*n*x^(3*m)*Log[f - f/x^m] - 12*b*f^3*k*m*x^(3*m)*Log[c* x^n]*Log[f - f/x^m] - 12*a*e^3*m*Log[d*(e + f*x^m)^k] - 4*b*e^3*n*Log[d*(e + f*x^m)^k] - 12*b*e^3*m*Log[c*x^n]*Log[d*(e + f*x^m)^k] + 4*f^3*k*m*x^(3 *m)*Log[x]*(3*a*m + b*n + 3*b*m*Log[c*x^n] + 3*b*n*Log[f - f/x^m] - 3*b*n* Log[1 + (f*x^m)/e]) - 12*b*f^3*k*n*x^(3*m)*PolyLog[2, -((f*x^m)/e)])/(36*e ^3*g*m^2*(g*x)^(3*m))
Time = 0.87 (sec) , antiderivative size = 473, 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 |
| \(\displaystyle -b n \int \left (-\frac {f k (g x)^{-3 m} x^{m-1}}{6 e g m}+\frac {f^2 k (g x)^{-3 m} x^{2 m-1}}{3 e^2 g m}+\frac {f^3 k (g x)^{-3 m} \log (x) x^{3 m-1}}{3 e^3 g}-\frac {f^3 k (g x)^{-3 m} \log \left (f x^m+e\right ) x^{3 m-1}}{3 e^3 g m}-\frac {(g x)^{-3 m} \log \left (d \left (f x^m+e\right )^k\right )}{3 g m x}\right )dx-\frac {(g x)^{-3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}+\frac {f^3 k x^{3 m} \log (x) (g x)^{-3 m} \left (a+b \log \left (c x^n\right )\right )}{3 e^3 g}-\frac {f^3 k x^{3 m} (g x)^{-3 m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3 g m}+\frac {f^2 k x^{2 m} (g x)^{-3 m} \left (a+b \log \left (c x^n\right )\right )}{3 e^2 g m}-\frac {f k x^m (g x)^{-3 m} \left (a+b \log \left (c x^n\right )\right )}{6 e g m}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(g x)^{-3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}+\frac {f^3 k x^{3 m} \log (x) (g x)^{-3 m} \left (a+b \log \left (c x^n\right )\right )}{3 e^3 g}-\frac {f^3 k x^{3 m} (g x)^{-3 m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3 g m}+\frac {f^2 k x^{2 m} (g x)^{-3 m} \left (a+b \log \left (c x^n\right )\right )}{3 e^2 g m}-\frac {f k x^m (g x)^{-3 m} \left (a+b \log \left (c x^n\right )\right )}{6 e g m}-b n \left (\frac {(g x)^{-3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}-\frac {f^3 k x^{3 m} (g x)^{-3 m} \operatorname {PolyLog}\left (2,\frac {f x^m}{e}+1\right )}{3 e^3 g m^2}+\frac {f^3 k x^{3 m} (g x)^{-3 m} \log \left (e+f x^m\right )}{9 e^3 g m^2}-\frac {f^3 k x^{3 m} (g x)^{-3 m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 e^3 g m^2}+\frac {f^3 k x^{3 m} \log ^2(x) (g x)^{-3 m}}{6 e^3 g}-\frac {f^3 k x^{3 m} \log (x) (g x)^{-3 m}}{9 e^3 g m}-\frac {4 f^2 k x^{2 m} (g x)^{-3 m}}{9 e^2 g m^2}+\frac {5 f k x^m (g x)^{-3 m}}{36 e g m^2}\right )\) |
-1/6*(f*k*x^m*(a + b*Log[c*x^n]))/(e*g*m*(g*x)^(3*m)) + (f^2*k*x^(2*m)*(a + b*Log[c*x^n]))/(3*e^2*g*m*(g*x)^(3*m)) + (f^3*k*x^(3*m)*Log[x]*(a + b*Lo g[c*x^n]))/(3*e^3*g*(g*x)^(3*m)) - (f^3*k*x^(3*m)*(a + b*Log[c*x^n])*Log[e + f*x^m])/(3*e^3*g*m*(g*x)^(3*m)) - ((a + b*Log[c*x^n])*Log[d*(e + f*x^m) ^k])/(3*g*m*(g*x)^(3*m)) - b*n*((5*f*k*x^m)/(36*e*g*m^2*(g*x)^(3*m)) - (4* f^2*k*x^(2*m))/(9*e^2*g*m^2*(g*x)^(3*m)) - (f^3*k*x^(3*m)*Log[x])/(9*e^3*g *m*(g*x)^(3*m)) + (f^3*k*x^(3*m)*Log[x]^2)/(6*e^3*g*(g*x)^(3*m)) + (f^3*k* x^(3*m)*Log[e + f*x^m])/(9*e^3*g*m^2*(g*x)^(3*m)) - (f^3*k*x^(3*m)*Log[-(( f*x^m)/e)]*Log[e + f*x^m])/(3*e^3*g*m^2*(g*x)^(3*m)) + Log[d*(e + f*x^m)^k ]/(9*g*m^2*(g*x)^(3*m)) - (f^3*k*x^(3*m)*PolyLog[2, 1 + (f*x^m)/e])/(3*e^3 *g*m^2*(g*x)^(3*m)))
3.2.55.3.1 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 = 0.28 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.83 \[ \int (g x)^{-1-3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=-\frac {12 \, b f^{3} g^{-3 \, m - 1} k m n x^{3 \, m} \log \left (x\right ) \log \left (\frac {f x^{m} + e}{e}\right ) + 12 \, b f^{3} g^{-3 \, m - 1} k n x^{3 \, m} {\rm Li}_2\left (-\frac {f x^{m} + e}{e} + 1\right ) - 2 \, {\left (3 \, b f^{3} k m^{2} n \log \left (x\right )^{2} + 2 \, {\left (3 \, b f^{3} k m^{2} \log \left (c\right ) + 3 \, a f^{3} k m^{2} + b f^{3} k m n\right )} \log \left (x\right )\right )} g^{-3 \, m - 1} x^{3 \, m} - 4 \, {\left (3 \, b e f^{2} k m n \log \left (x\right ) + 3 \, b e f^{2} k m \log \left (c\right ) + 3 \, a e f^{2} k m + 4 \, b e f^{2} k n\right )} g^{-3 \, m - 1} x^{2 \, m} + {\left (6 \, b e^{2} f k m n \log \left (x\right ) + 6 \, b e^{2} f k m \log \left (c\right ) + 6 \, a e^{2} f k m + 5 \, b e^{2} f k n\right )} g^{-3 \, m - 1} x^{m} + 4 \, {\left (3 \, b e^{3} m n \log \left (d\right ) \log \left (x\right ) + {\left (3 \, b e^{3} m \log \left (c\right ) + 3 \, a e^{3} m + b e^{3} n\right )} \log \left (d\right )\right )} g^{-3 \, m - 1} + 4 \, {\left ({\left (3 \, b f^{3} k m \log \left (c\right ) + 3 \, a f^{3} k m + b f^{3} k n\right )} g^{-3 \, m - 1} x^{3 \, m} + {\left (3 \, b e^{3} k m n \log \left (x\right ) + 3 \, b e^{3} k m \log \left (c\right ) + 3 \, a e^{3} k m + b e^{3} k n\right )} g^{-3 \, m - 1}\right )} \log \left (f x^{m} + e\right )}{36 \, e^{3} m^{2} x^{3 \, m}} \]
-1/36*(12*b*f^3*g^(-3*m - 1)*k*m*n*x^(3*m)*log(x)*log((f*x^m + e)/e) + 12* b*f^3*g^(-3*m - 1)*k*n*x^(3*m)*dilog(-(f*x^m + e)/e + 1) - 2*(3*b*f^3*k*m^ 2*n*log(x)^2 + 2*(3*b*f^3*k*m^2*log(c) + 3*a*f^3*k*m^2 + b*f^3*k*m*n)*log( x))*g^(-3*m - 1)*x^(3*m) - 4*(3*b*e*f^2*k*m*n*log(x) + 3*b*e*f^2*k*m*log(c ) + 3*a*e*f^2*k*m + 4*b*e*f^2*k*n)*g^(-3*m - 1)*x^(2*m) + (6*b*e^2*f*k*m*n *log(x) + 6*b*e^2*f*k*m*log(c) + 6*a*e^2*f*k*m + 5*b*e^2*f*k*n)*g^(-3*m - 1)*x^m + 4*(3*b*e^3*m*n*log(d)*log(x) + (3*b*e^3*m*log(c) + 3*a*e^3*m + b* e^3*n)*log(d))*g^(-3*m - 1) + 4*((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*n*log(x) + 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))/(e^3*m^2*x^(3*m))
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*m*log(x^n) + (3*m*log(c) + n)*b + 3*a*m)*g^(-3*m - 1)*log((f*x^m + e)^k)/(m^2*x^(3*m)) + integrate(1/9*(9*b*e*m*log(c)*log(d) + 9*a*e*m*lo g(d) + (3*(f*k*m + 3*f*m*log(d))*a + (f*k*n + 3*(f*k*m + 3*f*m*log(d))*log (c))*b)*x^m + 3*(3*b*e*m*log(d) + (f*k*m + 3*f*m*log(d))*b*x^m)*log(x^n))/ (f*g^(3*m + 1)*m*x*x^(4*m) + e*g^(3*m + 1)*m*x*x^(3*m)), 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. \[ \int (g x)^{-1-3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\int \frac {\ln \left (d\,{\left (e+f\,x^m\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (g\,x\right )}^{3\,m+1}} \,d x \]