\(\int (g x)^{-1-m} (a+b \log (c x^n)) \log (d (e+f x^m)^k) \, dx\) [159]

Optimal result

Integrand size = 32, antiderivative size = 304 \[ \int (g x)^{-1-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\frac {b f k n x^m (g x)^{-m} \log (x)}{e g m}-\frac {b f k n x^m (g x)^{-m} \log ^2(x)}{2 e g}+\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {b f k n x^m (g x)^{-m} \log \left (e+f x^m\right )}{e g m^2}+\frac {b f k n x^m (g x)^{-m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {b f k n x^m (g x)^{-m} \operatorname {PolyLog}\left (2,1+\frac {f x^m}{e}\right )}{e g m^2} \] Output:

b*f*k*n*x^m*ln(x)/e/g/m/((g*x)^m)-1/2*b*f*k*n*x^m*ln(x)^2/e/g/((g*x)^m)+f* 
k*x^m*ln(x)*(a+b*ln(c*x^n))/e/g/((g*x)^m)-b*f*k*n*x^m*ln(e+f*x^m)/e/g/m^2/ 
((g*x)^m)+b*f*k*n*x^m*ln(-f*x^m/e)*ln(e+f*x^m)/e/g/m^2/((g*x)^m)-f*k*x^m*( 
a+b*ln(c*x^n))*ln(e+f*x^m)/e/g/m/((g*x)^m)-b*n*ln(d*(e+f*x^m)^k)/g/m^2/((g 
*x)^m)-(a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k)/g/m/((g*x)^m)+b*f*k*n*x^m*polylog 
(2,1+f*x^m/e)/e/g/m^2/((g*x)^m)
 

Mathematica [A] (warning: unable to verify)

Time = 0.63 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.53 \[ \int (g x)^{-1-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\frac {(g x)^{-m} \left (-b f k m^2 n x^m \log ^2(x)-2 \left (a m+b n+b m \log \left (c x^n\right )\right ) \left (f k x^m \log \left (f-f x^{-m}\right )+e \log \left (d \left (e+f x^m\right )^k\right )\right )+2 f k m x^m \log (x) \left (a m+b n+b m \log \left (c x^n\right )+b n \log \left (f-f x^{-m}\right )-b n \log \left (1+\frac {f x^m}{e}\right )\right )-2 b f k n x^m \operatorname {PolyLog}\left (2,-\frac {f x^m}{e}\right )\right )}{2 e g m^2} \] Input:

Integrate[(g*x)^(-1 - m)*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]
 

Output:

(-(b*f*k*m^2*n*x^m*Log[x]^2) - 2*(a*m + b*n + b*m*Log[c*x^n])*(f*k*x^m*Log 
[f - f/x^m] + e*Log[d*(e + f*x^m)^k]) + 2*f*k*m*x^m*Log[x]*(a*m + b*n + b* 
m*Log[c*x^n] + b*n*Log[f - f/x^m] - b*n*Log[1 + (f*x^m)/e]) - 2*b*f*k*n*x^ 
m*PolyLog[2, -((f*x^m)/e)])/(2*e*g*m^2*(g*x)^m)
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 298, 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.

\(\displaystyle \int (g x)^{-m-1} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx\)

\(\Big \downarrow \) 2823

\(\displaystyle -b n \int \left (\frac {f k (g x)^{-m} \log (x) x^{m-1}}{e g}-\frac {f k (g x)^{-m} \log \left (f x^m+e\right ) x^{m-1}}{e g m}-\frac {(g x)^{-m} \log \left (d \left (f x^m+e\right )^k\right )}{g m x}\right )dx-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {f k x^m \log (x) (g x)^{-m} \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {f k x^m (g x)^{-m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{e g m}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {f k x^m \log (x) (g x)^{-m} \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {f k x^m (g x)^{-m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{e g m}-b n \left (\frac {(g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {f k x^m (g x)^{-m} \operatorname {PolyLog}\left (2,\frac {f x^m}{e}+1\right )}{e g m^2}+\frac {f k x^m (g x)^{-m} \log \left (e+f x^m\right )}{e g m^2}-\frac {f k x^m (g x)^{-m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}+\frac {f k x^m \log ^2(x) (g x)^{-m}}{2 e g}-\frac {f k x^m \log (x) (g x)^{-m}}{e g m}\right )\)

Input:

Int[(g*x)^(-1 - m)*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]
 

Output:

(f*k*x^m*Log[x]*(a + b*Log[c*x^n]))/(e*g*(g*x)^m) - (f*k*x^m*(a + b*Log[c* 
x^n])*Log[e + f*x^m])/(e*g*m*(g*x)^m) - ((a + b*Log[c*x^n])*Log[d*(e + f*x 
^m)^k])/(g*m*(g*x)^m) - b*n*(-((f*k*x^m*Log[x])/(e*g*m*(g*x)^m)) + (f*k*x^ 
m*Log[x]^2)/(2*e*g*(g*x)^m) + (f*k*x^m*Log[e + f*x^m])/(e*g*m^2*(g*x)^m) - 
 (f*k*x^m*Log[-((f*x^m)/e)]*Log[e + f*x^m])/(e*g*m^2*(g*x)^m) + Log[d*(e + 
 f*x^m)^k]/(g*m^2*(g*x)^m) - (f*k*x^m*PolyLog[2, 1 + (f*x^m)/e])/(e*g*m^2* 
(g*x)^m))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2823

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]
 

Maple [F]

\[\int \left (g x \right )^{-1-m} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \,x^{m}\right )^{k}\right )d x\]

Input:

int((g*x)^(-1-m)*(a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k),x)
 

Output:

int((g*x)^(-1-m)*(a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k),x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.79 \[ \int (g x)^{-1-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=-\frac {2 \, b f g^{-m - 1} k m n x^{m} \log \left (x\right ) \log \left (\frac {f x^{m} + e}{e}\right ) + 2 \, b f g^{-m - 1} k n x^{m} {\rm Li}_2\left (-\frac {f x^{m} + e}{e} + 1\right ) - {\left (b f k m^{2} n \log \left (x\right )^{2} + 2 \, {\left (b f k m^{2} \log \left (c\right ) + a f k m^{2} + b f k m n\right )} \log \left (x\right )\right )} g^{-m - 1} x^{m} + 2 \, {\left (b e m n \log \left (d\right ) \log \left (x\right ) + {\left (b e m \log \left (c\right ) + a e m + b e n\right )} \log \left (d\right )\right )} g^{-m - 1} + 2 \, {\left ({\left (b f k m \log \left (c\right ) + a f k m + b f k n\right )} g^{-m - 1} x^{m} + {\left (b e k m n \log \left (x\right ) + b e k m \log \left (c\right ) + a e k m + b e k n\right )} g^{-m - 1}\right )} \log \left (f x^{m} + e\right )}{2 \, e m^{2} x^{m}} \] Input:

integrate((g*x)^(-1-m)*(a+b*log(c*x^n))*log(d*(e+f*x^m)^k),x, algorithm="f 
ricas")
 

Output:

-1/2*(2*b*f*g^(-m - 1)*k*m*n*x^m*log(x)*log((f*x^m + e)/e) + 2*b*f*g^(-m - 
 1)*k*n*x^m*dilog(-(f*x^m + e)/e + 1) - (b*f*k*m^2*n*log(x)^2 + 2*(b*f*k*m 
^2*log(c) + a*f*k*m^2 + b*f*k*m*n)*log(x))*g^(-m - 1)*x^m + 2*(b*e*m*n*log 
(d)*log(x) + (b*e*m*log(c) + a*e*m + b*e*n)*log(d))*g^(-m - 1) + 2*((b*f*k 
*m*log(c) + a*f*k*m + b*f*k*n)*g^(-m - 1)*x^m + (b*e*k*m*n*log(x) + b*e*k* 
m*log(c) + a*e*k*m + b*e*k*n)*g^(-m - 1))*log(f*x^m + e))/(e*m^2*x^m)
 

Sympy [F(-1)]

Timed out. \[ \int (g x)^{-1-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\text {Timed out} \] Input:

integrate((g*x)**(-1-m)*(a+b*ln(c*x**n))*ln(d*(e+f*x**m)**k),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (g x)^{-1-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \left (g x\right )^{-m - 1} \log \left ({\left (f x^{m} + e\right )}^{k} d\right ) \,d x } \] Input:

integrate((g*x)^(-1-m)*(a+b*log(c*x^n))*log(d*(e+f*x^m)^k),x, algorithm="m 
axima")
 

Output:

-(b*m*log(x^n) + (m*log(c) + n)*b + a*m)*g^(-m - 1)*log((f*x^m + e)^k)/(m^ 
2*x^m) + integrate((b*e*m*log(c)*log(d) + a*e*m*log(d) + ((f*k*m + f*m*log 
(d))*a + (f*k*n + (f*k*m + f*m*log(d))*log(c))*b)*x^m + (b*e*m*log(d) + (f 
*k*m + f*m*log(d))*b*x^m)*log(x^n))/(f*g^(m + 1)*m*x*x^(2*m) + e*g^(m + 1) 
*m*x*x^m), x)
 

Giac [F]

\[ \int (g x)^{-1-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \left (g x\right )^{-m - 1} \log \left ({\left (f x^{m} + e\right )}^{k} d\right ) \,d x } \] Input:

integrate((g*x)^(-1-m)*(a+b*log(c*x^n))*log(d*(e+f*x^m)^k),x, algorithm="g 
iac")
 

Output:

integrate((b*log(c*x^n) + a)*(g*x)^(-m - 1)*log((f*x^m + e)^k*d), x)
 

Mupad [F(-1)]

Timed out. \[ \int (g x)^{-1-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\int \frac {\ln \left (d\,{\left (e+f\,x^m\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (g\,x\right )}^{m+1}} \,d x \] Input:

int((log(d*(e + f*x^m)^k)*(a + b*log(c*x^n)))/(g*x)^(m + 1),x)
 

Output:

int((log(d*(e + f*x^m)^k)*(a + b*log(c*x^n)))/(g*x)^(m + 1), x)
 

Reduce [F]

\[ \int (g x)^{-1-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx=\frac {-x^{m} \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{x^{2 m} f x +x^{m} e x}d x \right ) b \,e^{2} k \,m^{2}-x^{m} \mathrm {log}\left (x^{m} f +e \right ) a f k m -x^{m} \mathrm {log}\left (x^{m} f +e \right ) b f k n +x^{m} \mathrm {log}\left (x \right ) a f k \,m^{2}+x^{m} \mathrm {log}\left (x \right ) b f k m n -\mathrm {log}\left (\left (x^{m} f +e \right )^{k} d \right ) \mathrm {log}\left (x^{n} c \right ) b e m -\mathrm {log}\left (\left (x^{m} f +e \right )^{k} d \right ) a e m -\mathrm {log}\left (\left (x^{m} f +e \right )^{k} d \right ) b e n -\mathrm {log}\left (x^{n} c \right ) b e k m -b e k n}{x^{m} g^{m} e g \,m^{2}} \] Input:

int((g*x)^(-1-m)*(a+b*log(c*x^n))*log(d*(e+f*x^m)^k),x)
 

Output:

( - x**m*int(log(x**n*c)/(x**(2*m)*f*x + x**m*e*x),x)*b*e**2*k*m**2 - x**m 
*log(x**m*f + e)*a*f*k*m - x**m*log(x**m*f + e)*b*f*k*n + x**m*log(x)*a*f* 
k*m**2 + x**m*log(x)*b*f*k*m*n - log((x**m*f + e)**k*d)*log(x**n*c)*b*e*m 
- log((x**m*f + e)**k*d)*a*e*m - log((x**m*f + e)**k*d)*b*e*n - log(x**n*c 
)*b*e*k*m - b*e*k*n)/(x**m*g**m*e*g*m**2)