\(\int x (a+b \log (c x^n)) (d+e \log (f (g+h x)^q)) \, dx\) [405]

Optimal result

Integrand size = 26, antiderivative size = 202 \[ \int x \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\frac {a e g q x}{2 h}-\frac {3 b e g n q x}{4 h}+\frac {1}{4} b e n q x^2+\frac {b e g q x \log \left (c x^n\right )}{2 h}-\frac {1}{4} e q x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {b e g^2 n q \log (g+h x)}{4 h^2}-\frac {1}{4} b n x^2 \left (d+e \log \left (f (g+h x)^q\right )\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )-\frac {e g^2 q \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {h x}{g}\right )}{2 h^2}-\frac {b e g^2 n q \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{2 h^2} \] Output:

1/2*a*e*g*q*x/h-3/4*b*e*g*n*q*x/h+1/4*b*e*n*q*x^2+1/2*b*e*g*q*x*ln(c*x^n)/ 
h-1/4*e*q*x^2*(a+b*ln(c*x^n))+1/4*b*e*g^2*n*q*ln(h*x+g)/h^2-1/4*b*n*x^2*(d 
+e*ln(f*(h*x+g)^q))+1/2*x^2*(a+b*ln(c*x^n))*(d+e*ln(f*(h*x+g)^q))-1/2*e*g^ 
2*q*(a+b*ln(c*x^n))*ln(1+h*x/g)/h^2-1/2*b*e*g^2*n*q*polylog(2,-h*x/g)/h^2
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.23 \[ \int x \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\frac {2 a e g h q x-3 b e g h n q x+2 a d h^2 x^2-b d h^2 n x^2-a e h^2 q x^2+b e h^2 n q x^2-2 a e g^2 q \log (g+h x)+b e g^2 n q \log (g+h x)+2 b e g^2 n q \log (x) \log (g+h x)+2 a e h^2 x^2 \log \left (f (g+h x)^q\right )-b e h^2 n x^2 \log \left (f (g+h x)^q\right )+b \log \left (c x^n\right ) \left (-2 e g^2 q \log (g+h x)+h x \left (2 e g q+2 d h x-e h q x+2 e h x \log \left (f (g+h x)^q\right )\right )\right )-2 b e g^2 n q \log (x) \log \left (1+\frac {h x}{g}\right )-2 b e g^2 n q \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{4 h^2} \] Input:

Integrate[x*(a + b*Log[c*x^n])*(d + e*Log[f*(g + h*x)^q]),x]
 

Output:

(2*a*e*g*h*q*x - 3*b*e*g*h*n*q*x + 2*a*d*h^2*x^2 - b*d*h^2*n*x^2 - a*e*h^2 
*q*x^2 + b*e*h^2*n*q*x^2 - 2*a*e*g^2*q*Log[g + h*x] + b*e*g^2*n*q*Log[g + 
h*x] + 2*b*e*g^2*n*q*Log[x]*Log[g + h*x] + 2*a*e*h^2*x^2*Log[f*(g + h*x)^q 
] - b*e*h^2*n*x^2*Log[f*(g + h*x)^q] + b*Log[c*x^n]*(-2*e*g^2*q*Log[g + h* 
x] + h*x*(2*e*g*q + 2*d*h*x - e*h*q*x + 2*e*h*x*Log[f*(g + h*x)^q])) - 2*b 
*e*g^2*n*q*Log[x]*Log[1 + (h*x)/g] - 2*b*e*g^2*n*q*PolyLog[2, -((h*x)/g)]) 
/(4*h^2)
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2889, 2793, 2009, 2842, 49, 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.

\(\displaystyle \int x \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx\)

\(\Big \downarrow \) 2889

\(\displaystyle -\frac {1}{2} e h q \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{g+h x}dx-\frac {1}{2} b n \int x \left (d+e \log \left (f (g+h x)^q\right )\right )dx+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )\)

\(\Big \downarrow \) 2793

\(\displaystyle -\frac {1}{2} e h q \int \left (\frac {\left (a+b \log \left (c x^n\right )\right ) g^2}{h^2 (g+h x)}-\frac {\left (a+b \log \left (c x^n\right )\right ) g}{h^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{h}\right )dx-\frac {1}{2} b n \int x \left (d+e \log \left (f (g+h x)^q\right )\right )dx+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{2} b n \int x \left (d+e \log \left (f (g+h x)^q\right )\right )dx+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )-\frac {1}{2} e h q \left (\frac {g^2 \log \left (\frac {h x}{g}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{h^3}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 h}-\frac {a g x}{h^2}-\frac {b g x \log \left (c x^n\right )}{h^2}+\frac {b g^2 n \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{h^3}+\frac {b g n x}{h^2}-\frac {b n x^2}{4 h}\right )\)

\(\Big \downarrow \) 2842

\(\displaystyle -\frac {1}{2} b n \left (\frac {1}{2} x^2 \left (d+e \log \left (f (g+h x)^q\right )\right )-\frac {1}{2} e h q \int \frac {x^2}{g+h x}dx\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )-\frac {1}{2} e h q \left (\frac {g^2 \log \left (\frac {h x}{g}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{h^3}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 h}-\frac {a g x}{h^2}-\frac {b g x \log \left (c x^n\right )}{h^2}+\frac {b g^2 n \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{h^3}+\frac {b g n x}{h^2}-\frac {b n x^2}{4 h}\right )\)

\(\Big \downarrow \) 49

\(\displaystyle -\frac {1}{2} b n \left (\frac {1}{2} x^2 \left (d+e \log \left (f (g+h x)^q\right )\right )-\frac {1}{2} e h q \int \left (\frac {g^2}{h^2 (g+h x)}-\frac {g}{h^2}+\frac {x}{h}\right )dx\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )-\frac {1}{2} e h q \left (\frac {g^2 \log \left (\frac {h x}{g}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{h^3}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 h}-\frac {a g x}{h^2}-\frac {b g x \log \left (c x^n\right )}{h^2}+\frac {b g^2 n \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{h^3}+\frac {b g n x}{h^2}-\frac {b n x^2}{4 h}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )-\frac {1}{2} e h q \left (\frac {g^2 \log \left (\frac {h x}{g}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{h^3}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 h}-\frac {a g x}{h^2}-\frac {b g x \log \left (c x^n\right )}{h^2}+\frac {b g^2 n \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )}{h^3}+\frac {b g n x}{h^2}-\frac {b n x^2}{4 h}\right )-\frac {1}{2} b n \left (\frac {1}{2} x^2 \left (d+e \log \left (f (g+h x)^q\right )\right )-\frac {1}{2} e h q \left (\frac {g^2 \log (g+h x)}{h^3}-\frac {g x}{h^2}+\frac {x^2}{2 h}\right )\right )\)

Input:

Int[x*(a + b*Log[c*x^n])*(d + e*Log[f*(g + h*x)^q]),x]
 

Output:

(x^2*(a + b*Log[c*x^n])*(d + e*Log[f*(g + h*x)^q]))/2 - (b*n*(-1/2*(e*h*q* 
(-((g*x)/h^2) + x^2/(2*h) + (g^2*Log[g + h*x])/h^3)) + (x^2*(d + e*Log[f*( 
g + h*x)^q]))/2))/2 - (e*h*q*(-((a*g*x)/h^2) + (b*g*n*x)/h^2 - (b*n*x^2)/( 
4*h) - (b*g*x*Log[c*x^n])/h^2 + (x^2*(a + b*Log[c*x^n]))/(2*h) + (g^2*(a + 
 b*Log[c*x^n])*Log[1 + (h*x)/g])/h^3 + (b*g^2*n*PolyLog[2, -((h*x)/g)])/h^ 
3))/2
 

Defintions of rubi rules used

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 2009

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

rule 2793

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[a + b*Log[c*x^n], 
 (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, 
 f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && Integer 
Q[r]))
 

rule 2842

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_ 
))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/( 
g*(q + 1))), x] - Simp[b*e*(n/(g*(q + 1)))   Int[(f + g*x)^(q + 1)/(d + e*x 
), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && 
NeQ[q, -1]
 

rule 2889

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log 
[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*(x_)^(r_.), x_Symbol] :> Simp[x^( 
r + 1)*(a + b*Log[c*(d + e*x)^n])^p*((f + g*Log[h*(i + j*x)^m])/(r + 1)), x 
] + (-Simp[g*j*(m/(r + 1))   Int[x^(r + 1)*((a + b*Log[c*(d + e*x)^n])^p/(i 
 + j*x)), x], x] - Simp[b*e*n*(p/(r + 1))   Int[x^(r + 1)*(a + b*Log[c*(d + 
 e*x)^n])^(p - 1)*((f + g*Log[h*(i + j*x)^m])/(d + e*x)), x], x]) /; FreeQ[ 
{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0] && IntegerQ[r] && (E 
qQ[p, 1] || GtQ[r, 0]) && NeQ[r, -1]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 79.83 (sec) , antiderivative size = 922, normalized size of antiderivative = 4.56

method	result	size
risch	\(\text {Expression too large to display}\)	\(922\)

Input:

int(x*(a+b*ln(c*x^n))*(d+e*ln(f*(h*x+g)^q)),x,method=_RETURNVERBOSE)
 

Output:

-1/2*q*a*e*g^2/h^2*ln(h*x+g)+(1/4*I*e*Pi*csgn(I*(h*x+g)^q)*csgn(I*f*(h*x+g 
)^q)^2-1/4*I*e*Pi*csgn(I*(h*x+g)^q)*csgn(I*f*(h*x+g)^q)*csgn(I*f)-1/4*I*e* 
Pi*csgn(I*f*(h*x+g)^q)^3+1/4*I*e*Pi*csgn(I*f*(h*x+g)^q)^2*csgn(I*f)+1/2*ln 
(f)*e+1/2*d)*(1/2*(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*c 
sgn(I*c*x^n)*csgn(I*c)-I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn( 
I*c)+2*b*ln(c)+2*a)*x^2+b*x^2*ln(x^n)-1/2*b*n*x^2)-1/4*I*q/h^2*e*g^2*ln(h* 
x+g)*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/4*I*q/h^2*e*g^2*ln(h*x+g)*Pi*b*csg 
n(I*c*x^n)^2*csgn(I*c)+(1/2*b*e*x^2*ln(x^n)+1/4*e*x^2*(I*Pi*b*csgn(I*x^n)* 
csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*csgn(I*c 
*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)-n*b+2*a))*ln((h*x+g)^q) 
-1/4*x^2*a*e*q+1/2*n*b/h^2*e*g^2*q*ln(h*x+g)*ln(-h*x/g)-1/4*I*q/h*e*g*x*Pi 
*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/4*I*q/h^2*e*g^2*ln(h*x+g)*Pi*b*cs 
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/2*q/h^2*b*e*ln(x^n)*g^2*ln(h*x+g)-1/2* 
q/h^2*e*g^2*ln(h*x+g)*b*ln(c)+1/8*I*q*e*x^2*Pi*b*csgn(I*c*x^n)^3-1/4*I*q/h 
*e*g*x*Pi*b*csgn(I*c*x^n)^3+1/4*I*q/h^2*e*g^2*ln(h*x+g)*Pi*b*csgn(I*c*x^n) 
^3+1/8*I*q*e*x^2*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/8*I*q*e*x^2*Pi 
*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/8*I*q*e*x^2*Pi*b*csgn(I*c*x^n)^2*csgn(I*c 
)+1/2*n*b/h^2*e*g^2*q*dilog(-h*x/g)+1/4*I*q/h*e*g*x*Pi*b*csgn(I*x^n)*csgn( 
I*c*x^n)^2+1/4*I*q/h*e*g*x*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+1/2*q/h*b*e*ln(x 
^n)*g*x-5/8*q/h^2*b*e*n*g^2-1/4*q*b*e*ln(x^n)*x^2+1/2*q/h*e*g*x*b*ln(c)...
 

Fricas [F]

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

integrate(x*(a+b*log(c*x^n))*(d+e*log(f*(h*x+g)^q)),x, algorithm="fricas")
 

Output:

integral(b*d*x*log(c*x^n) + a*d*x + (b*e*x*log(c*x^n) + a*e*x)*log((h*x + 
g)^q*f), x)
 

Sympy [F(-1)]

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

integrate(x*(a+b*ln(c*x**n))*(d+e*ln(f*(h*x+g)**q)),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x*(a+b*log(c*x^n))*(d+e*log(f*(h*x+g)^q)),x, algorithm="maxima")
 

Output:

-1/4*b*d*n*x^2 - 1/4*a*e*h*q*(2*g^2*log(h*x + g)/h^3 + (h*x^2 - 2*g*x)/h^2 
) + 1/2*a*e*x^2*log((h*x + g)^q*f) + 1/2*b*d*x^2*log(c*x^n) + 1/2*a*d*x^2 
+ 1/4*b*e*((2*g^2*n*q*log(h*x + g)*log(x) + (2*h^2*x^2*log(x^n) - (h^2*n - 
 2*h^2*log(c))*x^2)*log((h*x + g)^q) + (2*g*h*q*x - 2*g^2*q*log(h*x + g) - 
 (h^2*q - 2*h^2*log(f))*x^2)*log(x^n))/h^2 - 4*integrate(1/4*(2*g^2*n*q*lo 
g(x) + 2*g^2*n*q - 2*(h^2*n*q - h^2*n*log(f) - (h^2*q - 2*h^2*log(f))*log( 
c))*x^2 + (g*h*n*q + 2*g*h*n*log(f) - 4*g*h*log(c)*log(f))*x)/(h^2*x + g*h 
), x))
 

Giac [F]

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

integrate(x*(a+b*log(c*x^n))*(d+e*log(f*(h*x+g)^q)),x, algorithm="giac")
 

Output:

integrate((e*log((h*x + g)^q*f) + d)*(b*log(c*x^n) + a)*x, x)
 

Mupad [F(-1)]

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

int(x*(d + e*log(f*(g + h*x)^q))*(a + b*log(c*x^n)),x)
 

Output:

int(x*(d + e*log(f*(g + h*x)^q))*(a + b*log(c*x^n)), x)
 

Reduce [F]

\[ \int x \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right ) \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{h \,x^{2}+g x}d x \right ) b e \,g^{3} n q +2 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) \mathrm {log}\left (x^{n} c \right ) b e \,h^{2} n \,x^{2}-2 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) a e \,g^{2} n +2 \,\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) a e \,h^{2} n \,x^{2}+\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) b e \,g^{2} n^{2}-\mathrm {log}\left (\left (h x +g \right )^{q} f \right ) b e \,h^{2} n^{2} x^{2}-\mathrm {log}\left (x^{n} c \right )^{2} b e \,g^{2} q +2 \,\mathrm {log}\left (x^{n} c \right ) b d \,h^{2} n \,x^{2}+2 \,\mathrm {log}\left (x^{n} c \right ) b e g h n q x -\mathrm {log}\left (x^{n} c \right ) b e \,h^{2} n q \,x^{2}+2 a d \,h^{2} n \,x^{2}+2 a e g h n q x -a e \,h^{2} n q \,x^{2}-b d \,h^{2} n^{2} x^{2}-3 b e g h \,n^{2} q x +b e \,h^{2} n^{2} q \,x^{2}}{4 h^{2} n} \] Input:

int(x*(a+b*log(c*x^n))*(d+e*log(f*(h*x+g)^q)),x)
 

Output:

(2*int(log(x**n*c)/(g*x + h*x**2),x)*b*e*g**3*n*q + 2*log((g + h*x)**q*f)* 
log(x**n*c)*b*e*h**2*n*x**2 - 2*log((g + h*x)**q*f)*a*e*g**2*n + 2*log((g 
+ h*x)**q*f)*a*e*h**2*n*x**2 + log((g + h*x)**q*f)*b*e*g**2*n**2 - log((g 
+ h*x)**q*f)*b*e*h**2*n**2*x**2 - log(x**n*c)**2*b*e*g**2*q + 2*log(x**n*c 
)*b*d*h**2*n*x**2 + 2*log(x**n*c)*b*e*g*h*n*q*x - log(x**n*c)*b*e*h**2*n*q 
*x**2 + 2*a*d*h**2*n*x**2 + 2*a*e*g*h*n*q*x - a*e*h**2*n*q*x**2 - b*d*h**2 
*n**2*x**2 - 3*b*e*g*h*n**2*q*x + b*e*h**2*n**2*q*x**2)/(4*h**2*n)