\(\int x^2 \log (f x^m) (a+b \log (c (d+e x)^n))^2 \, dx\) [367]

Optimal result

Integrand size = 26, antiderivative size = 705 \[ \int x^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\frac {2 a b d^2 m n x}{9 e^2}-\frac {71 b^2 d^2 m n^2 x}{54 e^2}+\frac {b d^2 m n (6 a-11 b n) x}{9 e^2}+\frac {19 b^2 d m n^2 x^2}{54 e}-\frac {2}{27} b^2 m n^2 x^3-\frac {2 a b d^2 n x \log \left (f x^m\right )}{3 e^2}+\frac {11 b^2 d^2 n^2 x \log \left (f x^m\right )}{9 e^2}-\frac {5 b^2 d n^2 x^2 \log \left (f x^m\right )}{18 e}+\frac {2}{27} b^2 n^2 x^3 \log \left (f x^m\right )+\frac {23 b^2 d^3 m n^2 \log (d+e x)}{54 e^3}+\frac {5 b^2 d^3 m n^2 \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{9 e^3}-\frac {5 b^2 d^3 n^2 \log \left (f x^m\right ) \log (d+e x)}{9 e^3}+\frac {8 b^2 d^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{9 e^3}+\frac {2 b^2 d^3 m n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{3 e^3}-\frac {2 b^2 d^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{3 e^3}-\frac {5 b d m n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{18 e}+\frac {4}{27} b m n x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b d n x^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e}-\frac {2}{9} b n x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {d^3 m \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{9 e^3}-\frac {1}{9} m x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2-\frac {d^3 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac {d^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac {1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac {11 b^2 d^3 m n^2 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{9 e^3}-\frac {2 b d^3 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{3 e^3}+\frac {2 b^2 d^3 m n^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )}{3 e^3} \] Output:

1/3*x^3*ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^2-1/9*m*x^3*(a+b*ln(c*(e*x+d)^n))^ 
2+5/9*b^2*d^3*m*n^2*ln(-e*x/d)*ln(e*x+d)/e^3-2/3*a*b*d^2*n*x*ln(f*x^m)/e^2 
-2/3*b^2*d^2*n*(e*x+d)*ln(f*x^m)*ln(c*(e*x+d)^n)/e^3+1/3*b*d*n*x^2*ln(f*x^ 
m)*(a+b*ln(c*(e*x+d)^n))/e+2/9*a*b*d^2*m*n*x/e^2+1/9*b*d^2*m*n*(-11*b*n+6* 
a)*x/e^2+8/9*b^2*d^2*m*n*(e*x+d)*ln(c*(e*x+d)^n)/e^3-5/18*b*d*m*n*x^2*(a+b 
*ln(c*(e*x+d)^n))/e-2/3*b*d^3*m*n*(a+b*ln(c*(e*x+d)^n))*polylog(2,1+e*x/d) 
/e^3+2/3*b^2*d^3*m*n*ln(-e*x/d)*ln(c*(e*x+d)^n)/e^3-2/9*b*n*x^3*ln(f*x^m)* 
(a+b*ln(c*(e*x+d)^n))+4/27*b*m*n*x^3*(a+b*ln(c*(e*x+d)^n))-1/3*d^3*m*ln(-e 
*x/d)*(a+b*ln(c*(e*x+d)^n))^2/e^3-1/9*d^3*m*(a+b*ln(c*(e*x+d)^n))^2/e^3+1/ 
3*d^3*ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^2/e^3+2/27*b^2*n^2*x^3*ln(f*x^m)-2/2 
7*b^2*m*n^2*x^3-71/54*b^2*d^2*m*n^2*x/e^2+19/54*b^2*d*m*n^2*x^2/e+2/3*b^2* 
d^3*m*n^2*polylog(3,1+e*x/d)/e^3+11/9*b^2*d^3*m*n^2*polylog(2,1+e*x/d)/e^3 
+23/54*b^2*d^3*m*n^2*ln(e*x+d)/e^3+11/9*b^2*d^2*n^2*x*ln(f*x^m)/e^2-5/18*b 
^2*d*n^2*x^2*ln(f*x^m)/e-5/9*b^2*d^3*n^2*ln(f*x^m)*ln(e*x+d)/e^3
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 976, normalized size of antiderivative = 1.38 \[ \int x^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:

Integrate[x^2*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2,x]
 

Output:

(48*a*b*d^2*e*m*n*x - 137*b^2*d^2*e*m*n^2*x - 15*a*b*d*e^2*m*n*x^2 + 19*b^ 
2*d*e^2*m*n^2*x^2 - 6*a^2*e^3*m*x^3 + 8*a*b*e^3*m*n*x^3 - 4*b^2*e^3*m*n^2* 
x^3 - 36*a*b*d^2*e*n*x*Log[f*x^m] + 66*b^2*d^2*e*n^2*x*Log[f*x^m] + 18*a*b 
*d*e^2*n*x^2*Log[f*x^m] - 15*b^2*d*e^2*n^2*x^2*Log[f*x^m] + 18*a^2*e^3*x^3 
*Log[f*x^m] - 12*a*b*e^3*n*x^3*Log[f*x^m] + 4*b^2*e^3*n^2*x^3*Log[f*x^m] - 
 12*a*b*d^3*m*n*Log[d + e*x] + 71*b^2*d^3*m*n^2*Log[d + e*x] - 36*a*b*d^3* 
m*n*Log[x]*Log[d + e*x] + 66*b^2*d^3*m*n^2*Log[x]*Log[d + e*x] + 36*a*b*d^ 
3*n*Log[f*x^m]*Log[d + e*x] - 66*b^2*d^3*n^2*Log[f*x^m]*Log[d + e*x] + 6*b 
^2*d^3*m*n^2*Log[d + e*x]^2 + 36*b^2*d^3*m*n^2*Log[x]*Log[d + e*x]^2 - 18* 
b^2*d^3*m*n^2*Log[-((e*x)/d)]*Log[d + e*x]^2 - 18*b^2*d^3*n^2*Log[f*x^m]*L 
og[d + e*x]^2 + 48*b^2*d^2*e*m*n*x*Log[c*(d + e*x)^n] - 15*b^2*d*e^2*m*n*x 
^2*Log[c*(d + e*x)^n] - 12*a*b*e^3*m*x^3*Log[c*(d + e*x)^n] + 8*b^2*e^3*m* 
n*x^3*Log[c*(d + e*x)^n] - 36*b^2*d^2*e*n*x*Log[f*x^m]*Log[c*(d + e*x)^n] 
+ 18*b^2*d*e^2*n*x^2*Log[f*x^m]*Log[c*(d + e*x)^n] + 36*a*b*e^3*x^3*Log[f* 
x^m]*Log[c*(d + e*x)^n] - 12*b^2*e^3*n*x^3*Log[f*x^m]*Log[c*(d + e*x)^n] - 
 12*b^2*d^3*m*n*Log[d + e*x]*Log[c*(d + e*x)^n] - 36*b^2*d^3*m*n*Log[x]*Lo 
g[d + e*x]*Log[c*(d + e*x)^n] + 36*b^2*d^3*n*Log[f*x^m]*Log[d + e*x]*Log[c 
*(d + e*x)^n] - 6*b^2*e^3*m*x^3*Log[c*(d + e*x)^n]^2 + 18*b^2*e^3*x^3*Log[ 
f*x^m]*Log[c*(d + e*x)^n]^2 + 36*a*b*d^3*m*n*Log[x]*Log[1 + (e*x)/d] - 66* 
b^2*d^3*m*n^2*Log[x]*Log[1 + (e*x)/d] - 36*b^2*d^3*m*n^2*Log[x]*Log[d +...
 

Rubi [A] (verified)

Time = 3.79 (sec) , antiderivative size = 845, normalized size of antiderivative = 1.20, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2875, 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.

\(\displaystyle \int x^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx\)

\(\Big \downarrow \) 2875

\(\displaystyle -m \int \left (-\frac {b^2 n^2 \log ^2(d+e x) d^3}{3 e^3 x}+\frac {2 b n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^3}{3 e^3 x}+\frac {2 b^2 n^2 d^2}{e^2}-\frac {2 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^2}{e^3 x}-\frac {b^2 n^2 (d+e x)^2 d}{2 e^3 x}+\frac {b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) d}{e^3 x}+\frac {2 b^2 n^2 (d+e x)^3}{27 e^3 x}+\frac {1}{3} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2-\frac {2 b n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3 x}\right )dx+\frac {2 b d^3 n \log (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^3}-\frac {2 b d^2 n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3}+\frac {b d n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3}-\frac {2 b n (d+e x)^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3}+\frac {1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-\frac {b^2 d^3 n^2 \log ^2(d+e x) \log \left (f x^m\right )}{3 e^3}+\frac {2 b^2 d^2 n^2 x \log \left (f x^m\right )}{e^2}-\frac {b^2 d n^2 (d+e x)^2 \log \left (f x^m\right )}{2 e^3}+\frac {2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^2 n^2 \log \left (f x^m\right ) \log ^2(d+e x) d^3}{3 e^3}+\frac {2 b n \log \left (f x^m\right ) \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^3}{3 e^3}+\frac {2 b^2 n^2 x \log \left (f x^m\right ) d^2}{e^2}-\frac {2 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^2}{e^3}-\frac {b^2 n^2 (d+e x)^2 \log \left (f x^m\right ) d}{2 e^3}+\frac {b n (d+e x)^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) d}{e^3}+\frac {1}{3} x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac {2 b^2 n^2 (d+e x)^3 \log \left (f x^m\right )}{27 e^3}-\frac {2 b n (d+e x)^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3}-m \left (-\frac {b^2 n^2 \log ^2(d+e x) d^3}{9 e^3}-\frac {b^2 n^2 \log (x) \log ^2(d+e x) d^3}{3 e^3}-\frac {\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 d^3}{3 e^3}+\frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 d^3}{3 e^3}-\frac {23 b^2 n^2 \log (x) d^3}{54 e^3}-\frac {11 b n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^3}{9 e^3}+\frac {2 b n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^3}{9 e^3}+\frac {2 b n \log (x) \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^3}{3 e^3}-\frac {11 b^2 n^2 \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right ) d^3}{9 e^3}+\frac {2 b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right ) d^3}{3 e^3}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,\frac {e x}{d}+1\right ) d^3}{3 e^3}+\frac {28 b^2 n^2 x d^2}{9 e^2}-\frac {11 a b n x d^2}{9 e^2}-\frac {11 b^2 n (d+e x) \log \left (c (d+e x)^n\right ) d^2}{9 e^3}-\frac {2 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^2}{3 e^3}-\frac {5 b^2 n^2 x^2 d}{36 e}-\frac {13 b^2 n^2 (d+e x)^2 d}{36 e^3}+\frac {13 b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) d}{18 e^3}+\frac {2}{81} b^2 n^2 x^3+\frac {4 b^2 n^2 (d+e x)^3}{81 e^3}+\frac {1}{9} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2-\frac {4 b n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{27 e^3}\right )\)

Input:

Int[x^2*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2,x]
 

Output:

(2*b^2*d^2*n^2*x*Log[f*x^m])/e^2 - (b^2*d*n^2*(d + e*x)^2*Log[f*x^m])/(2*e 
^3) + (2*b^2*n^2*(d + e*x)^3*Log[f*x^m])/(27*e^3) - (b^2*d^3*n^2*Log[f*x^m 
]*Log[d + e*x]^2)/(3*e^3) - (2*b*d^2*n*(d + e*x)*Log[f*x^m]*(a + b*Log[c*( 
d + e*x)^n]))/e^3 + (b*d*n*(d + e*x)^2*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n 
]))/e^3 - (2*b*n*(d + e*x)^3*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/(9*e^3 
) + (2*b*d^3*n*Log[f*x^m]*Log[d + e*x]*(a + b*Log[c*(d + e*x)^n]))/(3*e^3) 
 + (x^3*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/3 - m*((-11*a*b*d^2*n*x)/ 
(9*e^2) + (28*b^2*d^2*n^2*x)/(9*e^2) - (5*b^2*d*n^2*x^2)/(36*e) + (2*b^2*n 
^2*x^3)/81 - (13*b^2*d*n^2*(d + e*x)^2)/(36*e^3) + (4*b^2*n^2*(d + e*x)^3) 
/(81*e^3) - (23*b^2*d^3*n^2*Log[x])/(54*e^3) - (b^2*d^3*n^2*Log[d + e*x]^2 
)/(9*e^3) - (b^2*d^3*n^2*Log[x]*Log[d + e*x]^2)/(3*e^3) - (11*b^2*d^2*n*(d 
 + e*x)*Log[c*(d + e*x)^n])/(9*e^3) - (2*b*d^2*n*(d + e*x)*(a + b*Log[c*(d 
 + e*x)^n]))/(3*e^3) + (13*b*d*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/( 
18*e^3) - (4*b*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(27*e^3) - (11*b* 
d^3*n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/(9*e^3) + (2*b*d^3*n*Log 
[d + e*x]*(a + b*Log[c*(d + e*x)^n]))/(9*e^3) + (2*b*d^3*n*Log[x]*Log[d + 
e*x]*(a + b*Log[c*(d + e*x)^n]))/(3*e^3) + (x^3*(a + b*Log[c*(d + e*x)^n]) 
^2)/9 - (d^3*Log[x]*(a + b*Log[c*(d + e*x)^n])^2)/(3*e^3) + (d^3*Log[-((e* 
x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/(3*e^3) - (11*b^2*d^3*n^2*PolyLog[2, 
1 + (e*x)/d])/(9*e^3) + (2*b*d^3*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2...
 

Defintions of rubi rules used

rule 2009

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

rule 2875

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

Maple [F]

\[\int x^{2} \ln \left (f \,x^{m}\right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}d x\]

Input:

int(x^2*ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^2,x)
 

Output:

int(x^2*ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^2,x)
 

Fricas [F]

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

integrate(x^2*log(f*x^m)*(a+b*log(c*(e*x+d)^n))^2,x, algorithm="fricas")
 

Output:

integral(b^2*x^2*log((e*x + d)^n*c)^2*log(f*x^m) + 2*a*b*x^2*log((e*x + d) 
^n*c)*log(f*x^m) + a^2*x^2*log(f*x^m), x)
 

Sympy [F(-1)]

Timed out. \[ \int x^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\text {Timed out} \] Input:

integrate(x**2*ln(f*x**m)*(a+b*ln(c*(e*x+d)**n))**2,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^2*log(f*x^m)*(a+b*log(c*(e*x+d)^n))^2,x, algorithm="maxima")
 

Output:

-1/9*(b^2*(m - 3*log(f))*x^3 - 3*b^2*x^3*log(x^m))*log((e*x + d)^n)^2 + in 
tegrate(1/9*(9*(b^2*e*log(c)^2*log(f) + 2*a*b*e*log(c)*log(f) + a^2*e*log( 
f))*x^3 + 9*(b^2*d*log(c)^2*log(f) + 2*a*b*d*log(c)*log(f) + a^2*d*log(f)) 
*x^2 + 2*((9*a*b*e*log(f) + (9*e*log(c)*log(f) + (m*n - 3*n*log(f))*e)*b^2 
)*x^3 + 9*(b^2*d*log(c)*log(f) + a*b*d*log(f))*x^2 - 3*(((e*n - 3*e*log(c) 
)*b^2 - 3*a*b*e)*x^3 - 3*(b^2*d*log(c) + a*b*d)*x^2)*log(x^m))*log((e*x + 
d)^n) + 9*((b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x^3 + (b^2*d*log(c)^2 
 + 2*a*b*d*log(c) + a^2*d)*x^2)*log(x^m))/(e*x + d), x)
 

Giac [F]

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

integrate(x^2*log(f*x^m)*(a+b*log(c*(e*x+d)^n))^2,x, algorithm="giac")
 

Output:

integrate((b*log((e*x + d)^n*c) + a)^2*x^2*log(f*x^m), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int x^2\,\ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2 \,d x \] Input:

int(x^2*log(f*x^m)*(a + b*log(c*(d + e*x)^n))^2,x)
 

Output:

int(x^2*log(f*x^m)*(a + b*log(c*(d + e*x)^n))^2, x)
 

Reduce [F]

\[ \int x^2 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\frac {-36 \left (\int \frac {\mathrm {log}\left (x^{m} f \right )}{e \,x^{2}+d x}d x \right ) a b \,d^{4} m n +54 \left (\int \mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} \mathrm {log}\left (x^{m} f \right ) x^{2}d x \right ) b^{2} e^{3} m +36 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) \mathrm {log}\left (x^{m} f \right ) a b \,e^{3} m \,x^{3}-12 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) a b \,d^{3} m^{2}-12 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) a b \,e^{3} m^{2} x^{3}+18 \mathrm {log}\left (x^{m} f \right )^{2} a b \,d^{3} n +18 \,\mathrm {log}\left (x^{m} f \right ) a^{2} e^{3} m \,x^{3}-36 \,\mathrm {log}\left (x^{m} f \right ) a b \,d^{2} e m n x +18 \,\mathrm {log}\left (x^{m} f \right ) a b d \,e^{2} m n \,x^{2}-12 \,\mathrm {log}\left (x^{m} f \right ) a b \,e^{3} m n \,x^{3}-6 a^{2} e^{3} m^{2} x^{3}+48 a b \,d^{2} e \,m^{2} n x -15 a b d \,e^{2} m^{2} n \,x^{2}+8 a b \,e^{3} m^{2} n \,x^{3}}{54 e^{3} m} \] Input:

int(x^2*log(f*x^m)*(a+b*log(c*(e*x+d)^n))^2,x)
 

Output:

( - 36*int(log(x**m*f)/(d*x + e*x**2),x)*a*b*d**4*m*n + 54*int(log((d + e* 
x)**n*c)**2*log(x**m*f)*x**2,x)*b**2*e**3*m + 36*log((d + e*x)**n*c)*log(x 
**m*f)*a*b*e**3*m*x**3 - 12*log((d + e*x)**n*c)*a*b*d**3*m**2 - 12*log((d 
+ e*x)**n*c)*a*b*e**3*m**2*x**3 + 18*log(x**m*f)**2*a*b*d**3*n + 18*log(x* 
*m*f)*a**2*e**3*m*x**3 - 36*log(x**m*f)*a*b*d**2*e*m*n*x + 18*log(x**m*f)* 
a*b*d*e**2*m*n*x**2 - 12*log(x**m*f)*a*b*e**3*m*n*x**3 - 6*a**2*e**3*m**2* 
x**3 + 48*a*b*d**2*e*m**2*n*x - 15*a*b*d*e**2*m**2*n*x**2 + 8*a*b*e**3*m** 
2*n*x**3)/(54*e**3*m)