\(\int x^2 (a+b \log (c x^n))^2 \log (d (e+f x^2)^m) \, dx\) [110]

Optimal result

Integrand size = 28, antiderivative size = 576 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=-\frac {16 a b e m n x}{9 f}+\frac {52 b^2 e m n^2 x}{27 f}-\frac {4}{27} b^2 m n^2 x^3-\frac {4 b^2 e^{3/2} m n^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}-\frac {16 b^2 e m n x \log \left (c x^n\right )}{9 f}+\frac {8}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b e^{3/2} m n \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {2 e^{3/2} m \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^{3/2}}+\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2 b^2 (-e)^{3/2} m n^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{9 f^{3/2}}+\frac {2 b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {2 b^2 (-e)^{3/2} m n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{9 f^{3/2}}-\frac {2 b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {2 b^2 (-e)^{3/2} m n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {2 b^2 (-e)^{3/2} m n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}} \] Output:

-16/9*a*b*e*m*n*x/f+52/27*b^2*e*m*n^2*x/f-4/27*b^2*m*n^2*x^3-4/27*b^2*e^(3 
/2)*m*n^2*arctan(f^(1/2)*x/e^(1/2))/f^(3/2)-16/9*b^2*e*m*n*x*ln(c*x^n)/f+8 
/27*b*m*n*x^3*(a+b*ln(c*x^n))+4/9*b*e^(3/2)*m*n*arctan(f^(1/2)*x/e^(1/2))* 
(a+b*ln(c*x^n))/f^(3/2)+2/3*e*m*x*(a+b*ln(c*x^n))^2/f-2/9*m*x^3*(a+b*ln(c* 
x^n))^2-2/3*e^(3/2)*m*arctan(f^(1/2)*x/e^(1/2))*(a+b*ln(c*x^n))^2/f^(3/2)+ 
2/27*b^2*n^2*x^3*ln(d*(f*x^2+e)^m)-2/9*b*n*x^3*(a+b*ln(c*x^n))*ln(d*(f*x^2 
+e)^m)+1/3*x^3*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m)-2/9*b^2*(-e)^(3/2)*m*n^ 
2*polylog(2,-f^(1/2)*x/(-e)^(1/2))/f^(3/2)+2/3*b*(-e)^(3/2)*m*n*(a+b*ln(c* 
x^n))*polylog(2,-f^(1/2)*x/(-e)^(1/2))/f^(3/2)+2/9*b^2*(-e)^(3/2)*m*n^2*po 
lylog(2,f^(1/2)*x/(-e)^(1/2))/f^(3/2)-2/3*b*(-e)^(3/2)*m*n*(a+b*ln(c*x^n)) 
*polylog(2,f^(1/2)*x/(-e)^(1/2))/f^(3/2)-2/3*b^2*(-e)^(3/2)*m*n^2*polylog( 
3,-f^(1/2)*x/(-e)^(1/2))/f^(3/2)+2/3*b^2*(-e)^(3/2)*m*n^2*polylog(3,f^(1/2 
)*x/(-e)^(1/2))/f^(3/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 1128, normalized size of antiderivative = 1.96 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(18*a^2*e*Sqrt[f]*m*x - 48*a*b*e*Sqrt[f]*m*n*x + 52*b^2*e*Sqrt[f]*m*n^2*x 
- 6*a^2*f^(3/2)*m*x^3 + 8*a*b*f^(3/2)*m*n*x^3 - 4*b^2*f^(3/2)*m*n^2*x^3 - 
18*a^2*e^(3/2)*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 12*a*b*e^(3/2)*m*n*ArcTan[( 
Sqrt[f]*x)/Sqrt[e]] - 4*b^2*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 36 
*a*b*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 12*b^2*e^(3/2)*m*n^2 
*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 18*b^2*e^(3/2)*m*n^2*ArcTan[(Sqrt[f] 
*x)/Sqrt[e]]*Log[x]^2 + 36*a*b*e*Sqrt[f]*m*x*Log[c*x^n] - 48*b^2*e*Sqrt[f] 
*m*n*x*Log[c*x^n] - 12*a*b*f^(3/2)*m*x^3*Log[c*x^n] + 8*b^2*f^(3/2)*m*n*x^ 
3*Log[c*x^n] - 36*a*b*e^(3/2)*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 1 
2*b^2*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 36*b^2*e^(3/2)* 
m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] + 18*b^2*e*Sqrt[f]*m*x*L 
og[c*x^n]^2 - 6*b^2*f^(3/2)*m*x^3*Log[c*x^n]^2 - 18*b^2*e^(3/2)*m*ArcTan[( 
Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 - (18*I)*a*b*e^(3/2)*m*n*Log[x]*Log[1 - ( 
I*Sqrt[f]*x)/Sqrt[e]] + (6*I)*b^2*e^(3/2)*m*n^2*Log[x]*Log[1 - (I*Sqrt[f]* 
x)/Sqrt[e]] + (9*I)*b^2*e^(3/2)*m*n^2*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[ 
e]] - (18*I)*b^2*e^(3/2)*m*n*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[ 
e]] + (18*I)*a*b*e^(3/2)*m*n*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (6*I) 
*b^2*e^(3/2)*m*n^2*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (9*I)*b^2*e^(3/ 
2)*m*n^2*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (18*I)*b^2*e^(3/2)*m*n* 
Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 9*a^2*f^(3/2)*x^3*Lo...
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.24 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.09, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2825, 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.

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

\(\Big \downarrow \) 2825

\(\displaystyle -2 f m \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2 x^4}{3 \left (f x^2+e\right )}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) x^4}{9 \left (f x^2+e\right )}+\frac {2 b^2 n^2 x^4}{27 \left (f x^2+e\right )}\right )dx+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 f m \left (-\frac {2 b e^{3/2} n \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{5/2}}-\frac {b (-e)^{3/2} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{5/2}}+\frac {b (-e)^{3/2} n \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{5/2}}+\frac {(-e)^{3/2} \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{6 f^{5/2}}-\frac {(-e)^{3/2} \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{6 f^{5/2}}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{9 f}-\frac {4 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{27 f}+\frac {8 a b e n x}{9 f^2}+\frac {2 b^2 e^{3/2} n^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{5/2}}+\frac {8 b^2 e n x \log \left (c x^n\right )}{9 f^2}+\frac {i b^2 e^{3/2} n^2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{5/2}}-\frac {i b^2 e^{3/2} n^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{5/2}}+\frac {b^2 (-e)^{3/2} n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{5/2}}-\frac {b^2 (-e)^{3/2} n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{5/2}}-\frac {26 b^2 e n^2 x}{27 f^2}+\frac {2 b^2 n^2 x^3}{27 f}\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )\)

Input:

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

Output:

(2*b^2*n^2*x^3*Log[d*(e + f*x^2)^m])/27 - (2*b*n*x^3*(a + b*Log[c*x^n])*Lo 
g[d*(e + f*x^2)^m])/9 + (x^3*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/3 
- 2*f*m*((8*a*b*e*n*x)/(9*f^2) - (26*b^2*e*n^2*x)/(27*f^2) + (2*b^2*n^2*x^ 
3)/(27*f) + (2*b^2*e^(3/2)*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(27*f^(5/2)) + 
 (8*b^2*e*n*x*Log[c*x^n])/(9*f^2) - (4*b*n*x^3*(a + b*Log[c*x^n]))/(27*f) 
- (2*b*e^(3/2)*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(9*f^(5/2 
)) - (e*x*(a + b*Log[c*x^n])^2)/(3*f^2) + (x^3*(a + b*Log[c*x^n])^2)/(9*f) 
 + ((-e)^(3/2)*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(6*f^(5 
/2)) - ((-e)^(3/2)*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(6* 
f^(5/2)) - (b*(-e)^(3/2)*n*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqr 
t[-e])])/(3*f^(5/2)) + (b*(-e)^(3/2)*n*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt 
[f]*x)/Sqrt[-e]])/(3*f^(5/2)) + ((I/9)*b^2*e^(3/2)*n^2*PolyLog[2, ((-I)*Sq 
rt[f]*x)/Sqrt[e]])/f^(5/2) - ((I/9)*b^2*e^(3/2)*n^2*PolyLog[2, (I*Sqrt[f]* 
x)/Sqrt[e]])/f^(5/2) + (b^2*(-e)^(3/2)*n^2*PolyLog[3, -((Sqrt[f]*x)/Sqrt[- 
e])])/(3*f^(5/2)) - (b^2*(-e)^(3/2)*n^2*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/ 
(3*f^(5/2)))
 

Defintions of rubi rules used

rule 2009

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

rule 2825

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\frac {-18 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a^{2} e m +12 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a b e m n -4 \sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) b^{2} e m \,n^{2}-18 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{f \,x^{2}+e}d x \right ) b^{2} e^{2} f m -36 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{f \,x^{2}+e}d x \right ) a b \,e^{2} f m +12 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{f \,x^{2}+e}d x \right ) b^{2} e^{2} f m n +9 \,\mathrm {log}\left (\left (f \,x^{2}+e \right )^{m} d \right ) \mathrm {log}\left (x^{n} c \right )^{2} b^{2} f^{2} x^{3}+18 \,\mathrm {log}\left (\left (f \,x^{2}+e \right )^{m} d \right ) \mathrm {log}\left (x^{n} c \right ) a b \,f^{2} x^{3}-6 \,\mathrm {log}\left (\left (f \,x^{2}+e \right )^{m} d \right ) \mathrm {log}\left (x^{n} c \right ) b^{2} f^{2} n \,x^{3}+9 \,\mathrm {log}\left (\left (f \,x^{2}+e \right )^{m} d \right ) a^{2} f^{2} x^{3}-6 \,\mathrm {log}\left (\left (f \,x^{2}+e \right )^{m} d \right ) a b \,f^{2} n \,x^{3}+2 \,\mathrm {log}\left (\left (f \,x^{2}+e \right )^{m} d \right ) b^{2} f^{2} n^{2} x^{3}+18 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} e f m x -6 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} f^{2} m \,x^{3}+36 \,\mathrm {log}\left (x^{n} c \right ) a b e f m x -12 \,\mathrm {log}\left (x^{n} c \right ) a b \,f^{2} m \,x^{3}-48 \,\mathrm {log}\left (x^{n} c \right ) b^{2} e f m n x +8 \,\mathrm {log}\left (x^{n} c \right ) b^{2} f^{2} m n \,x^{3}+18 a^{2} e f m x -6 a^{2} f^{2} m \,x^{3}-48 a b e f m n x +8 a b \,f^{2} m n \,x^{3}+52 b^{2} e f m \,n^{2} x -4 b^{2} f^{2} m \,n^{2} x^{3}}{27 f^{2}} \] Input:

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

Output:

( - 18*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*e*m + 12*sqrt(f) 
*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*e*m*n - 4*sqrt(f)*sqrt(e)*atan( 
(f*x)/(sqrt(f)*sqrt(e)))*b**2*e*m*n**2 - 18*int(log(x**n*c)**2/(e + f*x**2 
),x)*b**2*e**2*f*m - 36*int(log(x**n*c)/(e + f*x**2),x)*a*b*e**2*f*m + 12* 
int(log(x**n*c)/(e + f*x**2),x)*b**2*e**2*f*m*n + 9*log((e + f*x**2)**m*d) 
*log(x**n*c)**2*b**2*f**2*x**3 + 18*log((e + f*x**2)**m*d)*log(x**n*c)*a*b 
*f**2*x**3 - 6*log((e + f*x**2)**m*d)*log(x**n*c)*b**2*f**2*n*x**3 + 9*log 
((e + f*x**2)**m*d)*a**2*f**2*x**3 - 6*log((e + f*x**2)**m*d)*a*b*f**2*n*x 
**3 + 2*log((e + f*x**2)**m*d)*b**2*f**2*n**2*x**3 + 18*log(x**n*c)**2*b** 
2*e*f*m*x - 6*log(x**n*c)**2*b**2*f**2*m*x**3 + 36*log(x**n*c)*a*b*e*f*m*x 
 - 12*log(x**n*c)*a*b*f**2*m*x**3 - 48*log(x**n*c)*b**2*e*f*m*n*x + 8*log( 
x**n*c)*b**2*f**2*m*n*x**3 + 18*a**2*e*f*m*x - 6*a**2*f**2*m*x**3 - 48*a*b 
*e*f*m*n*x + 8*a*b*f**2*m*n*x**3 + 52*b**2*e*f*m*n**2*x - 4*b**2*f**2*m*n* 
*2*x**3)/(27*f**2)