\(\int \frac {(f+g x)^3}{(a+b \log (c (d+e x)^n))^2} \, dx\) [94]

Optimal result

Integrand size = 24, antiderivative size = 339 \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {e^{-\frac {a}{b n}} (e f-d g)^3 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {6 e^{-\frac {2 a}{b n}} g (e f-d g)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {9 e^{-\frac {3 a}{b n}} g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {4 e^{-\frac {4 a}{b n}} g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \operatorname {ExpIntegralEi}\left (\frac {4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}-\frac {(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \] Output:

(-d*g+e*f)^3*(e*x+d)*Ei((a+b*ln(c*(e*x+d)^n))/b/n)/b^2/e^4/exp(a/b/n)/n^2/ 
((c*(e*x+d)^n)^(1/n))+6*g*(-d*g+e*f)^2*(e*x+d)^2*Ei(2*(a+b*ln(c*(e*x+d)^n) 
)/b/n)/b^2/e^4/exp(2*a/b/n)/n^2/((c*(e*x+d)^n)^(2/n))+9*g^2*(-d*g+e*f)*(e* 
x+d)^3*Ei(3*(a+b*ln(c*(e*x+d)^n))/b/n)/b^2/e^4/exp(3*a/b/n)/n^2/((c*(e*x+d 
)^n)^(3/n))+4*g^3*(e*x+d)^4*Ei(4*(a+b*ln(c*(e*x+d)^n))/b/n)/b^2/e^4/exp(4* 
a/b/n)/n^2/((c*(e*x+d)^n)^(4/n))-(e*x+d)*(g*x+f)^3/b/e/n/(a+b*ln(c*(e*x+d) 
^n))
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1674\) vs. \(2(339)=678\).

Time = 0.80 (sec) , antiderivative size = 1674, normalized size of antiderivative = 4.94 \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx =\text {Too large to display} \] Input:

Integrate[(f + g*x)^3/(a + b*Log[c*(d + e*x)^n])^2,x]
 

Output:

(-(b*d*e^3*E^((4*a)/(b*n))*f^3*n*(c*(d + e*x)^n)^(4/n)) - b*e^4*E^((4*a)/( 
b*n))*f^3*n*x*(c*(d + e*x)^n)^(4/n) - 3*b*d*e^3*E^((4*a)/(b*n))*f^2*g*n*x* 
(c*(d + e*x)^n)^(4/n) - 3*b*e^4*E^((4*a)/(b*n))*f^2*g*n*x^2*(c*(d + e*x)^n 
)^(4/n) - 3*b*d*e^3*E^((4*a)/(b*n))*f*g^2*n*x^2*(c*(d + e*x)^n)^(4/n) - 3* 
b*e^4*E^((4*a)/(b*n))*f*g^2*n*x^3*(c*(d + e*x)^n)^(4/n) - b*d*e^3*E^((4*a) 
/(b*n))*g^3*n*x^3*(c*(d + e*x)^n)^(4/n) - b*e^4*E^((4*a)/(b*n))*g^3*n*x^4* 
(c*(d + e*x)^n)^(4/n) + a*e^3*E^((3*a)/(b*n))*f^3*(d + e*x)*(c*(d + e*x)^n 
)^(3/n)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)] - 3*a*d*e^2*E^((3* 
a)/(b*n))*f^2*g*(d + e*x)*(c*(d + e*x)^n)^(3/n)*ExpIntegralEi[(a + b*Log[c 
*(d + e*x)^n])/(b*n)] + 3*a*d^2*e*E^((3*a)/(b*n))*f*g^2*(d + e*x)*(c*(d + 
e*x)^n)^(3/n)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)] - a*d^3*E^(( 
3*a)/(b*n))*g^3*(d + e*x)*(c*(d + e*x)^n)^(3/n)*ExpIntegralEi[(a + b*Log[c 
*(d + e*x)^n])/(b*n)] + 6*a*e^2*E^((2*a)/(b*n))*f^2*g*(d + e*x)^2*(c*(d + 
e*x)^n)^(2/n)*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)] - 12*a*d 
*e*E^((2*a)/(b*n))*f*g^2*(d + e*x)^2*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[( 
2*(a + b*Log[c*(d + e*x)^n]))/(b*n)] + 6*a*d^2*E^((2*a)/(b*n))*g^3*(d + e* 
x)^2*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b 
*n)] + 9*a*e*E^(a/(b*n))*f*g^2*(d + e*x)^3*(c*(d + e*x)^n)^n^(-1)*ExpInteg 
ralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n)] - 9*a*d*E^(a/(b*n))*g^3*(d + e 
*x)^3*(c*(d + e*x)^n)^n^(-1)*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]...
 

Rubi [A] (verified)

Time = 2.13 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.73, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2847, 2846, 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.

\(\displaystyle \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx\)

\(\Big \downarrow \) 2847

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

\(\Big \downarrow \) 2846

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

\(\Big \downarrow \) 2009

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

Input:

Int[(f + g*x)^3/(a + b*Log[c*(d + e*x)^n])^2,x]
 

Output:

(-3*(e*f - d*g)*(((e*f - d*g)^2*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + 
e*x)^n])/(b*n)])/(b*e^3*E^(a/(b*n))*n*(c*(d + e*x)^n)^n^(-1)) + (2*g*(e*f 
- d*g)*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(b 
*e^3*E^((2*a)/(b*n))*n*(c*(d + e*x)^n)^(2/n)) + (g^2*(d + e*x)^3*ExpIntegr 
alEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(b*e^3*E^((3*a)/(b*n))*n*(c*(d 
 + e*x)^n)^(3/n))))/(b*e*n) + (4*(((e*f - d*g)^3*(d + e*x)*ExpIntegralEi[( 
a + b*Log[c*(d + e*x)^n])/(b*n)])/(b*e^4*E^(a/(b*n))*n*(c*(d + e*x)^n)^n^( 
-1)) + (3*g*(e*f - d*g)^2*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e 
*x)^n]))/(b*n)])/(b*e^4*E^((2*a)/(b*n))*n*(c*(d + e*x)^n)^(2/n)) + (3*g^2* 
(e*f - d*g)*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n) 
])/(b*e^4*E^((3*a)/(b*n))*n*(c*(d + e*x)^n)^(3/n)) + (g^3*(d + e*x)^4*ExpI 
ntegralEi[(4*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(b*e^4*E^((4*a)/(b*n))*n* 
(c*(d + e*x)^n)^(4/n))))/(b*n) - ((d + e*x)*(f + g*x)^3)/(b*e*n*(a + b*Log 
[c*(d + e*x)^n]))
 

Defintions of rubi rules used

rule 2009

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

rule 2846

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

rule 2847

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

Maple [C] (warning: unable to verify)

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

Time = 3.34 (sec) , antiderivative size = 9517, normalized size of antiderivative = 28.07

method	result	size
risch	\(\text {Expression too large to display}\)	\(9517\)

Input:

int((g*x+f)^3/(a+b*ln(c*(e*x+d)^n))^2,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 681, normalized size of antiderivative = 2.01 \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx =\text {Too large to display} \] Input:

integrate((g*x+f)^3/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="fricas")
 

Output:

(9*(a*e*f*g^2 - a*d*g^3 + (b*e*f*g^2 - b*d*g^3)*n*log(e*x + d) + (b*e*f*g^ 
2 - b*d*g^3)*log(c))*e^((b*log(c) + a)/(b*n))*log_integral((e^3*x^3 + 3*d* 
e^2*x^2 + 3*d^2*e*x + d^3)*e^(3*(b*log(c) + a)/(b*n))) + 6*(a*e^2*f^2*g - 
2*a*d*e*f*g^2 + a*d^2*g^3 + (b*e^2*f^2*g - 2*b*d*e*f*g^2 + b*d^2*g^3)*n*lo 
g(e*x + d) + (b*e^2*f^2*g - 2*b*d*e*f*g^2 + b*d^2*g^3)*log(c))*e^(2*(b*log 
(c) + a)/(b*n))*log_integral((e^2*x^2 + 2*d*e*x + d^2)*e^(2*(b*log(c) + a) 
/(b*n))) + (a*e^3*f^3 - 3*a*d*e^2*f^2*g + 3*a*d^2*e*f*g^2 - a*d^3*g^3 + (b 
*e^3*f^3 - 3*b*d*e^2*f^2*g + 3*b*d^2*e*f*g^2 - b*d^3*g^3)*n*log(e*x + d) + 
 (b*e^3*f^3 - 3*b*d*e^2*f^2*g + 3*b*d^2*e*f*g^2 - b*d^3*g^3)*log(c))*e^(3* 
(b*log(c) + a)/(b*n))*log_integral((e*x + d)*e^((b*log(c) + a)/(b*n))) - ( 
b*e^4*g^3*n*x^4 + b*d*e^3*f^3*n + (3*b*e^4*f*g^2 + b*d*e^3*g^3)*n*x^3 + 3* 
(b*e^4*f^2*g + b*d*e^3*f*g^2)*n*x^2 + (b*e^4*f^3 + 3*b*d*e^3*f^2*g)*n*x)*e 
^(4*(b*log(c) + a)/(b*n)) + 4*(b*g^3*n*log(e*x + d) + b*g^3*log(c) + a*g^3 
)*log_integral((e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4)*e 
^(4*(b*log(c) + a)/(b*n))))*e^(-4*(b*log(c) + a)/(b*n))/(b^3*e^4*n^3*log(e 
*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2)
 

Sympy [F]

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

integrate((g*x+f)**3/(a+b*ln(c*(e*x+d)**n))**2,x)
 

Output:

Integral((f + g*x)**3/(a + b*log(c*(d + e*x)**n))**2, x)
 

Maxima [F]

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

integrate((g*x+f)^3/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="maxima")
 

Output:

-(e*g^3*x^4 + d*f^3 + (3*e*f*g^2 + d*g^3)*x^3 + 3*(e*f^2*g + d*f*g^2)*x^2 
+ (e*f^3 + 3*d*f^2*g)*x)/(b^2*e*n*log((e*x + d)^n) + b^2*e*n*log(c) + a*b* 
e*n) + integrate((4*e*g^3*x^3 + e*f^3 + 3*d*f^2*g + 3*(3*e*f*g^2 + d*g^3)* 
x^2 + 6*(e*f^2*g + d*f*g^2)*x)/(b^2*e*n*log((e*x + d)^n) + b^2*e*n*log(c) 
+ a*b*e*n), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3473 vs. \(2 (341) = 682\).

Time = 0.21 (sec) , antiderivative size = 3473, normalized size of antiderivative = 10.24 \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^3/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="giac")
 

Output:

b*e^3*f^3*n*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(e*x + d 
)/((b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2)*c^(1/n) 
) - 3*b*d*e^2*f^2*g*n*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*l 
og(e*x + d)/((b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^ 
2)*c^(1/n)) + 3*b*d^2*e*f*g^2*n*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(- 
a/(b*n))*log(e*x + d)/((b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a* 
b^2*e^4*n^2)*c^(1/n)) - b*d^3*g^3*n*Ei(log(c)/n + a/(b*n) + log(e*x + d))* 
e^(-a/(b*n))*log(e*x + d)/((b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) 
+ a*b^2*e^4*n^2)*c^(1/n)) - (e*x + d)*b*e^3*f^3*n/(b^3*e^4*n^3*log(e*x + d 
) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2) - 3*(e*x + d)^2*b*e^2*f^2*g*n/(b^3 
*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2) + 3*(e*x + d)* 
b*d*e^2*f^2*g*n/(b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4 
*n^2) - 3*(e*x + d)^3*b*e*f*g^2*n/(b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2* 
log(c) + a*b^2*e^4*n^2) + 6*(e*x + d)^2*b*d*e*f*g^2*n/(b^3*e^4*n^3*log(e*x 
 + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2) - 3*(e*x + d)*b*d^2*e*f*g^2*n/ 
(b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2) - (e*x + d 
)^4*b*g^3*n/(b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2 
) + 3*(e*x + d)^3*b*d*g^3*n/(b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) 
 + a*b^2*e^4*n^2) - 3*(e*x + d)^2*b*d^2*g^3*n/(b^3*e^4*n^3*log(e*x + d) + 
b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2) + (e*x + d)*b*d^3*g^3*n/(b^3*e^4*n^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {{\left (f+g\,x\right )}^3}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \] Input:

int((f + g*x)^3/(a + b*log(c*(d + e*x)^n))^2,x)
 

Output:

int((f + g*x)^3/(a + b*log(c*(d + e*x)^n))^2, x)
 

Reduce [F]

\[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx =\text {Too large to display} \] Input:

int((g*x+f)^3/(a+b*log(c*(e*x+d)^n))^2,x)
 

Output:

(int(x**4/(log((d + e*x)**n*c)**2*b**2*d + log((d + e*x)**n*c)**2*b**2*e*x 
 + 2*log((d + e*x)**n*c)*a*b*d + 2*log((d + e*x)**n*c)*a*b*e*x + a**2*d + 
a**2*e*x),x)*log((d + e*x)**n*c)*a*b*e**2*g**3*n + int(x**4/(log((d + e*x) 
**n*c)**2*b**2*d + log((d + e*x)**n*c)**2*b**2*e*x + 2*log((d + e*x)**n*c) 
*a*b*d + 2*log((d + e*x)**n*c)*a*b*e*x + a**2*d + a**2*e*x),x)*a**2*e**2*g 
**3*n + int(x**3/(log((d + e*x)**n*c)**2*b**2*d + log((d + e*x)**n*c)**2*b 
**2*e*x + 2*log((d + e*x)**n*c)*a*b*d + 2*log((d + e*x)**n*c)*a*b*e*x + a* 
*2*d + a**2*e*x),x)*log((d + e*x)**n*c)*a*b*d*e*g**3*n + 3*int(x**3/(log(( 
d + e*x)**n*c)**2*b**2*d + log((d + e*x)**n*c)**2*b**2*e*x + 2*log((d + e* 
x)**n*c)*a*b*d + 2*log((d + e*x)**n*c)*a*b*e*x + a**2*d + a**2*e*x),x)*log 
((d + e*x)**n*c)*a*b*e**2*f*g**2*n + int(x**3/(log((d + e*x)**n*c)**2*b**2 
*d + log((d + e*x)**n*c)**2*b**2*e*x + 2*log((d + e*x)**n*c)*a*b*d + 2*log 
((d + e*x)**n*c)*a*b*e*x + a**2*d + a**2*e*x),x)*a**2*d*e*g**3*n + 3*int(x 
**3/(log((d + e*x)**n*c)**2*b**2*d + log((d + e*x)**n*c)**2*b**2*e*x + 2*l 
og((d + e*x)**n*c)*a*b*d + 2*log((d + e*x)**n*c)*a*b*e*x + a**2*d + a**2*e 
*x),x)*a**2*e**2*f*g**2*n + 3*int(x**2/(log((d + e*x)**n*c)**2*b**2*d + lo 
g((d + e*x)**n*c)**2*b**2*e*x + 2*log((d + e*x)**n*c)*a*b*d + 2*log((d + e 
*x)**n*c)*a*b*e*x + a**2*d + a**2*e*x),x)*log((d + e*x)**n*c)*a*b*d*e*f*g* 
*2*n + 3*int(x**2/(log((d + e*x)**n*c)**2*b**2*d + log((d + e*x)**n*c)**2* 
b**2*e*x + 2*log((d + e*x)**n*c)*a*b*d + 2*log((d + e*x)**n*c)*a*b*e*x ...