\(\int \frac {(a+b \log (c (d+e x)^n))^2}{(f+g x)^3} \, dx\) [50]

Optimal result

Integrand size = 24, antiderivative size = 202 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^3} \, dx=-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 (f+g x)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (f+g x)^2}+\frac {b^2 e^2 n^2 \log (f+g x)}{g (e f-d g)^2}-\frac {b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1+\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^2}+\frac {b^2 e^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^3} \, dx=\frac {-\left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac {e (f+g x) \left (2 b (e f-d g) n \left (a+b \log \left (c (d+e x)^n\right )\right )+e (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b^2 e n^2 (f+g x) (\log (d+e x)-\log (f+g x))-2 b e n (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )-2 b^2 e n^2 (f+g x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )}{(e f-d g)^2}}{2 g (f+g x)^2} \] Input:

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

Output:

(-(a + b*Log[c*(d + e*x)^n])^2 + (e*(f + g*x)*(2*b*(e*f - d*g)*n*(a + b*Lo 
g[c*(d + e*x)^n]) + e*(f + g*x)*(a + b*Log[c*(d + e*x)^n])^2 - 2*b^2*e*n^2 
*(f + g*x)*(Log[d + e*x] - Log[f + g*x]) - 2*b*e*n*(f + g*x)*(a + b*Log[c* 
(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)] - 2*b^2*e*n^2*(f + g*x)*PolyL 
og[2, (g*(d + e*x))/(-(e*f) + d*g)]))/(e*f - d*g)^2)/(2*g*(f + g*x)^2)
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2845, 2858, 27, 2789, 2751, 16, 2779, 2838}

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.

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

\(\Big \downarrow \) 2845

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

\(\Big \downarrow \) 2858

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2751

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x 
_Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* 
(n/d)   Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, 
x] && EqQ[r*(q + 1) + 1, 0]
 

rule 2779

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

rule 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2845

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. 
)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ 
n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1)))   Int[(f + g*x)^(q + 1) 
*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, 
d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In 
tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
 

rule 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

Maple [C] (warning: unable to verify)

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

Time = 2.58 (sec) , antiderivative size = 661, normalized size of antiderivative = 3.27

method	result	size
risch	\(-\frac {b^{2} \ln \left (\left (e x +d \right )^{n}\right )^{2}}{2 \left (g x +f \right )^{2} g}-\frac {b^{2} n e \ln \left (\left (e x +d \right )^{n}\right )}{g \left (d g -e f \right ) \left (g x +f \right )}-\frac {b^{2} n \,e^{2} \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{g \left (d g -e f \right )^{2}}+\frac {b^{2} n \,e^{2} \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (e x +d \right )}{g \left (d g -e f \right )^{2}}-\frac {b^{2} n^{2} e^{2} \ln \left (e x +d \right )^{2}}{2 g \left (d g -e f \right )^{2}}+\frac {b^{2} n^{2} e^{2} \ln \left (g x +f \right )}{g \left (d g -e f \right )^{2}}-\frac {b^{2} n^{2} e^{2} \ln \left (e x +d \right )}{g \left (d g -e f \right )^{2}}+\frac {b^{2} n^{2} e^{2} \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{g \left (d g -e f \right )^{2}}+\frac {b^{2} n^{2} e^{2} \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{g \left (d g -e f \right )^{2}}+\left (i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 a \right ) b \left (-\frac {\ln \left (\left (e x +d \right )^{n}\right )}{2 \left (g x +f \right )^{2} g}+\frac {n e \left (-\frac {1}{\left (d g -e f \right ) \left (g x +f \right )}-\frac {e \ln \left (g x +f \right )}{\left (d g -e f \right )^{2}}+\frac {e \ln \left (e x +d \right )}{\left (d g -e f \right )^{2}}\right )}{2 g}\right )-\frac {{\left (i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 a \right )}^{2}}{8 \left (g x +f \right )^{2} g}\)	\(661\)

Input:

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

Output:

-1/2*b^2*ln((e*x+d)^n)^2/(g*x+f)^2/g-b^2/g*n*e*ln((e*x+d)^n)/(d*g-e*f)/(g* 
x+f)-b^2/g*n*e^2*ln((e*x+d)^n)/(d*g-e*f)^2*ln(g*x+f)+b^2/g*n*e^2*ln((e*x+d 
)^n)/(d*g-e*f)^2*ln(e*x+d)-1/2*b^2/g*n^2*e^2/(d*g-e*f)^2*ln(e*x+d)^2+b^2/g 
*n^2*e^2/(d*g-e*f)^2*ln(g*x+f)-b^2/g*n^2*e^2/(d*g-e*f)^2*ln(e*x+d)+b^2/g*n 
^2*e^2/(d*g-e*f)^2*dilog(((g*x+f)*e+d*g-e*f)/(d*g-e*f))+b^2/g*n^2*e^2/(d*g 
-e*f)^2*ln(g*x+f)*ln(((g*x+f)*e+d*g-e*f)/(d*g-e*f))+(I*b*Pi*csgn(I*(e*x+d) 
^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csg 
n(I*c)-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+I*b*Pi*csgn(I*c*(e*x+d)^n)^2*csgn(I*c) 
+2*b*ln(c)+2*a)*b*(-1/2*ln((e*x+d)^n)/(g*x+f)^2/g+1/2/g*n*e*(-1/(d*g-e*f)/ 
(g*x+f)-e/(d*g-e*f)^2*ln(g*x+f)+e/(d*g-e*f)^2*ln(e*x+d)))-1/8*(I*b*Pi*csgn 
(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x 
+d)^n)*csgn(I*c)-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+I*b*Pi*csgn(I*c*(e*x+d)^n)^2 
*csgn(I*c)+2*b*ln(c)+2*a)^2/(g*x+f)^2/g
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(4*int((log((d + e*x)**n*c)*x)/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 + 
d*g**3*x**3 + e*f**3*x + 3*e*f**2*g*x**2 + 3*e*f*g**2*x**3 + e*g**3*x**4), 
x)*b**2*d**3*e*f**4*g**4*n + 8*int((log((d + e*x)**n*c)*x)/(d*f**3 + 3*d*f 
**2*g*x + 3*d*f*g**2*x**2 + d*g**3*x**3 + e*f**3*x + 3*e*f**2*g*x**2 + 3*e 
*f*g**2*x**3 + e*g**3*x**4),x)*b**2*d**3*e*f**3*g**5*n*x + 4*int((log((d + 
 e*x)**n*c)*x)/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 + d*g**3*x**3 + e* 
f**3*x + 3*e*f**2*g*x**2 + 3*e*f*g**2*x**3 + e*g**3*x**4),x)*b**2*d**3*e*f 
**2*g**6*n*x**2 - 12*int((log((d + e*x)**n*c)*x)/(d*f**3 + 3*d*f**2*g*x + 
3*d*f*g**2*x**2 + d*g**3*x**3 + e*f**3*x + 3*e*f**2*g*x**2 + 3*e*f*g**2*x* 
*3 + e*g**3*x**4),x)*b**2*d**2*e**2*f**5*g**3*n - 24*int((log((d + e*x)**n 
*c)*x)/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 + d*g**3*x**3 + e*f**3*x + 
 3*e*f**2*g*x**2 + 3*e*f*g**2*x**3 + e*g**3*x**4),x)*b**2*d**2*e**2*f**4*g 
**4*n*x - 12*int((log((d + e*x)**n*c)*x)/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g* 
*2*x**2 + d*g**3*x**3 + e*f**3*x + 3*e*f**2*g*x**2 + 3*e*f*g**2*x**3 + e*g 
**3*x**4),x)*b**2*d**2*e**2*f**3*g**5*n*x**2 + 12*int((log((d + e*x)**n*c) 
*x)/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 + d*g**3*x**3 + e*f**3*x + 3* 
e*f**2*g*x**2 + 3*e*f*g**2*x**3 + e*g**3*x**4),x)*b**2*d*e**3*f**6*g**2*n 
+ 24*int((log((d + e*x)**n*c)*x)/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 
+ d*g**3*x**3 + e*f**3*x + 3*e*f**2*g*x**2 + 3*e*f*g**2*x**3 + e*g**3*x**4 
),x)*b**2*d*e**3*f**5*g**3*n*x + 12*int((log((d + e*x)**n*c)*x)/(d*f**3...