\(\int \frac {\log (d (e+f \sqrt {x})) (a+b \log (c x^n))^3}{x^2} \, dx\) [137]

Optimal result

Integrand size = 28, antiderivative size = 673 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=-\frac {90 b^3 f n^3}{e \sqrt {x}}+\frac {6 b^3 f^2 n^3 \log \left (e+f \sqrt {x}\right )}{e^2}-\frac {6 b^3 n^3 \log \left (d \left (e+f \sqrt {x}\right )\right )}{x}-\frac {12 b^3 f^2 n^3 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {3 b^3 f^2 n^3 \log (x)}{e^2}+\frac {3 b^3 f^2 n^3 \log ^2(x)}{2 e^2}-\frac {42 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {6 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {6 b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 b^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {9 b f n \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}-\frac {3 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {3 b f^2 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+\frac {f^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}-\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}-\frac {12 b^3 f^2 n^3 \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {12 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {6 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {24 b^3 f^2 n^3 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {24 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {48 b^3 f^2 n^3 \operatorname {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right )}{e^2} \] Output:

-90*b^3*f*n^3/e/x^(1/2)+6*b^3*f^2*n^3*ln(e+f*x^(1/2))/e^2-6*b^3*n^3*ln(d*( 
e+f*x^(1/2)))/x-12*b^3*f^2*n^3*ln(e+f*x^(1/2))*ln(-f*x^(1/2)/e)/e^2-3*b^3* 
f^2*n^3*ln(x)/e^2+3/2*b^3*f^2*n^3*ln(x)^2/e^2-42*b^2*f*n^2*(a+b*ln(c*x^n)) 
/e/x^(1/2)+6*b^2*f^2*n^2*ln(e+f*x^(1/2))*(a+b*ln(c*x^n))/e^2-6*b^2*n^2*ln( 
d*(e+f*x^(1/2)))*(a+b*ln(c*x^n))/x-3*b^2*f^2*n^2*ln(x)*(a+b*ln(c*x^n))/e^2 
-9*b*f*n*(a+b*ln(c*x^n))^2/e/x^(1/2)-3*b*n*ln(d*(e+f*x^(1/2)))*(a+b*ln(c*x 
^n))^2/x+3*b*f^2*n*ln(1+f*x^(1/2)/e)*(a+b*ln(c*x^n))^2/e^2-1/2*f^2*(a+b*ln 
(c*x^n))^3/e^2-f*(a+b*ln(c*x^n))^3/e/x^(1/2)-ln(d*(e+f*x^(1/2)))*(a+b*ln(c 
*x^n))^3/x+f^2*ln(1+f*x^(1/2)/e)*(a+b*ln(c*x^n))^3/e^2-1/8*f^2*(a+b*ln(c*x 
^n))^4/b/e^2/n-12*b^3*f^2*n^3*polylog(2,1+f*x^(1/2)/e)/e^2+12*b^2*f^2*n^2* 
(a+b*ln(c*x^n))*polylog(2,-f*x^(1/2)/e)/e^2+6*b*f^2*n*(a+b*ln(c*x^n))^2*po 
lylog(2,-f*x^(1/2)/e)/e^2-24*b^3*f^2*n^3*polylog(3,-f*x^(1/2)/e)/e^2-24*b^ 
2*f^2*n^2*(a+b*ln(c*x^n))*polylog(3,-f*x^(1/2)/e)/e^2+48*b^3*f^2*n^3*polyl 
og(4,-f*x^(1/2)/e)/e^2
 

Mathematica [A] (verified)

Time = 1.36 (sec) , antiderivative size = 976, normalized size of antiderivative = 1.45 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-((e^2*Log[d*(e + f*Sqrt[x])]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3*n^3 + 
 3*b*(a^2 + 2*a*b*n + 2*b^2*n^2)*Log[c*x^n] + 3*b^2*(a + b*n)*Log[c*x^n]^2 
 + b^3*Log[c*x^n]^3) + e*f*Sqrt[x]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3* 
n^3 + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*a*b^2*n*(-(n*Log[x]) + Log[c* 
x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[ 
c*x^n])^2 + 3*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + Log[ 
c*x^n])^3) - f^2*x*Log[e + f*Sqrt[x]]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b 
^3*n^3 + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*a*b^2*n*(-(n*Log[x]) + Log 
[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + L 
og[c*x^n])^2 + 3*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + L 
og[c*x^n])^3) + (f^2*x*Log[x]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3*n^3 + 
 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) 
 + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^n 
])^2 + 3*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + Log[c*x^n 
])^3))/2 + 3*b*f*n*Sqrt[x]*(a^2 + 2*a*b*n + 2*b^2*n^2 + 2*a*b*(-(n*Log[x]) 
 + Log[c*x^n]) + 2*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + b^2*(-(n*Log[x]) + L 
og[c*x^n])^2)*(2*e + (e - f*Sqrt[x]*Log[1 + (f*Sqrt[x])/e])*Log[x] + (f*Sq 
rt[x]*Log[x]^2)/4 - 2*f*Sqrt[x]*PolyLog[2, -((f*Sqrt[x])/e)]) + b^2*f*n^2* 
Sqrt[x]*(a + b*n - b*n*Log[x] + b*Log[c*x^n])*(24*e + 12*e*Log[x] + 3*e*Lo 
g[x]^2 - 3*f*Sqrt[x]*Log[1 + (f*Sqrt[x])/e]*Log[x]^2 + (f*Sqrt[x]*Log[x...
 

Rubi [A] (verified)

Time = 1.58 (sec) , antiderivative size = 779, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2824, 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.

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

\(\Big \downarrow \) 2824

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

\(\Big \downarrow \) 2009

\(\displaystyle -3 b n \left (\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^4}{24 b^2 e^2 n^2}+\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {2 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {2 f^2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {4 b f^2 n \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {8 b f^2 n \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b e^2 n}-\frac {f^2 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b e^2 n}-\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}-\frac {f^2 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {2 b f^2 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {3 f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}+\frac {14 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {2 b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{x}+\frac {4 b^2 f^2 n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{e^2}+\frac {8 b^2 f^2 n^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {16 b^2 f^2 n^2 \operatorname {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {b^2 f^2 n^2 \log ^2(x)}{2 e^2}-\frac {2 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right )}{e^2}+\frac {4 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {b^2 f^2 n^2 \log (x)}{e^2}+\frac {30 b^2 f n^2}{e \sqrt {x}}\right )-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}-\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt {x}}\)

Input:

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

Output:

-((f*(a + b*Log[c*x^n])^3)/(e*Sqrt[x])) + (f^2*Log[e + f*Sqrt[x]]*(a + b*L 
og[c*x^n])^3)/e^2 - (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3)/x - (f^2 
*Log[x]*(a + b*Log[c*x^n])^3)/(2*e^2) - 3*b*n*((30*b^2*f*n^2)/(e*Sqrt[x]) 
- (2*b^2*f^2*n^2*Log[e + f*Sqrt[x]])/e^2 + (2*b^2*n^2*Log[d*(e + f*Sqrt[x] 
)])/x + (4*b^2*f^2*n^2*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^2 + (b^ 
2*f^2*n^2*Log[x])/e^2 - (b^2*f^2*n^2*Log[x]^2)/(2*e^2) + (14*b*f*n*(a + b* 
Log[c*x^n]))/(e*Sqrt[x]) - (2*b*f^2*n*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n] 
))/e^2 + (2*b*n*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x + (b*f^2*n*Lo 
g[x]*(a + b*Log[c*x^n]))/e^2 + (3*f*(a + b*Log[c*x^n])^2)/(e*Sqrt[x]) + (L 
og[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x - (f^2*Log[1 + (f*Sqrt[x])/e 
]*(a + b*Log[c*x^n])^2)/e^2 + (f^2*(a + b*Log[c*x^n])^3)/(6*b*e^2*n) + (f^ 
2*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/(3*b*e^2*n) - (f^2*Log[1 + (f*S 
qrt[x])/e]*(a + b*Log[c*x^n])^3)/(3*b*e^2*n) - (f^2*Log[x]*(a + b*Log[c*x^ 
n])^3)/(6*b*e^2*n) + (f^2*(a + b*Log[c*x^n])^4)/(24*b^2*e^2*n^2) + (4*b^2* 
f^2*n^2*PolyLog[2, 1 + (f*Sqrt[x])/e])/e^2 - (4*b*f^2*n*(a + b*Log[c*x^n]) 
*PolyLog[2, -((f*Sqrt[x])/e)])/e^2 - (2*f^2*(a + b*Log[c*x^n])^2*PolyLog[2 
, -((f*Sqrt[x])/e)])/e^2 + (8*b^2*f^2*n^2*PolyLog[3, -((f*Sqrt[x])/e)])/e^ 
2 + (8*b*f^2*n*(a + b*Log[c*x^n])*PolyLog[3, -((f*Sqrt[x])/e)])/e^2 - (16* 
b^2*f^2*n^2*PolyLog[4, -((f*Sqrt[x])/e)])/e^2)
 

Defintions of rubi rules used

rule 2009

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

rule 2824

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

Maple [F]

\[\int \frac {\ln \left (d \left (e +f \sqrt {x}\right )\right ) {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{x^{2}}d x\]

Input:

int(ln(d*(e+f*x^(1/2)))*(a+b*ln(c*x^n))^3/x^2,x)
 

Output:

int(ln(d*(e+f*x^(1/2)))*(a+b*ln(c*x^n))^3/x^2,x)
 

Fricas [F]

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

integrate(log(d*(e+f*x^(1/2)))*(a+b*log(c*x^n))^3/x^2,x, algorithm="fricas 
")
 

Output:

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

Sympy [F(-1)]

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

integrate(ln(d*(e+f*x**(1/2)))*(a+b*ln(c*x**n))**3/x**2,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(log(d*(e+f*x^(1/2)))*(a+b*log(c*x^n))^3/x^2,x, algorithm="maxima 
")
 

Output:

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

Giac [F]

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

integrate(log(d*(e+f*x^(1/2)))*(a+b*log(c*x^n))^3/x^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((log(d*(e + f*x^(1/2)))*(a + b*log(c*x^n))^3)/x^2,x)
 

Output:

int((log(d*(e + f*x^(1/2)))*(a + b*log(c*x^n))^3)/x^2, x)
 

Reduce [F]

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

int(log(d*(e+f*x^(1/2)))*(a+b*log(c*x^n))^3/x^2,x)
 

Output:

( - 2*sqrt(x)*a**3*e*f - 6*sqrt(x)*a**2*b*e*f*n - 12*sqrt(x)*a*b**2*e*f*n* 
*2 - 12*sqrt(x)*b**3*e*f*n**3 - int(log(x**n*c)**3/(e**2*x**2 - f**2*x**3) 
,x)*b**3*e**4*x - 3*int(log(x**n*c)**2/(e**2*x**2 - f**2*x**3),x)*a*b**2*e 
**4*x - 3*int(log(x**n*c)**2/(e**2*x**2 - f**2*x**3),x)*b**3*e**4*n*x - 3* 
int(log(x**n*c)/(e**2*x**2 - f**2*x**3),x)*a**2*b*e**4*x - 6*int(log(x**n* 
c)/(e**2*x**2 - f**2*x**3),x)*a*b**2*e**4*n*x - 6*int(log(x**n*c)/(e**2*x* 
*2 - f**2*x**3),x)*b**3*e**4*n**2*x + int((sqrt(x)*log(x**n*c)**3)/(e**2*x 
**2 - f**2*x**3),x)*b**3*e**3*f*x + 3*int((sqrt(x)*log(x**n*c)**2)/(e**2*x 
**2 - f**2*x**3),x)*a*b**2*e**3*f*x + 3*int((sqrt(x)*log(x**n*c)**2)/(e**2 
*x**2 - f**2*x**3),x)*b**3*e**3*f*n*x + 3*int((sqrt(x)*log(x**n*c))/(e**2* 
x**2 - f**2*x**3),x)*a**2*b*e**3*f*x + 6*int((sqrt(x)*log(x**n*c))/(e**2*x 
**2 - f**2*x**3),x)*a*b**2*e**3*f*n*x + 6*int((sqrt(x)*log(x**n*c))/(e**2* 
x**2 - f**2*x**3),x)*b**3*e**3*f*n**2*x - 2*log(sqrt(x)*d*f + d*e)*log(x** 
n*c)**3*b**3*e**2 - 6*log(sqrt(x)*d*f + d*e)*log(x**n*c)**2*a*b**2*e**2 - 
6*log(sqrt(x)*d*f + d*e)*log(x**n*c)**2*b**3*e**2*n - 6*log(sqrt(x)*d*f + 
d*e)*log(x**n*c)*a**2*b*e**2 - 12*log(sqrt(x)*d*f + d*e)*log(x**n*c)*a*b** 
2*e**2*n - 12*log(sqrt(x)*d*f + d*e)*log(x**n*c)*b**3*e**2*n**2 - 2*log(sq 
rt(x)*d*f + d*e)*a**3*e**2 + 2*log(sqrt(x)*d*f + d*e)*a**3*f**2*x - 6*log( 
sqrt(x)*d*f + d*e)*a**2*b*e**2*n + 6*log(sqrt(x)*d*f + d*e)*a**2*b*f**2*n* 
x - 12*log(sqrt(x)*d*f + d*e)*a*b**2*e**2*n**2 + 12*log(sqrt(x)*d*f + d...