\(\int \frac {(a+b \log (c x^n))^2}{(d+e x^3)^2} \, dx\) [325]

Optimal result

Integrand size = 22, antiderivative size = 860 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^3\right )^2} \, dx =\text {Too large to display} \] Output:

1/9*x*(a+b*ln(c*x^n))^2/d^(5/3)/(d^(1/3)+e^(1/3)*x)-(-1)^(1/3)*x*(a+b*ln(c 
*x^n))^2/(1+(-1)^(1/3))^4/d^(5/3)/((-1)^(2/3)*d^(1/3)+e^(1/3)*x)+1/9*x*(a+ 
b*ln(c*x^n))^2/d^(5/3)/(d^(1/3)+(-1)^(2/3)*e^(1/3)*x)-2/9*b*n*(a+b*ln(c*x^ 
n))*ln(1+e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)+2/9*(a+b*ln(c*x^n))^2*ln(1+e^( 
1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)+2*(-1)^(1/3)*b*n*(a+b*ln(c*x^n))*ln(1-(-1) 
^(1/3)*e^(1/3)*x/d^(1/3))/(1+(-1)^(1/3))^4/d^(5/3)/e^(1/3)-2*I*3^(1/2)*(a+ 
b*ln(c*x^n))^2*ln(1-(-1)^(1/3)*e^(1/3)*x/d^(1/3))/(1+(-1)^(1/3))^5/d^(5/3) 
/e^(1/3)+2/9*(-1)^(1/3)*b*n*(a+b*ln(c*x^n))*ln(1+(-1)^(2/3)*e^(1/3)*x/d^(1 
/3))/d^(5/3)/e^(1/3)+2*(a+b*ln(c*x^n))^2*ln(1+(-1)^(2/3)*e^(1/3)*x/d^(1/3) 
)/(1+(-1)^(1/3))^4/d^(5/3)/e^(1/3)-2/9*b^2*n^2*polylog(2,-e^(1/3)*x/d^(1/3 
))/d^(5/3)/e^(1/3)+4/9*b*n*(a+b*ln(c*x^n))*polylog(2,-e^(1/3)*x/d^(1/3))/d 
^(5/3)/e^(1/3)+2*(-1)^(1/3)*b^2*n^2*polylog(2,(-1)^(1/3)*e^(1/3)*x/d^(1/3) 
)/(1+(-1)^(1/3))^4/d^(5/3)/e^(1/3)+4*I*3^(1/2)*b^2*n^2*polylog(3,(-1)^(1/3 
)*e^(1/3)*x/d^(1/3))/(1+(-1)^(1/3))^5/d^(5/3)/e^(1/3)+2/9*(-1)^(1/3)*b^2*n 
^2*polylog(2,-(-1)^(2/3)*e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)+4*b*n*(a+b*ln( 
c*x^n))*polylog(2,-(-1)^(2/3)*e^(1/3)*x/d^(1/3))/(1+(-1)^(1/3))^4/d^(5/3)/ 
e^(1/3)-4/9*b^2*n^2*polylog(3,-e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)-4*I*3^(1 
/2)*b*n*(a+b*ln(c*x^n))*polylog(2,(-1)^(1/3)*e^(1/3)*x/d^(1/3))/(1+(-1)^(1 
/3))^5/d^(5/3)/e^(1/3)-4*b^2*n^2*polylog(3,-(-1)^(2/3)*e^(1/3)*x/d^(1/3))/ 
(1+(-1)^(1/3))^4/d^(5/3)/e^(1/3)
 

Mathematica [A] (verified)

Time = 6.23 (sec) , antiderivative size = 1379, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^3\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(x*(a + b*(-(n*Log[x]) + Log[c*x^n]))^2)/(3*d*(d + e*x^3)) + (2*ArcTan[(-d 
^(1/3) + 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))]*(a + b*(-(n*Log[x]) + Log[c*x^n]) 
)^2)/(3*Sqrt[3]*d^(5/3)*e^(1/3)) + (2*(a + b*(-(n*Log[x]) + Log[c*x^n]))^2 
*Log[d^(1/3) + e^(1/3)*x])/(9*d^(5/3)*e^(1/3)) - ((a + b*(-(n*Log[x]) + Lo 
g[c*x^n]))^2*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(9*d^(5/3)*e^ 
(1/3)) + 2*b*n*(a + b*(-(n*Log[x]) + Log[c*x^n]))*(-1/3*((-1 + (-1)^(1/3)) 
*((-((-1)^(1/3)/d^(1/3)) - ((-1)^(2/3)*d^(1/3) + e^(1/3)*x)^(-1))*Log[x] + 
 ((-1)^(1/3)*Log[-((-1)^(2/3)*d^(1/3)) - e^(1/3)*x])/d^(1/3)))/((1 + (-1)^ 
(1/3))^2*d^(4/3)*e^(1/3)) + ((-1)^(1/3)*((d^(-1/3) - (d^(1/3) + e^(1/3)*x) 
^(-1))*Log[x] - Log[d^(1/3) + e^(1/3)*x]/d^(1/3)))/(3*(1 + (-1)^(1/3))^2*d 
^(4/3)*e^(1/3)) - (Log[x]/(e^(1/3)*((-1)^(1/3)*d^(1/3) - e^(1/3)*x)) - (-( 
((-1)^(2/3)*Log[x])/d^(1/3)) + ((-1)^(2/3)*Log[d^(1/3) + (-1)^(2/3)*e^(1/3 
)*x])/d^(1/3))/e^(1/3))/(3*(1 + (-1)^(1/3))^2*d^(4/3)) + (2*(-1)^(1/3)*(Lo 
g[x]*Log[1 + (e^(1/3)*x)/d^(1/3)] + PolyLog[2, -((e^(1/3)*x)/d^(1/3))]))/( 
3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3)) - (2*(Log[x]*Log[1 - ((-1)^(1/3)*e^( 
1/3)*x)/d^(1/3)] + PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)]))/(3*(1 + (- 
1)^(1/3))^2*d^(5/3)*e^(1/3)) - (2*(-1 + (-1)^(1/3))*(Log[x]*Log[1 + ((-1)^ 
(2/3)*e^(1/3)*x)/d^(1/3)] + PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))]) 
)/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3))) + b^2*n^2*(((-1)^(1/3)*(Log[x]*( 
(e^(1/3)*x*Log[x])/(d^(1/3) + e^(1/3)*x) - 2*Log[1 + (e^(1/3)*x)/d^(1/3...
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 860, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2767, 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.

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

\(\Big \downarrow \) 2767

\(\displaystyle \int \left (\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac {2 (-1)^{5/6} \sqrt {3} \left (a+b \log \left (c x^n\right )\right )^2}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right )}+\frac {2 (-1)^{2/3} \left (a+b \log \left (c x^n\right )\right )^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{9 d^{4/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )^2}+\frac {(-1)^{2/3} \left (a+b \log \left (c x^n\right )\right )^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right )^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (\sqrt [3]{-1}-1\right )^2 \left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 b^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{9 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b^2 \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{9 d^{5/3} \sqrt [3]{e}}-\frac {4 b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{9 d^{5/3} \sqrt [3]{e}}+\frac {4 i \sqrt {3} b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac {4 b^2 \operatorname {PolyLog}\left (3,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {2 b \left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) n}{9 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b \left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) n}{9 d^{5/3} \sqrt [3]{e}}+\frac {4 b \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{9 d^{5/3} \sqrt [3]{e}}-\frac {4 i \sqrt {3} b \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {4 b \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{9 d^{5/3} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}-\frac {\sqrt [3]{-1} x \left (a+b \log \left (c x^n\right )\right )^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left (\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}\)

Input:

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

Output:

(x*(a + b*Log[c*x^n])^2)/(9*d^(5/3)*(d^(1/3) + e^(1/3)*x)) - ((-1)^(1/3)*x 
*(a + b*Log[c*x^n])^2)/((1 + (-1)^(1/3))^4*d^(5/3)*((-1)^(2/3)*d^(1/3) + e 
^(1/3)*x)) + (x*(a + b*Log[c*x^n])^2)/(9*d^(5/3)*(d^(1/3) + (-1)^(2/3)*e^( 
1/3)*x)) - (2*b*n*(a + b*Log[c*x^n])*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5 
/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])^2*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^ 
(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b*n*(a + b*Log[c*x^n])*Log[1 - ((-1)^(1/3)* 
e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - ((2*I)*Sqrt[3] 
*(a + b*Log[c*x^n])^2*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^ 
(1/3))^5*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b*n*(a + b*Log[c*x^n])*Log[1 + ( 
(-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n] 
)^2*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e 
^(1/3)) - (2*b^2*n^2*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3 
)) + (4*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5 
/3)*e^(1/3)) + (2*(-1)^(1/3)*b^2*n^2*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^( 
1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - ((4*I)*Sqrt[3]*b*n*(a + b*Lo 
g[c*x^n])*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5* 
d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b^2*n^2*PolyLog[2, -(((-1)^(2/3)*e^(1/3)* 
x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) + (4*b*n*(a + b*Log[c*x^n])*PolyLog[2, - 
(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - 
(4*b^2*n^2*PolyLog[3, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) + ((...
 

Defintions of rubi rules used

rule 2009

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

rule 2767

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^( 
q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (d + e*x 
^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x] 
&& IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e^2*x^6 + 2*d*e*x^3 
+ d^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^3\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^3\right )^2} \, dx=\frac {-2 d^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) a^{2}-2 d^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) a^{2} e \,x^{3}-d^{\frac {4}{3}} \mathrm {log}\left (d^{\frac {2}{3}}-e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) a^{2}-d^{\frac {1}{3}} \mathrm {log}\left (d^{\frac {2}{3}}-e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) a^{2} e \,x^{3}+2 d^{\frac {4}{3}} \mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) a^{2}+2 d^{\frac {1}{3}} \mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) a^{2} e \,x^{3}+9 e^{\frac {1}{3}} \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e^{2} x^{6}+2 d e \,x^{3}+d^{2}}d x \right ) b^{2} d^{3}+9 e^{\frac {4}{3}} \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e^{2} x^{6}+2 d e \,x^{3}+d^{2}}d x \right ) b^{2} d^{2} x^{3}+18 e^{\frac {1}{3}} \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{2} x^{6}+2 d e \,x^{3}+d^{2}}d x \right ) a b \,d^{3}+18 e^{\frac {4}{3}} \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{2} x^{6}+2 d e \,x^{3}+d^{2}}d x \right ) a b \,d^{2} x^{3}+3 e^{\frac {1}{3}} a^{2} d x}{9 e^{\frac {1}{3}} d^{2} \left (e \,x^{3}+d \right )} \] Input:

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

Output:

( - 2*d**(1/3)*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3)*sqrt(3)))* 
a**2*d - 2*d**(1/3)*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3)*sqrt( 
3)))*a**2*e*x**3 - d**(1/3)*log(d**(2/3) - e**(1/3)*d**(1/3)*x + e**(2/3)* 
x**2)*a**2*d - d**(1/3)*log(d**(2/3) - e**(1/3)*d**(1/3)*x + e**(2/3)*x**2 
)*a**2*e*x**3 + 2*d**(1/3)*log(d**(1/3) + e**(1/3)*x)*a**2*d + 2*d**(1/3)* 
log(d**(1/3) + e**(1/3)*x)*a**2*e*x**3 + 9*e**(1/3)*int(log(x**n*c)**2/(d* 
*2 + 2*d*e*x**3 + e**2*x**6),x)*b**2*d**3 + 9*e**(1/3)*int(log(x**n*c)**2/ 
(d**2 + 2*d*e*x**3 + e**2*x**6),x)*b**2*d**2*e*x**3 + 18*e**(1/3)*int(log( 
x**n*c)/(d**2 + 2*d*e*x**3 + e**2*x**6),x)*a*b*d**3 + 18*e**(1/3)*int(log( 
x**n*c)/(d**2 + 2*d*e*x**3 + e**2*x**6),x)*a*b*d**2*e*x**3 + 3*e**(1/3)*a* 
*2*d*x)/(9*e**(1/3)*d**2*(d + e*x**3))