\(\int x^2 (a+b \log (c (d+e x^{2/3})^n))^2 \, dx\) [476]

Optimal result

Integrand size = 24, antiderivative size = 547 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx=-\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac {1984 b^2 d^3 n^2 x}{2835 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac {128 b^2 d n^2 x^{7/3}}{1323 e}+\frac {8}{243} b^2 n^2 x^3-\frac {4504 b^2 d^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{945 e^{9/2}}+\frac {4 i b^2 d^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 e^{9/2}}+\frac {8 b^2 d^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 e^{9/2}}-\frac {4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {4 b d^{9/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {4 i b^2 d^{9/2} n^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 e^{9/2}} \] Output:

-4/3*a*b*d^4*n*x^(1/3)/e^4+4504/945*b^2*d^4*n^2*x^(1/3)/e^4-1984/2835*b^2* 
d^3*n^2*x/e^3+1144/4725*b^2*d^2*n^2*x^(5/3)/e^2-128/1323*b^2*d*n^2*x^(7/3) 
/e+8/243*b^2*n^2*x^3-4504/945*b^2*d^(9/2)*n^2*arctan(e^(1/2)*x^(1/3)/d^(1/ 
2))/e^(9/2)+4/3*I*b^2*d^(9/2)*n^2*arctan(e^(1/2)*x^(1/3)/d^(1/2))^2/e^(9/2 
)+8/3*b^2*d^(9/2)*n^2*arctan(e^(1/2)*x^(1/3)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2 
)+I*e^(1/2)*x^(1/3)))/e^(9/2)-4/3*b^2*d^4*n*x^(1/3)*ln(c*(d+e*x^(2/3))^n)/ 
e^4+4/9*b*d^3*n*x*(a+b*ln(c*(d+e*x^(2/3))^n))/e^3-4/15*b*d^2*n*x^(5/3)*(a+ 
b*ln(c*(d+e*x^(2/3))^n))/e^2+4/21*b*d*n*x^(7/3)*(a+b*ln(c*(d+e*x^(2/3))^n) 
)/e-4/27*b*n*x^3*(a+b*ln(c*(d+e*x^(2/3))^n))+4/3*b*d^(9/2)*n*arctan(e^(1/2 
)*x^(1/3)/d^(1/2))*(a+b*ln(c*(d+e*x^(2/3))^n))/e^(9/2)+1/3*x^3*(a+b*ln(c*( 
d+e*x^(2/3))^n))^2+4/3*I*b^2*d^(9/2)*n^2*polylog(2,1-2*d^(1/2)/(d^(1/2)+I* 
e^(1/2)*x^(1/3)))/e^(9/2)
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 438, normalized size of antiderivative = 0.80 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx=\frac {396900 i b^2 d^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2+1260 b d^{9/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (315 a-1126 b n+630 b n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )+315 b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\sqrt {e} \sqrt [3]{x} \left (99225 a^2 e^4 x^{8/3}-1260 a b n \left (315 d^4-105 d^3 e x^{2/3}+63 d^2 e^2 x^{4/3}-45 d e^3 x^2+35 e^4 x^{8/3}\right )+8 b^2 n^2 \left (177345 d^4-26040 d^3 e x^{2/3}+9009 d^2 e^2 x^{4/3}-3600 d e^3 x^2+1225 e^4 x^{8/3}\right )-630 b \left (-315 a e^4 x^{8/3}+2 b n \left (315 d^4-105 d^3 e x^{2/3}+63 d^2 e^2 x^{4/3}-45 d e^3 x^2+35 e^4 x^{8/3}\right )\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )+99225 b^2 e^4 x^{8/3} \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )\right )+396900 i b^2 d^{9/2} n^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {d}+\sqrt {e} \sqrt [3]{x}}{-i \sqrt {d}+\sqrt {e} \sqrt [3]{x}}\right )}{297675 e^{9/2}} \] Input:

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

Output:

((396900*I)*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2 + 1260*b*d 
^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(315*a - 1126*b*n + 630*b*n*Log 
[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))] + 315*b*Log[c*(d + e*x^(2/3))^ 
n]) + Sqrt[e]*x^(1/3)*(99225*a^2*e^4*x^(8/3) - 1260*a*b*n*(315*d^4 - 105*d 
^3*e*x^(2/3) + 63*d^2*e^2*x^(4/3) - 45*d*e^3*x^2 + 35*e^4*x^(8/3)) + 8*b^2 
*n^2*(177345*d^4 - 26040*d^3*e*x^(2/3) + 9009*d^2*e^2*x^(4/3) - 3600*d*e^3 
*x^2 + 1225*e^4*x^(8/3)) - 630*b*(-315*a*e^4*x^(8/3) + 2*b*n*(315*d^4 - 10 
5*d^3*e*x^(2/3) + 63*d^2*e^2*x^(4/3) - 45*d*e^3*x^2 + 35*e^4*x^(8/3)))*Log 
[c*(d + e*x^(2/3))^n] + 99225*b^2*e^4*x^(8/3)*Log[c*(d + e*x^(2/3))^n]^2) 
+ (396900*I)*b^2*d^(9/2)*n^2*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x^(1/3))/((-I 
)*Sqrt[d] + Sqrt[e]*x^(1/3))])/(297675*e^(9/2))
 

Rubi [A] (verified)

Time = 1.53 (sec) , antiderivative size = 498, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2908, 2907, 2926, 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.

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

\(\Big \downarrow \) 2908

\(\displaystyle 3 \int x^{8/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2d\sqrt [3]{x}\)

\(\Big \downarrow \) 2907

\(\displaystyle 3 \left (\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {4}{9} b e n \int \frac {x^{10/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d+e x^{2/3}}d\sqrt [3]{x}\right )\)

\(\Big \downarrow \) 2926

\(\displaystyle 3 \left (\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {4}{9} b e n \int \left (-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) d^5}{e^5 \left (d+e x^{2/3}\right )}+\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) d^4}{e^5}-\frac {x^{2/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) d^3}{e^4}+\frac {x^{4/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) d^2}{e^3}-\frac {x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) d}{e^2}+\frac {x^{8/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e}\right )d\sqrt [3]{x}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 \left (\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {4}{9} b e n \left (-\frac {d^{9/2} \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{11/2}}-\frac {d^3 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^4}+\frac {d^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 e^3}-\frac {d x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{7 e^2}+\frac {x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e}+\frac {a d^4 \sqrt [3]{x}}{e^5}-\frac {i b d^{9/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{e^{11/2}}+\frac {1126 b d^{9/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{315 e^{11/2}}-\frac {2 b d^{9/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{e^{11/2}}+\frac {b d^4 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e^5}-\frac {i b d^{9/2} n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{e^{11/2}}-\frac {1126 b d^4 n \sqrt [3]{x}}{315 e^5}+\frac {496 b d^3 n x}{945 e^4}-\frac {286 b d^2 n x^{5/3}}{1575 e^3}+\frac {32 b d n x^{7/3}}{441 e^2}-\frac {2 b n x^3}{81 e}\right )\right )\)

Input:

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

Output:

3*((x^3*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/9 - (4*b*e*n*((a*d^4*x^(1/3))/ 
e^5 - (1126*b*d^4*n*x^(1/3))/(315*e^5) + (496*b*d^3*n*x)/(945*e^4) - (286* 
b*d^2*n*x^(5/3))/(1575*e^3) + (32*b*d*n*x^(7/3))/(441*e^2) - (2*b*n*x^3)/( 
81*e) + (1126*b*d^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(315*e^(11/2) 
) - (I*b*d^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/e^(11/2) - (2*b*d^ 
(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqr 
t[e]*x^(1/3))])/e^(11/2) + (b*d^4*x^(1/3)*Log[c*(d + e*x^(2/3))^n])/e^5 - 
(d^3*x*(a + b*Log[c*(d + e*x^(2/3))^n]))/(3*e^4) + (d^2*x^(5/3)*(a + b*Log 
[c*(d + e*x^(2/3))^n]))/(5*e^3) - (d*x^(7/3)*(a + b*Log[c*(d + e*x^(2/3))^ 
n]))/(7*e^2) + (x^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/(9*e) - (d^(9/2)*Arc 
Tan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/e^(11/2) 
- (I*b*d^(9/2)*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))] 
)/e^(11/2)))/9)
 

Defintions of rubi rules used

rule 2009

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

rule 2907

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

rule 2908

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_)*(x_)^(m_ 
.), x_Symbol] :> With[{k = Denominator[n]}, Simp[k   Subst[Int[x^(k*(m + 1) 
 - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, 
 b, c, d, e, m, p, q}, x] && FractionQ[n]
 

rule 2926

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[(a + b 
*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c, d, e 
, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] & 
& IntegerQ[s]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx=\frac {396900 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} e}{\sqrt {e}\, \sqrt {d}}\right ) a b \,d^{4} n -1418760 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} e}{\sqrt {e}\, \sqrt {d}}\right ) b^{2} d^{4} n^{2}-79380 x^{\frac {5}{3}} \mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) b^{2} d^{2} e^{3} n -79380 x^{\frac {5}{3}} a b \,d^{2} e^{3} n +72072 x^{\frac {5}{3}} b^{2} d^{2} e^{3} n^{2}-396900 x^{\frac {1}{3}} \mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) b^{2} d^{4} e n +56700 x^{\frac {7}{3}} \mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) b^{2} d \,e^{4} n -396900 x^{\frac {1}{3}} a b \,d^{4} e n +56700 x^{\frac {7}{3}} a b d \,e^{4} n +1418760 x^{\frac {1}{3}} b^{2} d^{4} e \,n^{2}-28800 x^{\frac {7}{3}} b^{2} d \,e^{4} n^{2}+132300 \left (\int \frac {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}{x^{\frac {2}{3}} d +x^{\frac {4}{3}} e}d x \right ) b^{2} d^{5} e n +99225 {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}^{2} b^{2} e^{5} x^{3}+198450 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) a b \,e^{5} x^{3}+132300 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) b^{2} d^{3} e^{2} n x -44100 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) b^{2} e^{5} n \,x^{3}+99225 a^{2} e^{5} x^{3}+132300 a b \,d^{3} e^{2} n x -44100 a b \,e^{5} n \,x^{3}-208320 b^{2} d^{3} e^{2} n^{2} x +9800 b^{2} e^{5} n^{2} x^{3}}{297675 e^{5}} \] Input:

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

Output:

(396900*sqrt(e)*sqrt(d)*atan((x**(1/3)*e)/(sqrt(e)*sqrt(d)))*a*b*d**4*n - 
1418760*sqrt(e)*sqrt(d)*atan((x**(1/3)*e)/(sqrt(e)*sqrt(d)))*b**2*d**4*n** 
2 - 79380*x**(2/3)*log((x**(2/3)*e + d)**n*c)*b**2*d**2*e**3*n*x - 79380*x 
**(2/3)*a*b*d**2*e**3*n*x + 72072*x**(2/3)*b**2*d**2*e**3*n**2*x - 396900* 
x**(1/3)*log((x**(2/3)*e + d)**n*c)*b**2*d**4*e*n + 56700*x**(1/3)*log((x* 
*(2/3)*e + d)**n*c)*b**2*d*e**4*n*x**2 - 396900*x**(1/3)*a*b*d**4*e*n + 56 
700*x**(1/3)*a*b*d*e**4*n*x**2 + 1418760*x**(1/3)*b**2*d**4*e*n**2 - 28800 
*x**(1/3)*b**2*d*e**4*n**2*x**2 + 132300*int(log((x**(2/3)*e + d)**n*c)/(x 
**(2/3)*d + x**(1/3)*e*x),x)*b**2*d**5*e*n + 99225*log((x**(2/3)*e + d)**n 
*c)**2*b**2*e**5*x**3 + 198450*log((x**(2/3)*e + d)**n*c)*a*b*e**5*x**3 + 
132300*log((x**(2/3)*e + d)**n*c)*b**2*d**3*e**2*n*x - 44100*log((x**(2/3) 
*e + d)**n*c)*b**2*e**5*n*x**3 + 99225*a**2*e**5*x**3 + 132300*a*b*d**3*e* 
*2*n*x - 44100*a*b*e**5*n*x**3 - 208320*b**2*d**3*e**2*n**2*x + 9800*b**2* 
e**5*n**2*x**3)/(297675*e**5)