\(\int x^2 (a+b \log (c (d+\frac {e}{x^{2/3}})^n))^2 \, dx\) [521]

Optimal result

Integrand size = 24, antiderivative size = 490 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx=-\frac {4 a b e^4 n \sqrt [3]{x}}{3 d^4}+\frac {568 b^2 e^4 n^2 \sqrt [3]{x}}{315 d^4}-\frac {32 b^2 e^3 n^2 x}{105 d^3}+\frac {8 b^2 e^2 n^2 x^{5/3}}{105 d^2}-\frac {1408 b^2 e^{9/2} n^2 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{315 d^{9/2}}-\frac {4 i b^2 e^{9/2} n^2 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{3 d^{9/2}}+\frac {8 b^2 e^{9/2} n^2 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{3 d^{9/2}}-\frac {4 b^2 e^4 n \sqrt [3]{x} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 d^4}+\frac {4 b e^3 n x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{9 d^3}-\frac {4 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{15 d^2}+\frac {4 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{21 d}+\frac {4 b e^{9/2} n \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 d^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2-\frac {4 i b^2 e^{9/2} n^2 \operatorname {PolyLog}\left (2,-1+\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{3 d^{9/2}} \] Output:

-4/3*a*b*e^4*n*x^(1/3)/d^4+568/315*b^2*e^4*n^2*x^(1/3)/d^4-32/105*b^2*e^3* 
n^2*x/d^3+8/105*b^2*e^2*n^2*x^(5/3)/d^2-1408/315*b^2*e^(9/2)*n^2*arctan(d^ 
(1/2)*x^(1/3)/e^(1/2))/d^(9/2)-4/3*I*b^2*e^(9/2)*n^2*arctan(d^(1/2)*x^(1/3 
)/e^(1/2))^2/d^(9/2)+8/3*b^2*e^(9/2)*n^2*arctan(d^(1/2)*x^(1/3)/e^(1/2))*l 
n(2-2*e^(1/2)/(e^(1/2)-I*d^(1/2)*x^(1/3)))/d^(9/2)-4/3*b^2*e^4*n*x^(1/3)*l 
n(c*(d+e/x^(2/3))^n)/d^4+4/9*b*e^3*n*x*(a+b*ln(c*(d+e/x^(2/3))^n))/d^3-4/1 
5*b*e^2*n*x^(5/3)*(a+b*ln(c*(d+e/x^(2/3))^n))/d^2+4/21*b*e*n*x^(7/3)*(a+b* 
ln(c*(d+e/x^(2/3))^n))/d+4/3*b*e^(9/2)*n*arctan(d^(1/2)*x^(1/3)/e^(1/2))*( 
a+b*ln(c*(d+e/x^(2/3))^n))/d^(9/2)+1/3*x^3*(a+b*ln(c*(d+e/x^(2/3))^n))^2-4 
/3*I*b^2*e^(9/2)*n^2*polylog(2,-1+2*e^(1/2)/(e^(1/2)-I*d^(1/2)*x^(1/3)))/d 
^(9/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 2.41 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.50 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

(x^3*(a + b*Log[c*(d + e/x^(2/3))^n])^2 - 4*b*e*n*((a*e^3*x^(1/3))/d^4 - ( 
2*b*e^(7/2)*n*ArcTan[Sqrt[e]/(Sqrt[d]*x^(1/3))])/d^(9/2) - (2*b*e*n*x^(5/3 
)*Hypergeometric2F1[-5/2, 1, -3/2, -(e/(d*x^(2/3)))])/(35*d^2) + (2*b*e^2* 
n*x*Hypergeometric2F1[-3/2, 1, -1/2, -(e/(d*x^(2/3)))])/(15*d^3) - (2*b*e^ 
3*n*x^(1/3)*Hypergeometric2F1[-1/2, 1, 1/2, -(e/(d*x^(2/3)))])/(3*d^4) + ( 
b*e^3*x^(1/3)*Log[c*(d + e/x^(2/3))^n])/d^4 - (e^2*x*(a + b*Log[c*(d + e/x 
^(2/3))^n]))/(3*d^3) + (e*x^(5/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(5*d^2 
) - (x^(7/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(7*d) + (e^(7/2)*(a + b*Log 
[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)])/(2*(-d)^(9/2)) - ( 
e^(7/2)*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)])/ 
(2*(-d)^(9/2)) - (b*e^(7/2)*n*(Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*(Log[Sqrt[e 
] - Sqrt[-d]*x^(1/3)] + 2*Log[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2] - 4*Log[ 
(Sqrt[-d]*x^(1/3))/Sqrt[e]]) - 4*PolyLog[2, 1 - (Sqrt[-d]*x^(1/3))/Sqrt[e] 
] + 2*PolyLog[2, 1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])]))/(4*(-d)^(9/2)) + 
(b*e^(7/2)*n*(Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*(Log[Sqrt[e] + Sqrt[-d]*x^(1 
/3)] + 2*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])] - 4*Log[-((Sqrt[-d]*x^( 
1/3))/Sqrt[e])]) + 2*PolyLog[2, (1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2] - 4*Po 
lyLog[2, 1 + (Sqrt[-d]*x^(1/3))/Sqrt[e]]))/(4*(-d)^(9/2))))/3
 

Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 441, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2908, 2907, 2005, 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.

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

\(\Big \downarrow \) 2908

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

\(\Big \downarrow \) 2907

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 2926

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

\(\Big \downarrow \) 2009

\(\displaystyle 3 \left (\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+\frac {4}{9} b e n \left (\frac {e^{7/2} \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{d^{9/2}}+\frac {e^2 x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 d^3}-\frac {e x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{5 d^2}+\frac {x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{7 d}-\frac {a e^3 \sqrt [3]{x}}{d^4}-\frac {i b e^{7/2} n \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{d^{9/2}}-\frac {352 b e^{7/2} n \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{105 d^{9/2}}+\frac {2 b e^{7/2} n \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{d^{9/2}}-\frac {b e^3 \sqrt [3]{x} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{d^4}-\frac {i b e^{7/2} n \operatorname {PolyLog}\left (2,\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}-1\right )}{d^{9/2}}+\frac {142 b e^3 n \sqrt [3]{x}}{105 d^4}-\frac {8 b e^2 n x}{35 d^3}+\frac {2 b e n x^{5/3}}{35 d^2}\right )\right )\)

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*e^3*x^(1/3) 
)/d^4) + (142*b*e^3*n*x^(1/3))/(105*d^4) - (8*b*e^2*n*x)/(35*d^3) + (2*b*e 
*n*x^(5/3))/(35*d^2) - (352*b*e^(7/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]) 
/(105*d^(9/2)) - (I*b*e^(7/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/d^(9/ 
2) + (2*b*e^(7/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e])/ 
(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/d^(9/2) - (b*e^3*x^(1/3)*Log[c*(d + e/x^(2 
/3))^n])/d^4 + (e^2*x*(a + b*Log[c*(d + e/x^(2/3))^n]))/(3*d^3) - (e*x^(5/ 
3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(5*d^2) + (x^(7/3)*(a + b*Log[c*(d + 
e/x^(2/3))^n]))/(7*d) + (e^(7/2)*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b* 
Log[c*(d + e/x^(2/3))^n]))/d^(9/2) - (I*b*e^(7/2)*n*PolyLog[2, -1 + (2*Sqr 
t[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/d^(9/2)))/9)
 

Defintions of rubi rules used

rule 2005

Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m 
+ n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg 
Q[n]
 

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]

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

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]

\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{2} x^{2} \,d x } \] 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(c*((d*x + e*x^(1/3))/x)^n)^2 + 2*a*b*x^2*log(c*((d*x 
+ e*x^(1/3))/x)^n) + a^2*x^2, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

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

Output:

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

Mupad [F(-1)]

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

\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx=\frac {420 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} d}{\sqrt {e}\, \sqrt {d}}\right ) a b \,e^{4} n -1408 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} d}{\sqrt {e}\, \sqrt {d}}\right ) b^{2} e^{4} n^{2}-84 x^{\frac {5}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b^{2} d^{3} e^{2} n -84 x^{\frac {5}{3}} a b \,d^{3} e^{2} n +24 x^{\frac {5}{3}} b^{2} d^{3} e^{2} n^{2}-105 x^{\frac {1}{3}} {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{2} b^{2} d \,e^{4}+60 x^{\frac {7}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b^{2} d^{4} e n -420 x^{\frac {1}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b^{2} d \,e^{4} n +60 x^{\frac {7}{3}} a b \,d^{4} e n -420 x^{\frac {1}{3}} a b d \,e^{4} n +568 x^{\frac {1}{3}} b^{2} d \,e^{4} n^{2}+35 \left (\int \frac {{\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{2}}{x^{\frac {2}{3}}}d x \right ) b^{2} d \,e^{4}+105 {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{2} b^{2} d^{5} x^{3}+210 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) a b \,d^{5} x^{3}+140 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b^{2} d^{2} e^{3} n x +105 a^{2} d^{5} x^{3}+140 a b \,d^{2} e^{3} n x -96 b^{2} d^{2} e^{3} n^{2} x}{315 d^{5}} \] Input:

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

Output:

(420*sqrt(e)*sqrt(d)*atan((x**(1/3)*d)/(sqrt(e)*sqrt(d)))*a*b*e**4*n - 140 
8*sqrt(e)*sqrt(d)*atan((x**(1/3)*d)/(sqrt(e)*sqrt(d)))*b**2*e**4*n**2 - 84 
*x**(2/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b**2*d**3*e**2*n*x - 8 
4*x**(2/3)*a*b*d**3*e**2*n*x + 24*x**(2/3)*b**2*d**3*e**2*n**2*x - 105*x** 
(1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**2*b**2*d*e**4 + 60*x**(1/ 
3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b**2*d**4*e*n*x**2 - 420*x**( 
1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b**2*d*e**4*n + 60*x**(1/3) 
*a*b*d**4*e*n*x**2 - 420*x**(1/3)*a*b*d*e**4*n + 568*x**(1/3)*b**2*d*e**4* 
n**2 + 35*int(log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**2/x**(2/3),x)*b** 
2*d*e**4 + 105*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**2*b**2*d**5*x**3 
 + 210*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*a*b*d**5*x**3 + 140*log(( 
(x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b**2*d**2*e**3*n*x + 105*a**2*d**5*x* 
*3 + 140*a*b*d**2*e**3*n*x - 96*b**2*d**2*e**3*n**2*x)/(315*d**5)