Integrand size = 20, antiderivative size = 267 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=-\frac {3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2}{2 e^3}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac {6 b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac {b^2 d^3 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{e^3}-\frac {6 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}-\frac {2 b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}+\frac {2 b d^3 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \] Output:
-3/2*b^2*d*n^2*(d+e*x^(1/3))^2/e^3+2/9*b^2*n^2*(d+e*x^(1/3))^3/e^3+6*b^2*d ^2*n^2*x^(1/3)/e^2-b^2*d^3*n^2*ln(d+e*x^(1/3))^2/e^3-6*b*d^2*n*(d+e*x^(1/3 ))*(a+b*ln(c*(d+e*x^(1/3))^n))/e^3+3*b*d*n*(d+e*x^(1/3))^2*(a+b*ln(c*(d+e* x^(1/3))^n))/e^3-2/3*b*n*(d+e*x^(1/3))^3*(a+b*ln(c*(d+e*x^(1/3))^n))/e^3+2 *b*d^3*n*ln(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n))/e^3+x*(a+b*ln(c*(d+e* x^(1/3))^n))^2
Time = 0.14 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.74 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {b^2 e n^2 \left (66 d^2-15 d e \sqrt [3]{x}+4 e^2 x^{2/3}\right ) \sqrt [3]{x}+6 a b n \left (7 d^3-6 d^2 e \sqrt [3]{x}+3 d e^2 x^{2/3}-2 e^3 x\right )+18 a^2 \left (d^3+e^3 x\right )+6 b \left (6 a \left (d^3+e^3 x\right )-b n \left (11 d^3+6 d^2 e \sqrt [3]{x}-3 d e^2 x^{2/3}+2 e^3 x\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+18 b^2 \left (d^3+e^3 x\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{18 e^3} \] Input:
Integrate[(a + b*Log[c*(d + e*x^(1/3))^n])^2,x]
Output:
(b^2*e*n^2*(66*d^2 - 15*d*e*x^(1/3) + 4*e^2*x^(2/3))*x^(1/3) + 6*a*b*n*(7* d^3 - 6*d^2*e*x^(1/3) + 3*d*e^2*x^(2/3) - 2*e^3*x) + 18*a^2*(d^3 + e^3*x) + 6*b*(6*a*(d^3 + e^3*x) - b*n*(11*d^3 + 6*d^2*e*x^(1/3) - 3*d*e^2*x^(2/3) + 2*e^3*x))*Log[c*(d + e*x^(1/3))^n] + 18*b^2*(d^3 + e^3*x)*Log[c*(d + e* x^(1/3))^n]^2)/(18*e^3)
Time = 0.84 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.71, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2901, 2845, 2858, 25, 27, 2772, 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.
| \(\displaystyle \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2901 |
| \(\displaystyle 3 \int x^{2/3} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle 3 \left (\frac {1}{3} x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-\frac {2}{3} b e n \int \frac {x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d+e \sqrt [3]{x}}d\sqrt [3]{x}\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle 3 \left (\frac {1}{3} x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-\frac {2}{3} b n \int x^{2/3} \left (a+b \log \left (c x^{n/3}\right )\right )d\left (d+e \sqrt [3]{x}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \left (\frac {2}{3} b n \int -x^{2/3} \left (a+b \log \left (c x^{n/3}\right )\right )d\left (d+e \sqrt [3]{x}\right )+\frac {1}{3} x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \left (\frac {2 b n \int -e^3 x^{2/3} \left (a+b \log \left (c x^{n/3}\right )\right )d\left (d+e \sqrt [3]{x}\right )}{3 e^3}+\frac {1}{3} x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle 3 \left (\frac {2 b n \left (-b n \int \left (\frac {\log \left (d+e \sqrt [3]{x}\right ) d^3}{\sqrt [3]{x}}-3 d^2+\frac {3}{2} \left (d+e \sqrt [3]{x}\right ) d-\frac {x^{2/3}}{3}\right )d\left (d+e \sqrt [3]{x}\right )+d^3 \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )-3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )+\frac {3}{2} d x^{2/3} \left (a+b \log \left (c x^{n/3}\right )\right )-\frac {1}{3} x \left (a+b \log \left (c x^{n/3}\right )\right )\right )}{3 e^3}+\frac {1}{3} x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {2 b n \left (d^3 \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )-3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )+\frac {3}{2} d x^{2/3} \left (a+b \log \left (c x^{n/3}\right )\right )-\frac {1}{3} x \left (a+b \log \left (c x^{n/3}\right )\right )-b n \left (\frac {1}{2} d^3 \log ^2\left (d+e \sqrt [3]{x}\right )-3 d^2 \left (d+e \sqrt [3]{x}\right )+\frac {3}{4} d x^{2/3}-\frac {x}{9}\right )\right )}{3 e^3}+\frac {1}{3} x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2\right )\) |
Input:
Int[(a + b*Log[c*(d + e*x^(1/3))^n])^2,x]
Output:
3*((x*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/3 + (2*b*n*(-(b*n*(-3*d^2*(d + e *x^(1/3)) + (3*d*x^(2/3))/4 - x/9 + (d^3*Log[d + e*x^(1/3)]^2)/2)) - 3*d^2 *(d + e*x^(1/3))*(a + b*Log[c*x^(n/3)]) + (3*d*x^(2/3)*(a + b*Log[c*x^(n/3 )]))/2 - (x*(a + b*Log[c*x^(n/3)]))/3 + d^3*Log[d + e*x^(1/3)]*(a + b*Log[ c*x^(n/3)])))/(3*e^3))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ .))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[m, -1])
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]))
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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbo l] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*Log[c* (d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && FractionQ[n]
\[\int {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )}^{2}d x\]
Input:
int((a+b*ln(c*(d+e*x^(1/3))^n))^2,x)
Output:
int((a+b*ln(c*(d+e*x^(1/3))^n))^2,x)
Time = 0.13 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.07 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {18 \, b^{2} e^{3} x \log \left (c\right )^{2} + 18 \, {\left (b^{2} e^{3} n^{2} x + b^{2} d^{3} n^{2}\right )} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 12 \, {\left (b^{2} e^{3} n - 3 \, a b e^{3}\right )} x \log \left (c\right ) + 2 \, {\left (2 \, b^{2} e^{3} n^{2} - 6 \, a b e^{3} n + 9 \, a^{2} e^{3}\right )} x + 6 \, {\left (3 \, b^{2} d e^{2} n^{2} x^{\frac {2}{3}} - 6 \, b^{2} d^{2} e n^{2} x^{\frac {1}{3}} - 11 \, b^{2} d^{3} n^{2} + 6 \, a b d^{3} n - 2 \, {\left (b^{2} e^{3} n^{2} - 3 \, a b e^{3} n\right )} x + 6 \, {\left (b^{2} e^{3} n x + b^{2} d^{3} n\right )} \log \left (c\right )\right )} \log \left (e x^{\frac {1}{3}} + d\right ) - 3 \, {\left (5 \, b^{2} d e^{2} n^{2} - 6 \, b^{2} d e^{2} n \log \left (c\right ) - 6 \, a b d e^{2} n\right )} x^{\frac {2}{3}} + 6 \, {\left (11 \, b^{2} d^{2} e n^{2} - 6 \, b^{2} d^{2} e n \log \left (c\right ) - 6 \, a b d^{2} e n\right )} x^{\frac {1}{3}}}{18 \, e^{3}} \] Input:
integrate((a+b*log(c*(d+e*x^(1/3))^n))^2,x, algorithm="fricas")
Output:
1/18*(18*b^2*e^3*x*log(c)^2 + 18*(b^2*e^3*n^2*x + b^2*d^3*n^2)*log(e*x^(1/ 3) + d)^2 - 12*(b^2*e^3*n - 3*a*b*e^3)*x*log(c) + 2*(2*b^2*e^3*n^2 - 6*a*b *e^3*n + 9*a^2*e^3)*x + 6*(3*b^2*d*e^2*n^2*x^(2/3) - 6*b^2*d^2*e*n^2*x^(1/ 3) - 11*b^2*d^3*n^2 + 6*a*b*d^3*n - 2*(b^2*e^3*n^2 - 3*a*b*e^3*n)*x + 6*(b ^2*e^3*n*x + b^2*d^3*n)*log(c))*log(e*x^(1/3) + d) - 3*(5*b^2*d*e^2*n^2 - 6*b^2*d*e^2*n*log(c) - 6*a*b*d*e^2*n)*x^(2/3) + 6*(11*b^2*d^2*e*n^2 - 6*b^ 2*d^2*e*n*log(c) - 6*a*b*d^2*e*n)*x^(1/3))/e^3
\[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\int \left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{2}\, dx \] Input:
integrate((a+b*ln(c*(d+e*x**(1/3))**n))**2,x)
Output:
Integral((a + b*log(c*(d + e*x**(1/3))**n))**2, x)
Time = 0.04 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.81 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {1}{3} \, {\left (e n {\left (\frac {6 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{4}} - \frac {2 \, e^{2} x - 3 \, d e x^{\frac {2}{3}} + 6 \, d^{2} x^{\frac {1}{3}}}{e^{3}}\right )} + 6 \, x \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )\right )} a b + \frac {1}{18} \, {\left (6 \, e n {\left (\frac {6 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{4}} - \frac {2 \, e^{2} x - 3 \, d e x^{\frac {2}{3}} + 6 \, d^{2} x^{\frac {1}{3}}}{e^{3}}\right )} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + 18 \, x \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{2} - \frac {{\left (18 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 4 \, e^{3} x + 66 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right ) + 15 \, d e^{2} x^{\frac {2}{3}} - 66 \, d^{2} e x^{\frac {1}{3}}\right )} n^{2}}{e^{3}}\right )} b^{2} + a^{2} x \] Input:
integrate((a+b*log(c*(d+e*x^(1/3))^n))^2,x, algorithm="maxima")
Output:
1/3*(e*n*(6*d^3*log(e*x^(1/3) + d)/e^4 - (2*e^2*x - 3*d*e*x^(2/3) + 6*d^2* x^(1/3))/e^3) + 6*x*log((e*x^(1/3) + d)^n*c))*a*b + 1/18*(6*e*n*(6*d^3*log (e*x^(1/3) + d)/e^4 - (2*e^2*x - 3*d*e*x^(2/3) + 6*d^2*x^(1/3))/e^3)*log(( e*x^(1/3) + d)^n*c) + 18*x*log((e*x^(1/3) + d)^n*c)^2 - (18*d^3*log(e*x^(1 /3) + d)^2 - 4*e^3*x + 66*d^3*log(e*x^(1/3) + d) + 15*d*e^2*x^(2/3) - 66*d ^2*e*x^(1/3))*n^2/e^3)*b^2 + a^2*x
Time = 0.13 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.74 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:
integrate((a+b*log(c*(d+e*x^(1/3))^n))^2,x, algorithm="giac")
Output:
1/18*(18*b^2*e*x*log(c)^2 + (18*(e*x^(1/3) + d)^3*log(e*x^(1/3) + d)^2/e^2 - 54*(e*x^(1/3) + d)^2*d*log(e*x^(1/3) + d)^2/e^2 + 54*(e*x^(1/3) + d)*d^ 2*log(e*x^(1/3) + d)^2/e^2 - 12*(e*x^(1/3) + d)^3*log(e*x^(1/3) + d)/e^2 + 54*(e*x^(1/3) + d)^2*d*log(e*x^(1/3) + d)/e^2 - 108*(e*x^(1/3) + d)*d^2*l og(e*x^(1/3) + d)/e^2 + 4*(e*x^(1/3) + d)^3/e^2 - 27*(e*x^(1/3) + d)^2*d/e ^2 + 108*(e*x^(1/3) + d)*d^2/e^2)*b^2*n^2 + 6*(6*(e*x^(1/3) + d)^3*log(e*x ^(1/3) + d)/e^2 - 18*(e*x^(1/3) + d)^2*d*log(e*x^(1/3) + d)/e^2 + 18*(e*x^ (1/3) + d)*d^2*log(e*x^(1/3) + d)/e^2 - 2*(e*x^(1/3) + d)^3/e^2 + 9*(e*x^( 1/3) + d)^2*d/e^2 - 18*(e*x^(1/3) + d)*d^2/e^2)*b^2*n*log(c) + 36*a*b*e*x* log(c) + 6*(6*(e*x^(1/3) + d)^3*log(e*x^(1/3) + d)/e^2 - 18*(e*x^(1/3) + d )^2*d*log(e*x^(1/3) + d)/e^2 + 18*(e*x^(1/3) + d)*d^2*log(e*x^(1/3) + d)/e ^2 - 2*(e*x^(1/3) + d)^3/e^2 + 9*(e*x^(1/3) + d)^2*d/e^2 - 18*(e*x^(1/3) + d)*d^2/e^2)*a*b*n + 18*a^2*e*x)/e
Time = 14.72 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.09 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )\,\left (\frac {2\,b\,x\,\left (3\,a-b\,n\right )}{3}-x^{2/3}\,\left (\frac {b\,d\,\left (3\,a-b\,n\right )}{e}-\frac {3\,a\,b\,d}{e}\right )+\frac {d\,x^{1/3}\,\left (\frac {2\,b\,d\,\left (3\,a-b\,n\right )}{e}-\frac {6\,a\,b\,d}{e}\right )}{e}\right )-x^{2/3}\,\left (\frac {d\,\left (3\,a^2-2\,a\,b\,n+\frac {2\,b^2\,n^2}{3}\right )}{2\,e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{2\,e}\right )+x^{1/3}\,\left (\frac {d\,\left (\frac {d\,\left (3\,a^2-2\,a\,b\,n+\frac {2\,b^2\,n^2}{3}\right )}{e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{e}\right )}{e}+\frac {2\,b^2\,d^2\,n^2}{e^2}\right )+x\,\left (a^2-\frac {2\,a\,b\,n}{3}+\frac {2\,b^2\,n^2}{9}\right )+{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2\,\left (b^2\,x+\frac {b^2\,d^3}{e^3}\right )-\frac {\ln \left (d+e\,x^{1/3}\right )\,\left (11\,b^2\,d^3\,n^2-6\,a\,b\,d^3\,n\right )}{3\,e^3} \] Input:
int((a + b*log(c*(d + e*x^(1/3))^n))^2,x)
Output:
log(c*(d + e*x^(1/3))^n)*((2*b*x*(3*a - b*n))/3 - x^(2/3)*((b*d*(3*a - b*n ))/e - (3*a*b*d)/e) + (d*x^(1/3)*((2*b*d*(3*a - b*n))/e - (6*a*b*d)/e))/e) - x^(2/3)*((d*(3*a^2 + (2*b^2*n^2)/3 - 2*a*b*n))/(2*e) - (d*(3*a^2 - b^2* n^2))/(2*e)) + x^(1/3)*((d*((d*(3*a^2 + (2*b^2*n^2)/3 - 2*a*b*n))/e - (d*( 3*a^2 - b^2*n^2))/e))/e + (2*b^2*d^2*n^2)/e^2) + x*(a^2 + (2*b^2*n^2)/9 - (2*a*b*n)/3) + log(c*(d + e*x^(1/3))^n)^2*(b^2*x + (b^2*d^3)/e^3) - (log(d + e*x^(1/3))*(11*b^2*d^3*n^2 - 6*a*b*d^3*n))/(3*e^3)
Time = 0.17 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {18 x^{\frac {2}{3}} \mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right ) b^{2} d \,e^{2} n +18 x^{\frac {2}{3}} a b d \,e^{2} n -15 x^{\frac {2}{3}} b^{2} d \,e^{2} n^{2}-36 x^{\frac {1}{3}} \mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right ) b^{2} d^{2} e n -36 x^{\frac {1}{3}} a b \,d^{2} e n +66 x^{\frac {1}{3}} b^{2} d^{2} e \,n^{2}+18 {\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right )}^{2} b^{2} d^{3}+18 {\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right )}^{2} b^{2} e^{3} x +36 \,\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right ) a b \,d^{3}+36 \,\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right ) a b \,e^{3} x -66 \,\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right ) b^{2} d^{3} n -12 \,\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right ) b^{2} e^{3} n x +18 a^{2} e^{3} x -12 a b \,e^{3} n x +4 b^{2} e^{3} n^{2} x}{18 e^{3}} \] Input:
int((a+b*log(c*(d+e*x^(1/3))^n))^2,x)
Output:
(18*x**(2/3)*log((x**(1/3)*e + d)**n*c)*b**2*d*e**2*n + 18*x**(2/3)*a*b*d* e**2*n - 15*x**(2/3)*b**2*d*e**2*n**2 - 36*x**(1/3)*log((x**(1/3)*e + d)** n*c)*b**2*d**2*e*n - 36*x**(1/3)*a*b*d**2*e*n + 66*x**(1/3)*b**2*d**2*e*n* *2 + 18*log((x**(1/3)*e + d)**n*c)**2*b**2*d**3 + 18*log((x**(1/3)*e + d)* *n*c)**2*b**2*e**3*x + 36*log((x**(1/3)*e + d)**n*c)*a*b*d**3 + 36*log((x* *(1/3)*e + d)**n*c)*a*b*e**3*x - 66*log((x**(1/3)*e + d)**n*c)*b**2*d**3*n - 12*log((x**(1/3)*e + d)**n*c)*b**2*e**3*n*x + 18*a**2*e**3*x - 12*a*b*e **3*n*x + 4*b**2*e**3*n**2*x)/(18*e**3)