Integrand size = 24, antiderivative size = 231 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x^2} \, dx=-\frac {b^2 e^2 n^2}{d^2 \sqrt [3]{x}}+\frac {b^2 e^3 n^2 \log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d x^{2/3}}+\frac {2 b e^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3 \sqrt [3]{x}}+\frac {2 b e^3 n \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x}-\frac {b^2 e^3 n^2 \log (x)}{d^3}-\frac {2 b^2 e^3 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt [3]{x}}\right )}{d^3} \] Output:
-b^2*e^2*n^2/d^2/x^(1/3)+b^2*e^3*n^2*ln(d+e*x^(1/3))/d^3-b*e*n*(a+b*ln(c*( d+e*x^(1/3))^n))/d/x^(2/3)+2*b*e^2*n*(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3)) ^n))/d^3/x^(1/3)+2*b*e^3*n*ln(1-d/(d+e*x^(1/3)))*(a+b*ln(c*(d+e*x^(1/3))^n ))/d^3-(a+b*ln(c*(d+e*x^(1/3))^n))^2/x-b^2*e^3*n^2*ln(x)/d^3-2*b^2*e^3*n^2 *polylog(2,d/(d+e*x^(1/3)))/d^3
Time = 0.25 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x^2} \, dx=-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x}-\frac {e \left (3 b d^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-6 b d e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+3 e^2 x^{2/3} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-6 b e^2 n x^{2/3} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )-2 b^2 e^2 n^2 x^{2/3} \left (3 \log \left (d+e \sqrt [3]{x}\right )-\log (x)\right )+b^2 e n^2 \sqrt [3]{x} \left (3 d-3 e \sqrt [3]{x} \log \left (d+e \sqrt [3]{x}\right )+e \sqrt [3]{x} \log (x)\right )-6 b^2 e^2 n^2 x^{2/3} \operatorname {PolyLog}\left (2,1+\frac {e \sqrt [3]{x}}{d}\right )\right )}{3 d^3 x^{2/3}} \] Input:
Integrate[(a + b*Log[c*(d + e*x^(1/3))^n])^2/x^2,x]
Output:
-((a + b*Log[c*(d + e*x^(1/3))^n])^2/x) - (e*(3*b*d^2*n*(a + b*Log[c*(d + e*x^(1/3))^n]) - 6*b*d*e*n*x^(1/3)*(a + b*Log[c*(d + e*x^(1/3))^n]) + 3*e^ 2*x^(2/3)*(a + b*Log[c*(d + e*x^(1/3))^n])^2 - 6*b*e^2*n*x^(2/3)*(a + b*Lo g[c*(d + e*x^(1/3))^n])*Log[-((e*x^(1/3))/d)] - 2*b^2*e^2*n^2*x^(2/3)*(3*L og[d + e*x^(1/3)] - Log[x]) + b^2*e*n^2*x^(1/3)*(3*d - 3*e*x^(1/3)*Log[d + e*x^(1/3)] + e*x^(1/3)*Log[x]) - 6*b^2*e^2*n^2*x^(2/3)*PolyLog[2, 1 + (e* x^(1/3))/d]))/(3*d^3*x^(2/3))
Time = 1.62 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {2904, 2845, 2858, 25, 27, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}
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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle 3 \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x^{4/3}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle 3 \left (\frac {2}{3} b e n \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{\left (d+e \sqrt [3]{x}\right ) x}d\sqrt [3]{x}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 x}\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle 3 \left (\frac {2}{3} b n \int \frac {a+b \log \left (c x^{n/3}\right )}{x^{4/3}}d\left (d+e \sqrt [3]{x}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 x}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \left (-\frac {2}{3} b n \int -\frac {a+b \log \left (c x^{n/3}\right )}{x^{4/3}}d\left (d+e \sqrt [3]{x}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \left (-\frac {2}{3} b e^3 n \int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 x}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle 3 \left (-\frac {2}{3} b e^3 n \left (\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 x}\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle 3 \left (-\frac {2}{3} b e^3 n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \int \frac {1}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 x}\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle 3 \left (-\frac {2}{3} b e^3 n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \int \left (-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{d^2 \sqrt [3]{x}}+\frac {1}{d e^2 x^{2/3}}\right )d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 x}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (-\frac {2}{3} b e^3 n \left (\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 x}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle 3 \left (-\frac {2}{3} b e^3 n \left (\frac {\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 x}\right )\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle 3 \left (-\frac {2}{3} b e^3 n \left (\frac {\frac {-\frac {b n \int -\frac {1}{e \sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 x}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle 3 \left (-\frac {2}{3} b e^3 n \left (\frac {\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 x}\right )\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle 3 \left (-\frac {2}{3} b e^3 n \left (\frac {\frac {\frac {b n \int \frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d}}{d}+\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 x}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle 3 \left (-\frac {2}{3} b e^3 n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,\frac {d}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 x}\right )\) |
Input:
Int[(a + b*Log[c*(d + e*x^(1/3))^n])^2/x^2,x]
Output:
3*(-1/3*(a + b*Log[c*(d + e*x^(1/3))^n])^2/x - (2*b*e^3*n*((-1/2*(b*n*(-(1 /(d*e*x^(1/3))) + Log[d + e*x^(1/3)]/d^2 - Log[-(e*x^(1/3))]/d^2)) + (a + b*Log[c*x^(n/3)])/(2*e^2*x^(2/3)))/d + (((b*n*Log[-(e*x^(1/3))])/d - ((d + e*x^(1/3))*(a + b*Log[c*x^(n/3)]))/(d*e*x^(1/3)))/d + (-((Log[1 - d/x^(1/ 3)]*(a + b*Log[c*x^(n/3)]))/d) + (b*n*PolyLog[2, d/x^(1/3)])/d)/d)/d))/3)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 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_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & & !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )}^{2}}{x^{2}}d x\]
Input:
int((a+b*ln(c*(d+e*x^(1/3))^n))^2/x^2,x)
Output:
int((a+b*ln(c*(d+e*x^(1/3))^n))^2/x^2,x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{2}} \,d x } \] Input:
integrate((a+b*log(c*(d+e*x^(1/3))^n))^2/x^2,x, algorithm="fricas")
Output:
integral((b^2*log((e*x^(1/3) + d)^n*c)^2 + 2*a*b*log((e*x^(1/3) + d)^n*c) + a^2)/x^2, x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{2}}{x^{2}}\, dx \] Input:
integrate((a+b*ln(c*(d+e*x**(1/3))**n))**2/x**2,x)
Output:
Integral((a + b*log(c*(d + e*x**(1/3))**n))**2/x**2, x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{2}} \,d x } \] Input:
integrate((a+b*log(c*(d+e*x^(1/3))^n))^2/x^2,x, algorithm="maxima")
Output:
-2*(log(e*x^(1/3)/d + 1)*log(x^(1/3)) + dilog(-e*x^(1/3)/d))*b^2*e^3*n^2/d ^3 - (2*a*b*e^3*n - (3*e^3*n^2 - 2*e^3*n*log(c))*b^2)*log(e*x^(1/3) + d)/d ^3 + 2*(b^2*e^3*n*log(c) + a*b*e^3*n)*log(x^(1/3))/d^3 + integrate((b^2*e^ 6*n^2*x - b^2*d^3*e^3*n^2)/x, x)/d^6 - 1/20*(12*b^2*e^8*n^2*x^(5/3) - 15*b ^2*d*e^7*n^2*x^(4/3) + 20*b^2*d^2*e^6*n^2*x - 40*b^2*d^3*e^5*n^2*x^(2/3) + 100*b^2*d^4*e^4*n^2*x^(1/3) + 20*(b^2*d^3*e^5*n^2*x^(2/3) - 2*b^2*d^4*e^4 *n^2*x^(1/3))*log(x^(1/3)))/d^8 + 1/60*(60*b^2*d^5*e^3*n^2*x^(5/3)*log(e*x ^(1/3) + d)^2 - 45*b^2*d*e^7*n^2*x^3 - 40*b^2*d^4*e^4*n^2*x^2*log(x) + 300 *b^2*d^4*e^4*n^2*x^2 - 60*b^2*d^8*x^(2/3)*log((e*x^(1/3) + d)^n)^2 - 60*(b ^2*d^7*e*n*log(c) + a*b*d^7*e*n)*x - 20*(6*b^2*d^5*e^3*n*x^(5/3)*log(e*x^( 1/3) + d) - 6*b^2*d^6*e^2*n*x^(4/3) + 3*b^2*d^7*e*n*x - 2*(b^2*d^5*e^3*n*x *log(x) - 3*b^2*d^8*log(c) - 3*a*b*d^8)*x^(2/3))*log((e*x^(1/3) + d)^n) - 60*(b^2*d^8*log(c)^2 + 2*a*b*d^8*log(c) + a^2*d^8)*x^(2/3) + 4*(9*b^2*e^8* n^2*x^3 + 5*b^2*d^3*e^5*n^2*x^2*log(x) - 15*b^2*d^3*e^5*n^2*x^2 + 30*(b^2* d^6*e^2*n*log(c) + a*b*d^6*e^2*n)*x)*x^(1/3) - 60*(b^2*d^3*e^5*n^2*x^3 + b ^2*d^6*e^2*n^2*x^2)/x^(2/3))/(d^8*x^(5/3))
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{2}} \,d x } \] Input:
integrate((a+b*log(c*(d+e*x^(1/3))^n))^2/x^2,x, algorithm="giac")
Output:
integrate((b*log((e*x^(1/3) + d)^n*c) + a)^2/x^2, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )\right )}^2}{x^2} \,d x \] Input:
int((a + b*log(c*(d + e*x^(1/3))^n))^2/x^2,x)
Output:
int((a + b*log(c*(d + e*x^(1/3))^n))^2/x^2, x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x^2} \, dx=\frac {6 x^{\frac {2}{3}} \mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right ) b^{2} d \,e^{2} n +6 x^{\frac {2}{3}} a b d \,e^{2} n -3 x^{\frac {2}{3}} b^{2} d \,e^{2} n^{2}-3 x^{\frac {1}{3}} \mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right ) b^{2} d^{2} e n -3 x^{\frac {1}{3}} a b \,d^{2} e n +2 \left (\int \frac {\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right )}{x^{\frac {4}{3}} e +d x}d x \right ) b^{2} d \,e^{3} n x +6 \,\mathrm {log}\left (x^{\frac {1}{3}}\right ) a b \,e^{3} n x -9 \,\mathrm {log}\left (x^{\frac {1}{3}}\right ) b^{2} e^{3} n^{2} x -3 {\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right )}^{2} b^{2} d^{3}-6 \,\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right ) a b \,d^{3}-6 \,\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right ) a b \,e^{3} x +9 \,\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right ) b^{2} e^{3} n x -3 a^{2} d^{3}}{3 d^{3} x} \] Input:
int((a+b*log(c*(d+e*x^(1/3))^n))^2/x^2,x)
Output:
(6*x**(2/3)*log((x**(1/3)*e + d)**n*c)*b**2*d*e**2*n + 6*x**(2/3)*a*b*d*e* *2*n - 3*x**(2/3)*b**2*d*e**2*n**2 - 3*x**(1/3)*log((x**(1/3)*e + d)**n*c) *b**2*d**2*e*n - 3*x**(1/3)*a*b*d**2*e*n + 2*int(log((x**(1/3)*e + d)**n*c )/(x**(1/3)*e*x + d*x),x)*b**2*d*e**3*n*x + 6*log(x**(1/3))*a*b*e**3*n*x - 9*log(x**(1/3))*b**2*e**3*n**2*x - 3*log((x**(1/3)*e + d)**n*c)**2*b**2*d **3 - 6*log((x**(1/3)*e + d)**n*c)*a*b*d**3 - 6*log((x**(1/3)*e + d)**n*c) *a*b*e**3*x + 9*log((x**(1/3)*e + d)**n*c)*b**2*e**3*n*x - 3*a**2*d**3)/(3 *d**3*x)