Integrand size = 24, antiderivative size = 95 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x} \, dx=\frac {3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \log \left (-\frac {e x^{2/3}}{d}\right )+3 b n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x^{2/3}}{d}\right )-3 b^2 n^2 \operatorname {PolyLog}\left (3,1+\frac {e x^{2/3}}{d}\right ) \]
3/2*(a+b*ln(c*(d+e*x^(2/3))^n))^2*ln(-e*x^(2/3)/d)+3*b*n*(a+b*ln(c*(d+e*x^ (2/3))^n))*polylog(2,1+e*x^(2/3)/d)-3*b^2*n^2*polylog(3,1+e*x^(2/3)/d)
Leaf count is larger than twice the leaf count of optimal. \(199\) vs. \(2(95)=190\).
Time = 0.09 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.09 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x} \, dx=\left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \log (x)+2 b n \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (\left (\log \left (d+e x^{2/3}\right )-\log \left (1+\frac {e x^{2/3}}{d}\right )\right ) \log (x)-\frac {3}{2} \operatorname {PolyLog}\left (2,-\frac {e x^{2/3}}{d}\right )\right )+\frac {3}{2} b^2 n^2 \left (\log ^2\left (d+e x^{2/3}\right ) \log \left (-\frac {e x^{2/3}}{d}\right )+2 \log \left (d+e x^{2/3}\right ) \operatorname {PolyLog}\left (2,1+\frac {e x^{2/3}}{d}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {e x^{2/3}}{d}\right )\right ) \]
(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2*Log[x] + 2*b*n *(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])*((Log[d + e*x^( 2/3)] - Log[1 + (e*x^(2/3))/d])*Log[x] - (3*PolyLog[2, -((e*x^(2/3))/d)])/ 2) + (3*b^2*n^2*(Log[d + e*x^(2/3)]^2*Log[-((e*x^(2/3))/d)] + 2*Log[d + e* x^(2/3)]*PolyLog[2, 1 + (e*x^(2/3))/d] - 2*PolyLog[3, 1 + (e*x^(2/3))/d])) /2
Time = 0.44 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2904, 2843, 2881, 2821, 7143}
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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle \frac {3}{2} \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^{2/3}}dx^{2/3}\) |
| \(\Big \downarrow \) 2843 |
| \(\displaystyle \frac {3}{2} \left (\log \left (-\frac {e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-2 b e n \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \log \left (-\frac {e x^{2/3}}{d}\right )}{d+e x^{2/3}}dx^{2/3}\right )\) |
| \(\Big \downarrow \) 2881 |
| \(\displaystyle \frac {3}{2} \left (\log \left (-\frac {e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-2 b n \int \frac {\log \left (-\frac {e x^{2/3}}{d}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{x^{2/3}}d\left (d+e x^{2/3}\right )\right )\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {3}{2} \left (\log \left (-\frac {e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-2 b n \left (b n \int \frac {\operatorname {PolyLog}\left (2,\frac {d+e x^{2/3}}{d}\right )}{x^{2/3}}d\left (d+e x^{2/3}\right )-\operatorname {PolyLog}\left (2,\frac {d+e x^{2/3}}{d}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {3}{2} \left (\log \left (-\frac {e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-2 b n \left (b n \operatorname {PolyLog}\left (3,\frac {d+e x^{2/3}}{d}\right )-\operatorname {PolyLog}\left (2,\frac {d+e x^{2/3}}{d}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )\right )\right )\) |
(3*((a + b*Log[c*(d + e*x^(2/3))^n])^2*Log[-((e*x^(2/3))/d)] - 2*b*n*(-((a + b*Log[c*x^((2*n)/3)])*PolyLog[2, (d + e*x^(2/3))/d]) + b*n*PolyLog[3, ( d + e*x^(2/3))/d])))/2
3.5.73.3.1 Defintions of rubi rules used
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b _.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c *x^n])^p/m), x] + Simp[b*n*(p/m) Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c *x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_. )*(x_)), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])^p/g), x] - Simp[b*e*n*(p/g) Int[Log[(e*(f + g*x))/(e*f - d*g)] *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[p, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log [(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym bol] :> Simp[1/e Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[h* ((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 0]
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[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )}^{2}}{x}d x\]
\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x} \,d x } \]
\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x^{\frac {2}{3}}\right )^{n} \right )}\right )^{2}}{x}\, dx \]
\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x} \,d x } \]
b^2*log((e*x^(2/3) + d)^n)^2*log(x) + integrate(1/3*(3*(b^2*e*log(c)^2 + 2 *a*b*e*log(c) + a^2*e)*x - 2*(2*b^2*e*n*x*log(x) - 3*(b^2*e*log(c) + a*b*e )*x - 3*(b^2*d*log(c) + a*b*d)*x^(1/3))*log((e*x^(2/3) + d)^n) + 3*(b^2*d* log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x^(1/3))/(e*x^2 + d*x^(4/3)), x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^2}{x} \,d x \]