Integrand size = 24, antiderivative size = 135 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x} \, dx=2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \log \left (-\frac {e \sqrt {x}}{d}\right )+6 b n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )-12 b^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \operatorname {PolyLog}\left (3,1+\frac {e \sqrt {x}}{d}\right )+12 b^3 n^3 \operatorname {PolyLog}\left (4,1+\frac {e \sqrt {x}}{d}\right ) \]
2*ln(-e*x^(1/2)/d)*(a+b*ln(c*(d+e*x^(1/2))^n))^3+6*b*n*(a+b*ln(c*(d+e*x^(1 /2))^n))^2*polylog(2,1+e*x^(1/2)/d)-12*b^2*n^2*(a+b*ln(c*(d+e*x^(1/2))^n)) *polylog(3,1+e*x^(1/2)/d)+12*b^3*n^3*polylog(4,1+e*x^(1/2)/d)
Leaf count is larger than twice the leaf count of optimal. \(333\) vs. \(2(135)=270\).
Time = 0.11 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.47 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x} \, dx=\left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \log (x)+3 b n \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \left (\left (\log \left (d+e \sqrt {x}\right )-\log \left (1+\frac {e \sqrt {x}}{d}\right )\right ) \log (x)-2 \operatorname {PolyLog}\left (2,-\frac {e \sqrt {x}}{d}\right )\right )+6 b^2 n^2 \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (\log ^2\left (d+e \sqrt {x}\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+2 \log \left (d+e \sqrt {x}\right ) \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {e \sqrt {x}}{d}\right )\right )+2 b^3 n^3 \left (\log ^3\left (d+e \sqrt {x}\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+3 \log ^2\left (d+e \sqrt {x}\right ) \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )-6 \log \left (d+e \sqrt {x}\right ) \operatorname {PolyLog}\left (3,1+\frac {e \sqrt {x}}{d}\right )+6 \operatorname {PolyLog}\left (4,1+\frac {e \sqrt {x}}{d}\right )\right ) \]
(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])^3*Log[x] + 3*b*n *(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])^2*((Log[d + e*S qrt[x]] - Log[1 + (e*Sqrt[x])/d])*Log[x] - 2*PolyLog[2, -((e*Sqrt[x])/d)]) + 6*b^2*n^2*(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])*(Lo g[d + e*Sqrt[x]]^2*Log[-((e*Sqrt[x])/d)] + 2*Log[d + e*Sqrt[x]]*PolyLog[2, 1 + (e*Sqrt[x])/d] - 2*PolyLog[3, 1 + (e*Sqrt[x])/d]) + 2*b^3*n^3*(Log[d + e*Sqrt[x]]^3*Log[-((e*Sqrt[x])/d)] + 3*Log[d + e*Sqrt[x]]^2*PolyLog[2, 1 + (e*Sqrt[x])/d] - 6*Log[d + e*Sqrt[x]]*PolyLog[3, 1 + (e*Sqrt[x])/d] + 6 *PolyLog[4, 1 + (e*Sqrt[x])/d])
Time = 0.52 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2904, 2843, 2881, 2821, 2830, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle 2 \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{\sqrt {x}}d\sqrt {x}\) |
| \(\Big \downarrow \) 2843 |
| \(\displaystyle 2 \left (\log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3-3 b e n \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \log \left (-\frac {e \sqrt {x}}{d}\right )}{d+e \sqrt {x}}d\sqrt {x}\right )\) |
| \(\Big \downarrow \) 2881 |
| \(\displaystyle 2 \left (\log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3-3 b n \int \frac {\log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{\sqrt {x}}d\left (d+e \sqrt {x}\right )\right )\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle 2 \left (\log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3-3 b n \left (2 b n \int \frac {\left (a+b \log \left (c x^{n/2}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d+e \sqrt {x}}{d}\right )}{\sqrt {x}}d\left (d+e \sqrt {x}\right )-\operatorname {PolyLog}\left (2,\frac {d+e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2\right )\right )\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle 2 \left (\log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3-3 b n \left (2 b n \left (\operatorname {PolyLog}\left (3,\frac {d+e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (3,\frac {d+e \sqrt {x}}{d}\right )}{\sqrt {x}}d\left (d+e \sqrt {x}\right )\right )-\operatorname {PolyLog}\left (2,\frac {d+e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2\right )\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle 2 \left (\log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3-3 b n \left (2 b n \left (\operatorname {PolyLog}\left (3,\frac {d+e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-b n \operatorname {PolyLog}\left (4,\frac {d+e \sqrt {x}}{d}\right )\right )-\operatorname {PolyLog}\left (2,\frac {d+e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2\right )\right )\) |
2*((a + b*Log[c*(d + e*Sqrt[x])^n])^3*Log[-((e*Sqrt[x])/d)] - 3*b*n*(-((a + b*Log[c*x^(n/2)])^2*PolyLog[2, (d + e*Sqrt[x])/d]) + 2*b*n*((a + b*Log[c *x^(n/2)])*PolyLog[3, (d + e*Sqrt[x])/d] - b*n*PolyLog[4, (d + e*Sqrt[x])/ d])))
3.5.18.3.1 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_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ .)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) , x] - Simp[b*n*(p/q) Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
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]
\[\int \frac {{\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )}^{3}}{x}d x\]
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x} \,d x } \]
integral((b^3*log((e*sqrt(x) + d)^n*c)^3 + 3*a*b^2*log((e*sqrt(x) + d)^n*c )^2 + 3*a^2*b*log((e*sqrt(x) + d)^n*c) + a^3)/x, x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{3}}{x}\, dx \]
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x} \,d x } \]
b^3*log((e*sqrt(x) + d)^n)^3*log(x) + integrate(-1/2*(3*(b^3*e*n*x*log(x) - 2*(b^3*e*log(c) + a*b^2*e)*x - 2*(b^3*d*log(c) + a*b^2*d)*sqrt(x))*log(( e*sqrt(x) + d)^n)^2 - 2*(b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2 + 3*a^2*b*e*l og(c) + a^3*e)*x - 6*((b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x + (b ^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*sqrt(x))*log((e*sqrt(x) + d)^n ) - 2*(b^3*d*log(c)^3 + 3*a*b^2*d*log(c)^2 + 3*a^2*b*d*log(c) + a^3*d)*sqr t(x))/(e*x^2 + d*x^(3/2)), x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\right )}^3}{x} \,d x \]