Integrand size = 20, antiderivative size = 260 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=\frac {3 b e n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{d^2}+\frac {3 b e^2 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{d^2}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )}{d^2}-\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^2}-\frac {6 b^3 e^2 n^3 \operatorname {PolyLog}\left (2,1+\frac {e}{d \sqrt {x}}\right )}{d^2}-\frac {6 b^3 e^2 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^2} \]
3*b*e^2*n*ln(1-d/(d+e/x^(1/2)))*(a+b*ln(c*(d+e/x^(1/2))^n))^2/d^2+x*(a+b*l n(c*(d+e/x^(1/2))^n))^3-6*b^2*e^2*n^2*(a+b*ln(c*(d+e/x^(1/2))^n))*ln(-e/d/ x^(1/2))/d^2-6*b^2*e^2*n^2*(a+b*ln(c*(d+e/x^(1/2))^n))*polylog(2,d/(d+e/x^ (1/2)))/d^2-6*b^3*e^2*n^3*polylog(2,1+e/d/x^(1/2))/d^2-6*b^3*e^2*n^3*polyl og(3,d/(d+e/x^(1/2)))/d^2+3*b*e*n*(a+b*ln(c*(d+e/x^(1/2))^n))^2*(d+e/x^(1/ 2))*x^(1/2)/d^2
\[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=\int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx \]
Time = 1.13 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.91, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2901, 2904, 2845, 2858, 27, 2789, 2755, 2754, 2779, 2821, 2838, 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.
| \(\displaystyle \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 2901 |
| \(\displaystyle 2 \int \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3d\sqrt {x}\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -2 \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^{3/2}}d\frac {1}{\sqrt {x}}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle -2 \left (\frac {3}{2} b e n \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{\left (d+\frac {e}{\sqrt {x}}\right ) x}d\frac {1}{\sqrt {x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle -2 \left (\frac {3}{2} b n \int \left (d+\frac {e}{\sqrt {x}}\right ) x \left (a+b \log \left (c x^{n/2}\right )\right )^2d\left (d+\frac {e}{\sqrt {x}}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\frac {3}{2} b e^2 n \int \frac {\left (d+\frac {e}{\sqrt {x}}\right ) x \left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -2 \left (\frac {3}{2} b e^2 n \left (\frac {\int \frac {x \left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\int -\frac {\left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c x^{n/2}\right )\right )^2}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 2755 |
| \(\displaystyle -2 \left (\frac {3}{2} b e^2 n \left (\frac {-\frac {2 b n \int -\frac {\sqrt {x} \left (a+b \log \left (c x^{n/2}\right )\right )}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e}}{d}+\frac {\int -\frac {\left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c x^{n/2}\right )\right )^2}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle -2 \left (\frac {3}{2} b e^2 n \left (\frac {-\frac {2 b n \left (b n \int \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (1-\frac {d+\frac {e}{\sqrt {x}}}{d}\right )d\left (d+\frac {e}{\sqrt {x}}\right )-\log \left (1-\frac {d+\frac {e}{\sqrt {x}}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e}}{d}+\frac {\int -\frac {\left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c x^{n/2}\right )\right )^2}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle -2 \left (\frac {3}{2} b e^2 n \left (\frac {-\frac {2 b n \left (b n \int \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (1-\frac {d+\frac {e}{\sqrt {x}}}{d}\right )d\left (d+\frac {e}{\sqrt {x}}\right )-\log \left (1-\frac {d+\frac {e}{\sqrt {x}}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e}}{d}+\frac {\frac {2 b n \int \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (1-d \left (d+\frac {e}{\sqrt {x}}\right )\right ) \left (a+b \log \left (c x^{n/2}\right )\right )d\left (d+\frac {e}{\sqrt {x}}\right )}{d}-\frac {\log \left (1-d \left (d+\frac {e}{\sqrt {x}}\right )\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle -2 \left (\frac {3}{2} b e^2 n \left (\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,d \left (d+\frac {e}{\sqrt {x}}\right )\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-b n \int \left (d+\frac {e}{\sqrt {x}}\right ) \operatorname {PolyLog}\left (2,d \left (d+\frac {e}{\sqrt {x}}\right )\right )d\left (d+\frac {e}{\sqrt {x}}\right )\right )}{d}-\frac {\log \left (1-d \left (d+\frac {e}{\sqrt {x}}\right )\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d}}{d}+\frac {-\frac {2 b n \left (b n \int \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (1-\frac {d+\frac {e}{\sqrt {x}}}{d}\right )d\left (d+\frac {e}{\sqrt {x}}\right )-\log \left (1-\frac {d+\frac {e}{\sqrt {x}}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -2 \left (\frac {3}{2} b e^2 n \left (\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,d \left (d+\frac {e}{\sqrt {x}}\right )\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-b n \int \left (d+\frac {e}{\sqrt {x}}\right ) \operatorname {PolyLog}\left (2,d \left (d+\frac {e}{\sqrt {x}}\right )\right )d\left (d+\frac {e}{\sqrt {x}}\right )\right )}{d}-\frac {\log \left (1-d \left (d+\frac {e}{\sqrt {x}}\right )\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d}}{d}+\frac {-\frac {2 b n \left (-\log \left (1-\frac {d+\frac {e}{\sqrt {x}}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-b n \operatorname {PolyLog}\left (2,\frac {d+\frac {e}{\sqrt {x}}}{d}\right )\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -2 \left (\frac {3}{2} b e^2 n \left (\frac {-\frac {2 b n \left (-\log \left (1-\frac {d+\frac {e}{\sqrt {x}}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-b n \operatorname {PolyLog}\left (2,\frac {d+\frac {e}{\sqrt {x}}}{d}\right )\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e}}{d}+\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,d \left (d+\frac {e}{\sqrt {x}}\right )\right ) \left (a+b \log \left (c x^{n/2}\right )\right )+b n \operatorname {PolyLog}\left (3,d \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}{d}-\frac {\log \left (1-d \left (d+\frac {e}{\sqrt {x}}\right )\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 x}\right )\) |
-2*(-1/2*(a + b*Log[c*(d + e/Sqrt[x])^n])^3/x + (3*b*e^2*n*((-(((d + e/Sqr t[x])*Sqrt[x]*(a + b*Log[c*x^(n/2)])^2)/(d*e)) - (2*b*n*(-(Log[1 - (d + e/ Sqrt[x])/d]*(a + b*Log[c*x^(n/2)])) - b*n*PolyLog[2, (d + e/Sqrt[x])/d]))/ d)/d + (-((Log[1 - d*(d + e/Sqrt[x])]*(a + b*Log[c*x^(n/2)])^2)/d) + (2*b* n*((a + b*Log[c*x^(n/2)])*PolyLog[2, d*(d + e/Sqrt[x])] + b*n*PolyLog[3, d *(d + e/Sqrt[x])]))/d)/d))/2)
3.5.37.3.1 Defintions of rubi rules used
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_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[b*n*(p/d) Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[p, 0]
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[(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[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_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[((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]
\[\int {\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )}^{3}d x\]
\[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{3} \,d x } \]
integral(b^3*log(c*((d*x + e*sqrt(x))/x)^n)^3 + 3*a*b^2*log(c*((d*x + e*sq rt(x))/x)^n)^2 + 3*a^2*b*log(c*((d*x + e*sqrt(x))/x)^n) + a^3, x)
\[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=\int \left (a + b \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right )^{3}\, dx \]
\[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{3} \,d x } \]
b^3*x*log((d*sqrt(x) + e)^n)^3 - 3*(e*n*(e*log(d*sqrt(x) + e)/d^2 - sqrt(x )/d) - x*log(c*(d + e/sqrt(x))^n))*a^2*b + a^3*x - integrate(1/2*(2*(b^3*d *x + b^3*e*sqrt(x))*log(x^(1/2*n))^3 + 3*(b^3*d*n*x - 2*(b^3*d*log(c) + a* b^2*d)*x + 2*(b^3*d*x + b^3*e*sqrt(x))*log(x^(1/2*n)) - 2*(b^3*e*log(c) + a*b^2*e)*sqrt(x))*log((d*sqrt(x) + e)^n)^2 - 6*((b^3*d*log(c) + a*b^2*d)*x + (b^3*e*log(c) + a*b^2*e)*sqrt(x))*log(x^(1/2*n))^2 - 2*(b^3*d*log(c)^3 + 3*a*b^2*d*log(c)^2)*x - 6*((b^3*d*x + b^3*e*sqrt(x))*log(x^(1/2*n))^2 + (b^3*d*log(c)^2 + 2*a*b^2*d*log(c))*x - 2*((b^3*d*log(c) + a*b^2*d)*x + (b ^3*e*log(c) + a*b^2*e)*sqrt(x))*log(x^(1/2*n)) + (b^3*e*log(c)^2 + 2*a*b^2 *e*log(c))*sqrt(x))*log((d*sqrt(x) + e)^n) + 6*((b^3*d*log(c)^2 + 2*a*b^2* d*log(c))*x + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c))*sqrt(x))*log(x^(1/2*n)) - 2*(b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2)*sqrt(x))/(d*x + e*sqrt(x)), x)
\[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\right )}^3 \,d x \]