Integrand size = 20, antiderivative size = 152 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\frac {2 b e n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^2}+\frac {2 b e^2 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^2}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {b^2 e^2 n^2 \log (x)}{d^2}-\frac {2 b^2 e^2 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^2} \]
b^2*e^2*n^2*ln(x)/d^2+2*b*e^2*n*ln(1-d/(d+e/x^(1/2)))*(a+b*ln(c*(d+e/x^(1/ 2))^n))/d^2+x*(a+b*ln(c*(d+e/x^(1/2))^n))^2-2*b^2*e^2*n^2*polylog(2,d/(d+e /x^(1/2)))/d^2+2*b*e*n*(a+b*ln(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))*x^(1/2)/d ^2
Time = 0.08 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.13 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {b e n \left (2 a d \sqrt {x}+2 b e n \log \left (d+\frac {e}{\sqrt {x}}\right )+2 b d \sqrt {x} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-2 e \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \log \left (e+d \sqrt {x}\right )+b e n \log (x)+b e n \left (\log \left (e+d \sqrt {x}\right ) \left (\log \left (e+d \sqrt {x}\right )-2 \log \left (-\frac {d \sqrt {x}}{e}\right )\right )-2 \operatorname {PolyLog}\left (2,1+\frac {d \sqrt {x}}{e}\right )\right )\right )}{d^2} \]
x*(a + b*Log[c*(d + e/Sqrt[x])^n])^2 + (b*e*n*(2*a*d*Sqrt[x] + 2*b*e*n*Log [d + e/Sqrt[x]] + 2*b*d*Sqrt[x]*Log[c*(d + e/Sqrt[x])^n] - 2*e*(a + b*Log[ c*(d + e/Sqrt[x])^n])*Log[e + d*Sqrt[x]] + b*e*n*Log[x] + b*e*n*(Log[e + d *Sqrt[x]]*(Log[e + d*Sqrt[x]] - 2*Log[-((d*Sqrt[x])/e)]) - 2*PolyLog[2, 1 + (d*Sqrt[x])/e])))/d^2
Time = 0.65 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2901, 2904, 2845, 2858, 27, 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.
\(\displaystyle \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx\) |
\(\Big \downarrow \) 2901 |
\(\displaystyle 2 \int \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2d\sqrt {x}\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle -2 \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^{3/2}}d\frac {1}{\sqrt {x}}\) |
\(\Big \downarrow \) 2845 |
\(\displaystyle -2 \left (b e n \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{\left (d+\frac {e}{\sqrt {x}}\right ) x}d\frac {1}{\sqrt {x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x}\right )\) |
\(\Big \downarrow \) 2858 |
\(\displaystyle -2 \left (b n \int \left (d+\frac {e}{\sqrt {x}}\right ) x \left (a+b \log \left (c x^{n/2}\right )\right )d\left (d+\frac {e}{\sqrt {x}}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -2 \left (b e^2 n \int \frac {\left (d+\frac {e}{\sqrt {x}}\right ) x \left (a+b \log \left (c x^{n/2}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x}\right )\) |
\(\Big \downarrow \) 2789 |
\(\displaystyle -2 \left (b e^2 n \left (\frac {\int \frac {x \left (a+b \log \left (c x^{n/2}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\int -\frac {\left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c x^{n/2}\right )\right )}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x}\right )\) |
\(\Big \downarrow \) 2751 |
\(\displaystyle -2 \left (b e^2 n \left (\frac {-\frac {b n \int -\frac {\sqrt {x}}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e}}{d}+\frac {\int -\frac {\left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c x^{n/2}\right )\right )}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle -2 \left (b e^2 n \left (\frac {\int -\frac {\left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c x^{n/2}\right )\right )}{e}d\left (d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {\frac {b n \log \left (-\frac {e}{\sqrt {x}}\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x}\right )\) |
\(\Big \downarrow \) 2779 |
\(\displaystyle -2 \left (b e^2 n \left (\frac {\frac {b n \int \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (1-d \left (d+\frac {e}{\sqrt {x}}\right )\right )d\left (d+\frac {e}{\sqrt {x}}\right )}{d}-\frac {\log \left (1-d \left (d+\frac {e}{\sqrt {x}}\right )\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d}}{d}+\frac {\frac {b n \log \left (-\frac {e}{\sqrt {x}}\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -2 \left (b e^2 n \left (\frac {\frac {b n \operatorname {PolyLog}\left (2,d \left (d+\frac {e}{\sqrt {x}}\right )\right )}{d}-\frac {\log \left (1-d \left (d+\frac {e}{\sqrt {x}}\right )\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d}}{d}+\frac {\frac {b n \log \left (-\frac {e}{\sqrt {x}}\right )}{d}-\frac {\sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{d e}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x}\right )\) |
-2*(-1/2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2/x + b*e^2*n*(((b*n*Log[-(e/Sqr t[x])])/d - ((d + e/Sqrt[x])*Sqrt[x]*(a + b*Log[c*x^(n/2)]))/(d*e))/d + (- ((Log[1 - d*(d + e/Sqrt[x])]*(a + b*Log[c*x^(n/2)]))/d) + (b*n*PolyLog[2, d*(d + e/Sqrt[x])])/d)/d))
3.5.31.3.1 Defintions of rubi rules used
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_.) + 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_.)/((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_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])
\[\int {\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )}^{2}d x\]
\[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2} \,d x } \]
integral(b^2*log(c*((d*x + e*sqrt(x))/x)^n)^2 + 2*a*b*log(c*((d*x + e*sqrt (x))/x)^n) + a^2, x)
\[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int \left (a + b \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right )^{2}\, dx \]
\[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2} \,d x } \]
-2*(e*n*(e*log(d*sqrt(x) + e)/d^2 - sqrt(x)/d) - x*log(c*(d + e/sqrt(x))^n ))*a*b + (x*log((d*sqrt(x) + e)^n)^2 - integrate(-(d*x*log(c)^2 + e*sqrt(x )*log(c)^2 + (d*x + e*sqrt(x))*log(x^(1/2*n))^2 - (d*n*x - 2*d*x*log(c) - 2*e*sqrt(x)*log(c) + 2*(d*x + e*sqrt(x))*log(x^(1/2*n)))*log((d*sqrt(x) + e)^n) - 2*(d*x*log(c) + e*sqrt(x)*log(c))*log(x^(1/2*n)))/(d*x + e*sqrt(x) ), x))*b^2 + a^2*x
\[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\right )}^2 \,d x \]