Integrand size = 22, antiderivative size = 51 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+2 b n \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right ) \] Output:
2*(a+b*ln(c*(d+e*x^(1/2))^n))*ln(-e*x^(1/2)/d)+2*b*n*polylog(2,1+e*x^(1/2) /d)
Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.04 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=2 b \log \left (c \left (d+e \sqrt {x}\right )^n\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+a \log (x)+2 b n \operatorname {PolyLog}\left (2,\frac {d+e \sqrt {x}}{d}\right ) \] Input:
Integrate[(a + b*Log[c*(d + e*Sqrt[x])^n])/x,x]
Output:
2*b*Log[c*(d + e*Sqrt[x])^n]*Log[-((e*Sqrt[x])/d)] + a*Log[x] + 2*b*n*Poly Log[2, (d + e*Sqrt[x])/d]
Time = 0.42 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2904, 2841, 2752}
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.
\(\displaystyle \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle 2 \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{\sqrt {x}}d\sqrt {x}\) |
\(\Big \downarrow \) 2841 |
\(\displaystyle 2 \left (\log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-b e n \int \frac {\log \left (-\frac {e \sqrt {x}}{d}\right )}{d+e \sqrt {x}}d\sqrt {x}\right )\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle 2 \left (\log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {\sqrt {x} e}{d}+1\right )\right )\) |
Input:
Int[(a + b*Log[c*(d + e*Sqrt[x])^n])/x,x]
Output:
2*((a + b*Log[c*(d + e*Sqrt[x])^n])*Log[-((e*Sqrt[x])/d)] + b*n*PolyLog[2, 1 + (e*Sqrt[x])/d])
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_ )), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x )^n])/g), x] - Simp[b*e*(n/g) Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 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])
\[\int \frac {a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )}{x}d x\]
Input:
int((a+b*ln(c*(d+e*x^(1/2))^n))/x,x)
Output:
int((a+b*ln(c*(d+e*x^(1/2))^n))/x,x)
\[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=\int { \frac {b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a}{x} \,d x } \] Input:
integrate((a+b*log(c*(d+e*x^(1/2))^n))/x,x, algorithm="fricas")
Output:
integral((b*log((e*sqrt(x) + d)^n*c) + a)/x, x)
\[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=\int \frac {a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{x}\, dx \] Input:
integrate((a+b*ln(c*(d+e*x**(1/2))**n))/x,x)
Output:
Integral((a + b*log(c*(d + e*sqrt(x))**n))/x, x)
Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (44) = 88\).
Time = 0.17 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.12 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=-2 \, {\left (\log \left (\frac {e \sqrt {x}}{d} + 1\right ) \log \left (\sqrt {x}\right ) + {\rm Li}_2\left (-\frac {e \sqrt {x}}{d}\right )\right )} b n + \frac {2 \, {\left (b e n \sqrt {x} \log \left (\sqrt {x}\right ) - b e n \sqrt {x}\right )}}{d} + \frac {b d \log \left ({\left (e \sqrt {x} + d\right )}^{n}\right ) \log \left (x\right ) + {\left (b d \log \left (c\right ) + a d\right )} \log \left (x\right ) - \frac {b e n x \log \left (x\right ) - 2 \, b e n x}{\sqrt {x}}}{d} \] Input:
integrate((a+b*log(c*(d+e*x^(1/2))^n))/x,x, algorithm="maxima")
Output:
-2*(log(e*sqrt(x)/d + 1)*log(sqrt(x)) + dilog(-e*sqrt(x)/d))*b*n + 2*(b*e* n*sqrt(x)*log(sqrt(x)) - b*e*n*sqrt(x))/d + (b*d*log((e*sqrt(x) + d)^n)*lo g(x) + (b*d*log(c) + a*d)*log(x) - (b*e*n*x*log(x) - 2*b*e*n*x)/sqrt(x))/d
\[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=\int { \frac {b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a}{x} \,d x } \] Input:
integrate((a+b*log(c*(d+e*x^(1/2))^n))/x,x, algorithm="giac")
Output:
integrate((b*log((e*sqrt(x) + d)^n*c) + a)/x, x)
Timed out. \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{x} \,d x \] Input:
int((a + b*log(c*(d + e*x^(1/2))^n))/x,x)
Output:
int((a + b*log(c*(d + e*x^(1/2))^n))/x, x)
\[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx=\frac {\left (\int \frac {\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )}{-e^{2} x^{2}+d^{2} x}d x \right ) b \,d^{2} n -\left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )}{-e^{2} x^{2}+d^{2} x}d x \right ) b d e n +\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )^{2} b +\mathrm {log}\left (x \right ) a n}{n} \] Input:
int((a+b*log(c*(d+e*x^(1/2))^n))/x,x)
Output:
(int(log((sqrt(x)*e + d)**n*c)/(d**2*x - e**2*x**2),x)*b*d**2*n - int((sqr t(x)*log((sqrt(x)*e + d)**n*c))/(d**2*x - e**2*x**2),x)*b*d*e*n + log((sqr t(x)*e + d)**n*c)**2*b + log(x)*a*n)/n