Integrand size = 22, antiderivative size = 55 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=-\frac {3}{2} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac {e}{d x^{2/3}}\right )-\frac {3}{2} b n \operatorname {PolyLog}\left (2,1+\frac {e}{d x^{2/3}}\right ) \] Output:
-3/2*(a+b*ln(c*(d+e/x^(2/3))^n))*ln(-e/d/x^(2/3))-3/2*b*n*polylog(2,1+e/d/ x^(2/3))
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=a \log (x)-\frac {3}{2} b \left (\log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right ) \log \left (-\frac {e}{d x^{2/3}}\right )+n \operatorname {PolyLog}\left (2,\frac {d+\frac {e}{x^{2/3}}}{d}\right )\right ) \] Input:
Integrate[(a + b*Log[c*(d + e/x^(2/3))^n])/x,x]
Output:
a*Log[x] - (3*b*(Log[c*(d + e/x^(2/3))^n]*Log[-(e/(d*x^(2/3)))] + n*PolyLo g[2, (d + e/x^(2/3))/d]))/2
Time = 0.45 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96, 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.
\(\displaystyle \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle -\frac {3}{2} \int x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )d\frac {1}{x^{2/3}}\) |
\(\Big \downarrow \) 2841 |
\(\displaystyle -\frac {3}{2} \left (\log \left (-\frac {e}{d x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-b e n \int \frac {\log \left (-\frac {e}{d x^{2/3}}\right )}{d+\frac {e}{x^{2/3}}}d\frac {1}{x^{2/3}}\right )\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle -\frac {3}{2} \left (\log \left (-\frac {e}{d x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {e}{d x^{2/3}}+1\right )\right )\) |
Input:
Int[(a + b*Log[c*(d + e/x^(2/3))^n])/x,x]
Output:
(-3*((a + b*Log[c*(d + e/x^(2/3))^n])*Log[-(e/(d*x^(2/3)))] + b*n*PolyLog[ 2, 1 + e/(d*x^(2/3))]))/2
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])
\[\int \frac {a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )}{x}d x\]
Input:
int((a+b*ln(c*(d+e/x^(2/3))^n))/x,x)
Output:
int((a+b*ln(c*(d+e/x^(2/3))^n))/x,x)
\[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=\int { \frac {b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a}{x} \,d x } \] Input:
integrate((a+b*log(c*(d+e/x^(2/3))^n))/x,x, algorithm="fricas")
Output:
integral((b*log(c*((d*x + e*x^(1/3))/x)^n) + a)/x, x)
Timed out. \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=\text {Timed out} \] Input:
integrate((a+b*ln(c*(d+e/x**(2/3))**n))/x,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (44) = 88\).
Time = 0.40 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.31 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=-\frac {3}{2} \, {\left (2 \, \log \left (\frac {d x^{\frac {2}{3}}}{e} + 1\right ) \log \left (x^{\frac {1}{3}}\right ) + {\rm Li}_2\left (-\frac {d x^{\frac {2}{3}}}{e}\right )\right )} b n + \frac {2 \, b e n \log \left (x\right )^{2} + 6 \, b d n x^{\frac {2}{3}} \log \left (x\right ) + 6 \, b e \log \left ({\left (d x^{\frac {2}{3}} + e\right )}^{n}\right ) \log \left (x\right ) - 12 \, b e \log \left (x\right ) \log \left (x^{\frac {1}{3} \, n}\right ) - 9 \, b d n x^{\frac {2}{3}} + 6 \, {\left (b e \log \left (c\right ) + a e\right )} \log \left (x\right ) - \frac {3 \, {\left (2 \, b d n x \log \left (x\right ) - 3 \, b d n x\right )}}{x^{\frac {1}{3}}}}{6 \, e} \] Input:
integrate((a+b*log(c*(d+e/x^(2/3))^n))/x,x, algorithm="maxima")
Output:
-3/2*(2*log(d*x^(2/3)/e + 1)*log(x^(1/3)) + dilog(-d*x^(2/3)/e))*b*n + 1/6 *(2*b*e*n*log(x)^2 + 6*b*d*n*x^(2/3)*log(x) + 6*b*e*log((d*x^(2/3) + e)^n) *log(x) - 12*b*e*log(x)*log(x^(1/3*n)) - 9*b*d*n*x^(2/3) + 6*(b*e*log(c) + a*e)*log(x) - 3*(2*b*d*n*x*log(x) - 3*b*d*n*x)/x^(1/3))/e
\[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=\int { \frac {b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a}{x} \,d x } \] Input:
integrate((a+b*log(c*(d+e/x^(2/3))^n))/x,x, algorithm="giac")
Output:
integrate((b*log(c*(d + e/x^(2/3))^n) + a)/x, x)
Timed out. \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{x} \,d x \] Input:
int((a + b*log(c*(d + e/x^(2/3))^n))/x,x)
Output:
int((a + b*log(c*(d + e/x^(2/3))^n))/x, x)
\[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=\left (\int \frac {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}{x}d x \right ) b +\mathrm {log}\left (x \right ) a \] Input:
int((a+b*log(c*(d+e/x^(2/3))^n))/x,x)
Output:
int(log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))/x,x)*b + log(x)*a