\(\int \frac {a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^n)}{x} \, dx\) [493]

Optimal result

Integrand size = 22, antiderivative size = 51 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=-3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )-3 b n \operatorname {PolyLog}\left (2,1+\frac {e}{d \sqrt [3]{x}}\right ) \] Output:

-3*(a+b*ln(c*(d+e/x^(1/3))^n))*ln(-e/d/x^(1/3))-3*b*n*polylog(2,1+e/d/x^(1 
/3))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.04 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=-3 b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )+a \log (x)-3 b n \operatorname {PolyLog}\left (2,\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right ) \] Input:

Integrate[(a + b*Log[c*(d + e/x^(1/3))^n])/x,x]
 

Output:

-3*b*Log[c*(d + e/x^(1/3))^n]*Log[-(e/(d*x^(1/3)))] + a*Log[x] - 3*b*n*Pol 
yLog[2, (d + e/x^(1/3))/d]
 

Rubi [A] (verified)

Time = 0.43 (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.

\(\displaystyle \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx\)

\(\Big \downarrow \) 2904

\(\displaystyle -3 \int \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )d\frac {1}{\sqrt [3]{x}}\)

\(\Big \downarrow \) 2841

\(\displaystyle -3 \left (\log \left (-\frac {e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-b e n \int \frac {\log \left (-\frac {e}{d \sqrt [3]{x}}\right )}{d+\frac {e}{\sqrt [3]{x}}}d\frac {1}{\sqrt [3]{x}}\right )\)

\(\Big \downarrow \) 2752

\(\displaystyle -3 \left (\log \left (-\frac {e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {e}{d \sqrt [3]{x}}+1\right )\right )\)

Input:

Int[(a + b*Log[c*(d + e/x^(1/3))^n])/x,x]
 

Output:

-3*((a + b*Log[c*(d + e/x^(1/3))^n])*Log[-(e/(d*x^(1/3)))] + b*n*PolyLog[2 
, 1 + e/(d*x^(1/3))])
 

Defintions of rubi rules used

rule 2752

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]
 

rule 2841

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]
 

rule 2904

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])
 

Maple [F]

\[\int \frac {a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )}{x}d x\]

Input:

int((a+b*ln(c*(d+e/x^(1/3))^n))/x,x)
 

Output:

int((a+b*ln(c*(d+e/x^(1/3))^n))/x,x)
 

Fricas [F]

\[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=\int { \frac {b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a}{x} \,d x } \] Input:

integrate((a+b*log(c*(d+e/x^(1/3))^n))/x,x, algorithm="fricas")
 

Output:

integral((b*log(c*((d*x + e*x^(2/3))/x)^n) + a)/x, x)
 

Sympy [F]

\[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=\int \frac {a + b \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}}{x}\, dx \] Input:

integrate((a+b*ln(c*(d+e/x**(1/3))**n))/x,x)
 

Output:

Integral((a + b*log(c*(d + e/x**(1/3))**n))/x, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (44) = 88\).

Time = 0.44 (sec) , antiderivative size = 185, normalized size of antiderivative = 3.63 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=-3 \, {\left (\log \left (\frac {d x^{\frac {1}{3}}}{e} + 1\right ) \log \left (x^{\frac {1}{3}}\right ) + {\rm Li}_2\left (-\frac {d x^{\frac {1}{3}}}{e}\right )\right )} b n + \frac {2 \, b e^{2} n \log \left (x\right )^{2} + 12 \, b e^{2} \log \left ({\left (d x^{\frac {1}{3}} + e\right )}^{n}\right ) \log \left (x\right ) - 12 \, b e^{2} \log \left (x\right ) \log \left (x^{\frac {1}{3} \, n}\right ) + 9 \, b d^{2} n x^{\frac {2}{3}} - 36 \, b d e n x^{\frac {1}{3}} - 6 \, {\left (b d^{2} n x^{\frac {2}{3}} - 2 \, b d e n x^{\frac {1}{3}}\right )} \log \left (x\right ) + 12 \, {\left (b e^{2} \log \left (c\right ) + a e^{2}\right )} \log \left (x\right ) + \frac {3 \, {\left (2 \, b d^{2} n x \log \left (x\right ) - 3 \, b d^{2} n x\right )}}{x^{\frac {1}{3}}} - \frac {12 \, {\left (b d e n x \log \left (x\right ) - 3 \, b d e n x\right )}}{x^{\frac {2}{3}}}}{12 \, e^{2}} \] Input:

integrate((a+b*log(c*(d+e/x^(1/3))^n))/x,x, algorithm="maxima")
 

Output:

-3*(log(d*x^(1/3)/e + 1)*log(x^(1/3)) + dilog(-d*x^(1/3)/e))*b*n + 1/12*(2 
*b*e^2*n*log(x)^2 + 12*b*e^2*log((d*x^(1/3) + e)^n)*log(x) - 12*b*e^2*log( 
x)*log(x^(1/3*n)) + 9*b*d^2*n*x^(2/3) - 36*b*d*e*n*x^(1/3) - 6*(b*d^2*n*x^ 
(2/3) - 2*b*d*e*n*x^(1/3))*log(x) + 12*(b*e^2*log(c) + a*e^2)*log(x) + 3*( 
2*b*d^2*n*x*log(x) - 3*b*d^2*n*x)/x^(1/3) - 12*(b*d*e*n*x*log(x) - 3*b*d*e 
*n*x)/x^(2/3))/e^2
 

Giac [F]

\[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=\int { \frac {b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a}{x} \,d x } \] Input:

integrate((a+b*log(c*(d+e/x^(1/3))^n))/x,x, algorithm="giac")
 

Output:

integrate((b*log(c*(d + e/x^(1/3))^n) + a)/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{x} \,d x \] Input:

int((a + b*log(c*(d + e/x^(1/3))^n))/x,x)
 

Output:

int((a + b*log(c*(d + e/x^(1/3))^n))/x, x)
 

Reduce [F]

\[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=\left (\int \frac {\mathrm {log}\left (\frac {\left (x^{\frac {1}{3}} d +e \right )^{n} c}{x^{\frac {n}{3}}}\right )}{x}d x \right ) b +\mathrm {log}\left (x \right ) a \] Input:

int((a+b*log(c*(d+e/x^(1/3))^n))/x,x)
 

Output:

int(log(((x**(1/3)*d + e)**n*c)/x**(n/3))/x,x)*b + log(x)*a