\(\int \frac {\log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n))}{x} \, dx\) [55]

Optimal result

Integrand size = 28, antiderivative size = 39 \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )+4 b n \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right ) \] Output:

-2*(a+b*ln(c*x^n))*polylog(2,-d*f*x^(1/2))+4*b*n*polylog(3,-d*f*x^(1/2))
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.28 \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-2 a \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )-2 b \log \left (c x^n\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )+4 b n \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right ) \] Input:

Integrate[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x,x]
 

Output:

-2*a*PolyLog[2, -(d*f*Sqrt[x])] - 2*b*Log[c*x^n]*PolyLog[2, -(d*f*Sqrt[x]) 
] + 4*b*n*PolyLog[3, -(d*f*Sqrt[x])]
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2821, 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.

\(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\)

\(\Big \downarrow \) 2821

\(\displaystyle 2 b n \int \frac {\operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{x}dx-2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle 4 b n \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )-2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )\)

Input:

Int[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x,x]
 

Output:

-2*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] + 4*b*n*PolyLog[3, -(d*f* 
Sqrt[x])]
 

Defintions of rubi rules used

rule 2821

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]
 

rule 7143

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]
 

Maple [F]

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

Input:

int(ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x^n))/x,x)
 

Output:

int(ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x^n))/x,x)
 

Fricas [F]

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

integrate(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))/x,x, algorithm="fricas")
 

Output:

integral((b*log(c*x^n) + a)*log(d*f*sqrt(x) + 1)/x, x)
 

Sympy [F(-1)]

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

integrate(ln(d*(1/d+f*x**(1/2)))*(a+b*ln(c*x**n))/x,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))/x,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))/x,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((log(d*(f*x^(1/2) + 1/d))*(a + b*log(c*x^n)))/x,x)
 

Output:

int((log(d*(f*x^(1/2) + 1/d))*(a + b*log(c*x^n)))/x, x)
 

Reduce [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\left (\int \frac {\mathrm {log}\left (\sqrt {x}\, d f +1\right )}{d^{2} f^{2} x^{2}-x}d x \right ) a +\left (\int \frac {\mathrm {log}\left (\sqrt {x}\, d f +1\right ) \mathrm {log}\left (x^{n} c \right )}{x}d x \right ) b +\left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, d f +1\right )}{d^{2} f^{2} x^{2}-x}d x \right ) a d f +\mathrm {log}\left (\sqrt {x}\, d f +1\right )^{2} a \] Input:

int(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))/x,x)
 

Output:

 - int(log(sqrt(x)*d*f + 1)/(d**2*f**2*x**2 - x),x)*a + int((log(sqrt(x)*d 
*f + 1)*log(x**n*c))/x,x)*b + int((sqrt(x)*log(sqrt(x)*d*f + 1))/(d**2*f** 
2*x**2 - x),x)*a*d*f + log(sqrt(x)*d*f + 1)**2*a