Integrand size = 23, antiderivative size = 159 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d}-\frac {p \operatorname {PolyLog}\left (2,1+\frac {b}{a x}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{d}-\frac {p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{d} \] Output:
-ln(c*(a+b/x)^p)*ln(-b/a/x)/d-ln(c*(a+b/x)^p)*ln(e*x+d)/d-p*ln(-e*x/d)*ln( e*x+d)/d+p*ln(-e*(a*x+b)/(a*d-b*e))*ln(e*x+d)/d-p*polylog(2,1+b/a/x)/d+p*p olylog(2,a*(e*x+d)/(a*d-b*e))/d-p*polylog(2,1+e*x/d)/d
Time = 0.05 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.01 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d}-\frac {p \operatorname {PolyLog}\left (2,\frac {a+\frac {b}{x}}{a}\right )}{d}-\frac {p \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{d} \] Input:
Integrate[Log[c*(a + b/x)^p]/(x*(d + e*x)),x]
Output:
-((Log[c*(a + b/x)^p]*Log[-(b/(a*x))])/d) - (Log[c*(a + b/x)^p]*Log[d + e* x])/d - (p*Log[-((e*x)/d)]*Log[d + e*x])/d + (p*Log[-((e*(b + a*x))/(a*d - b*e))]*Log[d + e*x])/d - (p*PolyLog[2, (a + b/x)/a])/d - (p*PolyLog[2, (d + e*x)/d])/d + (p*PolyLog[2, (a*(d + e*x))/(a*d - b*e)])/d
Time = 0.81 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2916, 2009}
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.
| \(\displaystyle \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx\) |
| \(\Big \downarrow \) 2916 |
| \(\displaystyle \int \left (\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d x}-\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d (d+e x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d}-\frac {\log \left (-\frac {b}{a x}\right ) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{d}+\frac {p \log (d+e x) \log \left (-\frac {e (a x+b)}{a d-b e}\right )}{d}-\frac {p \operatorname {PolyLog}\left (2,\frac {b}{a x}+1\right )}{d}-\frac {p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}\) |
Input:
Int[Log[c*(a + b/x)^p]/(x*(d + e*x)),x]
Output:
-((Log[c*(a + b/x)^p]*Log[-(b/(a*x))])/d) - (Log[c*(a + b/x)^p]*Log[d + e* x])/d - (p*Log[-((e*x)/d)]*Log[d + e*x])/d + (p*Log[-((e*(b + a*x))/(a*d - b*e))]*Log[d + e*x])/d - (p*PolyLog[2, 1 + b/(a*x)])/d + (p*PolyLog[2, (a *(d + e*x))/(a*d - b*e)])/d - (p*PolyLog[2, 1 + (e*x)/d])/d
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log [c*(d + e*x^n)^p])^q, x^m*(f + g*x)^r, x], x] /; FreeQ[{a, b, c, d, e, f, g , n, p, q}, x] && IntegerQ[m] && IntegerQ[r]
Time = 1.18 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.31
| method | result | size |
| parts | \(-\frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right ) \ln \left (e x +d \right )}{d}+\frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right ) \ln \left (x \right )}{d}+p b \left (\frac {\ln \left (x \right )^{2}}{2 d b}-\frac {\operatorname {dilog}\left (\frac {a x +b}{b}\right )}{d b}-\frac {\ln \left (x \right ) \ln \left (\frac {a x +b}{b}\right )}{d b}+\frac {\operatorname {dilog}\left (\frac {-d a +a \left (e x +d \right )+b e}{-d a +b e}\right )}{d b}+\frac {\ln \left (e x +d \right ) \ln \left (\frac {-d a +a \left (e x +d \right )+b e}{-d a +b e}\right )}{d b}-\frac {\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d b}-\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )}{d b}\right )\) | \(209\) |
Input:
int(ln(c*(a+b/x)^p)/x/(e*x+d),x,method=_RETURNVERBOSE)
Output:
-ln(c*(a+b/x)^p)*ln(e*x+d)/d+ln(c*(a+b/x)^p)/d*ln(x)+p*b*(1/2/d*ln(x)^2/b- 1/d/b*dilog((a*x+b)/b)-1/d/b*ln(x)*ln((a*x+b)/b)+1/d/b*dilog((-d*a+a*(e*x+ d)+b*e)/(-a*d+b*e))+1/d/b*ln(e*x+d)*ln((-d*a+a*(e*x+d)+b*e)/(-a*d+b*e))-1/ d/b*ln(e*x+d)*ln(-e*x/d)-1/d/b*dilog(-e*x/d))
\[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{{\left (e x + d\right )} x} \,d x } \] Input:
integrate(log(c*(a+b/x)^p)/x/(e*x+d),x, algorithm="fricas")
Output:
integral(log(c*((a*x + b)/x)^p)/(e*x^2 + d*x), x)
\[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=\int \frac {\log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{x \left (d + e x\right )}\, dx \] Input:
integrate(ln(c*(a+b/x)**p)/x/(e*x+d),x)
Output:
Integral(log(c*(a + b/x)**p)/(x*(d + e*x)), x)
Time = 0.07 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.13 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=-\frac {1}{2} \, b p {\left (\frac {2 \, \log \left (e x + d\right ) \log \left (x\right ) - \log \left (x\right )^{2}}{b d} + \frac {2 \, {\left (\log \left (\frac {a x}{b} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {a x}{b}\right )\right )}}{b d} - \frac {2 \, {\left (\log \left (\frac {e x}{d} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {e x}{d}\right )\right )}}{b d} - \frac {2 \, {\left (\log \left (e x + d\right ) \log \left (-\frac {a e x + a d}{a d - b e} + 1\right ) + {\rm Li}_2\left (\frac {a e x + a d}{a d - b e}\right )\right )}}{b d}\right )} - {\left (\frac {\log \left (e x + d\right )}{d} - \frac {\log \left (x\right )}{d}\right )} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right ) \] Input:
integrate(log(c*(a+b/x)^p)/x/(e*x+d),x, algorithm="maxima")
Output:
-1/2*b*p*((2*log(e*x + d)*log(x) - log(x)^2)/(b*d) + 2*(log(a*x/b + 1)*log (x) + dilog(-a*x/b))/(b*d) - 2*(log(e*x/d + 1)*log(x) + dilog(-e*x/d))/(b* d) - 2*(log(e*x + d)*log(-(a*e*x + a*d)/(a*d - b*e) + 1) + dilog((a*e*x + a*d)/(a*d - b*e)))/(b*d)) - (log(e*x + d)/d - log(x)/d)*log((a + b/x)^p*c)
\[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{{\left (e x + d\right )} x} \,d x } \] Input:
integrate(log(c*(a+b/x)^p)/x/(e*x+d),x, algorithm="giac")
Output:
integrate(log((a + b/x)^p*c)/((e*x + d)*x), x)
Timed out. \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=\int \frac {\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{x\,\left (d+e\,x\right )} \,d x \] Input:
int(log(c*(a + b/x)^p)/(x*(d + e*x)),x)
Output:
int(log(c*(a + b/x)^p)/(x*(d + e*x)), x)
\[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=\frac {-2 \left (\int \frac {\mathrm {log}\left (\frac {\left (a x +b \right )^{p} c}{x^{p}}\right )}{a e \,x^{3}+a d \,x^{2}+b e \,x^{2}+b d x}d x \right ) a b d p +2 \left (\int \frac {\mathrm {log}\left (\frac {\left (a x +b \right )^{p} c}{x^{p}}\right )}{a e \,x^{3}+a d \,x^{2}+b e \,x^{2}+b d x}d x \right ) b^{2} e p -\mathrm {log}\left (\frac {\left (a x +b \right )^{p} c}{x^{p}}\right )^{2} a}{2 b e p} \] Input:
int(log(c*(a+b/x)^p)/x/(e*x+d),x)
Output:
( - 2*int(log(((a*x + b)**p*c)/x**p)/(a*d*x**2 + a*e*x**3 + b*d*x + b*e*x* *2),x)*a*b*d*p + 2*int(log(((a*x + b)**p*c)/x**p)/(a*d*x**2 + a*e*x**3 + b *d*x + b*e*x**2),x)*b**2*e*p - log(((a*x + b)**p*c)/x**p)**2*a)/(2*b*e*p)