Integrand size = 21, antiderivative size = 97 \[ \int \frac {\log \left (c (a+b x)^p\right )}{x (d+e x)} \, dx=\frac {\log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d}-\frac {\log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )}{d} \] Output:
ln(-b*x/a)*ln(c*(b*x+a)^p)/d-ln(c*(b*x+a)^p)*ln(b*(e*x+d)/(-a*e+b*d))/d-p* polylog(2,-e*(b*x+a)/(-a*e+b*d))/d+p*polylog(2,1+b*x/a)/d
Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01 \[ \int \frac {\log \left (c (a+b x)^p\right )}{x (d+e x)} \, dx=\frac {\log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d}-\frac {\log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {a+b x}{a}\right )}{d}-\frac {p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d} \] Input:
Integrate[Log[c*(a + b*x)^p]/(x*(d + e*x)),x]
Output:
(Log[-((b*x)/a)]*Log[c*(a + b*x)^p])/d - (Log[c*(a + b*x)^p]*Log[(b*(d + e *x))/(b*d - a*e)])/d + (p*PolyLog[2, (a + b*x)/a])/d - (p*PolyLog[2, -((e* (a + b*x))/(b*d - a*e))])/d
Time = 0.54 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2863, 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.
\(\displaystyle \int \frac {\log \left (c (a+b x)^p\right )}{x (d+e x)} \, dx\) |
\(\Big \downarrow \) 2863 |
\(\displaystyle \int \left (\frac {\log \left (c (a+b x)^p\right )}{d x}-\frac {e \log \left (c (a+b x)^p\right )}{d (d+e x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}+\frac {\log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d}-\frac {p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {b x}{a}+1\right )}{d}\) |
Input:
Int[Log[c*(a + b*x)^p]/(x*(d + e*x)),x]
Output:
(Log[-((b*x)/a)]*Log[c*(a + b*x)^p])/d - (Log[c*(a + b*x)^p]*Log[(b*(d + e *x))/(b*d - a*e)])/d - (p*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/d + (p *PolyLog[2, 1 + (b*x)/a])/d
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) ^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c , d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
Time = 1.03 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.61
method | result | size |
parts | \(-\frac {\ln \left (c \left (b x +a \right )^{p}\right ) \ln \left (e x +d \right )}{d}+\frac {\ln \left (c \left (b x +a \right )^{p}\right ) \ln \left (x \right )}{d}-p b \left (\frac {\operatorname {dilog}\left (\frac {b x +a}{a}\right )}{d b}+\frac {\ln \left (x \right ) \ln \left (\frac {b x +a}{a}\right )}{d b}-\frac {\operatorname {dilog}\left (\frac {\left (e x +d \right ) b +e a -b d}{e a -b d}\right )}{d b}-\frac {\ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +e a -b d}{e a -b d}\right )}{d b}\right )\) | \(156\) |
risch | \(-\frac {\ln \left (\left (b x +a \right )^{p}\right ) \ln \left (e x +d \right )}{d}+\frac {\ln \left (\left (b x +a \right )^{p}\right ) \ln \left (x \right )}{d}-\frac {p \operatorname {dilog}\left (\frac {b x +a}{a}\right )}{d}-\frac {p \ln \left (x \right ) \ln \left (\frac {b x +a}{a}\right )}{d}+\frac {p \operatorname {dilog}\left (\frac {\left (e x +d \right ) b +e a -b d}{e a -b d}\right )}{d}+\frac {p \ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +e a -b d}{e a -b d}\right )}{d}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2}+\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (-\frac {\ln \left (e x +d \right )}{d}+\frac {\ln \left (x \right )}{d}\right )\) | \(263\) |
Input:
int(ln(c*(b*x+a)^p)/x/(e*x+d),x,method=_RETURNVERBOSE)
Output:
-ln(c*(b*x+a)^p)/d*ln(e*x+d)+ln(c*(b*x+a)^p)/d*ln(x)-p*b*(1/d*dilog((b*x+a )/a)/b+1/d*ln(x)*ln((b*x+a)/a)/b-1/d*dilog(((e*x+d)*b+e*a-b*d)/(a*e-b*d))/ b-1/d*ln(e*x+d)*ln(((e*x+d)*b+e*a-b*d)/(a*e-b*d))/b)
\[ \int \frac {\log \left (c (a+b x)^p\right )}{x (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{p} c\right )}{{\left (e x + d\right )} x} \,d x } \] Input:
integrate(log(c*(b*x+a)^p)/x/(e*x+d),x, algorithm="fricas")
Output:
integral(log((b*x + a)^p*c)/(e*x^2 + d*x), x)
\[ \int \frac {\log \left (c (a+b x)^p\right )}{x (d+e x)} \, dx=\int \frac {\log {\left (c \left (a + b x\right )^{p} \right )}}{x \left (d + e x\right )}\, dx \] Input:
integrate(ln(c*(b*x+a)**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 = 123, normalized size of antiderivative = 1.27 \[ \int \frac {\log \left (c (a+b x)^p\right )}{x (d+e x)} \, dx=-b p {\left (\frac {\log \left (\frac {b x}{a} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x}{a}\right )}{b d} - \frac {\log \left (e x + d\right ) \log \left (-\frac {b e x + b d}{b d - a e} + 1\right ) + {\rm Li}_2\left (\frac {b e x + b d}{b d - a e}\right )}{b d}\right )} - {\left (\frac {\log \left (e x + d\right )}{d} - \frac {\log \left (x\right )}{d}\right )} \log \left ({\left (b x + a\right )}^{p} c\right ) \] Input:
integrate(log(c*(b*x+a)^p)/x/(e*x+d),x, algorithm="maxima")
Output:
-b*p*((log(b*x/a + 1)*log(x) + dilog(-b*x/a))/(b*d) - (log(e*x + d)*log(-( b*e*x + b*d)/(b*d - a*e) + 1) + dilog((b*e*x + b*d)/(b*d - a*e)))/(b*d)) - (log(e*x + d)/d - log(x)/d)*log((b*x + a)^p*c)
\[ \int \frac {\log \left (c (a+b x)^p\right )}{x (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{p} c\right )}{{\left (e x + d\right )} x} \,d x } \] Input:
integrate(log(c*(b*x+a)^p)/x/(e*x+d),x, algorithm="giac")
Output:
integrate(log((b*x + a)^p*c)/((e*x + d)*x), x)
Timed out. \[ \int \frac {\log \left (c (a+b x)^p\right )}{x (d+e x)} \, dx=\int \frac {\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{x\,\left (d+e\,x\right )} \,d x \] 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)
\[ \int \frac {\log \left (c (a+b x)^p\right )}{x (d+e x)} \, dx=\int \frac {\mathrm {log}\left (\left (b x +a \right )^{p} c \right )}{e \,x^{2}+d x}d x \] Input:
int(log(c*(b*x+a)^p)/x/(e*x+d),x)
Output:
int(log((a + b*x)**p*c)/(d*x + e*x**2),x)