Integrand size = 22, antiderivative size = 63 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g} \] Output:
(a+b*ln(c*(e*x+d)^n))*ln(e*(g*x+f)/(-d*g+e*f))/g+b*n*polylog(2,-g*(e*x+d)/ (-d*g+e*f))/g
Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g}+\frac {b n \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )}{g} \] Input:
Integrate[(a + b*Log[c*(d + e*x)^n])/(f + g*x),x]
Output:
((a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g + (b*n*PolyL og[2, (g*(d + e*x))/(-(e*f) + d*g)])/g
Time = 0.47 (sec) , antiderivative size = 63, 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 = {2841, 2840, 2838}
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.
\(\displaystyle \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx\) |
\(\Big \downarrow \) 2841 |
\(\displaystyle \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {b e n \int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x}dx}{g}\) |
\(\Big \downarrow \) 2840 |
\(\displaystyle \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {b n \int \frac {\log \left (\frac {g (d+e x)}{e f-d g}+1\right )}{d+e x}d(d+e x)}{g}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g}\) |
Input:
Int[(a + b*Log[c*(d + e*x)^n])/(f + g*x),x]
Output:
((a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g + (b*n*PolyL og[2, -((g*(d + e*x))/(e*f - d*g))])/g
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_ Symbol] :> Simp[1/g Subst[Int[(a + b*Log[1 + c*e*(x/g)])/x, x], x, f + g* x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g + c *(e*f - d*g), 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.40 (sec) , antiderivative size = 217, normalized size of antiderivative = 3.44
method | result | size |
risch | \(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{g}-\frac {b n \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{g}-\frac {b n \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{g}+\frac {\left (\frac {i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \ln \left (g x +f \right )}{g}\) | \(217\) |
Input:
int((a+b*ln(c*(e*x+d)^n))/(g*x+f),x,method=_RETURNVERBOSE)
Output:
b*ln((e*x+d)^n)*ln(g*x+f)/g-b/g*n*dilog(((g*x+f)*e+d*g-e*f)/(d*g-e*f))-b/g *n*ln(g*x+f)*ln(((g*x+f)*e+d*g-e*f)/(d*g-e*f))+(1/2*I*b*Pi*csgn(I*(e*x+d)^ n)*csgn(I*c*(e*x+d)^n)^2-1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)* csgn(I*c)-1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3+1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^ 2*csgn(I*c)+b*ln(c)+a)*ln(g*x+f)/g
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{g x + f} \,d x } \] Input:
integrate((a+b*log(c*(e*x+d)^n))/(g*x+f),x, algorithm="fricas")
Output:
integral((b*log((e*x + d)^n*c) + a)/(g*x + f), x)
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx=\int \frac {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}{f + g x}\, dx \] Input:
integrate((a+b*ln(c*(e*x+d)**n))/(g*x+f),x)
Output:
Integral((a + b*log(c*(d + e*x)**n))/(f + g*x), x)
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{g x + f} \,d x } \] Input:
integrate((a+b*log(c*(e*x+d)^n))/(g*x+f),x, algorithm="maxima")
Output:
b*integrate((log((e*x + d)^n) + log(c))/(g*x + f), x) + a*log(g*x + f)/g
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{g x + f} \,d x } \] Input:
integrate((a+b*log(c*(e*x+d)^n))/(g*x+f),x, algorithm="giac")
Output:
integrate((b*log((e*x + d)^n*c) + a)/(g*x + f), x)
Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{f+g\,x} \,d x \] Input:
int((a + b*log(c*(d + e*x)^n))/(f + g*x),x)
Output:
int((a + b*log(c*(d + e*x)^n))/(f + g*x), x)
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{e g \,x^{2}+d g x +e f x +d f}d x \right ) b d g n -2 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{e g \,x^{2}+d g x +e f x +d f}d x \right ) b e f n +2 \,\mathrm {log}\left (g x +f \right ) a n +\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} b}{2 g n} \] Input:
int((a+b*log(c*(e*x+d)^n))/(g*x+f),x)
Output:
(2*int(log((d + e*x)**n*c)/(d*f + d*g*x + e*f*x + e*g*x**2),x)*b*d*g*n - 2 *int(log((d + e*x)**n*c)/(d*f + d*g*x + e*f*x + e*g*x**2),x)*b*e*f*n + 2*l og(f + g*x)*a*n + log((d + e*x)**n*c)**2*b)/(2*g*n)