\(\int \frac {\log (c (a+b x)^n)}{d x+e x^2} \, dx\) [345]

Optimal result

Integrand size = 22, antiderivative size = 97 \[ \int \frac {\log \left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\frac {\log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^n\right )}{d}-\frac {\log \left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {n \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {n \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )}{d} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01 \[ \int \frac {\log \left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\frac {\log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^n\right )}{d}-\frac {\log \left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}+\frac {n \operatorname {PolyLog}\left (2,\frac {a+b x}{a}\right )}{d}-\frac {n \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d} \] Input:

Integrate[Log[c*(a + b*x)^n]/(d*x + e*x^2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2026, 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.

\(\displaystyle \int \frac {\log \left (c (a+b x)^n\right )}{d x+e x^2} \, dx\)

\(\Big \downarrow \) 2026

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

\(\Big \downarrow \) 2863

\(\displaystyle \int \left (\frac {\log \left (c (a+b x)^n\right )}{d x}-\frac {e \log \left (c (a+b x)^n\right )}{d (d+e x)}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[Log[c*(a + b*x)^n]/(d*x + e*x^2),x]
 

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 2863

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]
 

Maple [A] (verified)

Time = 1.32 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.61

method	result	size
parts	\(-\frac {\ln \left (c \left (b x +a \right )^{n}\right ) \ln \left (e x +d \right )}{d}+\frac {\ln \left (c \left (b x +a \right )^{n}\right ) \ln \left (x \right )}{d}-n b \left (\frac {\operatorname {dilog}\left (\frac {b x +a}{a}\right )}{d b}+\frac {\ln \left (x \right ) \ln \left (\frac {b x +a}{a}\right )}{d b}-\frac {\operatorname {dilog}\left (\frac {\left (e x +d \right ) b +e a -b d}{e a -b d}\right )}{d b}-\frac {\ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +e a -b d}{e a -b d}\right )}{d b}\right )\)	\(156\)
risch	\(-\frac {\ln \left (e x +d \right ) \ln \left (\left (b x +a \right )^{n}\right )}{d}+\frac {\ln \left (\left (b x +a \right )^{n}\right ) \ln \left (x \right )}{d}-\frac {n \operatorname {dilog}\left (\frac {b x +a}{a}\right )}{d}-\frac {n \ln \left (x \right ) \ln \left (\frac {b x +a}{a}\right )}{d}+\frac {n \operatorname {dilog}\left (\frac {\left (e x +d \right ) b +e a -b d}{e a -b d}\right )}{d}+\frac {n \ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +e a -b d}{e a -b d}\right )}{d}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{3}}{2}+\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (-\frac {\ln \left (e x +d \right )}{d}+\frac {\ln \left (x \right )}{d}\right )\)	\(263\)

Input:

int(ln(c*(b*x+a)^n)/(e*x^2+d*x),x,method=_RETURNVERBOSE)
 

Output:

-ln(c*(b*x+a)^n)/d*ln(e*x+d)+ln(c*(b*x+a)^n)/d*ln(x)-n*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)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.27 \[ \int \frac {\log \left (c (a+b x)^n\right )}{d x+e x^2} \, dx=-b n {\left (\frac {\log \left (\frac {b x}{a} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x}{a}\right )}{b d} - \frac {\log \left (e x + d\right ) \log \left (-\frac {b e x + b d}{b d - a e} + 1\right ) + {\rm Li}_2\left (\frac {b e x + b d}{b d - a e}\right )}{b d}\right )} - {\left (\frac {\log \left (e x + d\right )}{d} - \frac {\log \left (x\right )}{d}\right )} \log \left ({\left (b x + a\right )}^{n} c\right ) \] Input:

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

Output:

-b*n*((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)^n*c)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\int \frac {\ln \left (c\,{\left (a+b\,x\right )}^n\right )}{e\,x^2+d\,x} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

int(log((a + b*x)**n*c)/(d*x + e*x**2),x)