\(\int \frac {A+B \log (e (a+b x)^n (c+d x)^{-n})}{a+b x} \, dx\) [151]

Optimal result

Integrand size = 31, antiderivative size = 79 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx=-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b}+\frac {B n \operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{b} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.63 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx=\frac {-B n \log ^2\left (\frac {-b c+a d}{d (a+b x)}\right )+2 A \log (a+b x)-2 B \log \left (\frac {-b c+a d}{d (a+b x)}\right ) \left (n \log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+2 B n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{2 b} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2942, 2858, 27, 2778, 2005, 2752}

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.

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

\(\Big \downarrow \) 2942

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

\(\Big \downarrow \) 2858

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2778

\(\displaystyle -\frac {B n (b c-a d) \int \frac {(a+b x) \log \left (-\frac {b c-a d}{d (a+b x)}\right )}{b c-a d+d (a+b x)}d\frac {1}{a+b x}}{b}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b}\)

\(\Big \downarrow \) 2005

\(\displaystyle -\frac {B n (b c-a d) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{d+\frac {b c-a d}{a+b x}}d\frac {1}{a+b x}}{b}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b}\)

\(\Big \downarrow \) 2752

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2005

Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m 
+ n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg 
Q[n]
 

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2778

Int[((a_.) + Log[(c_.)*(x_)^(n_)]*(b_.))/((x_)*((d_) + (e_.)*(x_)^(r_.))), 
x_Symbol] :> Simp[1/n   Subst[Int[(a + b*Log[c*x])/(x*(d + e*x^(r/n))), x], 
 x, x^n], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[r/n]
 

rule 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

rule 2942

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(-Log[-(b*c - a*d)/(d*(a 
+ b*x))])*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/g), x] + Simp[B*n*((b*c 
 - a*d)/g)   Int[Log[-(b*c - a*d)/(d*(a + b*x))]/((a + b*x)*(c + d*x)), x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && NeQ[b* 
c - a*d, 0] && EqQ[b*f - a*g, 0]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.19 (sec) , antiderivative size = 462, normalized size of antiderivative = 5.85

method	result	size
risch	\(\frac {B \ln \left (b x +a \right ) \ln \left (\left (b x +a \right )^{n}\right )}{b}+\frac {\left (-i B \pi \,\operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )+i B \pi \,\operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2}+i B \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2}-i B \pi \operatorname {csgn}\left (i \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{3}+i B \pi \,\operatorname {csgn}\left (i \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i B \pi \,\operatorname {csgn}\left (i \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i e \right )-i B \pi \operatorname {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}+i B \pi \operatorname {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2} \operatorname {csgn}\left (i e \right )+2 B \ln \left (e \right )+2 A \right ) \ln \left (b x +a \right )}{2 b}-\frac {B \ln \left (b x +a \right ) \ln \left (\left (d x +c \right )^{n}\right )}{b}+\frac {B n \operatorname {dilog}\left (\frac {-d a +b c +d \left (b x +a \right )}{-d a +b c}\right )}{b}+\frac {B n \ln \left (b x +a \right ) \ln \left (\frac {-d a +b c +d \left (b x +a \right )}{-d a +b c}\right )}{b}-\frac {B \ln \left (b x +a \right )^{2} n}{2 b}\)	\(462\)

Input:

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

Output:

B/b*ln(b*x+a)*ln((b*x+a)^n)+1/2*(-I*B*Pi*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a 
)^n)*csgn(I/((d*x+c)^n)*(b*x+a)^n)+I*B*Pi*csgn(I/((d*x+c)^n))*csgn(I/((d*x 
+c)^n)*(b*x+a)^n)^2+I*B*Pi*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n)*(b*x+a)^n) 
^2-I*B*Pi*csgn(I/((d*x+c)^n)*(b*x+a)^n)^3+I*B*Pi*csgn(I/((d*x+c)^n)*(b*x+a 
)^n)*csgn(I*e*(b*x+a)^n/((d*x+c)^n))^2-I*B*Pi*csgn(I/((d*x+c)^n)*(b*x+a)^n 
)*csgn(I*e*(b*x+a)^n/((d*x+c)^n))*csgn(I*e)-I*B*Pi*csgn(I*e*(b*x+a)^n/((d* 
x+c)^n))^3+I*B*Pi*csgn(I*e*(b*x+a)^n/((d*x+c)^n))^2*csgn(I*e)+2*B*ln(e)+2* 
A)*ln(b*x+a)/b-B/b*ln(b*x+a)*ln((d*x+c)^n)+B/b*n*dilog((-d*a+b*c+d*(b*x+a) 
)/(-a*d+b*c))+B/b*n*ln(b*x+a)*ln((-d*a+b*c+d*(b*x+a))/(-a*d+b*c))-1/2/b*B* 
ln(b*x+a)^2*n
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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