\(\int \frac {A+B \log (e (a+b x)^n (c+d x)^{-n})}{g+h x} \, dx\) [298]

Optimal result

Integrand size = 31, antiderivative size = 148 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx=-\frac {B n \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac {B n \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log (g+h x)}{h}-\frac {B n \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {B n \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{h} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.01 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx=\frac {\left (A+B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right ) \log (g+h x)+B n \left (\log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )+\operatorname {PolyLog}\left (2,\frac {h (a+b x)}{-b g+a h}\right )\right )-B n \left (\log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )+\operatorname {PolyLog}\left (2,\frac {h (c+d x)}{-d g+c h}\right )\right )}{h} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2946, 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.

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

\(\Big \downarrow \) 2946

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

\(\Big \downarrow \) 2841

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

\(\Big \downarrow \) 2840

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2838

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]
 

rule 2840

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]
 

rule 2841

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]
 

rule 2946

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

Maple [C] (warning: unable to verify)

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

Time = 3.15 (sec) , antiderivative size = 521, normalized size of antiderivative = 3.52

method	result	size
risch	\(\frac {B \ln \left (\left (b x +a \right )^{n}\right ) \ln \left (h x +g \right )}{h}-\frac {B n \operatorname {dilog}\left (\frac {\left (h x +g \right ) b +a h -b g}{a h -b g}\right )}{h}-\frac {B n \ln \left (h x +g \right ) \ln \left (\frac {\left (h x +g \right ) b +a h -b g}{a h -b g}\right )}{h}+\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 (h x +g \right )}{2 h}-\frac {B \ln \left (\left (d x +c \right )^{n}\right ) \ln \left (h x +g \right )}{h}+\frac {B n \operatorname {dilog}\left (\frac {d \left (h x +g \right )+c h -d g}{c h -d g}\right )}{h}+\frac {B n \ln \left (h x +g \right ) \ln \left (\frac {d \left (h x +g \right )+c h -d g}{c h -d g}\right )}{h}\)	\(521\)

Input:

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

Output:

B*ln((b*x+a)^n)*ln(h*x+g)/h-B/h*n*dilog(((h*x+g)*b+a*h-b*g)/(a*h-b*g))-B/h 
*n*ln(h*x+g)*ln(((h*x+g)*b+a*h-b*g)/(a*h-b*g))+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*csg 
n(I*e)+2*B*ln(e)+2*A)*ln(h*x+g)/h-B*ln((d*x+c)^n)*ln(h*x+g)/h+B/h*n*dilog( 
(d*(h*x+g)+c*h-d*g)/(c*h-d*g))+B/h*n*ln(h*x+g)*ln((d*(h*x+g)+c*h-d*g)/(c*h 
-d*g))
 

Fricas [F]

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

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

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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