\(\int \frac {A+B \log (e (a+b x)^n (c+d x)^{-n})}{a f h+b g h x^2+h (b f x+a g x)} \, dx\) [261]

Optimal result

Integrand size = 49, antiderivative size = 123 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a f h+b g h x^2+h (b f x+a g x)} \, dx=-\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log \left (1-\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {B n \operatorname {PolyLog}\left (2,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(303\) vs. \(2(123)=246\).

Time = 0.34 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.46 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a f h+b g h x^2+h (b f x+a g x)} \, dx=-\frac {-2 A \log (a+b x)+B n \log ^2(a+b x)-2 B n \log (a+b x) \log (c+d x)+2 B n \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)-2 B \log (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 A \log (f+g x)-2 B n \log (a+b x) \log (f+g x)+2 B n \log (c+d x) \log (f+g x)+2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (f+g x)+2 B n \log (a+b x) \log \left (\frac {b (f+g x)}{b f-a g}\right )-2 B n \log (c+d x) \log \left (\frac {d (f+g x)}{d f-c g}\right )+2 B n \operatorname {PolyLog}\left (2,\frac {g (a+b x)}{-b f+a g}\right )+2 B n \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )-2 B n \operatorname {PolyLog}\left (2,\frac {g (c+d x)}{-d f+c g}\right )}{(2 b f-2 a g) h} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.67 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.23, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.102, Rules used = {2973, 2976, 2026, 2779, 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.

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

\(\Big \downarrow \) 2973

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

\(\Big \downarrow \) 2976

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

\(\Big \downarrow \) 2026

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

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 2779

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

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 2973

Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] 
 :> Subst[Int[w*(A + B*Log[e*(u/v)^n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; Fr 
eeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !Intege 
rQ[n]
 

rule 2976

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*(P2x_)^(m_.), x_Symbol] :> With[{f = Coeff[P2x, x, 0], g = Coef 
f[P2x, x, 1], h = Coeff[P2x, x, 2]}, Simp[(b*c - a*d)   Subst[Int[(b^2*f - 
a*b*g + a^2*h - (2*b*d*f - b*c*g - a*d*g + 2*a*c*h)*x + (d^2*f - c*d*g + c^ 
2*h)*x^2)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(2*(m + 1))), x], x, (a + b*x)/ 
(c + d*x)], x]] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && PolyQ[P2x, x, 2] & 
& NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
 

Maple [C] (warning: unable to verify)

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

Time = 32.82 (sec) , antiderivative size = 1447, normalized size of antiderivative = 11.76

method	result	size
risch	\(\text {Expression too large to display}\)	\(1447\)

Input:

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

Output:

1/h/(a*g-b*f)*ln(g*x+f)*A-1/h/(a*g-b*f)*ln(b*x+a)*A-1/h*B*n/(a*g-b*f)*ln(g 
*x+f)*ln((b*(g*x+f)+a*g-b*f)/(a*g-b*f))+1/h*B*n/(a*g-b*f)*ln(g*x+f)*ln((d* 
(g*x+f)+c*g-d*f)/(c*g-d*f))-1/h*B*ln((b*x+a)^n)/(a*g-b*f)*ln(b*x+a)-1/h*B* 
n/(a*g-b*f)*dilog((b*(g*x+f)+a*g-b*f)/(a*g-b*f))+1/2/h*B*n/(a*g-b*f)*ln(b* 
x+a)^2+1/h*B/(a*g-b*f)*ln(g*x+f)*ln((b*x+a)^n)-1/h*B*n/(a*g-b*f)*ln(b*x+a) 
*ln((-d*a+b*c+d*(b*x+a))/(-a*d+b*c))+1/h/(a*g-b*f)*ln(g*x+f)*B*ln(e)-1/h/( 
a*g-b*f)*ln(b*x+a)*B*ln(e)-1/h*B/(a*g-b*f)*ln(g*x+f)*ln((d*x+c)^n)+1/h*B*l 
n((d*x+c)^n)/(a*g-b*f)*ln(b*x+a)+1/h*B*n/(a*g-b*f)*dilog((d*(g*x+f)+c*g-d* 
f)/(c*g-d*f))-1/h*B*n/(a*g-b*f)*dilog((-d*a+b*c+d*(b*x+a))/(-a*d+b*c))-1/2 
*I/h/(a*g-b*f)*ln(g*x+f)*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)+1/2*I/h/(a*g-b*f)*ln(b*x+a)*B*Pi*csgn(I/((d*x+ 
c)^n))*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n)*(b*x+a)^n)-1/2*I/h/(a*g-b*f)*l 
n(g*x+f)*B*Pi*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n)*(b* 
x+a)^n)+1/2*I/h/(a*g-b*f)*ln(b*x+a)*B*Pi*csgn(I/((d*x+c)^n)*(b*x+a)^n)*csg 
n(I*e*(b*x+a)^n/((d*x+c)^n))*csgn(I*e)+1/2*I/h/(a*g-b*f)*ln(g*x+f)*B*Pi*cs 
gn(I/((d*x+c)^n))*csgn(I/((d*x+c)^n)*(b*x+a)^n)^2+1/2*I/h/(a*g-b*f)*ln(g*x 
+f)*B*Pi*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n)*(b*x+a)^n)^2+1/2*I/h/(a*g-b* 
f)*ln(g*x+f)*B*Pi*csgn(I/((d*x+c)^n)*(b*x+a)^n)*csgn(I*e*(b*x+a)^n/((d*x+c 
)^n))^2+1/2*I/h/(a*g-b*f)*ln(g*x+f)*B*Pi*csgn(I*e*(b*x+a)^n/((d*x+c)^n))^2 
*csgn(I*e)-1/2*I/h/(a*g-b*f)*ln(b*x+a)*B*Pi*csgn(I/((d*x+c)^n))*csgn(I/...
 

Fricas [F]

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

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

Output:

integral((B*log((b*x + a)^n*e/(d*x + c)^n) + A)/(b*g*h*x^2 + a*f*h + (b*f 
+ a*g)*h*x), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a f h+b g h x^2+h (b f x+a g x)} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))/(a*f*h+b*g*h*x**2+h*(a*g*x+b 
*f*x)),x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

A*(log(b*x + a)/((b*f - a*g)*h) - log(g*x + f)/((b*f - a*g)*h)) - B*integr 
ate(-(log((b*x + a)^n) - log((d*x + c)^n) + log(e))/(b*g*h*x^2 + a*f*h + ( 
b*f*h + a*g*h)*x), x)
 

Giac [F]

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

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

Output:

integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)/(b*g*h*x^2 + a*f*h + (b*f 
*x + a*g*x)*h), x)
 

Mupad [F(-1)]

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

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))/(h*(a*g*x + b*f*x) + a*f*h + 
b*g*h*x^2),x)
 

Output:

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))/(h*(a*g*x + b*f*x) + a*f*h + 
b*g*h*x^2), x)
 

Reduce [F]

\[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a f h+b g h x^2+h (b f x+a g x)} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )}{b d g \,x^{3}+a d g \,x^{2}+b c g \,x^{2}+b d f \,x^{2}+a c g x +a d f x +b c f x +a c f}d x \right ) a^{2} b c d \,g^{2} n -2 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )}{b d g \,x^{3}+a d g \,x^{2}+b c g \,x^{2}+b d f \,x^{2}+a c g x +a d f x +b c f x +a c f}d x \right ) a^{2} b \,d^{2} f g n -2 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )}{b d g \,x^{3}+a d g \,x^{2}+b c g \,x^{2}+b d f \,x^{2}+a c g x +a d f x +b c f x +a c f}d x \right ) a \,b^{2} c^{2} g^{2} n +2 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )}{b d g \,x^{3}+a d g \,x^{2}+b c g \,x^{2}+b d f \,x^{2}+a c g x +a d f x +b c f x +a c f}d x \right ) a \,b^{2} d^{2} f^{2} n +2 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )}{b d g \,x^{3}+a d g \,x^{2}+b c g \,x^{2}+b d f \,x^{2}+a c g x +a d f x +b c f x +a c f}d x \right ) b^{3} c^{2} f g n -2 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )}{b d g \,x^{3}+a d g \,x^{2}+b c g \,x^{2}+b d f \,x^{2}+a c g x +a d f x +b c f x +a c f}d x \right ) b^{3} c d \,f^{2} n -2 \,\mathrm {log}\left (b x +a \right ) a^{2} d g n +2 \,\mathrm {log}\left (b x +a \right ) a b c g n +2 \,\mathrm {log}\left (g x +f \right ) a^{2} d g n -2 \,\mathrm {log}\left (g x +f \right ) a b c g n -\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )^{2} a b d g +\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )^{2} b^{2} d f}{2 g h n \left (a^{2} d g -a b c g -a b d f +b^{2} c f \right )} \] Input:

int((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x)),x 
)
 

Output:

(2*int(log(((a + b*x)**n*e)/(c + d*x)**n)/(a*c*f + a*c*g*x + a*d*f*x + a*d 
*g*x**2 + b*c*f*x + b*c*g*x**2 + b*d*f*x**2 + b*d*g*x**3),x)*a**2*b*c*d*g* 
*2*n - 2*int(log(((a + b*x)**n*e)/(c + d*x)**n)/(a*c*f + a*c*g*x + a*d*f*x 
 + a*d*g*x**2 + b*c*f*x + b*c*g*x**2 + b*d*f*x**2 + b*d*g*x**3),x)*a**2*b* 
d**2*f*g*n - 2*int(log(((a + b*x)**n*e)/(c + d*x)**n)/(a*c*f + a*c*g*x + a 
*d*f*x + a*d*g*x**2 + b*c*f*x + b*c*g*x**2 + b*d*f*x**2 + b*d*g*x**3),x)*a 
*b**2*c**2*g**2*n + 2*int(log(((a + b*x)**n*e)/(c + d*x)**n)/(a*c*f + a*c* 
g*x + a*d*f*x + a*d*g*x**2 + b*c*f*x + b*c*g*x**2 + b*d*f*x**2 + b*d*g*x** 
3),x)*a*b**2*d**2*f**2*n + 2*int(log(((a + b*x)**n*e)/(c + d*x)**n)/(a*c*f 
 + a*c*g*x + a*d*f*x + a*d*g*x**2 + b*c*f*x + b*c*g*x**2 + b*d*f*x**2 + b* 
d*g*x**3),x)*b**3*c**2*f*g*n - 2*int(log(((a + b*x)**n*e)/(c + d*x)**n)/(a 
*c*f + a*c*g*x + a*d*f*x + a*d*g*x**2 + b*c*f*x + b*c*g*x**2 + b*d*f*x**2 
+ b*d*g*x**3),x)*b**3*c*d*f**2*n - 2*log(a + b*x)*a**2*d*g*n + 2*log(a + b 
*x)*a*b*c*g*n + 2*log(f + g*x)*a**2*d*g*n - 2*log(f + g*x)*a*b*c*g*n - log 
(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b*d*g + log(((a + b*x)**n*e)/(c + d*x 
)**n)**2*b**2*d*f)/(2*g*h*n*(a**2*d*g - a*b*c*g - a*b*d*f + b**2*c*f))