\(\int \frac {A+B \log (e (a+b x)^n (c+d x)^{-n})}{(f+g x) (a h+b h x)} \, dx\) [252]

Optimal result

Integrand size = 41, antiderivative size = 123 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(f+g x) (a h+b h x)} \, dx=-\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log \left (1-\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {B n \operatorname {PolyLog}\left (2,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h} \] 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. \(304\) vs. \(2(123)=246\).

Time = 0.46 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.47 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(f+g x) (a h+b h x)} \, dx=-\frac {-2 A \log (a+b x)+B n \log ^2(a+b x)-2 B n \log (a+b x) \log (c+d x)+2 B n \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)-2 B \log (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 A \log (f+g x)-2 B n \log (a+b x) \log (f+g x)+2 B n \log (c+d x) \log (f+g x)+2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (f+g x)+2 B n \log (a+b x) \log \left (\frac {b (f+g x)}{b f-a g}\right )-2 B n \log (c+d x) \log \left (\frac {d (f+g x)}{d f-c g}\right )+2 B n \operatorname {PolyLog}\left (2,\frac {g (a+b x)}{-b f+a g}\right )+2 B n \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )-2 B n \operatorname {PolyLog}\left (2,\frac {g (c+d x)}{-d f+c g}\right )}{2 (b f-a g) h} \] Input:

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

Output:

-1/2*(-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)])/((b*f - a*g)*h)
 

Rubi [A] (warning: unable to verify)

Time = 0.63 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2973, 2967, 27, 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.

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

\(\Big \downarrow \) 2973

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

\(\Big \downarrow \) 2967

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 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 2967

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(b*c - a*d)   Subst[Int[(b*f - a*g - (d*f - c*g)*x)^m*(b*h - a*i 
- (d*h - c*i)*x)^q*((A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a 
+ b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n}, x] && 
NeQ[b*c - a*d, 0] && IntegersQ[m, q] && IGtQ[p, 0]
 

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]
 

Maple [C] (warning: unable to verify)

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

Time = 27.07 (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)))/(g*x+f)/(b*h*x+a*h),x,method=_RETURN 
VERBOSE)
 

Output:

-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*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*n/(a*g-b*f)*ln(b*x+a)*ln(( 
-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))*csgn(I/((d*x+c)^n)*(b*x+a)^n)^2+1/2*I/h/(a*g-b*f)*ln(g*x+f)*B*P 
i*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/((d*x+c)^n 
)*(b*x+a)^n)^2-1/2*I/h/(a*g-b*f)*ln(b*x+a)*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(b*x+a)*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(b*x+a) 
*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(g*x 
+f)*B*Pi*csgn(I*e*(b*x+a)^n/((d*x+c)^n))^3+1/2*I/h/(a*g-b*f)*ln(b*x+a)*B*P 
i*csgn(I/((d*x+c)^n)*(b*x+a)^n)^3+1/2*I/h/(a*g-b*f)*ln(b*x+a)*B*Pi*csgn(I* 
e*(b*x+a)^n/((d*x+c)^n))^3-1/2*I/h/(a*g-b*f)*ln(g*x+f)*B*Pi*csgn(I/((d*x+c 
)^n)*(b*x+a)^n)^3-1/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/(a*g-b*f)*ln(g*x+f)*A-1/h/(a*g-b*f)*ln(b*x+a)*A+1/h*B/(a*g-b*f)* 
ln(g*x+f)*ln((b*x+a)^n)-1/h*B/(a*g-b*f)*ln(g*x+f)*ln((d*x+c)^n)+1/h*B*ln(( 
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*...
 

Fricas [F]

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

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))/(g*x+f)/(b*h*x+a*h),x, algori 
thm="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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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