\(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^2}{(f+g x) (a h+b h x)} \, dx\) [251]

Optimal result

Integrand size = 43, antiderivative size = 203 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(f+g x) (a h+b h x)} \, dx=-\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \log \left (1-\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {2 B n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \operatorname {PolyLog}\left (2,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {2 B^2 n^2 \operatorname {PolyLog}\left (3,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h} \] Output:

-(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2*ln(1-(-a*g+b*f)*(d*x+c)/(-c*g+d*f)/(b 
*x+a))/(-a*g+b*f)/h+2*B*n*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))*polylog(2,(-a* 
g+b*f)*(d*x+c)/(-c*g+d*f)/(b*x+a))/(-a*g+b*f)/h+2*B^2*n^2*polylog(3,(-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. \(1415\) vs. \(2(203)=406\).

Time = 1.44 (sec) , antiderivative size = 1415, normalized size of antiderivative = 6.97 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(f+g x) (a h+b h x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*Log[a + b*x]*(A + B*(-(n*Log[a + b*x]) + n*Log[c + d*x] + Log[(e*(a + b 
*x)^n)/(c + d*x)^n]))^2 - 3*(A + B*(-(n*Log[a + b*x]) + n*Log[c + d*x] + L 
og[(e*(a + b*x)^n)/(c + d*x)^n]))^2*Log[f + g*x] + 3*B*n*(A + B*(-(n*Log[a 
 + b*x]) + n*Log[c + d*x] + Log[(e*(a + b*x)^n)/(c + d*x)^n]))*(Log[a + b* 
x]^2 - 2*(Log[a + b*x]*Log[(b*(f + g*x))/(b*f - a*g)] + PolyLog[2, (g*(a + 
 b*x))/(-(b*f) + a*g)])) - 6*A*B*n*(Log[c + d*x]*(Log[(d*(a + b*x))/(-(b*c 
) + a*d)] - Log[(d*(f + g*x))/(d*f - c*g)]) + PolyLog[2, (b*(c + d*x))/(b* 
c - a*d)] - PolyLog[2, (g*(c + d*x))/(-(d*f) + c*g)]) + 6*B^2*n*(n*Log[a + 
 b*x] - n*Log[c + d*x] - Log[(e*(a + b*x)^n)/(c + d*x)^n])*(Log[c + d*x]*( 
Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[(d*(f + g*x))/(d*f - c*g)]) + Poly 
Log[2, (b*(c + d*x))/(b*c - a*d)] - PolyLog[2, (g*(c + d*x))/(-(d*f) + c*g 
)]) + B^2*n^2*(Log[a + b*x]^2*(Log[a + b*x] - 3*Log[(b*(f + g*x))/(b*f - a 
*g)]) - 6*Log[a + b*x]*PolyLog[2, (g*(a + b*x))/(-(b*f) + a*g)] + 6*PolyLo 
g[3, (g*(a + b*x))/(-(b*f) + a*g)]) + 3*B^2*n^2*(Log[(d*(a + b*x))/(-(b*c) 
 + a*d)]*Log[c + d*x]^2 - Log[c + d*x]^2*Log[(d*(f + g*x))/(d*f - c*g)] + 
2*Log[c + d*x]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] - 2*Log[c + d*x]*Poly 
Log[2, (g*(c + d*x))/(-(d*f) + c*g)] - 2*PolyLog[3, (b*(c + d*x))/(b*c - a 
*d)] + 2*PolyLog[3, (g*(c + d*x))/(-(d*f) + c*g)]) - 6*B^2*n^2*((Log[a + b 
*x]^2*(Log[c + d*x] - Log[(b*(c + d*x))/(b*c - a*d)]))/2 - Log[a + b*x]*Lo 
g[c + d*x]*Log[(b*(f + g*x))/(b*f - a*g)] - (Log[(g*(c + d*x))/(-(d*f) ...
 

Rubi [A] (warning: unable to verify)

Time = 0.94 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {2973, 2967, 27, 2779, 2821, 7143}

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.

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

\(\Big \downarrow \) 2973

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

\(\Big \downarrow \) 2967

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\frac {2 B n \left (\operatorname {PolyLog}\left (2,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+B n \operatorname {PolyLog}\left (3,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )\right )}{b f-a g}-\frac {\log \left (1-\frac {(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b f-a g}}{h}\)

Input:

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

Output:

(-(((A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - ((b*f - a*g)*(c + d*x 
))/((d*f - c*g)*(a + b*x))])/(b*f - a*g)) + (2*B*n*((A + B*Log[e*((a + b*x 
)/(c + d*x))^n])*PolyLog[2, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x) 
)] + B*n*PolyLog[3, ((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 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c 
*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 
0] && EqQ[d*e, 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]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(3*int(log(((a + b*x)**n*e)/(c + d*x)**n)**2/(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**2* 
c*d*g**2*n - 3*int(log(((a + b*x)**n*e)/(c + d*x)**n)**2/(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**2*d**2*f*g*n - 3*int(log(((a + b*x)**n*e)/(c + d*x)**n)**2/(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**3*c**2*g**2*n + 3*int(log(((a + b*x)**n*e)/(c + d*x)**n 
)**2/(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**3*d**2*f**2*n + 3*int(log(((a + b*x)**n*e)/(c 
 + d*x)**n)**2/(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**4*c**2*f*g*n - 3*int(log(((a + b*x)** 
n*e)/(c + d*x)**n)**2/(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**4*c*d*f**2*n + 6*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**3*b*c*d*g**2*n - 6*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**3*b*d**2*f*g*n - 6*i 
nt(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**2*c**2*g* 
*2*n + 6*int(log(((a + b*x)**n*e)/(c + d*x)**n)/(a*c*f + a*c*g*x + a*d*...