\(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^3}{a f h+b g h x^2+h (b f x+a g x)} \, dx\) [259]

Optimal result

Integrand size = 51, antiderivative size = 282 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{a f h+b g h x^2+h (b f x+a g x)} \, dx=-\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \log \left (1-\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {3 B n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {6 B^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \operatorname {PolyLog}\left (3,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {6 B^3 n^3 \operatorname {PolyLog}\left (4,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h} \] Output:

-(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3*ln(1-(-a*g+b*f)*(d*x+c)/(-c*g+d*f)/(b 
*x+a))/(-a*g+b*f)/h+3*B*n*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2*polylog(2,(- 
a*g+b*f)*(d*x+c)/(-c*g+d*f)/(b*x+a))/(-a*g+b*f)/h+6*B^2*n^2*(A+B*ln(e*(b*x 
+a)^n/((d*x+c)^n)))*polylog(3,(-a*g+b*f)*(d*x+c)/(-c*g+d*f)/(b*x+a))/(-a*g 
+b*f)/h+6*B^3*n^3*polylog(4,(-a*g+b*f)*(d*x+c)/(-c*g+d*f)/(b*x+a))/(-a*g+b 
*f)/h
 

Mathematica [F]

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

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(a*f*h + b*g*h*x^2 + 
h*(b*f*x + a*g*x)),x]
 

Output:

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(a*f*h + b*g*h*x^2 + 
h*(b*f*x + a*g*x)), x]
 

Rubi [A] (warning: unable to verify)

Time = 1.16 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.137, Rules used = {2973, 2976, 2026, 2779, 2821, 2830, 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.

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

\(\Big \downarrow \) 2973

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

\(\Big \downarrow \) 2976

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

\(\Big \downarrow \) 2026

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 7143

\(\displaystyle (b c-a d) \left (\frac {3 B n \left (\operatorname {PolyLog}\left (2,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2-2 B n \left (-\left (\operatorname {PolyLog}\left (3,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )\right )-B n \operatorname {PolyLog}\left (4,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )\right )\right )}{h (b c-a d) (b f-a g)}-\frac {\log \left (1-\frac {(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{h (b c-a d) (b f-a g)}\right )\)

Input:

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(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])^3*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)) + (3 
*B*n*((A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*PolyLog[2, ((b*f - a*g)*(c 
+ d*x))/((d*f - c*g)*(a + b*x))] - 2*B*n*(-((A + B*Log[e*((a + b*x)/(c + d 
*x))^n])*PolyLog[3, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]) - B* 
n*PolyLog[4, ((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 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 2830

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ 
.)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) 
, x] - Simp[b*n*(p/q)   Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(p - 
1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[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 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]
 

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]

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

Input:

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x)), 
x)
 

Output:

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x)), 
x)
 

Fricas [F]

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

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(a*f*h+b*g*h*x^2+h*(a*g*x+b 
*f*x)),x, algorithm="fricas")
 

Output:

integral((B^3*log((b*x + a)^n*e/(d*x + c)^n)^3 + 3*A*B^2*log((b*x + a)^n*e 
/(d*x + c)^n)^2 + 3*A^2*B*log((b*x + a)^n*e/(d*x + c)^n) + A^3)/(b*g*h*x^2 
 + a*f*h + (b*f + a*g)*h*x), x)
 

Sympy [F(-1)]

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

integrate((A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**3/(a*f*h+b*g*h*x**2+h*(a*g* 
x+b*f*x)),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(a*f*h+b*g*h*x^2+h*(a*g*x+b 
*f*x)),x, algorithm="maxima")
 

Output:

A^3*(log(b*x + a)/((b*f - a*g)*h) - log(g*x + f)/((b*f - a*g)*h)) - integr 
ate(-(B^3*log((b*x + a)^n)^3 - B^3*log((d*x + c)^n)^3 + B^3*log(e)^3 + 3*A 
*B^2*log(e)^2 + 3*A^2*B*log(e) + 3*(B^3*log(e) + A*B^2)*log((b*x + a)^n)^2 
 + 3*(B^3*log((b*x + a)^n) + B^3*log(e) + A*B^2)*log((d*x + c)^n)^2 + 3*(B 
^3*log(e)^2 + 2*A*B^2*log(e) + A^2*B)*log((b*x + a)^n) - 3*(B^3*log((b*x + 
 a)^n)^2 + B^3*log(e)^2 + 2*A*B^2*log(e) + A^2*B + 2*(B^3*log(e) + A*B^2)* 
log((b*x + a)^n))*log((d*x + c)^n))/(b*g*h*x^2 + a*f*h + (b*f*h + a*g*h)*x 
), x)
 

Giac [F]

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

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(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)^3/(b*g*h*x^2 + a*f*h + (b 
*f*x + a*g*x)*h), x)
 

Mupad [F(-1)]

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

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

Reduce [F]

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

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

Output:

(4*int(log(((a + b*x)**n*e)/(c + d*x)**n)**3/(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**3* 
c*d*g**2*n - 4*int(log(((a + b*x)**n*e)/(c + d*x)**n)**3/(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**3*d**2*f*g*n - 4*int(log(((a + b*x)**n*e)/(c + d*x)**n)**3/(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**4*c**2*g**2*n + 4*int(log(((a + b*x)**n*e)/(c + d*x)**n 
)**3/(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**4*d**2*f**2*n + 4*int(log(((a + b*x)**n*e)/(c 
 + d*x)**n)**3/(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**5*c**2*f*g*n - 4*int(log(((a + b*x)** 
n*e)/(c + d*x)**n)**3/(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**5*c*d*f**2*n + 12*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**3*b**2*c*d*g**2*n - 12 
*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**3*b**2*d* 
*2*f*g*n - 12*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**3*c**2*g**2*n + 12*int(log(((a + b*x)**n*e)/(c + d*x)**n)**2/(...