\(\int (A+B \log (e (a+b x)^n (c+d x)^{-n}))^3 \, dx\) [311]

Optimal result

Integrand size = 25, antiderivative size = 203 \[ \int \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\frac {3 B (b c-a d) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{b d}+\frac {(a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{b}+\frac {6 B^2 (b c-a d) n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b d}-\frac {6 B^3 (b c-a d) n^3 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{b d} \] Output:

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

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.86 \[ \int \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\frac {A^3 b d x-3 A^2 B (b c-a d) n \log (c+d x)+3 A^2 B d (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+3 A B^2 d (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+B^3 d (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )+3 A B^2 (b c-a d) n \left (-\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 n \log \left (\frac {d (a+b x)}{-b c+a d}\right )-2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+n \log \left (\frac {b c-a d}{b c+b d x}\right )\right )+2 n \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )+3 B^3 (b c-a d) n \left (\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )-2 n^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )\right )}{b d} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.72 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.86, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2936, 2973, 2951, 2754, 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.

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

\(\Big \downarrow \) 2936

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

\(\Big \downarrow \) 2951

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

\(\Big \downarrow \) 2754

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 7143

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

Input:

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3,x]
 

Output:

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

Defintions of rubi rules used

rule 2754

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb 
ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) 
  Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, 
b, c, d, e, n}, 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 2936

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

rule 2951

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F(-2)]

Exception generated. \[ \int \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

3*A^2*B*x*log((b*x + a)^n*e/(d*x + c)^n) + A^3*x + 3*(a*e*n*log(b*x + a)/b 
 - c*e*n*log(d*x + c)/d)*A^2*B/e - (B^3*b*d*x*log((d*x + c)^n)^3 - 3*(B^3* 
a*d*n*log(b*x + a) - B^3*b*c*n*log(d*x + c) + B^3*b*d*x*log((b*x + a)^n) + 
 (B^3*b*d*log(e) + A*B^2*b*d)*x)*log((d*x + c)^n)^2)/(b*d) - integrate(-(B 
^3*b*c*log(e)^3 + 3*A*B^2*b*c*log(e)^2 + (B^3*b*d*x + B^3*b*c)*log((b*x + 
a)^n)^3 + 3*(B^3*b*c*log(e) + A*B^2*b*c + (B^3*b*d*log(e) + A*B^2*b*d)*x)* 
log((b*x + a)^n)^2 + (B^3*b*d*log(e)^3 + 3*A*B^2*b*d*log(e)^2)*x + 3*(B^3* 
b*c*log(e)^2 + 2*A*B^2*b*c*log(e) + (B^3*b*d*log(e)^2 + 2*A*B^2*b*d*log(e) 
)*x)*log((b*x + a)^n) - 3*(2*B^3*a*d*n^2*log(b*x + a) - 2*B^3*b*c*n^2*log( 
d*x + c) + B^3*b*c*log(e)^2 + 2*A*B^2*b*c*log(e) + (B^3*b*d*x + B^3*b*c)*l 
og((b*x + a)^n)^2 + ((2*n*log(e) + log(e)^2)*B^3*b*d + 2*A*B^2*b*d*(n + lo 
g(e)))*x + 2*(B^3*b*c*log(e) + A*B^2*b*c + (B^3*b*d*(n + log(e)) + A*B^2*b 
*d)*x)*log((b*x + a)^n))*log((d*x + c)^n))/(b*d*x + b*c), x)
 

Giac [F]

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

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\frac {3 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )^{2} x}{b d \,x^{2}+a d x +b c x +a c}d x \right ) a \,b^{3} d^{2} n -3 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )^{2} x}{b d \,x^{2}+a d x +b c x +a c}d x \right ) b^{4} c d n +6 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) x}{b d \,x^{2}+a d x +b c x +a c}d x \right ) a^{2} b^{2} d^{2} n -6 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) x}{b d \,x^{2}+a d x +b c x +a c}d x \right ) a \,b^{3} c d n +3 \,\mathrm {log}\left (d x +c \right ) a^{3} d n -3 \,\mathrm {log}\left (d x +c \right ) a^{2} b c n +\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )^{3} b^{3} d x +3 \mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )^{2} a \,b^{2} d x +3 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{3} d +3 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{2} b d x +a^{3} d x}{d} \] Input:

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

Output:

(3*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x)/(a*c + a*d*x + b*c*x + b* 
d*x**2),x)*a*b**3*d**2*n - 3*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x) 
/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**4*c*d*n + 6*int((log(((a + b*x)**n 
*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**2*b**2*d**2*n 
- 6*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d* 
x**2),x)*a*b**3*c*d*n + 3*log(c + d*x)*a**3*d*n - 3*log(c + d*x)*a**2*b*c* 
n + log(((a + b*x)**n*e)/(c + d*x)**n)**3*b**3*d*x + 3*log(((a + b*x)**n*e 
)/(c + d*x)**n)**2*a*b**2*d*x + 3*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3* 
d + 3*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b*d*x + a**3*d*x)/d