Integrand size = 33, antiderivative size = 390 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^3} \, dx=\frac {6 B^3 d n^3 (c+d x)}{(b c-a d)^2 (a+b x)}-\frac {3 b B^3 n^3 (c+d x)^2}{8 (b c-a d)^2 (a+b x)^2}+\frac {6 B^2 d n^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{(b c-a d)^2 (a+b x)}-\frac {3 b B^2 n^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{4 (b c-a d)^2 (a+b x)^2}+\frac {3 B d n (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^2 (a+b x)}-\frac {3 b B n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{4 (b c-a d)^2 (a+b x)^2}+\frac {d (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^2 (a+b x)}-\frac {b (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{2 (b c-a d)^2 (a+b x)^2} \]
6*B^3*d*n^3*(d*x+c)/(-a*d+b*c)^2/(b*x+a)-3/8*b*B^3*n^3*(d*x+c)^2/(-a*d+b*c )^2/(b*x+a)^2+6*B^2*d*n^2*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+ b*c)^2/(b*x+a)-3/4*b*B^2*n^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/( -a*d+b*c)^2/(b*x+a)^2+3*B*d*n*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/ (-a*d+b*c)^2/(b*x+a)-3/4*b*B*n*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n))) ^2/(-a*d+b*c)^2/(b*x+a)^2+d*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(- a*d+b*c)^2/(b*x+a)-1/2*b*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(-a *d+b*c)^2/(b*x+a)^2
Time = 0.69 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.78 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^3} \, dx=-\frac {-4 B^3 d^2 n^3 (a+b x)^2 \log ^3(a+b x)+4 B^3 d^2 n^3 (a+b x)^2 \log ^3(c+d x)+6 B^2 d^2 n^2 (a+b x)^2 \log ^2(c+d x) \left (2 A+3 B n+2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+6 B^2 d^2 n^2 (a+b x)^2 \log ^2(a+b x) \left (2 A+3 B n+2 B n \log (c+d x)+2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+6 B d^2 n (a+b x)^2 \log (c+d x) \left (2 A^2+6 A B n+7 B^2 n^2+2 B (2 A+3 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right )+(b c-a d) \left (4 A^3 (b c-a d)+3 B^3 n^3 (-15 a d+b (c-14 d x))+6 A B^2 n^2 (-7 a d+b (c-6 d x))+6 A^2 B n (-3 a d+b (c-2 d x))+6 B \left (2 A^2 (b c-a d)+B^2 n^2 (-7 a d+b (c-6 d x))+2 A B n (-3 a d+b (c-2 d x))\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 B^2 (2 A (b c-a d)+B n (-3 a d+b (c-2 d x))) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+4 B^3 (b c-a d) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )\right )-6 B d^2 n (a+b x)^2 \log (a+b x) \left (2 A^2+6 A B n+7 B^2 n^2+2 B^2 n^2 \log ^2(c+d x)+2 B (2 A+3 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+2 B n \log (c+d x) \left (2 A+3 B n+2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )}{8 b (b c-a d)^2 (a+b x)^2} \]
-1/8*(-4*B^3*d^2*n^3*(a + b*x)^2*Log[a + b*x]^3 + 4*B^3*d^2*n^3*(a + b*x)^ 2*Log[c + d*x]^3 + 6*B^2*d^2*n^2*(a + b*x)^2*Log[c + d*x]^2*(2*A + 3*B*n + 2*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 6*B^2*d^2*n^2*(a + b*x)^2*Log[a + b*x]^2*(2*A + 3*B*n + 2*B*n*Log[c + d*x] + 2*B*Log[(e*(a + b*x)^n)/(c + d *x)^n]) + 6*B*d^2*n*(a + b*x)^2*Log[c + d*x]*(2*A^2 + 6*A*B*n + 7*B^2*n^2 + 2*B*(2*A + 3*B*n)*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 2*B^2*Log[(e*(a + b *x)^n)/(c + d*x)^n]^2) + (b*c - a*d)*(4*A^3*(b*c - a*d) + 3*B^3*n^3*(-15*a *d + b*(c - 14*d*x)) + 6*A*B^2*n^2*(-7*a*d + b*(c - 6*d*x)) + 6*A^2*B*n*(- 3*a*d + b*(c - 2*d*x)) + 6*B*(2*A^2*(b*c - a*d) + B^2*n^2*(-7*a*d + b*(c - 6*d*x)) + 2*A*B*n*(-3*a*d + b*(c - 2*d*x)))*Log[(e*(a + b*x)^n)/(c + d*x) ^n] + 6*B^2*(2*A*(b*c - a*d) + B*n*(-3*a*d + b*(c - 2*d*x)))*Log[(e*(a + b *x)^n)/(c + d*x)^n]^2 + 4*B^3*(b*c - a*d)*Log[(e*(a + b*x)^n)/(c + d*x)^n] ^3) - 6*B*d^2*n*(a + b*x)^2*Log[a + b*x]*(2*A^2 + 6*A*B*n + 7*B^2*n^2 + 2* B^2*n^2*Log[c + d*x]^2 + 2*B*(2*A + 3*B*n)*Log[(e*(a + b*x)^n)/(c + d*x)^n ] + 2*B^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 2*B*n*Log[c + d*x]*(2*A + 3 *B*n + 2*B*Log[(e*(a + b*x)^n)/(c + d*x)^n])))/(b*(b*c - a*d)^2*(a + b*x)^ 2)
Time = 0.55 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.81, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2973, 2949, 2795, 2009}
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.
| \(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 2973 |
| \(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x)^3}dx\) |
| \(\Big \downarrow \) 2949 |
| \(\displaystyle \frac {\int \frac {(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{(a+b x)^3}d\frac {a+b x}{c+d x}}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {\int \left (\frac {b (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{(a+b x)^3}-\frac {d (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{(a+b x)^2}\right )d\frac {a+b x}{c+d x}}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {3 b B^2 n^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 (a+b x)^2}+\frac {6 B^2 d n^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac {3 b B n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 (a+b x)^2}+\frac {3 B d n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{a+b x}-\frac {b (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{2 (a+b x)^2}+\frac {d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{a+b x}-\frac {3 b B^3 n^3 (c+d x)^2}{8 (a+b x)^2}+\frac {6 B^3 d n^3 (c+d x)}{a+b x}}{(b c-a d)^2}\) |
((6*B^3*d*n^3*(c + d*x))/(a + b*x) - (3*b*B^3*n^3*(c + d*x)^2)/(8*(a + b*x )^2) + (6*B^2*d*n^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) - (3*b*B^2*n^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/( 4*(a + b*x)^2) + (3*B*d*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) ^2)/(a + b*x) - (3*b*B*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n] )^2)/(4*(a + b*x)^2) + (d*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) ^3)/(a + b*x) - (b*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/( 2*(a + b*x)^2))/(b*c - a*d)^2
3.2.69.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b , c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 ] && IntegerQ[m] && IntegerQ[r]))
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/b)^m Subst[Int[x^m*((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] && Ne Q[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[b*f - a*g, 0] && (GtQ[p, 0] || Lt Q[m, -1])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1621\) vs. \(2(382)=764\).
Time = 48.69 (sec) , antiderivative size = 1622, normalized size of antiderivative = 4.16
| method | result | size |
| parallelrisch | \(\text {Expression too large to display}\) | \(1622\) |
| risch | \(\text {Expression too large to display}\) | \(120138\) |
-1/8*(24*A*B^2*x*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^4*d^3*n-24*A*B^2*ln(e*(b* x+a)^n/((d*x+c)^n))^2*a*b^4*c*d^2+36*A*B^2*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2 *b^3*d^3*n+12*A*B^2*ln(e*(b*x+a)^n/((d*x+c)^n))*b^5*c^2*d*n-24*A^2*B*ln(e* (b*x+a)^n/((d*x+c)^n))*a*b^4*c*d^2-48*A*B^2*a*b^4*c*d^2*n^2-24*A^2*B*a*b^4 *c*d^2*n+12*A^2*B*ln(d*x+c)*a^2*b^3*d^3*n-24*B^3*ln(e*(b*x+a)^n/((d*x+c)^n ))^2*a*b^4*c*d^2*n-36*A*B^2*ln(b*x+a)*a^2*b^3*d^3*n^2+36*A*B^2*ln(d*x+c)*a ^2*b^3*d^3*n^2-12*A^2*B*ln(b*x+a)*a^2*b^3*d^3*n+36*A*B^2*x*a*b^4*d^3*n^2-3 6*A*B^2*x*b^5*c*d^2*n^2-48*B^3*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^4*c*d^2*n^2 +12*A^2*B*x*a*b^4*d^3*n-12*A^2*B*x*b^5*c*d^2*n+84*B^3*ln(d*x+c)*x*a*b^4*d^ 3*n^3-84*B^3*ln(b*x+a)*x*a*b^4*d^3*n^3-36*A*B^2*ln(b*x+a)*x^2*b^5*d^3*n^2- 24*B^3*x*ln(e*(b*x+a)^n/((d*x+c)^n))^2*a*b^4*d^3*n-12*B^3*x*ln(e*(b*x+a)^n /((d*x+c)^n))^2*b^5*c*d^2*n+36*B^3*x*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^4*d^3 *n^2-36*B^3*x*ln(e*(b*x+a)^n/((d*x+c)^n))*b^5*c*d^2*n^2-24*A*B^2*x*ln(e*(b *x+a)^n/((d*x+c)^n))^2*a*b^4*d^3-12*A^2*B*ln(b*x+a)*x^2*b^5*d^3*n+12*A^2*B *ln(d*x+c)*x^2*b^5*d^3*n+36*A*B^2*ln(d*x+c)*x^2*b^5*d^3*n^2+4*A^3*a^2*b^3* d^3+4*A^3*b^5*c^2*d-8*B^3*ln(e*(b*x+a)^n/((d*x+c)^n))^3*a*b^4*c*d^2+6*B^3* ln(e*(b*x+a)^n/((d*x+c)^n))^2*b^5*c^2*d*n+42*B^3*ln(e*(b*x+a)^n/((d*x+c)^n ))*a^2*b^3*d^3*n^2+6*B^3*ln(e*(b*x+a)^n/((d*x+c)^n))*b^5*c^2*d*n^2+12*A*B^ 2*ln(e*(b*x+a)^n/((d*x+c)^n))^2*b^5*c^2*d+12*A^2*B*ln(e*(b*x+a)^n/((d*x+c) ^n))*a^2*b^3*d^3+12*A^2*B*ln(e*(b*x+a)^n/((d*x+c)^n))*b^5*c^2*d-42*B^3*...
Leaf count of result is larger than twice the leaf count of optimal. 2244 vs. \(2 (382) = 764\).
Time = 0.36 (sec) , antiderivative size = 2244, normalized size of antiderivative = 5.75 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^3} \, dx=\text {Too large to display} \]
-1/8*(4*A^3*b^2*c^2 - 8*A^3*a*b*c*d + 4*A^3*a^2*d^2 + 3*(B^3*b^2*c^2 - 16* B^3*a*b*c*d + 15*B^3*a^2*d^2)*n^3 - 4*(B^3*b^2*d^2*n^3*x^2 + 2*B^3*a*b*d^2 *n^3*x - (B^3*b^2*c^2 - 2*B^3*a*b*c*d)*n^3)*log(b*x + a)^3 + 4*(B^3*b^2*d^ 2*n^3*x^2 + 2*B^3*a*b*d^2*n^3*x - (B^3*b^2*c^2 - 2*B^3*a*b*c*d)*n^3)*log(d *x + c)^3 + 4*(B^3*b^2*c^2 - 2*B^3*a*b*c*d + B^3*a^2*d^2)*log(e)^3 + 6*(A* B^2*b^2*c^2 - 8*A*B^2*a*b*c*d + 7*A*B^2*a^2*d^2)*n^2 + 6*((B^3*b^2*c^2 - 4 *B^3*a*b*c*d)*n^3 + 2*(A*B^2*b^2*c^2 - 2*A*B^2*a*b*c*d)*n^2 - (3*B^3*b^2*d ^2*n^3 + 2*A*B^2*b^2*d^2*n^2)*x^2 - 2*(2*A*B^2*a*b*d^2*n^2 + (B^3*b^2*c*d + 2*B^3*a*b*d^2)*n^3)*x - 2*(B^3*b^2*d^2*n^2*x^2 + 2*B^3*a*b*d^2*n^2*x - ( B^3*b^2*c^2 - 2*B^3*a*b*c*d)*n^2)*log(e))*log(b*x + a)^2 + 6*((B^3*b^2*c^2 - 4*B^3*a*b*c*d)*n^3 + 2*(A*B^2*b^2*c^2 - 2*A*B^2*a*b*c*d)*n^2 - (3*B^3*b ^2*d^2*n^3 + 2*A*B^2*b^2*d^2*n^2)*x^2 - 2*(2*A*B^2*a*b*d^2*n^2 + (B^3*b^2* c*d + 2*B^3*a*b*d^2)*n^3)*x - 2*(B^3*b^2*d^2*n^3*x^2 + 2*B^3*a*b*d^2*n^3*x - (B^3*b^2*c^2 - 2*B^3*a*b*c*d)*n^3)*log(b*x + a) - 2*(B^3*b^2*d^2*n^2*x^ 2 + 2*B^3*a*b*d^2*n^2*x - (B^3*b^2*c^2 - 2*B^3*a*b*c*d)*n^2)*log(e))*log(d *x + c)^2 + 6*(2*A*B^2*b^2*c^2 - 4*A*B^2*a*b*c*d + 2*A*B^2*a^2*d^2 - 2*(B^ 3*b^2*c*d - B^3*a*b*d^2)*n*x + (B^3*b^2*c^2 - 4*B^3*a*b*c*d + 3*B^3*a^2*d^ 2)*n)*log(e)^2 + 6*(A^2*B*b^2*c^2 - 4*A^2*B*a*b*c*d + 3*A^2*B*a^2*d^2)*n - 6*(7*(B^3*b^2*c*d - B^3*a*b*d^2)*n^3 + 6*(A*B^2*b^2*c*d - A*B^2*a*b*d^2)* n^2 + 2*(A^2*B*b^2*c*d - A^2*B*a*b*d^2)*n)*x + 6*((B^3*b^2*c^2 - 8*B^3*...
Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2246 vs. \(2 (382) = 764\).
Time = 0.33 (sec) , antiderivative size = 2246, normalized size of antiderivative = 5.76 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^3} \, dx=\text {Too large to display} \]
-1/2*B^3*log((b*x + a)^n*e/(d*x + c)^n)^3/(b^3*x^2 + 2*a*b^2*x + a^2*b) + 1/8*(6*(2*d^2*e*n*log(b*x + a)/(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2) - 2*d^2 *e*n*log(d*x + c)/(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2) + (2*b*d*e*n*x - b*c *e*n + 3*a*d*e*n)/(a^2*b^2*c - a^3*b*d + (b^4*c - a*b^3*d)*x^2 + 2*(a*b^3* c - a^2*b^2*d)*x))*log((b*x + a)^n*e/(d*x + c)^n)^2/e - (6*(b^2*c^2*e^2*n^ 2 - 8*a*b*c*d*e^2*n^2 + 7*a^2*d^2*e^2*n^2 + 2*(b^2*d^2*e^2*n^2*x^2 + 2*a*b *d^2*e^2*n^2*x + a^2*d^2*e^2*n^2)*log(b*x + a)^2 + 2*(b^2*d^2*e^2*n^2*x^2 + 2*a*b*d^2*e^2*n^2*x + a^2*d^2*e^2*n^2)*log(d*x + c)^2 - 6*(b^2*c*d*e^2*n ^2 - a*b*d^2*e^2*n^2)*x - 6*(b^2*d^2*e^2*n^2*x^2 + 2*a*b*d^2*e^2*n^2*x + a ^2*d^2*e^2*n^2)*log(b*x + a) + 2*(3*b^2*d^2*e^2*n^2*x^2 + 6*a*b*d^2*e^2*n^ 2*x + 3*a^2*d^2*e^2*n^2 - 2*(b^2*d^2*e^2*n^2*x^2 + 2*a*b*d^2*e^2*n^2*x + a ^2*d^2*e^2*n^2)*log(b*x + a))*log(d*x + c))*log((b*x + a)^n*e/(d*x + c)^n) /((a^2*b^3*c^2 - 2*a^3*b^2*c*d + a^4*b*d^2 + (b^5*c^2 - 2*a*b^4*c*d + a^2* b^3*d^2)*x^2 + 2*(a*b^4*c^2 - 2*a^2*b^3*c*d + a^3*b^2*d^2)*x)*e) + (3*b^2* c^2*e^3*n^3 - 48*a*b*c*d*e^3*n^3 + 45*a^2*d^2*e^3*n^3 - 4*(b^2*d^2*e^3*n^3 *x^2 + 2*a*b*d^2*e^3*n^3*x + a^2*d^2*e^3*n^3)*log(b*x + a)^3 + 4*(b^2*d^2* e^3*n^3*x^2 + 2*a*b*d^2*e^3*n^3*x + a^2*d^2*e^3*n^3)*log(d*x + c)^3 + 18*( b^2*d^2*e^3*n^3*x^2 + 2*a*b*d^2*e^3*n^3*x + a^2*d^2*e^3*n^3)*log(b*x + a)^ 2 + 6*(3*b^2*d^2*e^3*n^3*x^2 + 6*a*b*d^2*e^3*n^3*x + 3*a^2*d^2*e^3*n^3 - 2 *(b^2*d^2*e^3*n^3*x^2 + 2*a*b*d^2*e^3*n^3*x + a^2*d^2*e^3*n^3)*log(b*x ...
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^3} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (b x + a\right )}^{3}} \,d x } \]
Time = 5.30 (sec) , antiderivative size = 966, normalized size of antiderivative = 2.48 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^3} \, dx=-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^3\,\left (\frac {B^3}{2\,b\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}-\frac {B^3\,d^2}{2\,b\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {\frac {4\,A^3\,a\,d-4\,A^3\,b\,c+45\,B^3\,a\,d\,n^3-3\,B^3\,b\,c\,n^3+18\,A^2\,B\,a\,d\,n-6\,A^2\,B\,b\,c\,n+42\,A\,B^2\,a\,d\,n^2-6\,A\,B^2\,b\,c\,n^2}{2\,\left (a\,d-b\,c\right )}+\frac {3\,x\,\left (2\,b\,d\,A^2\,B\,n+6\,b\,d\,A\,B^2\,n^2+7\,b\,d\,B^3\,n^3\right )}{a\,d-b\,c}}{4\,a^2\,b+8\,a\,b^2\,x+4\,b^3\,x^2}-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^2\,\left (\frac {3\,A\,B^2}{2\,\left (a^2\,b+2\,a\,b^2\,x+b^3\,x^2\right )}-\frac {3\,d^2\,\left (3\,n\,B^3+2\,A\,B^2\right )}{4\,b\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {3\,B^3\,d^2\,\left (\frac {b\,n\,\left (a\,d-b\,c\right )\,\left (2\,a\,d-b\,c\right )}{d^2}+\frac {2\,b^2\,n\,x\,\left (a\,d-b\,c\right )}{d}+\frac {a\,b\,n\,\left (a\,d-b\,c\right )}{d}\right )}{4\,b\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (a^2\,b+2\,a\,b^2\,x+b^3\,x^2\right )}\right )-\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\,\left (\frac {3\,B\,b\,d\,\left (A^2-B^2\,n^2\right )\,x^2+3\,B\,\left (a\,d+b\,c\right )\,\left (A^2-B^2\,n^2\right )\,x+3\,B\,a\,c\,\left (A^2-B^2\,n^2\right )}{2\,b\,{\left (a+b\,x\right )}^3\,\left (c+d\,x\right )}+\frac {3\,d^2\,\left (3\,n\,B^3+2\,A\,B^2\right )\,\left (x\,\left (\left (\frac {b\,n\,\left (a\,d-b\,c\right )\,\left (2\,a\,d-b\,c\right )}{d^2}+\frac {a\,b\,n\,\left (a\,d-b\,c\right )}{d}\right )\,\left (a\,d+b\,c\right )+\frac {2\,a\,b^2\,c\,n\,\left (a\,d-b\,c\right )}{d}\right )+x^2\,\left (b\,d\,\left (\frac {b\,n\,\left (a\,d-b\,c\right )\,\left (2\,a\,d-b\,c\right )}{d^2}+\frac {a\,b\,n\,\left (a\,d-b\,c\right )}{d}\right )+\frac {2\,b^2\,n\,\left (a\,d+b\,c\right )\,\left (a\,d-b\,c\right )}{d}\right )+a\,c\,\left (\frac {b\,n\,\left (a\,d-b\,c\right )\,\left (2\,a\,d-b\,c\right )}{d^2}+\frac {a\,b\,n\,\left (a\,d-b\,c\right )}{d}\right )+2\,b^3\,n\,x^3\,\left (a\,d-b\,c\right )\right )}{4\,b^2\,{\left (a+b\,x\right )}^3\,\left (c+d\,x\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {B\,d^2\,n\,\mathrm {atan}\left (\frac {B\,d^2\,n\,\left (2\,b\,d\,x-\frac {b^3\,c^2-a^2\,b\,d^2}{b\,\left (a\,d-b\,c\right )}\right )\,\left (2\,A^2+6\,A\,B\,n+7\,B^2\,n^2\right )\,3{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (6\,A^2\,B\,d^2\,n+18\,A\,B^2\,d^2\,n^2+21\,B^3\,d^2\,n^3\right )}\right )\,\left (2\,A^2+6\,A\,B\,n+7\,B^2\,n^2\right )\,3{}\mathrm {i}}{2\,b\,{\left (a\,d-b\,c\right )}^2} \]
- log((e*(a + b*x)^n)/(c + d*x)^n)^3*(B^3/(2*b*(a^2 + b^2*x^2 + 2*a*b*x)) - (B^3*d^2)/(2*b*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) - ((4*A^3*a*d - 4*A^3*b *c + 45*B^3*a*d*n^3 - 3*B^3*b*c*n^3 + 18*A^2*B*a*d*n - 6*A^2*B*b*c*n + 42* A*B^2*a*d*n^2 - 6*A*B^2*b*c*n^2)/(2*(a*d - b*c)) + (3*x*(7*B^3*b*d*n^3 + 2 *A^2*B*b*d*n + 6*A*B^2*b*d*n^2))/(a*d - b*c))/(4*a^2*b + 4*b^3*x^2 + 8*a*b ^2*x) - log((e*(a + b*x)^n)/(c + d*x)^n)^2*((3*A*B^2)/(2*(a^2*b + b^3*x^2 + 2*a*b^2*x)) - (3*d^2*(2*A*B^2 + 3*B^3*n))/(4*b*(a^2*d^2 + b^2*c^2 - 2*a* b*c*d)) + (3*B^3*d^2*((b*n*(a*d - b*c)*(2*a*d - b*c))/d^2 + (2*b^2*n*x*(a* d - b*c))/d + (a*b*n*(a*d - b*c))/d))/(4*b*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d) *(a^2*b + b^3*x^2 + 2*a*b^2*x))) - log((e*(a + b*x)^n)/(c + d*x)^n)*((3*B* a*c*(A^2 - B^2*n^2) + 3*B*x*(a*d + b*c)*(A^2 - B^2*n^2) + 3*B*b*d*x^2*(A^2 - B^2*n^2))/(2*b*(a + b*x)^3*(c + d*x)) + (3*d^2*(2*A*B^2 + 3*B^3*n)*(x*( ((b*n*(a*d - b*c)*(2*a*d - b*c))/d^2 + (a*b*n*(a*d - b*c))/d)*(a*d + b*c) + (2*a*b^2*c*n*(a*d - b*c))/d) + x^2*(b*d*((b*n*(a*d - b*c)*(2*a*d - b*c)) /d^2 + (a*b*n*(a*d - b*c))/d) + (2*b^2*n*(a*d + b*c)*(a*d - b*c))/d) + a*c *((b*n*(a*d - b*c)*(2*a*d - b*c))/d^2 + (a*b*n*(a*d - b*c))/d) + 2*b^3*n*x ^3*(a*d - b*c)))/(4*b^2*(a + b*x)^3*(c + d*x)*(a^2*d^2 + b^2*c^2 - 2*a*b*c *d))) - (B*d^2*n*atan((B*d^2*n*(2*b*d*x - (b^3*c^2 - a^2*b*d^2)/(b*(a*d - b*c)))*(2*A^2 + 7*B^2*n^2 + 6*A*B*n)*3i)/((a*d - b*c)*(21*B^3*d^2*n^3 + 6* A^2*B*d^2*n + 18*A*B^2*d^2*n^2)))*(2*A^2 + 7*B^2*n^2 + 6*A*B*n)*3i)/(2*...