Integrand size = 33, antiderivative size = 611 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx=-\frac {6 B^3 d^2 n^3 (c+d x)}{(b c-a d)^3 (a+b x)}+\frac {3 b B^3 d n^3 (c+d x)^2}{4 (b c-a d)^3 (a+b x)^2}-\frac {2 b^2 B^3 n^3 (c+d x)^3}{27 (b c-a d)^3 (a+b x)^3}-\frac {6 B^2 d^2 n^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{(b c-a d)^3 (a+b x)}+\frac {3 b B^2 d n^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{2 (b c-a d)^3 (a+b x)^2}-\frac {2 b^2 B^2 n^2 (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{9 (b c-a d)^3 (a+b x)^3}-\frac {3 B d^2 n (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^3 (a+b x)}+\frac {3 b B d n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 (b c-a d)^3 (a+b x)^2}-\frac {b^2 B n (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{3 (b c-a d)^3 (a+b x)^3}-\frac {d^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^3 (a+b x)}+\frac {b d (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^3 (a+b x)^2}-\frac {b^2 (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 (b c-a d)^3 (a+b x)^3} \]
-6*B^3*d^2*n^3*(d*x+c)/(-a*d+b*c)^3/(b*x+a)+3/4*b*B^3*d*n^3*(d*x+c)^2/(-a* d+b*c)^3/(b*x+a)^2-2/27*b^2*B^3*n^3*(d*x+c)^3/(-a*d+b*c)^3/(b*x+a)^3-6*B^2 *d^2*n^2*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^3/(b*x+a)+3/ 2*b*B^2*d*n^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^3/(b* x+a)^2-2/9*b^2*B^2*n^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b *c)^3/(b*x+a)^3-3*B*d^2*n*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(-a* d+b*c)^3/(b*x+a)+3/2*b*B*d*n*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2 /(-a*d+b*c)^3/(b*x+a)^2-1/3*b^2*B*n*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c) ^n)))^2/(-a*d+b*c)^3/(b*x+a)^3-d^2*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n) ))^3/(-a*d+b*c)^3/(b*x+a)+b*d*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^ 3/(-a*d+b*c)^3/(b*x+a)^2-1/3*b^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n) ))^3/(-a*d+b*c)^3/(b*x+a)^3
Time = 0.86 (sec) , antiderivative size = 1003, normalized size of antiderivative = 1.64 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx=\frac {-36 B^3 d^3 n^3 (a+b x)^3 \log ^3(a+b x)+36 B^3 d^3 n^3 (a+b x)^3 \log ^3(c+d x)+18 B^2 d^3 n^2 (a+b x)^3 \log ^2(c+d x) \left (6 A+11 B n+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+18 B^2 d^3 n^2 (a+b x)^3 \log ^2(a+b x) \left (6 A+11 B n+6 B n \log (c+d x)+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+6 B d^3 n (a+b x)^3 \log (c+d x) \left (18 A^2+66 A B n+85 B^2 n^2+6 B (6 A+11 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+18 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right )-(b c-a d) \left (36 A^3 b^2 c^2-72 a A^3 b c d+36 a^2 A^3 d^2+36 A^2 b^2 B c^2 n-126 a A^2 b B c d n+198 a^2 A^2 B d^2 n+24 A b^2 B^2 c^2 n^2-138 a A b B^2 c d n^2+510 a^2 A B^2 d^2 n^2+8 b^2 B^3 c^2 n^3-73 a b B^3 c d n^3+575 a^2 B^3 d^2 n^3-54 A^2 b^2 B c d n x+270 a A^2 b B d^2 n x-90 A b^2 B^2 c d n^2 x+882 a A b B^2 d^2 n^2 x-57 b^2 B^3 c d n^3 x+1077 a b B^3 d^2 n^3 x+108 A^2 b^2 B d^2 n x^2+396 A b^2 B^2 d^2 n^2 x^2+510 b^2 B^3 d^2 n^3 x^2+6 B \left (18 A^2 (b c-a d)^2+6 A B n \left (11 a^2 d^2+a b d (-7 c+15 d x)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )+B^2 n^2 \left (85 a^2 d^2+a b d (-23 c+147 d x)+b^2 \left (4 c^2-15 c d x+66 d^2 x^2\right )\right )\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+18 B^2 \left (6 A (b c-a d)^2+B n \left (11 a^2 d^2+a b d (-7 c+15 d x)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+36 B^3 (b c-a d)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )\right )-6 B d^3 n (a+b x)^3 \log (a+b x) \left (18 A^2+66 A B n+85 B^2 n^2+18 B^2 n^2 \log ^2(c+d x)+6 B (6 A+11 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+18 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+6 B n \log (c+d x) \left (6 A+11 B n+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )}{108 b (b c-a d)^3 (a+b x)^3} \]
(-36*B^3*d^3*n^3*(a + b*x)^3*Log[a + b*x]^3 + 36*B^3*d^3*n^3*(a + b*x)^3*L og[c + d*x]^3 + 18*B^2*d^3*n^2*(a + b*x)^3*Log[c + d*x]^2*(6*A + 11*B*n + 6*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 18*B^2*d^3*n^2*(a + b*x)^3*Log[a + b*x]^2*(6*A + 11*B*n + 6*B*n*Log[c + d*x] + 6*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 6*B*d^3*n*(a + b*x)^3*Log[c + d*x]*(18*A^2 + 66*A*B*n + 85*B^2* n^2 + 6*B*(6*A + 11*B*n)*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 18*B^2*Log[(e* (a + b*x)^n)/(c + d*x)^n]^2) - (b*c - a*d)*(36*A^3*b^2*c^2 - 72*a*A^3*b*c* d + 36*a^2*A^3*d^2 + 36*A^2*b^2*B*c^2*n - 126*a*A^2*b*B*c*d*n + 198*a^2*A^ 2*B*d^2*n + 24*A*b^2*B^2*c^2*n^2 - 138*a*A*b*B^2*c*d*n^2 + 510*a^2*A*B^2*d ^2*n^2 + 8*b^2*B^3*c^2*n^3 - 73*a*b*B^3*c*d*n^3 + 575*a^2*B^3*d^2*n^3 - 54 *A^2*b^2*B*c*d*n*x + 270*a*A^2*b*B*d^2*n*x - 90*A*b^2*B^2*c*d*n^2*x + 882* a*A*b*B^2*d^2*n^2*x - 57*b^2*B^3*c*d*n^3*x + 1077*a*b*B^3*d^2*n^3*x + 108* A^2*b^2*B*d^2*n*x^2 + 396*A*b^2*B^2*d^2*n^2*x^2 + 510*b^2*B^3*d^2*n^3*x^2 + 6*B*(18*A^2*(b*c - a*d)^2 + 6*A*B*n*(11*a^2*d^2 + a*b*d*(-7*c + 15*d*x) + b^2*(2*c^2 - 3*c*d*x + 6*d^2*x^2)) + B^2*n^2*(85*a^2*d^2 + a*b*d*(-23*c + 147*d*x) + b^2*(4*c^2 - 15*c*d*x + 66*d^2*x^2)))*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 18*B^2*(6*A*(b*c - a*d)^2 + B*n*(11*a^2*d^2 + a*b*d*(-7*c + 15 *d*x) + b^2*(2*c^2 - 3*c*d*x + 6*d^2*x^2)))*Log[(e*(a + b*x)^n)/(c + d*x)^ n]^2 + 36*B^3*(b*c - a*d)^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^3) - 6*B*d^3* n*(a + b*x)^3*Log[a + b*x]*(18*A^2 + 66*A*B*n + 85*B^2*n^2 + 18*B^2*n^2...
Time = 0.68 (sec) , antiderivative size = 493, 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.
| \(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x)^4} \, dx\) |
| \(\Big \downarrow \) 2973 |
| \(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x)^4}dx\) |
| \(\Big \downarrow \) 2949 |
| \(\displaystyle \frac {\int \frac {(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{(a+b x)^4}d\frac {a+b x}{c+d x}}{(b c-a d)^3}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {\int \left (\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3 (c+d x)^4}{(a+b x)^4}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3 (c+d x)^3}{(a+b x)^3}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3 (c+d x)^2}{(a+b x)^2}\right )d\frac {a+b x}{c+d x}}{(b c-a d)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {2 b^2 B^2 n^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 (a+b x)^3}-\frac {b^2 B n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 (a+b x)^3}-\frac {b^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 (a+b x)^3}-\frac {6 B^2 d^2 n^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}+\frac {3 b B^2 d n^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}-\frac {3 B d^2 n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{a+b x}-\frac {d^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{a+b x}+\frac {3 b B d n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 (a+b x)^2}+\frac {b d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{(a+b x)^2}-\frac {2 b^2 B^3 n^3 (c+d x)^3}{27 (a+b x)^3}-\frac {6 B^3 d^2 n^3 (c+d x)}{a+b x}+\frac {3 b B^3 d n^3 (c+d x)^2}{4 (a+b x)^2}}{(b c-a d)^3}\) |
((-6*B^3*d^2*n^3*(c + d*x))/(a + b*x) + (3*b*B^3*d*n^3*(c + d*x)^2)/(4*(a + b*x)^2) - (2*b^2*B^3*n^3*(c + d*x)^3)/(27*(a + b*x)^3) - (6*B^2*d^2*n^2* (c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) + (3*b*B^2*d*n ^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(a + b*x)^2) - ( 2*b^2*B^2*n^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(a + b*x)^3) - (3*B*d^2*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/( a + b*x) + (3*b*B*d*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 )/(2*(a + b*x)^2) - (b^2*B*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x) )^n])^2)/(3*(a + b*x)^3) - (d^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x ))^n])^3)/(a + b*x) + (b*d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^ n])^3)/(a + b*x)^2 - (b^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n ])^3)/(3*(a + b*x)^3))/(b*c - a*d)^3
3.2.70.3.1 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. \(2686\) vs. \(2(597)=1194\).
Time = 95.36 (sec) , antiderivative size = 2687, normalized size of antiderivative = 4.40
| method | result | size |
| parallelrisch | \(\text {Expression too large to display}\) | \(2687\) |
| risch | \(\text {Expression too large to display}\) | \(175812\) |
-1/108*(-648*A*B^2*x*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^6*c*d^3*n-216*A*B^2*x ^2*ln(e*(b*x+a)^n/((d*x+c)^n))*b^7*c*d^3*n-324*B^3*x*ln(e*(b*x+a)^n/((d*x+ c)^n))^2*a*b^6*c*d^3*n-972*B^3*x*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^6*c*d^3*n ^2+540*A*B^2*x*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2*b^5*d^4*n+108*A*B^2*x*ln(e* (b*x+a)^n/((d*x+c)^n))*b^7*c^2*d^2*n-972*A*B^2*x*a*b^6*c*d^3*n^2-324*A^2*B *x*a*b^6*c*d^3*n-648*A*B^2*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2*b^5*c*d^3*n+324 *A*B^2*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^6*c^2*d^2*n-1188*A*B^2*ln(b*x+a)*x^ 2*a*b^6*d^4*n^2+1188*A*B^2*ln(d*x+c)*x^2*a*b^6*d^4*n^2-324*A^2*B*ln(b*x+a) *x^2*a*b^6*d^4*n+324*A^2*B*ln(d*x+c)*x^2*a*b^6*d^4*n-1188*A*B^2*ln(b*x+a)* x*a^2*b^5*d^4*n^2+1188*A*B^2*ln(d*x+c)*x*a^2*b^5*d^4*n^2-324*A^2*B*ln(b*x+ a)*x*a^2*b^5*d^4*n+324*A^2*B*ln(d*x+c)*x*a^2*b^5*d^4*n+108*A^2*B*ln(d*x+c) *a^3*b^4*d^4*n-486*B^3*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))^2*a*b^6*d^4*n-108*B ^3*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))^2*b^7*c*d^3*n+396*B^3*x^2*ln(e*(b*x+a)^ n/((d*x+c)^n))*a*b^6*d^4*n^2-396*B^3*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))*b^7*c *d^3*n^2-324*A*B^2*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))^2*a*b^6*d^4+396*A*B^2*x ^2*a*b^6*d^4*n^2-396*A*B^2*x^2*b^7*c*d^3*n^2-324*B^3*x*ln(e*(b*x+a)^n/((d* x+c)^n))^2*a^2*b^5*d^4*n-396*A*B^2*ln(b*x+a)*x^3*b^7*d^4*n^2+396*A*B^2*ln( d*x+c)*x^3*b^7*d^4*n^2+54*B^3*x*ln(e*(b*x+a)^n/((d*x+c)^n))^2*b^7*c^2*d^2* n+882*B^3*x*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2*b^5*d^4*n^2+90*B^3*x*ln(e*(b*x +a)^n/((d*x+c)^n))*b^7*c^2*d^2*n^2-1134*B^3*x*a*b^6*c*d^3*n^3+108*A^2*B...
Leaf count of result is larger than twice the leaf count of optimal. 4008 vs. \(2 (597) = 1194\).
Time = 0.41 (sec) , antiderivative size = 4008, normalized size of antiderivative = 6.56 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx=\text {Too large to display} \]
-1/108*(36*A^3*b^3*c^3 - 108*A^3*a*b^2*c^2*d + 108*A^3*a^2*b*c*d^2 - 36*A^ 3*a^3*d^3 + (8*B^3*b^3*c^3 - 81*B^3*a*b^2*c^2*d + 648*B^3*a^2*b*c*d^2 - 57 5*B^3*a^3*d^3)*n^3 + 36*(B^3*b^3*d^3*n^3*x^3 + 3*B^3*a*b^2*d^3*n^3*x^2 + 3 *B^3*a^2*b*d^3*n^3*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b*c*d^ 2)*n^3)*log(b*x + a)^3 - 36*(B^3*b^3*d^3*n^3*x^3 + 3*B^3*a*b^2*d^3*n^3*x^2 + 3*B^3*a^2*b*d^3*n^3*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b* c*d^2)*n^3)*log(d*x + c)^3 + 36*(B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a ^2*b*c*d^2 - B^3*a^3*d^3)*log(e)^3 + 6*(4*A*B^2*b^3*c^3 - 27*A*B^2*a*b^2*c ^2*d + 108*A*B^2*a^2*b*c*d^2 - 85*A*B^2*a^3*d^3)*n^2 + 6*(85*(B^3*b^3*c*d^ 2 - B^3*a*b^2*d^3)*n^3 + 66*(A*B^2*b^3*c*d^2 - A*B^2*a*b^2*d^3)*n^2 + 18*( A^2*B*b^3*c*d^2 - A^2*B*a*b^2*d^3)*n)*x^2 + 18*((2*B^3*b^3*c^3 - 9*B^3*a*b ^2*c^2*d + 18*B^3*a^2*b*c*d^2)*n^3 + (11*B^3*b^3*d^3*n^3 + 6*A*B^2*b^3*d^3 *n^2)*x^3 + 6*(A*B^2*b^3*c^3 - 3*A*B^2*a*b^2*c^2*d + 3*A*B^2*a^2*b*c*d^2)* n^2 + 3*(6*A*B^2*a*b^2*d^3*n^2 + (2*B^3*b^3*c*d^2 + 9*B^3*a*b^2*d^3)*n^3)* x^2 + 3*(6*A*B^2*a^2*b*d^3*n^2 - (B^3*b^3*c^2*d - 6*B^3*a*b^2*c*d^2 - 6*B^ 3*a^2*b*d^3)*n^3)*x + 6*(B^3*b^3*d^3*n^2*x^3 + 3*B^3*a*b^2*d^3*n^2*x^2 + 3 *B^3*a^2*b*d^3*n^2*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b*c*d^ 2)*n^2)*log(e))*log(b*x + a)^2 + 18*((2*B^3*b^3*c^3 - 9*B^3*a*b^2*c^2*d + 18*B^3*a^2*b*c*d^2)*n^3 + (11*B^3*b^3*d^3*n^3 + 6*A*B^2*b^3*d^3*n^2)*x^3 + 6*(A*B^2*b^3*c^3 - 3*A*B^2*a*b^2*c^2*d + 3*A*B^2*a^2*b*c*d^2)*n^2 + 3*...
Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 3630 vs. \(2 (597) = 1194\).
Time = 0.43 (sec) , antiderivative size = 3630, normalized size of antiderivative = 5.94 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx=\text {Too large to display} \]
-1/3*B^3*log((b*x + a)^n*e/(d*x + c)^n)^3/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b ^2*x + a^3*b) - 1/108*(18*(6*d^3*e*n*log(b*x + a)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) - 6*d^3*e*n*log(d*x + c)/(b^4*c^3 - 3*a*b^ 3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) + (6*b^2*d^2*e*n*x^2 + 2*b^2*c^2*e* n - 7*a*b*c*d*e*n + 11*a^2*d^2*e*n - 3*(b^2*c*d*e*n - 5*a*b*d^2*e*n)*x)/(a ^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4* d^2)*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^2 + 3*(a^2*b^4*c^ 2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*x))*log((b*x + a)^n*e/(d*x + c)^n)^2/e + (6*(4*b^3*c^3*e^2*n^2 - 27*a*b^2*c^2*d*e^2*n^2 + 108*a^2*b*c*d^2*e^2*n^2 - 85*a^3*d^3*e^2*n^2 + 66*(b^3*c*d^2*e^2*n^2 - a*b^2*d^3*e^2*n^2)*x^2 - 18* (b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a ^3*d^3*e^2*n^2)*log(b*x + a)^2 - 18*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d^3*e^2 *n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*log(d*x + c)^2 - 3*(5* b^3*c^2*d*e^2*n^2 - 54*a*b^2*c*d^2*e^2*n^2 + 49*a^2*b*d^3*e^2*n^2)*x + 66* (b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a ^3*d^3*e^2*n^2)*log(b*x + a) - 6*(11*b^3*d^3*e^2*n^2*x^3 + 33*a*b^2*d^3*e^ 2*n^2*x^2 + 33*a^2*b*d^3*e^2*n^2*x + 11*a^3*d^3*e^2*n^2 - 6*(b^3*d^3*e^2*n ^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2 )*log(b*x + a))*log(d*x + c))*log((b*x + a)^n*e/(d*x + c)^n)/((a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3 + (b^7*c^3 - 3*a*b^6*c...
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (b x + a\right )}^{4}} \,d x } \]
Time = 7.73 (sec) , antiderivative size = 2069, normalized size of antiderivative = 3.39 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^4} \, dx=\text {Too large to display} \]
((36*A^3*a^2*d^2 + 36*A^3*b^2*c^2 + 575*B^3*a^2*d^2*n^3 + 8*B^3*b^2*c^2*n^ 3 + 198*A^2*B*a^2*d^2*n + 36*A^2*B*b^2*c^2*n - 72*A^3*a*b*c*d + 510*A*B^2* a^2*d^2*n^2 + 24*A*B^2*b^2*c^2*n^2 - 73*B^3*a*b*c*d*n^3 - 126*A^2*B*a*b*c* d*n - 138*A*B^2*a*b*c*d*n^2)/(6*(a*d - b*c)) + (x*(359*B^3*a*b*d^2*n^3 - 1 9*B^3*b^2*c*d*n^3 + 90*A^2*B*a*b*d^2*n - 18*A^2*B*b^2*c*d*n + 294*A*B^2*a* b*d^2*n^2 - 30*A*B^2*b^2*c*d*n^2))/(2*(a*d - b*c)) + (x^2*(85*B^3*b^2*d^2* n^3 + 18*A^2*B*b^2*d^2*n + 66*A*B^2*b^2*d^2*n^2))/(a*d - b*c))/(x^3*(18*b^ 5*c - 18*a*b^4*d) + x*(54*a^2*b^3*c - 54*a^3*b^2*d) - x^2*(54*a^2*b^3*d - 54*a*b^4*c) + 18*a^3*b^2*c - 18*a^4*b*d) - log((e*(a + b*x)^n)/(c + d*x)^n )^3*(B^3/(3*b*(a^3 + b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x)) - (B^3*d^3)/(3*b* (a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - log((e*(a + b*x)^n )/(c + d*x)^n)^2*((A*B^2)/(a^3*b + b^4*x^3 + 3*a^2*b^2*x + 3*a*b^3*x^2) - (d^3*(6*A*B^2 + 11*B^3*n))/(6*b*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2 *b*c*d^2)) + (B^3*d^3*(a*((b*n*(a*d - b*c)*(3*a*d - b*c))/(6*d^2) + (a*b*n *(a*d - b*c))/(3*d)) + x*(b*((b*n*(a*d - b*c)*(3*a*d - b*c))/(6*d^2) + (a* b*n*(a*d - b*c))/(3*d)) + (2*a*b^2*n*(a*d - b*c))/(3*d) + (b^2*n*(a*d - b* c)*(3*a*d - b*c))/(3*d^2)) + (b*n*(a*d - b*c)*(3*a^2*d^2 + b^2*c^2 - 3*a*b *c*d))/(3*d^3) + (b^3*n*x^2*(a*d - b*c))/d))/(b*(a^3*d^3 - b^3*c^3 + 3*a*b ^2*c^2*d - 3*a^2*b*c*d^2)*(a^3*b + b^4*x^3 + 3*a^2*b^2*x + 3*a*b^3*x^2))) - log((e*(a + b*x)^n)/(c + d*x)^n)*((x*((a*d + b*c)*(3*A^2*B*a*d - 3*A^...