Integrand size = 31, antiderivative size = 135 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {(B d-A e) (a+b x)^5}{7 e (b d-a e) (d+e x)^7}+\frac {(5 b B d+2 A b e-7 a B e) (a+b x)^5}{42 e (b d-a e)^2 (d+e x)^6}+\frac {b (5 b B d+2 A b e-7 a B e) (a+b x)^5}{210 e (b d-a e)^3 (d+e x)^5} \]
-1/7*(-A*e+B*d)*(b*x+a)^5/e/(-a*e+b*d)/(e*x+d)^7+1/42*(2*A*b*e-7*B*a*e+5*B *b*d)*(b*x+a)^5/e/(-a*e+b*d)^2/(e*x+d)^6+1/210*b*(2*A*b*e-7*B*a*e+5*B*b*d) *(b*x+a)^5/e/(-a*e+b*d)^3/(e*x+d)^5
Leaf count is larger than twice the leaf count of optimal. \(323\) vs. \(2(135)=270\).
Time = 0.10 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.39 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {5 a^4 e^4 (6 A e+B (d+7 e x))+4 a^3 b e^3 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+3 a^2 b^2 e^2 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 a b^3 e \left (3 A e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+b^4 \left (2 A e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 B \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )}{210 e^6 (d+e x)^7} \]
-1/210*(5*a^4*e^4*(6*A*e + B*(d + 7*e*x)) + 4*a^3*b*e^3*(5*A*e*(d + 7*e*x) + 2*B*(d^2 + 7*d*e*x + 21*e^2*x^2)) + 3*a^2*b^2*e^2*(4*A*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*B*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)) + 2*a* b^3*e*(3*A*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 4*B*(d^4 + 7* d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)) + b^4*(2*A*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 5*B*(d^5 + 7*d^ 4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)))/(e^ 6*(d + e*x)^7)
Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1184, 27, 87, 55, 48}
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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2 (A+B x)}{(d+e x)^8} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^4 (a+b x)^4 (A+B x)}{(d+e x)^8}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^4 (A+B x)}{(d+e x)^8}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-7 a B e+2 A b e+5 b B d) \int \frac {(a+b x)^4}{(d+e x)^7}dx}{7 e (b d-a e)}-\frac {(a+b x)^5 (B d-A e)}{7 e (d+e x)^7 (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-7 a B e+2 A b e+5 b B d) \left (\frac {b \int \frac {(a+b x)^4}{(d+e x)^6}dx}{6 (b d-a e)}+\frac {(a+b x)^5}{6 (d+e x)^6 (b d-a e)}\right )}{7 e (b d-a e)}-\frac {(a+b x)^5 (B d-A e)}{7 e (d+e x)^7 (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\left (\frac {b (a+b x)^5}{30 (d+e x)^5 (b d-a e)^2}+\frac {(a+b x)^5}{6 (d+e x)^6 (b d-a e)}\right ) (-7 a B e+2 A b e+5 b B d)}{7 e (b d-a e)}-\frac {(a+b x)^5 (B d-A e)}{7 e (d+e x)^7 (b d-a e)}\) |
-1/7*((B*d - A*e)*(a + b*x)^5)/(e*(b*d - a*e)*(d + e*x)^7) + ((5*b*B*d + 2 *A*b*e - 7*a*B*e)*((a + b*x)^5/(6*(b*d - a*e)*(d + e*x)^6) + (b*(a + b*x)^ 5)/(30*(b*d - a*e)^2*(d + e*x)^5)))/(7*e*(b*d - a*e))
3.17.88.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(412\) vs. \(2(129)=258\).
Time = 0.24 (sec) , antiderivative size = 413, normalized size of antiderivative = 3.06
| method | result | size |
| risch | \(\frac {-\frac {B \,b^{4} x^{5}}{2 e}-\frac {b^{3} \left (2 A b e +8 B a e +5 B b d \right ) x^{4}}{6 e^{2}}-\frac {b^{2} \left (6 A a b \,e^{2}+2 A \,b^{2} d e +9 a^{2} B \,e^{2}+8 B a b d e +5 B \,b^{2} d^{2}\right ) x^{3}}{6 e^{3}}-\frac {b \left (12 A \,a^{2} b \,e^{3}+6 A a \,b^{2} d \,e^{2}+2 A \,b^{3} d^{2} e +8 B \,e^{3} a^{3}+9 B \,a^{2} b d \,e^{2}+8 B a \,b^{2} d^{2} e +5 B \,b^{3} d^{3}\right ) x^{2}}{10 e^{4}}-\frac {\left (20 A \,a^{3} b \,e^{4}+12 A \,a^{2} b^{2} d \,e^{3}+6 A a \,b^{3} d^{2} e^{2}+2 A \,b^{4} d^{3} e +5 B \,a^{4} e^{4}+8 B \,a^{3} b d \,e^{3}+9 B \,a^{2} b^{2} d^{2} e^{2}+8 B a \,b^{3} d^{3} e +5 b^{4} B \,d^{4}\right ) x}{30 e^{5}}-\frac {30 A \,a^{4} e^{5}+20 A \,a^{3} b d \,e^{4}+12 A \,a^{2} b^{2} d^{2} e^{3}+6 A a \,b^{3} d^{3} e^{2}+2 A \,b^{4} d^{4} e +5 B \,a^{4} d \,e^{4}+8 B \,a^{3} b \,d^{2} e^{3}+9 B \,a^{2} b^{2} d^{3} e^{2}+8 B a \,b^{3} d^{4} e +5 b^{4} B \,d^{5}}{210 e^{6}}}{\left (e x +d \right )^{7}}\) | \(413\) |
| default | \(-\frac {4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +B \,a^{4} e^{4}-8 B \,a^{3} b d \,e^{3}+18 B \,a^{2} b^{2} d^{2} e^{2}-16 B a \,b^{3} d^{3} e +5 b^{4} B \,d^{4}}{6 e^{6} \left (e x +d \right )^{6}}-\frac {2 b \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +2 B \,e^{3} a^{3}-9 B \,a^{2} b d \,e^{2}+12 B a \,b^{2} d^{2} e -5 B \,b^{3} d^{3}\right )}{5 e^{6} \left (e x +d \right )^{5}}-\frac {b^{3} \left (A b e +4 B a e -5 B b d \right )}{3 e^{6} \left (e x +d \right )^{3}}-\frac {b^{2} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +3 a^{2} B \,e^{2}-8 B a b d e +5 B \,b^{2} d^{2}\right )}{2 e^{6} \left (e x +d \right )^{4}}-\frac {b^{4} B}{2 e^{6} \left (e x +d \right )^{2}}-\frac {A \,a^{4} e^{5}-4 A \,a^{3} b d \,e^{4}+6 A \,a^{2} b^{2} d^{2} e^{3}-4 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e -B \,a^{4} d \,e^{4}+4 B \,a^{3} b \,d^{2} e^{3}-6 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e -b^{4} B \,d^{5}}{7 e^{6} \left (e x +d \right )^{7}}\) | \(430\) |
| norman | \(\frac {-\frac {B \,b^{4} x^{5}}{2 e}-\frac {\left (2 A \,b^{4} e^{2}+8 B \,e^{2} b^{3} a +5 B \,b^{4} d e \right ) x^{4}}{6 e^{3}}-\frac {\left (6 A a \,b^{3} e^{3}+2 A \,b^{4} d \,e^{2}+9 B \,e^{3} b^{2} a^{2}+8 B a \,b^{3} d \,e^{2}+5 B \,b^{4} d^{2} e \right ) x^{3}}{6 e^{4}}-\frac {\left (12 A \,a^{2} b^{2} e^{4}+6 A a \,b^{3} d \,e^{3}+2 A \,b^{4} d^{2} e^{2}+8 B \,a^{3} b \,e^{4}+9 B \,a^{2} b^{2} d \,e^{3}+8 B a \,b^{3} d^{2} e^{2}+5 b^{4} B \,d^{3} e \right ) x^{2}}{10 e^{5}}-\frac {\left (20 A \,a^{3} b \,e^{5}+12 A \,a^{2} b^{2} d \,e^{4}+6 A a \,b^{3} d^{2} e^{3}+2 A \,b^{4} d^{3} e^{2}+5 B \,e^{5} a^{4}+8 B \,a^{3} b d \,e^{4}+9 B \,a^{2} b^{2} d^{2} e^{3}+8 B a \,b^{3} d^{3} e^{2}+5 b^{4} B \,d^{4} e \right ) x}{30 e^{6}}-\frac {30 A \,a^{4} e^{6}+20 A \,a^{3} b d \,e^{5}+12 A \,a^{2} b^{2} d^{2} e^{4}+6 A a \,b^{3} d^{3} e^{3}+2 A \,b^{4} d^{4} e^{2}+5 B \,a^{4} d \,e^{5}+8 B \,a^{3} b \,d^{2} e^{4}+9 B \,a^{2} b^{2} d^{3} e^{3}+8 B a \,b^{3} d^{4} e^{2}+5 b^{4} B \,d^{5} e}{210 e^{7}}}{\left (e x +d \right )^{7}}\) | \(450\) |
| gosper | \(-\frac {105 B \,x^{5} b^{4} e^{5}+70 A \,b^{4} e^{5} x^{4}+280 B \,x^{4} a \,b^{3} e^{5}+175 B \,x^{4} b^{4} d \,e^{4}+210 A \,x^{3} a \,b^{3} e^{5}+70 A \,x^{3} b^{4} d \,e^{4}+315 B \,x^{3} a^{2} b^{2} e^{5}+280 B \,x^{3} a \,b^{3} d \,e^{4}+175 B \,x^{3} b^{4} d^{2} e^{3}+252 A \,x^{2} a^{2} b^{2} e^{5}+126 A \,x^{2} a \,b^{3} d \,e^{4}+42 A \,x^{2} b^{4} d^{2} e^{3}+168 B \,x^{2} a^{3} b \,e^{5}+189 B \,x^{2} a^{2} b^{2} d \,e^{4}+168 B \,x^{2} a \,b^{3} d^{2} e^{3}+105 B \,x^{2} b^{4} d^{3} e^{2}+140 A x \,a^{3} b \,e^{5}+84 A x \,a^{2} b^{2} d \,e^{4}+42 A x a \,b^{3} d^{2} e^{3}+14 A x \,b^{4} d^{3} e^{2}+35 B x \,a^{4} e^{5}+56 B x \,a^{3} b d \,e^{4}+63 B x \,a^{2} b^{2} d^{2} e^{3}+56 B x a \,b^{3} d^{3} e^{2}+35 B x \,b^{4} d^{4} e +30 A \,a^{4} e^{5}+20 A \,a^{3} b d \,e^{4}+12 A \,a^{2} b^{2} d^{2} e^{3}+6 A a \,b^{3} d^{3} e^{2}+2 A \,b^{4} d^{4} e +5 B \,a^{4} d \,e^{4}+8 B \,a^{3} b \,d^{2} e^{3}+9 B \,a^{2} b^{2} d^{3} e^{2}+8 B a \,b^{3} d^{4} e +5 b^{4} B \,d^{5}}{210 e^{6} \left (e x +d \right )^{7}}\) | \(469\) |
| parallelrisch | \(-\frac {105 b^{4} B \,x^{5} e^{6}+70 A \,b^{4} e^{6} x^{4}+280 B a \,b^{3} e^{6} x^{4}+175 B \,b^{4} d \,e^{5} x^{4}+210 A a \,b^{3} e^{6} x^{3}+70 A \,b^{4} d \,e^{5} x^{3}+315 B \,a^{2} b^{2} e^{6} x^{3}+280 B a \,b^{3} d \,e^{5} x^{3}+175 B \,b^{4} d^{2} e^{4} x^{3}+252 A \,a^{2} b^{2} e^{6} x^{2}+126 A a \,b^{3} d \,e^{5} x^{2}+42 A \,b^{4} d^{2} e^{4} x^{2}+168 B \,a^{3} b \,e^{6} x^{2}+189 B \,a^{2} b^{2} d \,e^{5} x^{2}+168 B a \,b^{3} d^{2} e^{4} x^{2}+105 B \,b^{4} d^{3} e^{3} x^{2}+140 A \,a^{3} b \,e^{6} x +84 A \,a^{2} b^{2} d \,e^{5} x +42 A a \,b^{3} d^{2} e^{4} x +14 A \,b^{4} d^{3} e^{3} x +35 B \,a^{4} e^{6} x +56 B \,a^{3} b d \,e^{5} x +63 B \,a^{2} b^{2} d^{2} e^{4} x +56 B a \,b^{3} d^{3} e^{3} x +35 B \,b^{4} d^{4} e^{2} x +30 A \,a^{4} e^{6}+20 A \,a^{3} b d \,e^{5}+12 A \,a^{2} b^{2} d^{2} e^{4}+6 A a \,b^{3} d^{3} e^{3}+2 A \,b^{4} d^{4} e^{2}+5 B \,a^{4} d \,e^{5}+8 B \,a^{3} b \,d^{2} e^{4}+9 B \,a^{2} b^{2} d^{3} e^{3}+8 B a \,b^{3} d^{4} e^{2}+5 b^{4} B \,d^{5} e}{210 e^{7} \left (e x +d \right )^{7}}\) | \(476\) |
(-1/2*B*b^4/e*x^5-1/6*b^3/e^2*(2*A*b*e+8*B*a*e+5*B*b*d)*x^4-1/6*b^2/e^3*(6 *A*a*b*e^2+2*A*b^2*d*e+9*B*a^2*e^2+8*B*a*b*d*e+5*B*b^2*d^2)*x^3-1/10*b/e^4 *(12*A*a^2*b*e^3+6*A*a*b^2*d*e^2+2*A*b^3*d^2*e+8*B*a^3*e^3+9*B*a^2*b*d*e^2 +8*B*a*b^2*d^2*e+5*B*b^3*d^3)*x^2-1/30/e^5*(20*A*a^3*b*e^4+12*A*a^2*b^2*d* e^3+6*A*a*b^3*d^2*e^2+2*A*b^4*d^3*e+5*B*a^4*e^4+8*B*a^3*b*d*e^3+9*B*a^2*b^ 2*d^2*e^2+8*B*a*b^3*d^3*e+5*B*b^4*d^4)*x-1/210/e^6*(30*A*a^4*e^5+20*A*a^3* b*d*e^4+12*A*a^2*b^2*d^2*e^3+6*A*a*b^3*d^3*e^2+2*A*b^4*d^4*e+5*B*a^4*d*e^4 +8*B*a^3*b*d^2*e^3+9*B*a^2*b^2*d^3*e^2+8*B*a*b^3*d^4*e+5*B*b^4*d^5))/(e*x+ d)^7
Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (129) = 258\).
Time = 0.33 (sec) , antiderivative size = 478, normalized size of antiderivative = 3.54 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {105 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 30 \, A a^{4} e^{5} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 5 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 35 \, {\left (5 \, B b^{4} d e^{4} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 35 \, {\left (5 \, B b^{4} d^{2} e^{3} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 21 \, {\left (5 \, B b^{4} d^{3} e^{2} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 7 \, {\left (5 \, B b^{4} d^{4} e + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 5 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{210 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \]
-1/210*(105*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 30*A*a^4*e^5 + 2*(4*B*a*b^3 + A* b^4)*d^4*e + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 4*(2*B*a^3*b + 3*A*a^2* b^2)*d^2*e^3 + 5*(B*a^4 + 4*A*a^3*b)*d*e^4 + 35*(5*B*b^4*d*e^4 + 2*(4*B*a* b^3 + A*b^4)*e^5)*x^4 + 35*(5*B*b^4*d^2*e^3 + 2*(4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 21*(5*B*b^4*d^3*e^2 + 2*(4*B*a*b^ 3 + A*b^4)*d^2*e^3 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 4*(2*B*a^3*b + 3* A*a^2*b^2)*e^5)*x^2 + 7*(5*B*b^4*d^4*e + 2*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 3 *(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 5 *(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d^6*e^7*x + d^7*e^6)
Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (129) = 258\).
Time = 0.21 (sec) , antiderivative size = 478, normalized size of antiderivative = 3.54 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {105 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 30 \, A a^{4} e^{5} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 5 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 35 \, {\left (5 \, B b^{4} d e^{4} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 35 \, {\left (5 \, B b^{4} d^{2} e^{3} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 21 \, {\left (5 \, B b^{4} d^{3} e^{2} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 7 \, {\left (5 \, B b^{4} d^{4} e + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 5 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{210 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \]
-1/210*(105*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 30*A*a^4*e^5 + 2*(4*B*a*b^3 + A* b^4)*d^4*e + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 4*(2*B*a^3*b + 3*A*a^2* b^2)*d^2*e^3 + 5*(B*a^4 + 4*A*a^3*b)*d*e^4 + 35*(5*B*b^4*d*e^4 + 2*(4*B*a* b^3 + A*b^4)*e^5)*x^4 + 35*(5*B*b^4*d^2*e^3 + 2*(4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 21*(5*B*b^4*d^3*e^2 + 2*(4*B*a*b^ 3 + A*b^4)*d^2*e^3 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 4*(2*B*a^3*b + 3* A*a^2*b^2)*e^5)*x^2 + 7*(5*B*b^4*d^4*e + 2*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 3 *(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 5 *(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d^6*e^7*x + d^7*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (129) = 258\).
Time = 0.30 (sec) , antiderivative size = 468, normalized size of antiderivative = 3.47 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {105 \, B b^{4} e^{5} x^{5} + 175 \, B b^{4} d e^{4} x^{4} + 280 \, B a b^{3} e^{5} x^{4} + 70 \, A b^{4} e^{5} x^{4} + 175 \, B b^{4} d^{2} e^{3} x^{3} + 280 \, B a b^{3} d e^{4} x^{3} + 70 \, A b^{4} d e^{4} x^{3} + 315 \, B a^{2} b^{2} e^{5} x^{3} + 210 \, A a b^{3} e^{5} x^{3} + 105 \, B b^{4} d^{3} e^{2} x^{2} + 168 \, B a b^{3} d^{2} e^{3} x^{2} + 42 \, A b^{4} d^{2} e^{3} x^{2} + 189 \, B a^{2} b^{2} d e^{4} x^{2} + 126 \, A a b^{3} d e^{4} x^{2} + 168 \, B a^{3} b e^{5} x^{2} + 252 \, A a^{2} b^{2} e^{5} x^{2} + 35 \, B b^{4} d^{4} e x + 56 \, B a b^{3} d^{3} e^{2} x + 14 \, A b^{4} d^{3} e^{2} x + 63 \, B a^{2} b^{2} d^{2} e^{3} x + 42 \, A a b^{3} d^{2} e^{3} x + 56 \, B a^{3} b d e^{4} x + 84 \, A a^{2} b^{2} d e^{4} x + 35 \, B a^{4} e^{5} x + 140 \, A a^{3} b e^{5} x + 5 \, B b^{4} d^{5} + 8 \, B a b^{3} d^{4} e + 2 \, A b^{4} d^{4} e + 9 \, B a^{2} b^{2} d^{3} e^{2} + 6 \, A a b^{3} d^{3} e^{2} + 8 \, B a^{3} b d^{2} e^{3} + 12 \, A a^{2} b^{2} d^{2} e^{3} + 5 \, B a^{4} d e^{4} + 20 \, A a^{3} b d e^{4} + 30 \, A a^{4} e^{5}}{210 \, {\left (e x + d\right )}^{7} e^{6}} \]
-1/210*(105*B*b^4*e^5*x^5 + 175*B*b^4*d*e^4*x^4 + 280*B*a*b^3*e^5*x^4 + 70 *A*b^4*e^5*x^4 + 175*B*b^4*d^2*e^3*x^3 + 280*B*a*b^3*d*e^4*x^3 + 70*A*b^4* d*e^4*x^3 + 315*B*a^2*b^2*e^5*x^3 + 210*A*a*b^3*e^5*x^3 + 105*B*b^4*d^3*e^ 2*x^2 + 168*B*a*b^3*d^2*e^3*x^2 + 42*A*b^4*d^2*e^3*x^2 + 189*B*a^2*b^2*d*e ^4*x^2 + 126*A*a*b^3*d*e^4*x^2 + 168*B*a^3*b*e^5*x^2 + 252*A*a^2*b^2*e^5*x ^2 + 35*B*b^4*d^4*e*x + 56*B*a*b^3*d^3*e^2*x + 14*A*b^4*d^3*e^2*x + 63*B*a ^2*b^2*d^2*e^3*x + 42*A*a*b^3*d^2*e^3*x + 56*B*a^3*b*d*e^4*x + 84*A*a^2*b^ 2*d*e^4*x + 35*B*a^4*e^5*x + 140*A*a^3*b*e^5*x + 5*B*b^4*d^5 + 8*B*a*b^3*d ^4*e + 2*A*b^4*d^4*e + 9*B*a^2*b^2*d^3*e^2 + 6*A*a*b^3*d^3*e^2 + 8*B*a^3*b *d^2*e^3 + 12*A*a^2*b^2*d^2*e^3 + 5*B*a^4*d*e^4 + 20*A*a^3*b*d*e^4 + 30*A* a^4*e^5)/((e*x + d)^7*e^6)
Time = 10.95 (sec) , antiderivative size = 479, normalized size of antiderivative = 3.55 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx=-\frac {\frac {5\,B\,a^4\,d\,e^4+30\,A\,a^4\,e^5+8\,B\,a^3\,b\,d^2\,e^3+20\,A\,a^3\,b\,d\,e^4+9\,B\,a^2\,b^2\,d^3\,e^2+12\,A\,a^2\,b^2\,d^2\,e^3+8\,B\,a\,b^3\,d^4\,e+6\,A\,a\,b^3\,d^3\,e^2+5\,B\,b^4\,d^5+2\,A\,b^4\,d^4\,e}{210\,e^6}+\frac {x\,\left (5\,B\,a^4\,e^4+8\,B\,a^3\,b\,d\,e^3+20\,A\,a^3\,b\,e^4+9\,B\,a^2\,b^2\,d^2\,e^2+12\,A\,a^2\,b^2\,d\,e^3+8\,B\,a\,b^3\,d^3\,e+6\,A\,a\,b^3\,d^2\,e^2+5\,B\,b^4\,d^4+2\,A\,b^4\,d^3\,e\right )}{30\,e^5}+\frac {b^3\,x^4\,\left (2\,A\,b\,e+8\,B\,a\,e+5\,B\,b\,d\right )}{6\,e^2}+\frac {b\,x^2\,\left (8\,B\,a^3\,e^3+9\,B\,a^2\,b\,d\,e^2+12\,A\,a^2\,b\,e^3+8\,B\,a\,b^2\,d^2\,e+6\,A\,a\,b^2\,d\,e^2+5\,B\,b^3\,d^3+2\,A\,b^3\,d^2\,e\right )}{10\,e^4}+\frac {b^2\,x^3\,\left (9\,B\,a^2\,e^2+8\,B\,a\,b\,d\,e+6\,A\,a\,b\,e^2+5\,B\,b^2\,d^2+2\,A\,b^2\,d\,e\right )}{6\,e^3}+\frac {B\,b^4\,x^5}{2\,e}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]
-((30*A*a^4*e^5 + 5*B*b^4*d^5 + 2*A*b^4*d^4*e + 5*B*a^4*d*e^4 + 6*A*a*b^3* d^3*e^2 + 8*B*a^3*b*d^2*e^3 + 12*A*a^2*b^2*d^2*e^3 + 9*B*a^2*b^2*d^3*e^2 + 20*A*a^3*b*d*e^4 + 8*B*a*b^3*d^4*e)/(210*e^6) + (x*(5*B*a^4*e^4 + 5*B*b^4 *d^4 + 20*A*a^3*b*e^4 + 2*A*b^4*d^3*e + 6*A*a*b^3*d^2*e^2 + 12*A*a^2*b^2*d *e^3 + 9*B*a^2*b^2*d^2*e^2 + 8*B*a*b^3*d^3*e + 8*B*a^3*b*d*e^3))/(30*e^5) + (b^3*x^4*(2*A*b*e + 8*B*a*e + 5*B*b*d))/(6*e^2) + (b*x^2*(8*B*a^3*e^3 + 5*B*b^3*d^3 + 12*A*a^2*b*e^3 + 2*A*b^3*d^2*e + 6*A*a*b^2*d*e^2 + 8*B*a*b^2 *d^2*e + 9*B*a^2*b*d*e^2))/(10*e^4) + (b^2*x^3*(9*B*a^2*e^2 + 5*B*b^2*d^2 + 6*A*a*b*e^2 + 2*A*b^2*d*e + 8*B*a*b*d*e))/(6*e^3) + (B*b^4*x^5)/(2*e))/( d^7 + e^7*x^7 + 7*d*e^6*x^6 + 21*d^5*e^2*x^2 + 35*d^4*e^3*x^3 + 35*d^3*e^4 *x^4 + 21*d^2*e^5*x^5 + 7*d^6*e*x)