Integrand size = 31, antiderivative size = 185 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx=-\frac {(B d-A e) (a+b x)^5}{8 e (b d-a e) (d+e x)^8}+\frac {(5 b B d+3 A b e-8 a B e) (a+b x)^5}{56 e (b d-a e)^2 (d+e x)^7}+\frac {b (5 b B d+3 A b e-8 a B e) (a+b x)^5}{168 e (b d-a e)^3 (d+e x)^6}+\frac {b^2 (5 b B d+3 A b e-8 a B e) (a+b x)^5}{840 e (b d-a e)^4 (d+e x)^5} \]
-1/8*(-A*e+B*d)*(b*x+a)^5/e/(-a*e+b*d)/(e*x+d)^8+1/56*(3*A*b*e-8*B*a*e+5*B *b*d)*(b*x+a)^5/e/(-a*e+b*d)^2/(e*x+d)^7+1/168*b*(3*A*b*e-8*B*a*e+5*B*b*d) *(b*x+a)^5/e/(-a*e+b*d)^3/(e*x+d)^6+1/840*b^2*(3*A*b*e-8*B*a*e+5*B*b*d)*(b *x+a)^5/e/(-a*e+b*d)^4/(e*x+d)^5
Time = 0.09 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.73 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx=-\frac {15 a^4 e^4 (7 A e+B (d+8 e x))+20 a^3 b e^3 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+6 a^2 b^2 e^2 \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+12 a b^3 e \left (A e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+B \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+b^4 \left (3 A e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 B \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )}{840 e^6 (d+e x)^8} \]
-1/840*(15*a^4*e^4*(7*A*e + B*(d + 8*e*x)) + 20*a^3*b*e^3*(3*A*e*(d + 8*e* x) + B*(d^2 + 8*d*e*x + 28*e^2*x^2)) + 6*a^2*b^2*e^2*(5*A*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*B*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3)) + 12*a *b^3*e*(A*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + B*(d^4 + 8*d^3 *e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4)) + b^4*(3*A*e*(d^4 + 8* d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*B*(d^5 + 8*d^4*e *x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)))/(e^6*( d + e*x)^8)
Time = 0.28 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {1184, 27, 87, 55, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2 (A+B x)}{(d+e x)^9} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^4 (a+b x)^4 (A+B x)}{(d+e x)^9}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^4 (A+B x)}{(d+e x)^9}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-8 a B e+3 A b e+5 b B d) \int \frac {(a+b x)^4}{(d+e x)^8}dx}{8 e (b d-a e)}-\frac {(a+b x)^5 (B d-A e)}{8 e (d+e x)^8 (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-8 a B e+3 A b e+5 b B d) \left (\frac {2 b \int \frac {(a+b x)^4}{(d+e x)^7}dx}{7 (b d-a e)}+\frac {(a+b x)^5}{7 (d+e x)^7 (b d-a e)}\right )}{8 e (b d-a e)}-\frac {(a+b x)^5 (B d-A e)}{8 e (d+e x)^8 (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-8 a B e+3 A b e+5 b B d) \left (\frac {2 b \left (\frac {b \int \frac {(a+b x)^4}{(d+e x)^6}dx}{6 (b d-a e)}+\frac {(a+b x)^5}{6 (d+e x)^6 (b d-a e)}\right )}{7 (b d-a e)}+\frac {(a+b x)^5}{7 (d+e x)^7 (b d-a e)}\right )}{8 e (b d-a e)}-\frac {(a+b x)^5 (B d-A e)}{8 e (d+e x)^8 (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\left (\frac {(a+b x)^5}{7 (d+e x)^7 (b d-a e)}+\frac {2 b \left (\frac {b (a+b x)^5}{30 (d+e x)^5 (b d-a e)^2}+\frac {(a+b x)^5}{6 (d+e x)^6 (b d-a e)}\right )}{7 (b d-a e)}\right ) (-8 a B e+3 A b e+5 b B d)}{8 e (b d-a e)}-\frac {(a+b x)^5 (B d-A e)}{8 e (d+e x)^8 (b d-a e)}\) |
-1/8*((B*d - A*e)*(a + b*x)^5)/(e*(b*d - a*e)*(d + e*x)^8) + ((5*b*B*d + 3 *A*b*e - 8*a*B*e)*((a + b*x)^5/(7*(b*d - a*e)*(d + e*x)^7) + (2*b*((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*(b*d - a*e))))/(8*e*(b*d - a*e))
3.17.89.3.1 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. \(2(177)=354\).
Time = 0.24 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.23
| method | result | size |
| risch | \(\frac {-\frac {B \,b^{4} x^{5}}{3 e}-\frac {b^{3} \left (3 A b e +12 B a e +5 B b d \right ) x^{4}}{12 e^{2}}-\frac {b^{2} \left (12 A a b \,e^{2}+3 A \,b^{2} d e +18 a^{2} B \,e^{2}+12 B a b d e +5 B \,b^{2} d^{2}\right ) x^{3}}{15 e^{3}}-\frac {b \left (30 A \,a^{2} b \,e^{3}+12 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +20 B \,e^{3} a^{3}+18 B \,a^{2} b d \,e^{2}+12 B a \,b^{2} d^{2} e +5 B \,b^{3} d^{3}\right ) x^{2}}{30 e^{4}}-\frac {\left (60 A \,a^{3} b \,e^{4}+30 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}+3 A \,b^{4} d^{3} e +15 B \,a^{4} e^{4}+20 B \,a^{3} b d \,e^{3}+18 B \,a^{2} b^{2} d^{2} e^{2}+12 B a \,b^{3} d^{3} e +5 b^{4} B \,d^{4}\right ) x}{105 e^{5}}-\frac {105 A \,a^{4} e^{5}+60 A \,a^{3} b d \,e^{4}+30 A \,a^{2} b^{2} d^{2} e^{3}+12 A a \,b^{3} d^{3} e^{2}+3 A \,b^{4} d^{4} e +15 B \,a^{4} d \,e^{4}+20 B \,a^{3} b \,d^{2} e^{3}+18 B \,a^{2} b^{2} d^{3} e^{2}+12 B a \,b^{3} d^{4} e +5 b^{4} B \,d^{5}}{840 e^{6}}}{\left (e x +d \right )^{8}}\) | \(413\) |
| default | \(-\frac {b \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +2 B \,e^{3} a^{3}-9 B \,a^{2} b d \,e^{2}+12 B a \,b^{2} d^{2} e -5 B \,b^{3} d^{3}\right )}{3 e^{6} \left (e x +d \right )^{6}}-\frac {2 b^{2} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +3 a^{2} B \,e^{2}-8 B a b d e +5 B \,b^{2} d^{2}\right )}{5 e^{6} \left (e x +d \right )^{5}}-\frac {A \,a^{4} e^{5}-4 A \,a^{3} b d \,e^{4}+6 A \,a^{2} b^{2} d^{2} e^{3}-4 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e -B \,a^{4} d \,e^{4}+4 B \,a^{3} b \,d^{2} e^{3}-6 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e -b^{4} B \,d^{5}}{8 e^{6} \left (e x +d \right )^{8}}-\frac {b^{4} B}{3 e^{6} \left (e x +d \right )^{3}}-\frac {b^{3} \left (A b e +4 B a e -5 B b d \right )}{4 e^{6} \left (e x +d \right )^{4}}-\frac {4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +B \,a^{4} e^{4}-8 B \,a^{3} b d \,e^{3}+18 B \,a^{2} b^{2} d^{2} e^{2}-16 B a \,b^{3} d^{3} e +5 b^{4} B \,d^{4}}{7 e^{6} \left (e x +d \right )^{7}}\) | \(430\) |
| norman | \(\frac {-\frac {B \,b^{4} x^{5}}{3 e}-\frac {\left (3 A \,b^{4} e^{3}+12 B \,e^{3} b^{3} a +5 B \,b^{4} d \,e^{2}\right ) x^{4}}{12 e^{4}}-\frac {\left (12 A a \,b^{3} e^{4}+3 A \,b^{4} d \,e^{3}+18 B \,a^{2} b^{2} e^{4}+12 B a \,b^{3} d \,e^{3}+5 b^{4} B \,d^{2} e^{2}\right ) x^{3}}{15 e^{5}}-\frac {\left (30 A \,a^{2} b^{2} e^{5}+12 A a \,b^{3} d \,e^{4}+3 A \,b^{4} d^{2} e^{3}+20 B \,a^{3} b \,e^{5}+18 B \,a^{2} b^{2} d \,e^{4}+12 B a \,b^{3} d^{2} e^{3}+5 b^{4} B \,d^{3} e^{2}\right ) x^{2}}{30 e^{6}}-\frac {\left (60 A \,a^{3} b \,e^{6}+30 A \,a^{2} b^{2} d \,e^{5}+12 A a \,b^{3} d^{2} e^{4}+3 A \,b^{4} d^{3} e^{3}+15 B \,e^{6} a^{4}+20 B \,a^{3} b d \,e^{5}+18 B \,a^{2} b^{2} d^{2} e^{4}+12 B a \,b^{3} d^{3} e^{3}+5 b^{4} B \,d^{4} e^{2}\right ) x}{105 e^{7}}-\frac {105 A \,a^{4} e^{7}+60 A \,a^{3} b d \,e^{6}+30 A \,a^{2} b^{2} d^{2} e^{5}+12 A a \,b^{3} d^{3} e^{4}+3 A \,b^{4} d^{4} e^{3}+15 B \,a^{4} d \,e^{6}+20 B \,a^{3} b \,d^{2} e^{5}+18 B \,a^{2} b^{2} d^{3} e^{4}+12 B a \,b^{3} d^{4} e^{3}+5 b^{4} B \,d^{5} e^{2}}{840 e^{8}}}{\left (e x +d \right )^{8}}\) | \(460\) |
| gosper | \(-\frac {280 B \,x^{5} b^{4} e^{5}+210 A \,b^{4} e^{5} x^{4}+840 B \,x^{4} a \,b^{3} e^{5}+350 B \,x^{4} b^{4} d \,e^{4}+672 A \,x^{3} a \,b^{3} e^{5}+168 A \,x^{3} b^{4} d \,e^{4}+1008 B \,x^{3} a^{2} b^{2} e^{5}+672 B \,x^{3} a \,b^{3} d \,e^{4}+280 B \,x^{3} b^{4} d^{2} e^{3}+840 A \,x^{2} a^{2} b^{2} e^{5}+336 A \,x^{2} a \,b^{3} d \,e^{4}+84 A \,x^{2} b^{4} d^{2} e^{3}+560 B \,x^{2} a^{3} b \,e^{5}+504 B \,x^{2} a^{2} b^{2} d \,e^{4}+336 B \,x^{2} a \,b^{3} d^{2} e^{3}+140 B \,x^{2} b^{4} d^{3} e^{2}+480 A x \,a^{3} b \,e^{5}+240 A x \,a^{2} b^{2} d \,e^{4}+96 A x a \,b^{3} d^{2} e^{3}+24 A x \,b^{4} d^{3} e^{2}+120 B x \,a^{4} e^{5}+160 B x \,a^{3} b d \,e^{4}+144 B x \,a^{2} b^{2} d^{2} e^{3}+96 B x a \,b^{3} d^{3} e^{2}+40 B x \,b^{4} d^{4} e +105 A \,a^{4} e^{5}+60 A \,a^{3} b d \,e^{4}+30 A \,a^{2} b^{2} d^{2} e^{3}+12 A a \,b^{3} d^{3} e^{2}+3 A \,b^{4} d^{4} e +15 B \,a^{4} d \,e^{4}+20 B \,a^{3} b \,d^{2} e^{3}+18 B \,a^{2} b^{2} d^{3} e^{2}+12 B a \,b^{3} d^{4} e +5 b^{4} B \,d^{5}}{840 e^{6} \left (e x +d \right )^{8}}\) | \(469\) |
| parallelrisch | \(-\frac {280 b^{4} B \,x^{5} e^{7}+210 A \,b^{4} e^{7} x^{4}+840 B a \,b^{3} e^{7} x^{4}+350 B \,b^{4} d \,e^{6} x^{4}+672 A a \,b^{3} e^{7} x^{3}+168 A \,b^{4} d \,e^{6} x^{3}+1008 B \,a^{2} b^{2} e^{7} x^{3}+672 B a \,b^{3} d \,e^{6} x^{3}+280 B \,b^{4} d^{2} e^{5} x^{3}+840 A \,a^{2} b^{2} e^{7} x^{2}+336 A a \,b^{3} d \,e^{6} x^{2}+84 A \,b^{4} d^{2} e^{5} x^{2}+560 B \,a^{3} b \,e^{7} x^{2}+504 B \,a^{2} b^{2} d \,e^{6} x^{2}+336 B a \,b^{3} d^{2} e^{5} x^{2}+140 B \,b^{4} d^{3} e^{4} x^{2}+480 A \,a^{3} b \,e^{7} x +240 A \,a^{2} b^{2} d \,e^{6} x +96 A a \,b^{3} d^{2} e^{5} x +24 A \,b^{4} d^{3} e^{4} x +120 B \,a^{4} e^{7} x +160 B \,a^{3} b d \,e^{6} x +144 B \,a^{2} b^{2} d^{2} e^{5} x +96 B a \,b^{3} d^{3} e^{4} x +40 B \,b^{4} d^{4} e^{3} x +105 A \,a^{4} e^{7}+60 A \,a^{3} b d \,e^{6}+30 A \,a^{2} b^{2} d^{2} e^{5}+12 A a \,b^{3} d^{3} e^{4}+3 A \,b^{4} d^{4} e^{3}+15 B \,a^{4} d \,e^{6}+20 B \,a^{3} b \,d^{2} e^{5}+18 B \,a^{2} b^{2} d^{3} e^{4}+12 B a \,b^{3} d^{4} e^{3}+5 b^{4} B \,d^{5} e^{2}}{840 e^{8} \left (e x +d \right )^{8}}\) | \(478\) |
(-1/3*B*b^4/e*x^5-1/12*b^3/e^2*(3*A*b*e+12*B*a*e+5*B*b*d)*x^4-1/15*b^2/e^3 *(12*A*a*b*e^2+3*A*b^2*d*e+18*B*a^2*e^2+12*B*a*b*d*e+5*B*b^2*d^2)*x^3-1/30 *b/e^4*(30*A*a^2*b*e^3+12*A*a*b^2*d*e^2+3*A*b^3*d^2*e+20*B*a^3*e^3+18*B*a^ 2*b*d*e^2+12*B*a*b^2*d^2*e+5*B*b^3*d^3)*x^2-1/105/e^5*(60*A*a^3*b*e^4+30*A *a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2+3*A*b^4*d^3*e+15*B*a^4*e^4+20*B*a^3*b*d* e^3+18*B*a^2*b^2*d^2*e^2+12*B*a*b^3*d^3*e+5*B*b^4*d^4)*x-1/840/e^6*(105*A* a^4*e^5+60*A*a^3*b*d*e^4+30*A*a^2*b^2*d^2*e^3+12*A*a*b^3*d^3*e^2+3*A*b^4*d ^4*e+15*B*a^4*d*e^4+20*B*a^3*b*d^2*e^3+18*B*a^2*b^2*d^3*e^2+12*B*a*b^3*d^4 *e+5*B*b^4*d^5))/(e*x+d)^8
Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (177) = 354\).
Time = 0.32 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.64 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx=-\frac {280 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 105 \, A a^{4} e^{5} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 10 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 15 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 70 \, {\left (5 \, B b^{4} d e^{4} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 56 \, {\left (5 \, B b^{4} d^{2} e^{3} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 28 \, {\left (5 \, B b^{4} d^{3} e^{2} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 10 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 8 \, {\left (5 \, B b^{4} d^{4} e + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 10 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 15 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{840 \, {\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \]
-1/840*(280*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 105*A*a^4*e^5 + 3*(4*B*a*b^3 + A *b^4)*d^4*e + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 10*(2*B*a^3*b + 3*A*a^ 2*b^2)*d^2*e^3 + 15*(B*a^4 + 4*A*a^3*b)*d*e^4 + 70*(5*B*b^4*d*e^4 + 3*(4*B *a*b^3 + A*b^4)*e^5)*x^4 + 56*(5*B*b^4*d^2*e^3 + 3*(4*B*a*b^3 + A*b^4)*d*e ^4 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 28*(5*B*b^4*d^3*e^2 + 3*(4*B*a *b^3 + A*b^4)*d^2*e^3 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 10*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 8*(5*B*b^4*d^4*e + 3*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 10*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^ 4 + 15*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^14*x^8 + 8*d*e^13*x^7 + 28*d^2*e^12* x^6 + 56*d^3*e^11*x^5 + 70*d^4*e^10*x^4 + 56*d^5*e^9*x^3 + 28*d^6*e^8*x^2 + 8*d^7*e^7*x + d^8*e^6)
Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (177) = 354\).
Time = 0.21 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.64 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx=-\frac {280 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 105 \, A a^{4} e^{5} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 10 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 15 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 70 \, {\left (5 \, B b^{4} d e^{4} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 56 \, {\left (5 \, B b^{4} d^{2} e^{3} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 28 \, {\left (5 \, B b^{4} d^{3} e^{2} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 10 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 8 \, {\left (5 \, B b^{4} d^{4} e + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 10 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 15 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{840 \, {\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \]
-1/840*(280*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 105*A*a^4*e^5 + 3*(4*B*a*b^3 + A *b^4)*d^4*e + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 10*(2*B*a^3*b + 3*A*a^ 2*b^2)*d^2*e^3 + 15*(B*a^4 + 4*A*a^3*b)*d*e^4 + 70*(5*B*b^4*d*e^4 + 3*(4*B *a*b^3 + A*b^4)*e^5)*x^4 + 56*(5*B*b^4*d^2*e^3 + 3*(4*B*a*b^3 + A*b^4)*d*e ^4 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 28*(5*B*b^4*d^3*e^2 + 3*(4*B*a *b^3 + A*b^4)*d^2*e^3 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 10*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 8*(5*B*b^4*d^4*e + 3*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 10*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^ 4 + 15*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^14*x^8 + 8*d*e^13*x^7 + 28*d^2*e^12* x^6 + 56*d^3*e^11*x^5 + 70*d^4*e^10*x^4 + 56*d^5*e^9*x^3 + 28*d^6*e^8*x^2 + 8*d^7*e^7*x + d^8*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (177) = 354\).
Time = 0.28 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.53 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx=-\frac {280 \, B b^{4} e^{5} x^{5} + 350 \, B b^{4} d e^{4} x^{4} + 840 \, B a b^{3} e^{5} x^{4} + 210 \, A b^{4} e^{5} x^{4} + 280 \, B b^{4} d^{2} e^{3} x^{3} + 672 \, B a b^{3} d e^{4} x^{3} + 168 \, A b^{4} d e^{4} x^{3} + 1008 \, B a^{2} b^{2} e^{5} x^{3} + 672 \, A a b^{3} e^{5} x^{3} + 140 \, B b^{4} d^{3} e^{2} x^{2} + 336 \, B a b^{3} d^{2} e^{3} x^{2} + 84 \, A b^{4} d^{2} e^{3} x^{2} + 504 \, B a^{2} b^{2} d e^{4} x^{2} + 336 \, A a b^{3} d e^{4} x^{2} + 560 \, B a^{3} b e^{5} x^{2} + 840 \, A a^{2} b^{2} e^{5} x^{2} + 40 \, B b^{4} d^{4} e x + 96 \, B a b^{3} d^{3} e^{2} x + 24 \, A b^{4} d^{3} e^{2} x + 144 \, B a^{2} b^{2} d^{2} e^{3} x + 96 \, A a b^{3} d^{2} e^{3} x + 160 \, B a^{3} b d e^{4} x + 240 \, A a^{2} b^{2} d e^{4} x + 120 \, B a^{4} e^{5} x + 480 \, A a^{3} b e^{5} x + 5 \, B b^{4} d^{5} + 12 \, B a b^{3} d^{4} e + 3 \, A b^{4} d^{4} e + 18 \, B a^{2} b^{2} d^{3} e^{2} + 12 \, A a b^{3} d^{3} e^{2} + 20 \, B a^{3} b d^{2} e^{3} + 30 \, A a^{2} b^{2} d^{2} e^{3} + 15 \, B a^{4} d e^{4} + 60 \, A a^{3} b d e^{4} + 105 \, A a^{4} e^{5}}{840 \, {\left (e x + d\right )}^{8} e^{6}} \]
-1/840*(280*B*b^4*e^5*x^5 + 350*B*b^4*d*e^4*x^4 + 840*B*a*b^3*e^5*x^4 + 21 0*A*b^4*e^5*x^4 + 280*B*b^4*d^2*e^3*x^3 + 672*B*a*b^3*d*e^4*x^3 + 168*A*b^ 4*d*e^4*x^3 + 1008*B*a^2*b^2*e^5*x^3 + 672*A*a*b^3*e^5*x^3 + 140*B*b^4*d^3 *e^2*x^2 + 336*B*a*b^3*d^2*e^3*x^2 + 84*A*b^4*d^2*e^3*x^2 + 504*B*a^2*b^2* d*e^4*x^2 + 336*A*a*b^3*d*e^4*x^2 + 560*B*a^3*b*e^5*x^2 + 840*A*a^2*b^2*e^ 5*x^2 + 40*B*b^4*d^4*e*x + 96*B*a*b^3*d^3*e^2*x + 24*A*b^4*d^3*e^2*x + 144 *B*a^2*b^2*d^2*e^3*x + 96*A*a*b^3*d^2*e^3*x + 160*B*a^3*b*d*e^4*x + 240*A* a^2*b^2*d*e^4*x + 120*B*a^4*e^5*x + 480*A*a^3*b*e^5*x + 5*B*b^4*d^5 + 12*B *a*b^3*d^4*e + 3*A*b^4*d^4*e + 18*B*a^2*b^2*d^3*e^2 + 12*A*a*b^3*d^3*e^2 + 20*B*a^3*b*d^2*e^3 + 30*A*a^2*b^2*d^2*e^3 + 15*B*a^4*d*e^4 + 60*A*a^3*b*d *e^4 + 105*A*a^4*e^5)/((e*x + d)^8*e^6)
Time = 10.88 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.65 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx=-\frac {\frac {15\,B\,a^4\,d\,e^4+105\,A\,a^4\,e^5+20\,B\,a^3\,b\,d^2\,e^3+60\,A\,a^3\,b\,d\,e^4+18\,B\,a^2\,b^2\,d^3\,e^2+30\,A\,a^2\,b^2\,d^2\,e^3+12\,B\,a\,b^3\,d^4\,e+12\,A\,a\,b^3\,d^3\,e^2+5\,B\,b^4\,d^5+3\,A\,b^4\,d^4\,e}{840\,e^6}+\frac {x\,\left (15\,B\,a^4\,e^4+20\,B\,a^3\,b\,d\,e^3+60\,A\,a^3\,b\,e^4+18\,B\,a^2\,b^2\,d^2\,e^2+30\,A\,a^2\,b^2\,d\,e^3+12\,B\,a\,b^3\,d^3\,e+12\,A\,a\,b^3\,d^2\,e^2+5\,B\,b^4\,d^4+3\,A\,b^4\,d^3\,e\right )}{105\,e^5}+\frac {b^3\,x^4\,\left (3\,A\,b\,e+12\,B\,a\,e+5\,B\,b\,d\right )}{12\,e^2}+\frac {b\,x^2\,\left (20\,B\,a^3\,e^3+18\,B\,a^2\,b\,d\,e^2+30\,A\,a^2\,b\,e^3+12\,B\,a\,b^2\,d^2\,e+12\,A\,a\,b^2\,d\,e^2+5\,B\,b^3\,d^3+3\,A\,b^3\,d^2\,e\right )}{30\,e^4}+\frac {b^2\,x^3\,\left (18\,B\,a^2\,e^2+12\,B\,a\,b\,d\,e+12\,A\,a\,b\,e^2+5\,B\,b^2\,d^2+3\,A\,b^2\,d\,e\right )}{15\,e^3}+\frac {B\,b^4\,x^5}{3\,e}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \]
-((105*A*a^4*e^5 + 5*B*b^4*d^5 + 3*A*b^4*d^4*e + 15*B*a^4*d*e^4 + 12*A*a*b ^3*d^3*e^2 + 20*B*a^3*b*d^2*e^3 + 30*A*a^2*b^2*d^2*e^3 + 18*B*a^2*b^2*d^3* e^2 + 60*A*a^3*b*d*e^4 + 12*B*a*b^3*d^4*e)/(840*e^6) + (x*(15*B*a^4*e^4 + 5*B*b^4*d^4 + 60*A*a^3*b*e^4 + 3*A*b^4*d^3*e + 12*A*a*b^3*d^2*e^2 + 30*A*a ^2*b^2*d*e^3 + 18*B*a^2*b^2*d^2*e^2 + 12*B*a*b^3*d^3*e + 20*B*a^3*b*d*e^3) )/(105*e^5) + (b^3*x^4*(3*A*b*e + 12*B*a*e + 5*B*b*d))/(12*e^2) + (b*x^2*( 20*B*a^3*e^3 + 5*B*b^3*d^3 + 30*A*a^2*b*e^3 + 3*A*b^3*d^2*e + 12*A*a*b^2*d *e^2 + 12*B*a*b^2*d^2*e + 18*B*a^2*b*d*e^2))/(30*e^4) + (b^2*x^3*(18*B*a^2 *e^2 + 5*B*b^2*d^2 + 12*A*a*b*e^2 + 3*A*b^2*d*e + 12*B*a*b*d*e))/(15*e^3) + (B*b^4*x^5)/(3*e))/(d^8 + e^8*x^8 + 8*d*e^7*x^7 + 28*d^6*e^2*x^2 + 56*d^ 5*e^3*x^3 + 70*d^4*e^4*x^4 + 56*d^3*e^5*x^5 + 28*d^2*e^6*x^6 + 8*d^7*e*x)