Integrand size = 25, antiderivative size = 304 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^9} \, dx=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{8 e^6 (d+e x)^8}+\frac {\left (c d^2-b d e+a e^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{7 e^6 (d+e x)^7}+\frac {B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{6 e^6 (d+e x)^6}+\frac {2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{5 e^6 (d+e x)^5}+\frac {c (5 B c d-2 b B e-A c e)}{4 e^6 (d+e x)^4}-\frac {B c^2}{3 e^6 (d+e x)^3} \]
1/8*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)^2/e^6/(e*x+d)^8+1/7*(a*e^2-b*d*e+c*d^2) *(2*A*e*(-b*e+2*c*d)-B*(5*c*d^2-e*(-a*e+3*b*d)))/e^6/(e*x+d)^7+1/6*(B*(10* c^2*d^3+b*e^2*(-2*a*e+3*b*d)-6*c*d*e*(-a*e+2*b*d))-A*e*(6*c^2*d^2+b^2*e^2- 2*c*e*(-a*e+3*b*d)))/e^6/(e*x+d)^6+1/5*(2*A*c*e*(-b*e+2*c*d)-B*(10*c^2*d^2 +b^2*e^2-2*c*e*(-a*e+4*b*d)))/e^6/(e*x+d)^5+1/4*c*(-A*c*e-2*B*b*e+5*B*c*d) /e^6/(e*x+d)^4-1/3*B*c^2/e^6/(e*x+d)^3
Time = 0.14 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.23 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^9} \, dx=-\frac {A e \left (3 c^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 e^2 \left (21 a^2 e^2+6 a b e (d+8 e x)+b^2 \left (d^2+8 d e x+28 e^2 x^2\right )\right )+2 c e \left (5 a e \left (d^2+8 d e x+28 e^2 x^2\right )+3 b \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )\right )+B \left (5 c^2 \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+e^2 \left (15 a^2 e^2 (d+8 e x)+10 a b e \left (d^2+8 d e x+28 e^2 x^2\right )+3 b^2 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+6 c e \left (a e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+b \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )\right )}{840 e^6 (d+e x)^8} \]
-1/840*(A*e*(3*c^2*(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*e^2*(21*a^2*e^2 + 6*a*b*e*(d + 8*e*x) + b^2*(d^2 + 8*d*e*x + 2 8*e^2*x^2)) + 2*c*e*(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))) + B*(5*c^2*(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^2*(15*a^2*e^2*(d + 8*e*x) + 10*a*b*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*b^2*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3)) + 6*c*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))))/(e^6*(d + e*x)^8)
Time = 0.56 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1195, 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.
| \(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^9} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )-2 A c e (2 c d-b e)}{e^5 (d+e x)^6}+\frac {A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )-B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )}{e^5 (d+e x)^7}+\frac {\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{e^5 (d+e x)^8}+\frac {(A e-B d) \left (a e^2-b d e+c d^2\right )^2}{e^5 (d+e x)^9}+\frac {c (A c e+2 b B e-5 B c d)}{e^5 (d+e x)^5}+\frac {B c^2}{e^5 (d+e x)^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^6 (d+e x)^5}+\frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^6 (d+e x)^6}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{7 e^6 (d+e x)^7}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{8 e^6 (d+e x)^8}+\frac {c (-A c e-2 b B e+5 B c d)}{4 e^6 (d+e x)^4}-\frac {B c^2}{3 e^6 (d+e x)^3}\) |
((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2)/(8*e^6*(d + e*x)^8) - ((c*d^2 - b* d*e + a*e^2)*(5*B*c*d^2 - B*e*(3*b*d - a*e) - 2*A*e*(2*c*d - b*e)))/(7*e^6 *(d + e*x)^7) + (B*(10*c^2*d^3 + b*e^2*(3*b*d - 2*a*e) - 6*c*d*e*(2*b*d - a*e)) - A*e*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))/(6*e^6*(d + e*x)^ 6) + (2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(4*b*d - a*e )))/(5*e^6*(d + e*x)^5) + (c*(5*B*c*d - 2*b*B*e - A*c*e))/(4*e^6*(d + e*x) ^4) - (B*c^2)/(3*e^6*(d + e*x)^3)
3.24.32.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.37 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.44
| method | result | size |
| risch | \(\frac {-\frac {B \,c^{2} x^{5}}{3 e}-\frac {c \left (3 A c e +6 B b e +5 B c d \right ) x^{4}}{12 e^{2}}-\frac {\left (6 A b c \,e^{2}+3 A \,c^{2} d e +6 B \,e^{2} a c +3 B \,b^{2} e^{2}+6 B b c d e +5 B \,c^{2} d^{2}\right ) x^{3}}{15 e^{3}}-\frac {\left (10 A a c \,e^{3}+5 A \,b^{2} e^{3}+6 A b c d \,e^{2}+3 A \,c^{2} d^{2} e +10 B a b \,e^{3}+6 B a c d \,e^{2}+3 B \,b^{2} d \,e^{2}+6 B b c \,d^{2} e +5 B \,c^{2} d^{3}\right ) x^{2}}{30 e^{4}}-\frac {\left (30 A a b \,e^{4}+10 A a c d \,e^{3}+5 A \,b^{2} d \,e^{3}+6 A b c \,d^{2} e^{2}+3 A \,c^{2} d^{3} e +15 B \,e^{4} a^{2}+10 B a b d \,e^{3}+6 B a c \,d^{2} e^{2}+3 B \,b^{2} d^{2} e^{2}+6 B b c \,d^{3} e +5 B \,c^{2} d^{4}\right ) x}{105 e^{5}}-\frac {105 A \,a^{2} e^{5}+30 A a b d \,e^{4}+10 A a c \,d^{2} e^{3}+5 A \,b^{2} d^{2} e^{3}+6 A b c \,d^{3} e^{2}+3 A \,c^{2} d^{4} e +15 B \,a^{2} d \,e^{4}+10 B a b \,d^{2} e^{3}+6 B a c \,d^{3} e^{2}+3 B \,b^{2} d^{3} e^{2}+6 B b c \,d^{4} e +5 B \,c^{2} d^{5}}{840 e^{6}}}{\left (e x +d \right )^{8}}\) | \(439\) |
| default | \(-\frac {2 A a c \,e^{3}+A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e +2 B a b \,e^{3}-6 B a c d \,e^{2}-3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}}{6 e^{6} \left (e x +d \right )^{6}}-\frac {2 A b c \,e^{2}-4 A \,c^{2} d e +2 B \,e^{2} a c +B \,b^{2} e^{2}-8 B b c d e +10 B \,c^{2} d^{2}}{5 e^{6} \left (e x +d \right )^{5}}-\frac {2 A a b \,e^{4}-4 A a c d \,e^{3}-2 A \,b^{2} d \,e^{3}+6 A b c \,d^{2} e^{2}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}-4 B a b d \,e^{3}+6 B a c \,d^{2} e^{2}+3 B \,b^{2} d^{2} e^{2}-8 B b c \,d^{3} e +5 B \,c^{2} d^{4}}{7 e^{6} \left (e x +d \right )^{7}}-\frac {B \,c^{2}}{3 e^{6} \left (e x +d \right )^{3}}-\frac {c \left (A c e +2 B b e -5 B c d \right )}{4 e^{6} \left (e x +d \right )^{4}}-\frac {A \,a^{2} e^{5}-2 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}-2 A b c \,d^{3} e^{2}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}-2 B a c \,d^{3} e^{2}-B \,b^{2} d^{3} e^{2}+2 B b c \,d^{4} e -B \,c^{2} d^{5}}{8 e^{6} \left (e x +d \right )^{8}}\) | \(453\) |
| norman | \(\frac {-\frac {B \,c^{2} x^{5}}{3 e}-\frac {\left (3 A \,c^{2} e^{3}+6 B \,e^{3} b c +5 B \,c^{2} d \,e^{2}\right ) x^{4}}{12 e^{4}}-\frac {\left (6 A b c \,e^{4}+3 A \,c^{2} d \,e^{3}+6 B \,e^{4} a c +3 B \,b^{2} e^{4}+6 B b c d \,e^{3}+5 B \,c^{2} d^{2} e^{2}\right ) x^{3}}{15 e^{5}}-\frac {\left (10 A a c \,e^{5}+5 A \,b^{2} e^{5}+6 A b c d \,e^{4}+3 A \,c^{2} d^{2} e^{3}+10 B a b \,e^{5}+6 B a c d \,e^{4}+3 B \,b^{2} d \,e^{4}+6 B b c \,d^{2} e^{3}+5 B \,c^{2} d^{3} e^{2}\right ) x^{2}}{30 e^{6}}-\frac {\left (30 A b a \,e^{6}+10 A a c d \,e^{5}+5 A \,b^{2} d \,e^{5}+6 A b c \,d^{2} e^{4}+3 A \,c^{2} d^{3} e^{3}+15 B \,a^{2} e^{6}+10 B a b d \,e^{5}+6 B a c \,d^{2} e^{4}+3 B \,b^{2} d^{2} e^{4}+6 B b c \,d^{3} e^{3}+5 B \,c^{2} d^{4} e^{2}\right ) x}{105 e^{7}}-\frac {105 A \,a^{2} e^{7}+30 A a b d \,e^{6}+10 A a c \,d^{2} e^{5}+5 A \,b^{2} d^{2} e^{5}+6 A b c \,d^{3} e^{4}+3 A \,c^{2} d^{4} e^{3}+15 B \,a^{2} d \,e^{6}+10 B a b \,d^{2} e^{5}+6 B a c \,d^{3} e^{4}+3 B \,b^{2} d^{3} e^{4}+6 B b c \,d^{4} e^{3}+5 B \,c^{2} d^{5} e^{2}}{840 e^{8}}}{\left (e x +d \right )^{8}}\) | \(478\) |
| gosper | \(-\frac {280 B \,x^{5} c^{2} e^{5}+210 A \,x^{4} c^{2} e^{5}+420 B \,x^{4} b c \,e^{5}+350 B \,x^{4} c^{2} d \,e^{4}+336 A \,x^{3} b c \,e^{5}+168 A \,x^{3} c^{2} d \,e^{4}+336 B \,x^{3} a c \,e^{5}+168 B \,x^{3} b^{2} e^{5}+336 B \,x^{3} b c d \,e^{4}+280 B \,x^{3} c^{2} d^{2} e^{3}+280 A \,x^{2} a c \,e^{5}+140 A \,x^{2} b^{2} e^{5}+168 A \,x^{2} b c d \,e^{4}+84 A \,x^{2} c^{2} d^{2} e^{3}+280 B \,x^{2} a b \,e^{5}+168 B \,x^{2} a c d \,e^{4}+84 B \,x^{2} b^{2} d \,e^{4}+168 B \,x^{2} b c \,d^{2} e^{3}+140 B \,x^{2} c^{2} d^{3} e^{2}+240 A x a b \,e^{5}+80 A x a c d \,e^{4}+40 A x \,b^{2} d \,e^{4}+48 A x b c \,d^{2} e^{3}+24 A x \,c^{2} d^{3} e^{2}+120 B x \,a^{2} e^{5}+80 B x a b d \,e^{4}+48 B x a c \,d^{2} e^{3}+24 B x \,b^{2} d^{2} e^{3}+48 B x b c \,d^{3} e^{2}+40 B x \,c^{2} d^{4} e +105 A \,a^{2} e^{5}+30 A a b d \,e^{4}+10 A a c \,d^{2} e^{3}+5 A \,b^{2} d^{2} e^{3}+6 A b c \,d^{3} e^{2}+3 A \,c^{2} d^{4} e +15 B \,a^{2} d \,e^{4}+10 B a b \,d^{2} e^{3}+6 B a c \,d^{3} e^{2}+3 B \,b^{2} d^{3} e^{2}+6 B b c \,d^{4} e +5 B \,c^{2} d^{5}}{840 e^{6} \left (e x +d \right )^{8}}\) | \(498\) |
| parallelrisch | \(-\frac {280 B \,c^{2} x^{5} e^{7}+210 A \,c^{2} e^{7} x^{4}+420 B b c \,e^{7} x^{4}+350 B \,c^{2} d \,e^{6} x^{4}+336 A b c \,e^{7} x^{3}+168 A \,c^{2} d \,e^{6} x^{3}+336 B a c \,e^{7} x^{3}+168 B \,b^{2} e^{7} x^{3}+336 B b c d \,e^{6} x^{3}+280 B \,c^{2} d^{2} e^{5} x^{3}+280 A a c \,e^{7} x^{2}+140 A \,b^{2} e^{7} x^{2}+168 A b c d \,e^{6} x^{2}+84 A \,c^{2} d^{2} e^{5} x^{2}+280 B a b \,e^{7} x^{2}+168 B a c d \,e^{6} x^{2}+84 B \,b^{2} d \,e^{6} x^{2}+168 B b c \,d^{2} e^{5} x^{2}+140 B \,c^{2} d^{3} e^{4} x^{2}+240 A a b \,e^{7} x +80 A a c d \,e^{6} x +40 A \,b^{2} d \,e^{6} x +48 A b c \,d^{2} e^{5} x +24 A \,c^{2} d^{3} e^{4} x +120 B \,a^{2} e^{7} x +80 B a b d \,e^{6} x +48 B a c \,d^{2} e^{5} x +24 B \,b^{2} d^{2} e^{5} x +48 B b c \,d^{3} e^{4} x +40 B \,c^{2} d^{4} e^{3} x +105 A \,a^{2} e^{7}+30 A a b d \,e^{6}+10 A a c \,d^{2} e^{5}+5 A \,b^{2} d^{2} e^{5}+6 A b c \,d^{3} e^{4}+3 A \,c^{2} d^{4} e^{3}+15 B \,a^{2} d \,e^{6}+10 B a b \,d^{2} e^{5}+6 B a c \,d^{3} e^{4}+3 B \,b^{2} d^{3} e^{4}+6 B b c \,d^{4} e^{3}+5 B \,c^{2} d^{5} e^{2}}{840 e^{8} \left (e x +d \right )^{8}}\) | \(507\) |
(-1/3*B*c^2*x^5/e-1/12*c/e^2*(3*A*c*e+6*B*b*e+5*B*c*d)*x^4-1/15/e^3*(6*A*b *c*e^2+3*A*c^2*d*e+6*B*a*c*e^2+3*B*b^2*e^2+6*B*b*c*d*e+5*B*c^2*d^2)*x^3-1/ 30/e^4*(10*A*a*c*e^3+5*A*b^2*e^3+6*A*b*c*d*e^2+3*A*c^2*d^2*e+10*B*a*b*e^3+ 6*B*a*c*d*e^2+3*B*b^2*d*e^2+6*B*b*c*d^2*e+5*B*c^2*d^3)*x^2-1/105/e^5*(30*A *a*b*e^4+10*A*a*c*d*e^3+5*A*b^2*d*e^3+6*A*b*c*d^2*e^2+3*A*c^2*d^3*e+15*B*a ^2*e^4+10*B*a*b*d*e^3+6*B*a*c*d^2*e^2+3*B*b^2*d^2*e^2+6*B*b*c*d^3*e+5*B*c^ 2*d^4)*x-1/840/e^6*(105*A*a^2*e^5+30*A*a*b*d*e^4+10*A*a*c*d^2*e^3+5*A*b^2* d^2*e^3+6*A*b*c*d^3*e^2+3*A*c^2*d^4*e+15*B*a^2*d*e^4+10*B*a*b*d^2*e^3+6*B* a*c*d^3*e^2+3*B*b^2*d^3*e^2+6*B*b*c*d^4*e+5*B*c^2*d^5))/(e*x+d)^8
Time = 0.31 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^9} \, dx=-\frac {280 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 105 \, A a^{2} e^{5} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 70 \, {\left (5 \, B c^{2} d e^{4} + 3 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 56 \, {\left (5 \, B c^{2} d^{2} e^{3} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 28 \, {\left (5 \, B c^{2} d^{3} e^{2} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 8 \, {\left (5 \, B c^{2} d^{4} e + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{840 \, {\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \]
-1/840*(280*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 105*A*a^2*e^5 + 3*(2*B*b*c + A*c ^2)*d^4*e + 3*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 + 5*(2*B*a*b + A*b^2 + 2*A *a*c)*d^2*e^3 + 15*(B*a^2 + 2*A*a*b)*d*e^4 + 70*(5*B*c^2*d*e^4 + 3*(2*B*b* c + A*c^2)*e^5)*x^4 + 56*(5*B*c^2*d^2*e^3 + 3*(2*B*b*c + A*c^2)*d*e^4 + 3* (B*b^2 + 2*(B*a + A*b)*c)*e^5)*x^3 + 28*(5*B*c^2*d^3*e^2 + 3*(2*B*b*c + A* c^2)*d^2*e^3 + 3*(B*b^2 + 2*(B*a + A*b)*c)*d*e^4 + 5*(2*B*a*b + A*b^2 + 2* A*a*c)*e^5)*x^2 + 8*(5*B*c^2*d^4*e + 3*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^ 2 + 2*(B*a + A*b)*c)*d^2*e^3 + 5*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 + 15*(B *a^2 + 2*A*a*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. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^9} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^9} \, dx=-\frac {280 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 105 \, A a^{2} e^{5} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 70 \, {\left (5 \, B c^{2} d e^{4} + 3 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 56 \, {\left (5 \, B c^{2} d^{2} e^{3} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 28 \, {\left (5 \, B c^{2} d^{3} e^{2} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 8 \, {\left (5 \, B c^{2} d^{4} e + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{840 \, {\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \]
-1/840*(280*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 105*A*a^2*e^5 + 3*(2*B*b*c + A*c ^2)*d^4*e + 3*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 + 5*(2*B*a*b + A*b^2 + 2*A *a*c)*d^2*e^3 + 15*(B*a^2 + 2*A*a*b)*d*e^4 + 70*(5*B*c^2*d*e^4 + 3*(2*B*b* c + A*c^2)*e^5)*x^4 + 56*(5*B*c^2*d^2*e^3 + 3*(2*B*b*c + A*c^2)*d*e^4 + 3* (B*b^2 + 2*(B*a + A*b)*c)*e^5)*x^3 + 28*(5*B*c^2*d^3*e^2 + 3*(2*B*b*c + A* c^2)*d^2*e^3 + 3*(B*b^2 + 2*(B*a + A*b)*c)*d*e^4 + 5*(2*B*a*b + A*b^2 + 2* A*a*c)*e^5)*x^2 + 8*(5*B*c^2*d^4*e + 3*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^ 2 + 2*(B*a + A*b)*c)*d^2*e^3 + 5*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 + 15*(B *a^2 + 2*A*a*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)
Time = 0.28 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.63 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^9} \, dx=-\frac {280 \, B c^{2} e^{5} x^{5} + 350 \, B c^{2} d e^{4} x^{4} + 420 \, B b c e^{5} x^{4} + 210 \, A c^{2} e^{5} x^{4} + 280 \, B c^{2} d^{2} e^{3} x^{3} + 336 \, B b c d e^{4} x^{3} + 168 \, A c^{2} d e^{4} x^{3} + 168 \, B b^{2} e^{5} x^{3} + 336 \, B a c e^{5} x^{3} + 336 \, A b c e^{5} x^{3} + 140 \, B c^{2} d^{3} e^{2} x^{2} + 168 \, B b c d^{2} e^{3} x^{2} + 84 \, A c^{2} d^{2} e^{3} x^{2} + 84 \, B b^{2} d e^{4} x^{2} + 168 \, B a c d e^{4} x^{2} + 168 \, A b c d e^{4} x^{2} + 280 \, B a b e^{5} x^{2} + 140 \, A b^{2} e^{5} x^{2} + 280 \, A a c e^{5} x^{2} + 40 \, B c^{2} d^{4} e x + 48 \, B b c d^{3} e^{2} x + 24 \, A c^{2} d^{3} e^{2} x + 24 \, B b^{2} d^{2} e^{3} x + 48 \, B a c d^{2} e^{3} x + 48 \, A b c d^{2} e^{3} x + 80 \, B a b d e^{4} x + 40 \, A b^{2} d e^{4} x + 80 \, A a c d e^{4} x + 120 \, B a^{2} e^{5} x + 240 \, A a b e^{5} x + 5 \, B c^{2} d^{5} + 6 \, B b c d^{4} e + 3 \, A c^{2} d^{4} e + 3 \, B b^{2} d^{3} e^{2} + 6 \, B a c d^{3} e^{2} + 6 \, A b c d^{3} e^{2} + 10 \, B a b d^{2} e^{3} + 5 \, A b^{2} d^{2} e^{3} + 10 \, A a c d^{2} e^{3} + 15 \, B a^{2} d e^{4} + 30 \, A a b d e^{4} + 105 \, A a^{2} e^{5}}{840 \, {\left (e x + d\right )}^{8} e^{6}} \]
-1/840*(280*B*c^2*e^5*x^5 + 350*B*c^2*d*e^4*x^4 + 420*B*b*c*e^5*x^4 + 210* A*c^2*e^5*x^4 + 280*B*c^2*d^2*e^3*x^3 + 336*B*b*c*d*e^4*x^3 + 168*A*c^2*d* e^4*x^3 + 168*B*b^2*e^5*x^3 + 336*B*a*c*e^5*x^3 + 336*A*b*c*e^5*x^3 + 140* B*c^2*d^3*e^2*x^2 + 168*B*b*c*d^2*e^3*x^2 + 84*A*c^2*d^2*e^3*x^2 + 84*B*b^ 2*d*e^4*x^2 + 168*B*a*c*d*e^4*x^2 + 168*A*b*c*d*e^4*x^2 + 280*B*a*b*e^5*x^ 2 + 140*A*b^2*e^5*x^2 + 280*A*a*c*e^5*x^2 + 40*B*c^2*d^4*e*x + 48*B*b*c*d^ 3*e^2*x + 24*A*c^2*d^3*e^2*x + 24*B*b^2*d^2*e^3*x + 48*B*a*c*d^2*e^3*x + 4 8*A*b*c*d^2*e^3*x + 80*B*a*b*d*e^4*x + 40*A*b^2*d*e^4*x + 80*A*a*c*d*e^4*x + 120*B*a^2*e^5*x + 240*A*a*b*e^5*x + 5*B*c^2*d^5 + 6*B*b*c*d^4*e + 3*A*c ^2*d^4*e + 3*B*b^2*d^3*e^2 + 6*B*a*c*d^3*e^2 + 6*A*b*c*d^3*e^2 + 10*B*a*b* d^2*e^3 + 5*A*b^2*d^2*e^3 + 10*A*a*c*d^2*e^3 + 15*B*a^2*d*e^4 + 30*A*a*b*d *e^4 + 105*A*a^2*e^5)/((e*x + d)^8*e^6)
Time = 11.23 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^9} \, dx=-\frac {\frac {15\,B\,a^2\,d\,e^4+105\,A\,a^2\,e^5+10\,B\,a\,b\,d^2\,e^3+30\,A\,a\,b\,d\,e^4+6\,B\,a\,c\,d^3\,e^2+10\,A\,a\,c\,d^2\,e^3+3\,B\,b^2\,d^3\,e^2+5\,A\,b^2\,d^2\,e^3+6\,B\,b\,c\,d^4\,e+6\,A\,b\,c\,d^3\,e^2+5\,B\,c^2\,d^5+3\,A\,c^2\,d^4\,e}{840\,e^6}+\frac {x^3\,\left (3\,B\,b^2\,e^2+6\,B\,b\,c\,d\,e+6\,A\,b\,c\,e^2+5\,B\,c^2\,d^2+3\,A\,c^2\,d\,e+6\,B\,a\,c\,e^2\right )}{15\,e^3}+\frac {x^2\,\left (3\,B\,b^2\,d\,e^2+5\,A\,b^2\,e^3+6\,B\,b\,c\,d^2\,e+6\,A\,b\,c\,d\,e^2+10\,B\,a\,b\,e^3+5\,B\,c^2\,d^3+3\,A\,c^2\,d^2\,e+6\,B\,a\,c\,d\,e^2+10\,A\,a\,c\,e^3\right )}{30\,e^4}+\frac {x\,\left (15\,B\,a^2\,e^4+10\,B\,a\,b\,d\,e^3+30\,A\,a\,b\,e^4+6\,B\,a\,c\,d^2\,e^2+10\,A\,a\,c\,d\,e^3+3\,B\,b^2\,d^2\,e^2+5\,A\,b^2\,d\,e^3+6\,B\,b\,c\,d^3\,e+6\,A\,b\,c\,d^2\,e^2+5\,B\,c^2\,d^4+3\,A\,c^2\,d^3\,e\right )}{105\,e^5}+\frac {c\,x^4\,\left (3\,A\,c\,e+6\,B\,b\,e+5\,B\,c\,d\right )}{12\,e^2}+\frac {B\,c^2\,x^5}{3\,e}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \]
-((105*A*a^2*e^5 + 5*B*c^2*d^5 + 15*B*a^2*d*e^4 + 3*A*c^2*d^4*e + 5*A*b^2* d^2*e^3 + 3*B*b^2*d^3*e^2 + 30*A*a*b*d*e^4 + 6*B*b*c*d^4*e + 10*A*a*c*d^2* e^3 + 10*B*a*b*d^2*e^3 + 6*A*b*c*d^3*e^2 + 6*B*a*c*d^3*e^2)/(840*e^6) + (x ^3*(3*B*b^2*e^2 + 5*B*c^2*d^2 + 6*A*b*c*e^2 + 6*B*a*c*e^2 + 3*A*c^2*d*e + 6*B*b*c*d*e))/(15*e^3) + (x^2*(5*A*b^2*e^3 + 5*B*c^2*d^3 + 10*A*a*c*e^3 + 10*B*a*b*e^3 + 3*A*c^2*d^2*e + 3*B*b^2*d*e^2 + 6*A*b*c*d*e^2 + 6*B*a*c*d*e ^2 + 6*B*b*c*d^2*e))/(30*e^4) + (x*(15*B*a^2*e^4 + 5*B*c^2*d^4 + 30*A*a*b* e^4 + 5*A*b^2*d*e^3 + 3*A*c^2*d^3*e + 3*B*b^2*d^2*e^2 + 10*A*a*c*d*e^3 + 1 0*B*a*b*d*e^3 + 6*B*b*c*d^3*e + 6*A*b*c*d^2*e^2 + 6*B*a*c*d^2*e^2))/(105*e ^5) + (c*x^4*(3*A*c*e + 6*B*b*e + 5*B*c*d))/(12*e^2) + (B*c^2*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)