Integrand size = 25, antiderivative size = 555 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{9 e^8 (d+e x)^9}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{8 e^8 (d+e x)^8}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{7 e^8 (d+e x)^7}+\frac {A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{6 e^8 (d+e x)^6}+\frac {B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{5 e^8 (d+e x)^5}+\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right )}{4 e^8 (d+e x)^4}+\frac {c^2 (7 B c d-3 b B e-A c e)}{3 e^8 (d+e x)^3}-\frac {B c^3}{2 e^8 (d+e x)^2} \]
1/9*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)^3/e^8/(e*x+d)^9+1/8*(a*e^2-b*d*e+c*d^2) ^2*(3*A*e*(-b*e+2*c*d)-B*(7*c*d^2-e*(-a*e+4*b*d)))/e^8/(e*x+d)^8+3/7*(a*e^ 2-b*d*e+c*d^2)*(B*(7*c^2*d^3-c*d*e*(-3*a*e+8*b*d)+b*e^2*(-a*e+2*b*d))-A*e* (5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d)))/e^8/(e*x+d)^7+1/6*(A*e*(-b*e+2*c*d)* (10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))-B*(35*c^3*d^4-b^2*e^3*(-3*a*e+4* b*d)-30*c^2*d^2*e*(-a*e+2*b*d)+3*c*e^2*(a^2*e^2-8*a*b*d*e+10*b^2*d^2)))/e^ 8/(e*x+d)^6+1/5*(B*(35*c^3*d^3-b^3*e^3+3*b*c*e^2*(-2*a*e+5*b*d)-15*c^2*d*e *(-a*e+3*b*d))-3*A*c*e*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d)))/e^8/(e*x+d)^5 +3/4*c*(A*c*e*(-b*e+2*c*d)-B*(7*c^2*d^2+b^2*e^2-c*e*(-a*e+6*b*d)))/e^8/(e* x+d)^4+1/3*c^2*(-A*c*e-3*B*b*e+7*B*c*d)/e^8/(e*x+d)^3-1/2*B*c^3/e^8/(e*x+d )^2
Time = 0.34 (sec) , antiderivative size = 852, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {A e \left (10 c^3 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )+5 e^3 \left (56 a^3 e^3+21 a^2 b e^2 (d+9 e x)+6 a b^2 e \left (d^2+9 d e x+36 e^2 x^2\right )+b^3 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+6 c e^2 \left (5 a^2 e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+5 a b e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+2 b^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+3 c^2 e \left (4 a e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 b \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )\right )+B \left (35 c^3 \left (d^7+9 d^6 e x+36 d^5 e^2 x^2+84 d^4 e^3 x^3+126 d^3 e^4 x^4+126 d^2 e^5 x^5+84 d e^6 x^6+36 e^7 x^7\right )+e^3 \left (35 a^3 e^3 (d+9 e x)+30 a^2 b e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+15 a b^2 e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 b^3 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+3 c e^2 \left (5 a^2 e^2 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+8 a b e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 b^2 \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )+15 c^2 e \left (a e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+2 b \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )\right )}{2520 e^8 (d+e x)^9} \]
-1/2520*(A*e*(10*c^3*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6) + 5*e^3*(56*a^3*e^3 + 21*a^2 *b*e^2*(d + 9*e*x) + 6*a*b^2*e*(d^2 + 9*d*e*x + 36*e^2*x^2) + b^3*(d^3 + 9 *d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) + 6*c*e^2*(5*a^2*e^2*(d^2 + 9*d*e*x + 36*e^2*x^2) + 5*a*b*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 2 *b^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4)) + 3* c^2*e*(4*a*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^ 4) + 5*b*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^ 4 + 126*e^5*x^5))) + B*(35*c^3*(d^7 + 9*d^6*e*x + 36*d^5*e^2*x^2 + 84*d^4* e^3*x^3 + 126*d^3*e^4*x^4 + 126*d^2*e^5*x^5 + 84*d*e^6*x^6 + 36*e^7*x^7) + e^3*(35*a^3*e^3*(d + 9*e*x) + 30*a^2*b*e^2*(d^2 + 9*d*e*x + 36*e^2*x^2) + 15*a*b^2*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*b^3*(d^4 + 9 *d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4)) + 3*c*e^2*(5*a^2* e^2*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 8*a*b*e*(d^4 + 9*d^3*e *x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 5*b^2*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)) + 15*c^ 2*e*(a*e*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^ 4 + 126*e^5*x^5) + 2*b*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6))))/(e^8*(d + e*x)^9)
Time = 1.23 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.00, 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.
| \(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )-A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (d+e x)^7}+\frac {3 c \left (B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )-A c e (2 c d-b e)\right )}{e^7 (d+e x)^5}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )\right )}{e^7 (d+e x)^8}+\frac {3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )}{e^7 (d+e x)^6}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^7 (d+e x)^9}+\frac {(A e-B d) \left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)^{10}}+\frac {c^2 (A c e+3 b B e-7 B c d)}{e^7 (d+e x)^4}+\frac {B c^3}{e^7 (d+e x)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{6 e^8 (d+e x)^6}+\frac {3 c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{4 e^8 (d+e x)^4}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{7 e^8 (d+e x)^7}+\frac {B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^8 (d+e x)^5}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{8 e^8 (d+e x)^8}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{9 e^8 (d+e x)^9}+\frac {c^2 (-A c e-3 b B e+7 B c d)}{3 e^8 (d+e x)^3}-\frac {B c^3}{2 e^8 (d+e x)^2}\) |
((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3)/(9*e^8*(d + e*x)^9) - ((c*d^2 - b* d*e + a*e^2)^2*(7*B*c*d^2 - B*e*(4*b*d - a*e) - 3*A*e*(2*c*d - b*e)))/(8*e ^8*(d + e*x)^8) + (3*(c*d^2 - b*d*e + a*e^2)*(B*(7*c^2*d^3 - c*d*e*(8*b*d - 3*a*e) + b*e^2*(2*b*d - a*e)) - A*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))))/(7*e^8*(d + e*x)^7) + (A*e*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2 *c*e*(5*b*d - 3*a*e)) - B*(35*c^3*d^4 - b^2*e^3*(4*b*d - 3*a*e) - 30*c^2*d ^2*e*(2*b*d - a*e) + 3*c*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2)))/(6*e^8*( d + e*x)^6) + (B*(35*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(5*b*d - 2*a*e) - 15*c^ 2*d*e*(3*b*d - a*e)) - 3*A*c*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/ (5*e^8*(d + e*x)^5) + (3*c*(A*c*e*(2*c*d - b*e) - B*(7*c^2*d^2 + b^2*e^2 - c*e*(6*b*d - a*e))))/(4*e^8*(d + e*x)^4) + (c^2*(7*B*c*d - 3*b*B*e - A*c* e))/(3*e^8*(d + e*x)^3) - (B*c^3)/(2*e^8*(d + e*x)^2)
3.24.48.3.1 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 = 0.36 (sec) , antiderivative size = 1048, normalized size of antiderivative = 1.89
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1048\) |
| default | \(\text {Expression too large to display}\) | \(1067\) |
| norman | \(\text {Expression too large to display}\) | \(1093\) |
| gosper | \(\text {Expression too large to display}\) | \(1213\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1220\) |
(-1/2*B*c^3*x^7/e-1/6*c^2/e^2*(2*A*c*e+6*B*b*e+7*B*c*d)*x^6-1/4*c/e^3*(3*A *b*c*e^2+2*A*c^2*d*e+3*B*a*c*e^2+3*B*b^2*e^2+6*B*b*c*d*e+7*B*c^2*d^2)*x^5- 1/20/e^4*(12*A*a*c^2*e^3+12*A*b^2*c*e^3+15*A*b*c^2*d*e^2+10*A*c^3*d^2*e+24 *B*a*b*c*e^3+15*B*a*c^2*d*e^2+4*B*b^3*e^3+15*B*b^2*c*d*e^2+30*B*b*c^2*d^2* e+35*B*c^3*d^3)*x^4-1/30/e^5*(30*A*a*b*c*e^4+12*A*a*c^2*d*e^3+5*A*b^3*e^4+ 12*A*b^2*c*d*e^3+15*A*b*c^2*d^2*e^2+10*A*c^3*d^3*e+15*B*a^2*c*e^4+15*B*a*b ^2*e^4+24*B*a*b*c*d*e^3+15*B*a*c^2*d^2*e^2+4*B*b^3*d*e^3+15*B*b^2*c*d^2*e^ 2+30*B*b*c^2*d^3*e+35*B*c^3*d^4)*x^3-1/70/e^6*(30*A*a^2*c*e^5+30*A*a*b^2*e ^5+30*A*a*b*c*d*e^4+12*A*a*c^2*d^2*e^3+5*A*b^3*d*e^4+12*A*b^2*c*d^2*e^3+15 *A*b*c^2*d^3*e^2+10*A*c^3*d^4*e+30*B*a^2*b*e^5+15*B*a^2*c*d*e^4+15*B*a*b^2 *d*e^4+24*B*a*b*c*d^2*e^3+15*B*a*c^2*d^3*e^2+4*B*b^3*d^2*e^3+15*B*b^2*c*d^ 3*e^2+30*B*b*c^2*d^4*e+35*B*c^3*d^5)*x^2-1/280/e^7*(105*A*a^2*b*e^6+30*A*a ^2*c*d*e^5+30*A*a*b^2*d*e^5+30*A*a*b*c*d^2*e^4+12*A*a*c^2*d^3*e^3+5*A*b^3* d^2*e^4+12*A*b^2*c*d^3*e^3+15*A*b*c^2*d^4*e^2+10*A*c^3*d^5*e+35*B*a^3*e^6+ 30*B*a^2*b*d*e^5+15*B*a^2*c*d^2*e^4+15*B*a*b^2*d^2*e^4+24*B*a*b*c*d^3*e^3+ 15*B*a*c^2*d^4*e^2+4*B*b^3*d^3*e^3+15*B*b^2*c*d^4*e^2+30*B*b*c^2*d^5*e+35* B*c^3*d^6)*x-1/2520/e^8*(280*A*a^3*e^7+105*A*a^2*b*d*e^6+30*A*a^2*c*d^2*e^ 5+30*A*a*b^2*d^2*e^5+30*A*a*b*c*d^3*e^4+12*A*a*c^2*d^4*e^3+5*A*b^3*d^3*e^4 +12*A*b^2*c*d^4*e^3+15*A*b*c^2*d^5*e^2+10*A*c^3*d^6*e+35*B*a^3*d*e^6+30*B* a^2*b*d^2*e^5+15*B*a^2*c*d^3*e^4+15*B*a*b^2*d^3*e^4+24*B*a*b*c*d^4*e^3+...
Time = 0.30 (sec) , antiderivative size = 945, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {1260 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 280 \, A a^{3} e^{7} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4} e^{3} + 5 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3} e^{4} + 30 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6} + 420 \, {\left (7 \, B c^{3} d e^{6} + 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 630 \, {\left (7 \, B c^{3} d^{2} e^{5} + 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{6} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{7}\right )} x^{5} + 126 \, {\left (35 \, B c^{3} d^{3} e^{4} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{6} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{7}\right )} x^{4} + 84 \, {\left (35 \, B c^{3} d^{4} e^{3} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{5} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{6} + 5 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{7}\right )} x^{3} + 36 \, {\left (35 \, B c^{3} d^{5} e^{2} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{4} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{5} + 5 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{6} + 30 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 9 \, {\left (35 \, B c^{3} d^{6} e + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} e^{4} + 5 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} e^{5} + 30 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{6} + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x}{2520 \, {\left (e^{17} x^{9} + 9 \, d e^{16} x^{8} + 36 \, d^{2} e^{15} x^{7} + 84 \, d^{3} e^{14} x^{6} + 126 \, d^{4} e^{13} x^{5} + 126 \, d^{5} e^{12} x^{4} + 84 \, d^{6} e^{11} x^{3} + 36 \, d^{7} e^{10} x^{2} + 9 \, d^{8} e^{9} x + d^{9} e^{8}\right )}} \]
-1/2520*(1260*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 280*A*a^3*e^7 + 10*(3*B*b*c^2 + A*c^3)*d^6*e + 15*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 + 4*(B*b^3 + 3*A* a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + 5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e^4 + 30*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + 35*(B*a^ 3 + 3*A*a^2*b)*d*e^6 + 420*(7*B*c^3*d*e^6 + 2*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 630*(7*B*c^3*d^2*e^5 + 2*(3*B*b*c^2 + A*c^3)*d*e^6 + 3*(B*b^2*c + (B*a + A*b)*c^2)*e^7)*x^5 + 126*(35*B*c^3*d^3*e^4 + 10*(3*B*b*c^2 + A*c^3)*d^2* e^5 + 15*(B*b^2*c + (B*a + A*b)*c^2)*d*e^6 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B *a*b + A*b^2)*c)*e^7)*x^4 + 84*(35*B*c^3*d^4*e^3 + 10*(3*B*b*c^2 + A*c^3)* d^3*e^4 + 15*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^6 + 5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)* c)*e^7)*x^3 + 36*(35*B*c^3*d^5*e^2 + 10*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 15*( B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A *b^2)*c)*d^2*e^5 + 5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^6 + 3 0*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 + 9*(35*B*c^3*d^6*e + 10*(3*B*b*c ^2 + A*c^3)*d^5*e^2 + 15*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4 + 5*(3*B*a*b^2 + A*b^3 + 3*(B*a ^2 + 2*A*a*b)*c)*d^2*e^5 + 30*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^6 + 35*(B* a^3 + 3*A*a^2*b)*e^7)*x)/(e^17*x^9 + 9*d*e^16*x^8 + 36*d^2*e^15*x^7 + 84*d ^3*e^14*x^6 + 126*d^4*e^13*x^5 + 126*d^5*e^12*x^4 + 84*d^6*e^11*x^3 + 3...
Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 945, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {1260 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 280 \, A a^{3} e^{7} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4} e^{3} + 5 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3} e^{4} + 30 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6} + 420 \, {\left (7 \, B c^{3} d e^{6} + 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 630 \, {\left (7 \, B c^{3} d^{2} e^{5} + 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{6} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{7}\right )} x^{5} + 126 \, {\left (35 \, B c^{3} d^{3} e^{4} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{6} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{7}\right )} x^{4} + 84 \, {\left (35 \, B c^{3} d^{4} e^{3} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{5} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{6} + 5 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{7}\right )} x^{3} + 36 \, {\left (35 \, B c^{3} d^{5} e^{2} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{4} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{5} + 5 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{6} + 30 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 9 \, {\left (35 \, B c^{3} d^{6} e + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} e^{4} + 5 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} e^{5} + 30 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{6} + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x}{2520 \, {\left (e^{17} x^{9} + 9 \, d e^{16} x^{8} + 36 \, d^{2} e^{15} x^{7} + 84 \, d^{3} e^{14} x^{6} + 126 \, d^{4} e^{13} x^{5} + 126 \, d^{5} e^{12} x^{4} + 84 \, d^{6} e^{11} x^{3} + 36 \, d^{7} e^{10} x^{2} + 9 \, d^{8} e^{9} x + d^{9} e^{8}\right )}} \]
-1/2520*(1260*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 280*A*a^3*e^7 + 10*(3*B*b*c^2 + A*c^3)*d^6*e + 15*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 + 4*(B*b^3 + 3*A* a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + 5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e^4 + 30*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + 35*(B*a^ 3 + 3*A*a^2*b)*d*e^6 + 420*(7*B*c^3*d*e^6 + 2*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 630*(7*B*c^3*d^2*e^5 + 2*(3*B*b*c^2 + A*c^3)*d*e^6 + 3*(B*b^2*c + (B*a + A*b)*c^2)*e^7)*x^5 + 126*(35*B*c^3*d^3*e^4 + 10*(3*B*b*c^2 + A*c^3)*d^2* e^5 + 15*(B*b^2*c + (B*a + A*b)*c^2)*d*e^6 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B *a*b + A*b^2)*c)*e^7)*x^4 + 84*(35*B*c^3*d^4*e^3 + 10*(3*B*b*c^2 + A*c^3)* d^3*e^4 + 15*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^6 + 5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)* c)*e^7)*x^3 + 36*(35*B*c^3*d^5*e^2 + 10*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 15*( B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A *b^2)*c)*d^2*e^5 + 5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^6 + 3 0*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 + 9*(35*B*c^3*d^6*e + 10*(3*B*b*c ^2 + A*c^3)*d^5*e^2 + 15*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4 + 5*(3*B*a*b^2 + A*b^3 + 3*(B*a ^2 + 2*A*a*b)*c)*d^2*e^5 + 30*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^6 + 35*(B* a^3 + 3*A*a^2*b)*e^7)*x)/(e^17*x^9 + 9*d*e^16*x^8 + 36*d^2*e^15*x^7 + 84*d ^3*e^14*x^6 + 126*d^4*e^13*x^5 + 126*d^5*e^12*x^4 + 84*d^6*e^11*x^3 + 3...
Leaf count of result is larger than twice the leaf count of optimal. 1212 vs. \(2 (540) = 1080\).
Time = 0.28 (sec) , antiderivative size = 1212, normalized size of antiderivative = 2.18 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {1260 \, B c^{3} e^{7} x^{7} + 2940 \, B c^{3} d e^{6} x^{6} + 2520 \, B b c^{2} e^{7} x^{6} + 840 \, A c^{3} e^{7} x^{6} + 4410 \, B c^{3} d^{2} e^{5} x^{5} + 3780 \, B b c^{2} d e^{6} x^{5} + 1260 \, A c^{3} d e^{6} x^{5} + 1890 \, B b^{2} c e^{7} x^{5} + 1890 \, B a c^{2} e^{7} x^{5} + 1890 \, A b c^{2} e^{7} x^{5} + 4410 \, B c^{3} d^{3} e^{4} x^{4} + 3780 \, B b c^{2} d^{2} e^{5} x^{4} + 1260 \, A c^{3} d^{2} e^{5} x^{4} + 1890 \, B b^{2} c d e^{6} x^{4} + 1890 \, B a c^{2} d e^{6} x^{4} + 1890 \, A b c^{2} d e^{6} x^{4} + 504 \, B b^{3} e^{7} x^{4} + 3024 \, B a b c e^{7} x^{4} + 1512 \, A b^{2} c e^{7} x^{4} + 1512 \, A a c^{2} e^{7} x^{4} + 2940 \, B c^{3} d^{4} e^{3} x^{3} + 2520 \, B b c^{2} d^{3} e^{4} x^{3} + 840 \, A c^{3} d^{3} e^{4} x^{3} + 1260 \, B b^{2} c d^{2} e^{5} x^{3} + 1260 \, B a c^{2} d^{2} e^{5} x^{3} + 1260 \, A b c^{2} d^{2} e^{5} x^{3} + 336 \, B b^{3} d e^{6} x^{3} + 2016 \, B a b c d e^{6} x^{3} + 1008 \, A b^{2} c d e^{6} x^{3} + 1008 \, A a c^{2} d e^{6} x^{3} + 1260 \, B a b^{2} e^{7} x^{3} + 420 \, A b^{3} e^{7} x^{3} + 1260 \, B a^{2} c e^{7} x^{3} + 2520 \, A a b c e^{7} x^{3} + 1260 \, B c^{3} d^{5} e^{2} x^{2} + 1080 \, B b c^{2} d^{4} e^{3} x^{2} + 360 \, A c^{3} d^{4} e^{3} x^{2} + 540 \, B b^{2} c d^{3} e^{4} x^{2} + 540 \, B a c^{2} d^{3} e^{4} x^{2} + 540 \, A b c^{2} d^{3} e^{4} x^{2} + 144 \, B b^{3} d^{2} e^{5} x^{2} + 864 \, B a b c d^{2} e^{5} x^{2} + 432 \, A b^{2} c d^{2} e^{5} x^{2} + 432 \, A a c^{2} d^{2} e^{5} x^{2} + 540 \, B a b^{2} d e^{6} x^{2} + 180 \, A b^{3} d e^{6} x^{2} + 540 \, B a^{2} c d e^{6} x^{2} + 1080 \, A a b c d e^{6} x^{2} + 1080 \, B a^{2} b e^{7} x^{2} + 1080 \, A a b^{2} e^{7} x^{2} + 1080 \, A a^{2} c e^{7} x^{2} + 315 \, B c^{3} d^{6} e x + 270 \, B b c^{2} d^{5} e^{2} x + 90 \, A c^{3} d^{5} e^{2} x + 135 \, B b^{2} c d^{4} e^{3} x + 135 \, B a c^{2} d^{4} e^{3} x + 135 \, A b c^{2} d^{4} e^{3} x + 36 \, B b^{3} d^{3} e^{4} x + 216 \, B a b c d^{3} e^{4} x + 108 \, A b^{2} c d^{3} e^{4} x + 108 \, A a c^{2} d^{3} e^{4} x + 135 \, B a b^{2} d^{2} e^{5} x + 45 \, A b^{3} d^{2} e^{5} x + 135 \, B a^{2} c d^{2} e^{5} x + 270 \, A a b c d^{2} e^{5} x + 270 \, B a^{2} b d e^{6} x + 270 \, A a b^{2} d e^{6} x + 270 \, A a^{2} c d e^{6} x + 315 \, B a^{3} e^{7} x + 945 \, A a^{2} b e^{7} x + 35 \, B c^{3} d^{7} + 30 \, B b c^{2} d^{6} e + 10 \, A c^{3} d^{6} e + 15 \, B b^{2} c d^{5} e^{2} + 15 \, B a c^{2} d^{5} e^{2} + 15 \, A b c^{2} d^{5} e^{2} + 4 \, B b^{3} d^{4} e^{3} + 24 \, B a b c d^{4} e^{3} + 12 \, A b^{2} c d^{4} e^{3} + 12 \, A a c^{2} d^{4} e^{3} + 15 \, B a b^{2} d^{3} e^{4} + 5 \, A b^{3} d^{3} e^{4} + 15 \, B a^{2} c d^{3} e^{4} + 30 \, A a b c d^{3} e^{4} + 30 \, B a^{2} b d^{2} e^{5} + 30 \, A a b^{2} d^{2} e^{5} + 30 \, A a^{2} c d^{2} e^{5} + 35 \, B a^{3} d e^{6} + 105 \, A a^{2} b d e^{6} + 280 \, A a^{3} e^{7}}{2520 \, {\left (e x + d\right )}^{9} e^{8}} \]
-1/2520*(1260*B*c^3*e^7*x^7 + 2940*B*c^3*d*e^6*x^6 + 2520*B*b*c^2*e^7*x^6 + 840*A*c^3*e^7*x^6 + 4410*B*c^3*d^2*e^5*x^5 + 3780*B*b*c^2*d*e^6*x^5 + 12 60*A*c^3*d*e^6*x^5 + 1890*B*b^2*c*e^7*x^5 + 1890*B*a*c^2*e^7*x^5 + 1890*A* b*c^2*e^7*x^5 + 4410*B*c^3*d^3*e^4*x^4 + 3780*B*b*c^2*d^2*e^5*x^4 + 1260*A *c^3*d^2*e^5*x^4 + 1890*B*b^2*c*d*e^6*x^4 + 1890*B*a*c^2*d*e^6*x^4 + 1890* A*b*c^2*d*e^6*x^4 + 504*B*b^3*e^7*x^4 + 3024*B*a*b*c*e^7*x^4 + 1512*A*b^2* c*e^7*x^4 + 1512*A*a*c^2*e^7*x^4 + 2940*B*c^3*d^4*e^3*x^3 + 2520*B*b*c^2*d ^3*e^4*x^3 + 840*A*c^3*d^3*e^4*x^3 + 1260*B*b^2*c*d^2*e^5*x^3 + 1260*B*a*c ^2*d^2*e^5*x^3 + 1260*A*b*c^2*d^2*e^5*x^3 + 336*B*b^3*d*e^6*x^3 + 2016*B*a *b*c*d*e^6*x^3 + 1008*A*b^2*c*d*e^6*x^3 + 1008*A*a*c^2*d*e^6*x^3 + 1260*B* a*b^2*e^7*x^3 + 420*A*b^3*e^7*x^3 + 1260*B*a^2*c*e^7*x^3 + 2520*A*a*b*c*e^ 7*x^3 + 1260*B*c^3*d^5*e^2*x^2 + 1080*B*b*c^2*d^4*e^3*x^2 + 360*A*c^3*d^4* e^3*x^2 + 540*B*b^2*c*d^3*e^4*x^2 + 540*B*a*c^2*d^3*e^4*x^2 + 540*A*b*c^2* d^3*e^4*x^2 + 144*B*b^3*d^2*e^5*x^2 + 864*B*a*b*c*d^2*e^5*x^2 + 432*A*b^2* c*d^2*e^5*x^2 + 432*A*a*c^2*d^2*e^5*x^2 + 540*B*a*b^2*d*e^6*x^2 + 180*A*b^ 3*d*e^6*x^2 + 540*B*a^2*c*d*e^6*x^2 + 1080*A*a*b*c*d*e^6*x^2 + 1080*B*a^2* b*e^7*x^2 + 1080*A*a*b^2*e^7*x^2 + 1080*A*a^2*c*e^7*x^2 + 315*B*c^3*d^6*e* x + 270*B*b*c^2*d^5*e^2*x + 90*A*c^3*d^5*e^2*x + 135*B*b^2*c*d^4*e^3*x + 1 35*B*a*c^2*d^4*e^3*x + 135*A*b*c^2*d^4*e^3*x + 36*B*b^3*d^3*e^4*x + 216*B* a*b*c*d^3*e^4*x + 108*A*b^2*c*d^3*e^4*x + 108*A*a*c^2*d^3*e^4*x + 135*B...
Time = 11.35 (sec) , antiderivative size = 1142, normalized size of antiderivative = 2.06 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {\frac {35\,B\,a^3\,d\,e^6+280\,A\,a^3\,e^7+30\,B\,a^2\,b\,d^2\,e^5+105\,A\,a^2\,b\,d\,e^6+15\,B\,a^2\,c\,d^3\,e^4+30\,A\,a^2\,c\,d^2\,e^5+15\,B\,a\,b^2\,d^3\,e^4+30\,A\,a\,b^2\,d^2\,e^5+24\,B\,a\,b\,c\,d^4\,e^3+30\,A\,a\,b\,c\,d^3\,e^4+15\,B\,a\,c^2\,d^5\,e^2+12\,A\,a\,c^2\,d^4\,e^3+4\,B\,b^3\,d^4\,e^3+5\,A\,b^3\,d^3\,e^4+15\,B\,b^2\,c\,d^5\,e^2+12\,A\,b^2\,c\,d^4\,e^3+30\,B\,b\,c^2\,d^6\,e+15\,A\,b\,c^2\,d^5\,e^2+35\,B\,c^3\,d^7+10\,A\,c^3\,d^6\,e}{2520\,e^8}+\frac {x^4\,\left (4\,B\,b^3\,e^3+15\,B\,b^2\,c\,d\,e^2+12\,A\,b^2\,c\,e^3+30\,B\,b\,c^2\,d^2\,e+15\,A\,b\,c^2\,d\,e^2+24\,B\,a\,b\,c\,e^3+35\,B\,c^3\,d^3+10\,A\,c^3\,d^2\,e+15\,B\,a\,c^2\,d\,e^2+12\,A\,a\,c^2\,e^3\right )}{20\,e^4}+\frac {x\,\left (35\,B\,a^3\,e^6+30\,B\,a^2\,b\,d\,e^5+105\,A\,a^2\,b\,e^6+15\,B\,a^2\,c\,d^2\,e^4+30\,A\,a^2\,c\,d\,e^5+15\,B\,a\,b^2\,d^2\,e^4+30\,A\,a\,b^2\,d\,e^5+24\,B\,a\,b\,c\,d^3\,e^3+30\,A\,a\,b\,c\,d^2\,e^4+15\,B\,a\,c^2\,d^4\,e^2+12\,A\,a\,c^2\,d^3\,e^3+4\,B\,b^3\,d^3\,e^3+5\,A\,b^3\,d^2\,e^4+15\,B\,b^2\,c\,d^4\,e^2+12\,A\,b^2\,c\,d^3\,e^3+30\,B\,b\,c^2\,d^5\,e+15\,A\,b\,c^2\,d^4\,e^2+35\,B\,c^3\,d^6+10\,A\,c^3\,d^5\,e\right )}{280\,e^7}+\frac {x^2\,\left (30\,B\,a^2\,b\,e^5+15\,B\,a^2\,c\,d\,e^4+30\,A\,a^2\,c\,e^5+15\,B\,a\,b^2\,d\,e^4+30\,A\,a\,b^2\,e^5+24\,B\,a\,b\,c\,d^2\,e^3+30\,A\,a\,b\,c\,d\,e^4+15\,B\,a\,c^2\,d^3\,e^2+12\,A\,a\,c^2\,d^2\,e^3+4\,B\,b^3\,d^2\,e^3+5\,A\,b^3\,d\,e^4+15\,B\,b^2\,c\,d^3\,e^2+12\,A\,b^2\,c\,d^2\,e^3+30\,B\,b\,c^2\,d^4\,e+15\,A\,b\,c^2\,d^3\,e^2+35\,B\,c^3\,d^5+10\,A\,c^3\,d^4\,e\right )}{70\,e^6}+\frac {x^5\,\left (3\,B\,b^2\,c\,e^2+6\,B\,b\,c^2\,d\,e+3\,A\,b\,c^2\,e^2+7\,B\,c^3\,d^2+2\,A\,c^3\,d\,e+3\,B\,a\,c^2\,e^2\right )}{4\,e^3}+\frac {x^3\,\left (15\,B\,a^2\,c\,e^4+15\,B\,a\,b^2\,e^4+24\,B\,a\,b\,c\,d\,e^3+30\,A\,a\,b\,c\,e^4+15\,B\,a\,c^2\,d^2\,e^2+12\,A\,a\,c^2\,d\,e^3+4\,B\,b^3\,d\,e^3+5\,A\,b^3\,e^4+15\,B\,b^2\,c\,d^2\,e^2+12\,A\,b^2\,c\,d\,e^3+30\,B\,b\,c^2\,d^3\,e+15\,A\,b\,c^2\,d^2\,e^2+35\,B\,c^3\,d^4+10\,A\,c^3\,d^3\,e\right )}{30\,e^5}+\frac {c^2\,x^6\,\left (2\,A\,c\,e+6\,B\,b\,e+7\,B\,c\,d\right )}{6\,e^2}+\frac {B\,c^3\,x^7}{2\,e}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \]
-((280*A*a^3*e^7 + 35*B*c^3*d^7 + 35*B*a^3*d*e^6 + 10*A*c^3*d^6*e + 5*A*b^ 3*d^3*e^4 + 4*B*b^3*d^4*e^3 + 30*A*a*b^2*d^2*e^5 + 12*A*a*c^2*d^4*e^3 + 30 *A*a^2*c*d^2*e^5 + 15*B*a*b^2*d^3*e^4 + 30*B*a^2*b*d^2*e^5 + 15*A*b*c^2*d^ 5*e^2 + 12*A*b^2*c*d^4*e^3 + 15*B*a*c^2*d^5*e^2 + 15*B*a^2*c*d^3*e^4 + 15* B*b^2*c*d^5*e^2 + 105*A*a^2*b*d*e^6 + 30*B*b*c^2*d^6*e + 30*A*a*b*c*d^3*e^ 4 + 24*B*a*b*c*d^4*e^3)/(2520*e^8) + (x^4*(4*B*b^3*e^3 + 35*B*c^3*d^3 + 12 *A*a*c^2*e^3 + 12*A*b^2*c*e^3 + 10*A*c^3*d^2*e + 24*B*a*b*c*e^3 + 15*A*b*c ^2*d*e^2 + 15*B*a*c^2*d*e^2 + 30*B*b*c^2*d^2*e + 15*B*b^2*c*d*e^2))/(20*e^ 4) + (x*(35*B*a^3*e^6 + 35*B*c^3*d^6 + 105*A*a^2*b*e^6 + 10*A*c^3*d^5*e + 5*A*b^3*d^2*e^4 + 4*B*b^3*d^3*e^3 + 12*A*a*c^2*d^3*e^3 + 15*B*a*b^2*d^2*e^ 4 + 15*A*b*c^2*d^4*e^2 + 12*A*b^2*c*d^3*e^3 + 15*B*a*c^2*d^4*e^2 + 15*B*a^ 2*c*d^2*e^4 + 15*B*b^2*c*d^4*e^2 + 30*A*a*b^2*d*e^5 + 30*A*a^2*c*d*e^5 + 3 0*B*a^2*b*d*e^5 + 30*B*b*c^2*d^5*e + 30*A*a*b*c*d^2*e^4 + 24*B*a*b*c*d^3*e ^3))/(280*e^7) + (x^2*(35*B*c^3*d^5 + 30*A*a*b^2*e^5 + 30*A*a^2*c*e^5 + 30 *B*a^2*b*e^5 + 5*A*b^3*d*e^4 + 10*A*c^3*d^4*e + 4*B*b^3*d^2*e^3 + 12*A*a*c ^2*d^2*e^3 + 15*A*b*c^2*d^3*e^2 + 12*A*b^2*c*d^2*e^3 + 15*B*a*c^2*d^3*e^2 + 15*B*b^2*c*d^3*e^2 + 15*B*a*b^2*d*e^4 + 15*B*a^2*c*d*e^4 + 30*B*b*c^2*d^ 4*e + 24*B*a*b*c*d^2*e^3 + 30*A*a*b*c*d*e^4))/(70*e^6) + (x^5*(7*B*c^3*d^2 + 2*A*c^3*d*e + 3*A*b*c^2*e^2 + 3*B*a*c^2*e^2 + 3*B*b^2*c*e^2 + 6*B*b*c^2 *d*e))/(4*e^3) + (x^3*(5*A*b^3*e^4 + 35*B*c^3*d^4 + 15*B*a*b^2*e^4 + 15...