3.1095 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^7} \, dx\)
Optimal. Leaf size=447 \[ -\frac{b^9 (d+e x)^4 (-10 a B e-A b e+11 b B d)}{4 e^{12}}+\frac{5 b^8 (d+e x)^3 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{3 e^{12}}-\frac{15 b^7 (d+e x)^2 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{2 e^{12}}+\frac{30 b^6 x (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{11}}-\frac{42 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)}+\frac{15 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^2}-\frac{5 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{e^{12} (d+e x)^3}-\frac{42 b^5 (b d-a e)^4 \log (d+e x) (-6 a B e-5 A b e+11 b B d)}{e^{12}}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{4 e^{12} (d+e x)^4}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{5 e^{12} (d+e x)^5}+\frac{(b d-a e)^{10} (B d-A e)}{6 e^{12} (d+e x)^6}+\frac{b^{10} B (d+e x)^5}{5 e^{12}} \]
[Out]
(30*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(6*e^12*(d + e*x)^
6) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(5*e^12*(d + e*x)^5) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b
*e - 2*a*B*e))/(4*e^12*(d + e*x)^4) - (5*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(e^12*(d + e*x)^3)
+ (15*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(e^12*(d + e*x)^2) - (42*b^4*(b*d - a*e)^5*(11*b*B*d -
6*A*b*e - 5*a*B*e))/(e^12*(d + e*x)) - (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*(d + e*x)^2)/(2*e
^12) + (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^3)/(3*e^12) - (b^9*(11*b*B*d - A*b*e - 10*a
*B*e)*(d + e*x)^4)/(4*e^12) + (b^10*B*(d + e*x)^5)/(5*e^12) - (42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*
B*e)*Log[d + e*x])/e^12
________________________________________________________________________________________
Rubi [A] time = 1.15881, antiderivative size = 447, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used =
{77} \[ -\frac{b^9 (d+e x)^4 (-10 a B e-A b e+11 b B d)}{4 e^{12}}+\frac{5 b^8 (d+e x)^3 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{3 e^{12}}-\frac{15 b^7 (d+e x)^2 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{2 e^{12}}+\frac{30 b^6 x (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{e^{11}}-\frac{42 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)}+\frac{15 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^2}-\frac{5 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{e^{12} (d+e x)^3}-\frac{42 b^5 (b d-a e)^4 \log (d+e x) (-6 a B e-5 A b e+11 b B d)}{e^{12}}+\frac{5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{4 e^{12} (d+e x)^4}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{5 e^{12} (d+e x)^5}+\frac{(b d-a e)^{10} (B d-A e)}{6 e^{12} (d+e x)^6}+\frac{b^{10} B (d+e x)^5}{5 e^{12}} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^10*(A + B*x))/(d + e*x)^7,x]
[Out]
(30*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(6*e^12*(d + e*x)^
6) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(5*e^12*(d + e*x)^5) + (5*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b
*e - 2*a*B*e))/(4*e^12*(d + e*x)^4) - (5*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(e^12*(d + e*x)^3)
+ (15*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(e^12*(d + e*x)^2) - (42*b^4*(b*d - a*e)^5*(11*b*B*d -
6*A*b*e - 5*a*B*e))/(e^12*(d + e*x)) - (15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*(d + e*x)^2)/(2*e
^12) + (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^3)/(3*e^12) - (b^9*(11*b*B*d - A*b*e - 10*a
*B*e)*(d + e*x)^4)/(4*e^12) + (b^10*B*(d + e*x)^5)/(5*e^12) - (42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*
B*e)*Log[d + e*x])/e^12
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rubi steps
\begin{align*} \int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^7} \, dx &=\int \left (-\frac{30 b^6 (b d-a e)^3 (-11 b B d+4 A b e+7 a B e)}{e^{11}}+\frac{(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^7}+\frac{(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)^6}+\frac{5 b (b d-a e)^8 (-11 b B d+9 A b e+2 a B e)}{e^{11} (d+e x)^5}-\frac{15 b^2 (b d-a e)^7 (-11 b B d+8 A b e+3 a B e)}{e^{11} (d+e x)^4}+\frac{30 b^3 (b d-a e)^6 (-11 b B d+7 A b e+4 a B e)}{e^{11} (d+e x)^3}-\frac{42 b^4 (b d-a e)^5 (-11 b B d+6 A b e+5 a B e)}{e^{11} (d+e x)^2}+\frac{42 b^5 (b d-a e)^4 (-11 b B d+5 A b e+6 a B e)}{e^{11} (d+e x)}+\frac{15 b^7 (b d-a e)^2 (-11 b B d+3 A b e+8 a B e) (d+e x)}{e^{11}}-\frac{5 b^8 (b d-a e) (-11 b B d+2 A b e+9 a B e) (d+e x)^2}{e^{11}}+\frac{b^9 (-11 b B d+A b e+10 a B e) (d+e x)^3}{e^{11}}+\frac{b^{10} B (d+e x)^4}{e^{11}}\right ) \, dx\\ &=\frac{30 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) x}{e^{11}}+\frac{(b d-a e)^{10} (B d-A e)}{6 e^{12} (d+e x)^6}-\frac{(b d-a e)^9 (11 b B d-10 A b e-a B e)}{5 e^{12} (d+e x)^5}+\frac{5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{4 e^{12} (d+e x)^4}-\frac{5 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{e^{12} (d+e x)^3}+\frac{15 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^2}-\frac{42 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{e^{12} (d+e x)}-\frac{15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) (d+e x)^2}{2 e^{12}}+\frac{5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^3}{3 e^{12}}-\frac{b^9 (11 b B d-A b e-10 a B e) (d+e x)^4}{4 e^{12}}+\frac{b^{10} B (d+e x)^5}{5 e^{12}}-\frac{42 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e) \log (d+e x)}{e^{12}}\\ \end{align*}
Mathematica [A] time = 0.537822, size = 505, normalized size = 1.13 \[ \frac{30 b^7 e^2 x^2 \left (45 a^2 b e^2 (A e-7 B d)+120 a^3 B e^3+70 a b^2 d e (4 B d-A e)+28 b^3 d^2 (A e-3 B d)\right )-60 b^6 e x \left (-315 a^2 b^2 d e^2 (4 B d-A e)-120 a^3 b e^3 (A e-7 B d)-210 a^4 B e^4+280 a b^3 d^2 e (3 B d-A e)-42 b^4 d^3 (5 B d-2 A e)\right )-20 b^8 e^3 x^3 \left (-45 a^2 B e^2-10 a b e (A e-7 B d)+7 b^2 d (A e-4 B d)\right )+15 b^9 e^4 x^4 (10 a B e+A b e-7 b B d)-\frac{2520 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{d+e x}+\frac{900 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{(d+e x)^2}-\frac{300 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{(d+e x)^3}-2520 b^5 (b d-a e)^4 \log (d+e x) (-6 a B e-5 A b e+11 b B d)+\frac{75 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{(d+e x)^4}-\frac{12 (b d-a e)^9 (-a B e-10 A b e+11 b B d)}{(d+e x)^5}+\frac{10 (b d-a e)^{10} (B d-A e)}{(d+e x)^6}+12 b^{10} B e^5 x^5}{60 e^{12}} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^7,x]
[Out]
(-60*b^6*e*(-210*a^4*B*e^4 - 42*b^4*d^3*(5*B*d - 2*A*e) + 280*a*b^3*d^2*e*(3*B*d - A*e) - 315*a^2*b^2*d*e^2*(4
*B*d - A*e) - 120*a^3*b*e^3*(-7*B*d + A*e))*x + 30*b^7*e^2*(120*a^3*B*e^3 + 70*a*b^2*d*e*(4*B*d - A*e) + 45*a^
2*b*e^2*(-7*B*d + A*e) + 28*b^3*d^2*(-3*B*d + A*e))*x^2 - 20*b^8*e^3*(-45*a^2*B*e^2 - 10*a*b*e*(-7*B*d + A*e)
+ 7*b^2*d*(-4*B*d + A*e))*x^3 + 15*b^9*e^4*(-7*b*B*d + A*b*e + 10*a*B*e)*x^4 + 12*b^10*B*e^5*x^5 + (10*(b*d -
a*e)^10*(B*d - A*e))/(d + e*x)^6 - (12*(b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(d + e*x)^5 + (75*b*(b*d -
a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(d + e*x)^4 - (300*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(
d + e*x)^3 + (900*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(d + e*x)^2 - (2520*b^4*(b*d - a*e)^5*(11*
b*B*d - 6*A*b*e - 5*a*B*e))/(d + e*x) - 2520*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e)*Log[d + e*x])/(6
0*e^12)
________________________________________________________________________________________
Maple [B] time = 0.036, size = 2781, normalized size = 6.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^10*(B*x+A)/(e*x+d)^7,x)
[Out]
1/5*b^10/e^7*B*x^5+1/4*b^10/e^7*A*x^4-1/6/e/(e*x+d)^6*a^10*A-1/5/e^2/(e*x+d)^5*B*a^10+10/3*b^9/e^7*A*x^3*a+28/
3*b^10/e^9*B*x^3*d^2-7/3*b^10/e^8*A*x^3*d+15*b^8/e^7*B*x^3*a^2+14*b^10/e^9*A*x^2*d^2+45/2*b^8/e^7*A*x^2*a^2-45
/4*b^2/e^3/(e*x+d)^4*A*a^8-45/4*b^10/e^11/(e*x+d)^4*A*d^8-5/2*b/e^3/(e*x+d)^4*B*a^9+55/4*b^10/e^12/(e*x+d)^4*B
*d^9+210*b^10/e^11*B*d^4*x+5/2*b^9/e^7*B*x^4*a+60*b^7/e^7*B*x^2*a^3-42*b^10/e^10*B*x^2*d^3+120*b^7/e^7*A*a^3*x
-84*b^10/e^10*A*d^3*x-1/6/e^11/(e*x+d)^6*A*b^10*d^10+1/6/e^2/(e*x+d)^6*B*d*a^10+1/6/e^12/(e*x+d)^6*b^10*B*d^11
-40*b^3/e^4/(e*x+d)^3*A*a^7+40*b^10/e^11/(e*x+d)^3*A*d^7-15*b^2/e^4/(e*x+d)^3*B*a^8-55*b^10/e^12/(e*x+d)^3*B*d
^8-105*b^4/e^5/(e*x+d)^2*A*a^6-105*b^10/e^11/(e*x+d)^2*A*d^6-60*b^3/e^5/(e*x+d)^2*B*a^7+165*b^10/e^12/(e*x+d)^
2*B*d^7-252*b^5/e^6/(e*x+d)*A*a^5+252*b^10/e^11/(e*x+d)*A*d^5-210*b^4/e^6/(e*x+d)*B*a^6-462*b^10/e^12/(e*x+d)*
B*d^6-7/4*b^10/e^8*B*x^4*d+210*b^6/e^7*B*a^4*x+252/e^7/(e*x+d)^5*A*a^4*b^6*d^5-168/e^8/(e*x+d)^5*A*a^3*b^7*d^6
+72/e^9/(e*x+d)^5*A*a^2*b^8*d^7-18/e^10/(e*x+d)^5*A*a*b^9*d^8+4/e^3/(e*x+d)^5*B*a^9*b*d-27/e^4/(e*x+d)^5*B*a^8
*b^2*d^2+96/e^5/(e*x+d)^5*B*a^7*b^3*d^3-210/e^6/(e*x+d)^5*B*a^6*b^4*d^4+1512/5/e^7/(e*x+d)^5*B*a^5*b^5*d^5-294
/e^8/(e*x+d)^5*B*a^4*b^6*d^6+192/e^9/(e*x+d)^5*B*a^3*b^7*d^7-81/e^10/(e*x+d)^5*B*a^2*b^8*d^8+20/e^11/(e*x+d)^5
*B*a*b^9*d^9+90*b^3/e^4/(e*x+d)^4*A*a^7*d-315*b^4/e^5/(e*x+d)^4*A*a^6*d^2+630*b^5/e^6/(e*x+d)^4*A*a^5*d^3-1575
/2*b^6/e^7/(e*x+d)^4*A*a^4*d^4+630*b^7/e^8/(e*x+d)^4*A*a^3*d^5-315*b^8/e^9/(e*x+d)^4*A*a^2*d^6+90*b^9/e^10/(e*
x+d)^4*A*a*d^7+135/4*b^2/e^4/(e*x+d)^4*B*a^8*d-180*b^3/e^5/(e*x+d)^4*B*a^7*d^2+525*b^4/e^6/(e*x+d)^4*B*a^6*d^3
-945*b^5/e^7/(e*x+d)^4*B*a^5*d^4+2205/2*b^6/e^8/(e*x+d)^4*B*a^4*d^5-840*b^7/e^9/(e*x+d)^4*B*a^3*d^6+405*b^8/e^
10/(e*x+d)^4*B*a^2*d^7-225/2*b^9/e^11/(e*x+d)^4*B*a*d^8+15/2/e^10/(e*x+d)^6*B*a^2*b^8*d^9-5/3/e^11/(e*x+d)^6*B
*a*b^9*d^10+280*b^4/e^5/(e*x+d)^3*A*a^6*d-840*b^5/e^6/(e*x+d)^3*A*a^5*d^2+1400*b^6/e^7/(e*x+d)^3*A*a^4*d^3-140
0*b^7/e^8/(e*x+d)^3*A*a^3*d^4+840*b^8/e^9/(e*x+d)^3*A*a^2*d^5-280*b^9/e^10/(e*x+d)^3*A*a*d^6+160*b^3/e^5/(e*x+
d)^3*B*a^7*d-700*b^4/e^6/(e*x+d)^3*B*a^6*d^2+1680*b^5/e^7/(e*x+d)^3*B*a^5*d^3-315*b^8/e^8*A*a^2*d*x+1260*b^8/e
^9*B*a^2*d^2*x-840*b^9/e^10*B*a*d^3*x+280*b^9/e^9*A*a*d^2*x-840*b^7/e^8*B*a^3*d*x+140*b^9/e^9*B*x^2*a*d^2-35*b
^9/e^8*A*x^2*a*d-315/2*b^8/e^8*B*x^2*a^2*d-70/3*b^9/e^8*B*x^3*a*d+5/3/e^2/(e*x+d)^6*A*d*a^9*b-15/2/e^3/(e*x+d)
^6*A*d^2*a^8*b^2+20/e^4/(e*x+d)^6*A*d^3*a^7*b^3-35/e^5/(e*x+d)^6*A*d^4*a^6*b^4+42/e^6/(e*x+d)^6*A*d^5*a^5*b^5-
35/e^7/(e*x+d)^6*A*d^6*a^4*b^6+20/e^8/(e*x+d)^6*A*a^3*b^7*d^7-15/2/e^9/(e*x+d)^6*A*a^2*b^8*d^8+5/3/e^10/(e*x+d
)^6*A*a*b^9*d^9-5/3/e^3/(e*x+d)^6*B*d^2*a^9*b+15/2/e^4/(e*x+d)^6*B*d^3*a^8*b^2-20/e^5/(e*x+d)^6*B*d^4*a^7*b^3+
35/e^6/(e*x+d)^6*B*d^5*a^6*b^4-42/e^7/(e*x+d)^6*B*d^6*a^5*b^5+35/e^8/(e*x+d)^6*B*a^4*b^6*d^7-20/e^9/(e*x+d)^6*
B*a^3*b^7*d^8-840*b^7/e^8*ln(e*x+d)*A*a^3*d+1260*b^8/e^9*ln(e*x+d)*A*a^2*d^2-840*b^9/e^10*ln(e*x+d)*A*a*d^3-14
70*b^6/e^8*ln(e*x+d)*B*a^4*d+3360*b^7/e^9*ln(e*x+d)*B*a^3*d^2-3780*b^8/e^10*ln(e*x+d)*B*a^2*d^3+2100*b^9/e^11*
ln(e*x+d)*B*a*d^4+18/e^3/(e*x+d)^5*A*a^8*b^2*d-72/e^4/(e*x+d)^5*A*a^7*b^3*d^2+168/e^5/(e*x+d)^5*A*a^6*b^4*d^3-
252/e^6/(e*x+d)^5*A*a^5*b^5*d^4-2450*b^6/e^8/(e*x+d)^3*B*a^4*d^4+2240*b^7/e^9/(e*x+d)^3*B*a^3*d^5-1260*b^8/e^1
0/(e*x+d)^3*B*a^2*d^6+400*b^9/e^11/(e*x+d)^3*B*a*d^7+630*b^5/e^6/(e*x+d)^2*A*a^5*d-1575*b^6/e^7/(e*x+d)^2*A*a^
4*d^2+2100*b^7/e^8/(e*x+d)^2*A*a^3*d^3-1575*b^8/e^9/(e*x+d)^2*A*a^2*d^4+630*b^9/e^10/(e*x+d)^2*A*a*d^5+525*b^4
/e^6/(e*x+d)^2*B*a^6*d-1890*b^5/e^7/(e*x+d)^2*B*a^5*d^2+3675*b^6/e^8/(e*x+d)^2*B*a^4*d^3-4200*b^7/e^9/(e*x+d)^
2*B*a^3*d^4+2835*b^8/e^10/(e*x+d)^2*B*a^2*d^5-1050*b^9/e^11/(e*x+d)^2*B*a*d^6+1260*b^6/e^7/(e*x+d)*A*a^4*d-252
0*b^7/e^8/(e*x+d)*A*a^3*d^2+2520*b^8/e^9/(e*x+d)*A*a^2*d^3-1260*b^9/e^10/(e*x+d)*A*a*d^4+1512*b^5/e^7/(e*x+d)*
B*a^5*d-4410*b^6/e^8/(e*x+d)*B*a^4*d^2+6720*b^7/e^9/(e*x+d)*B*a^3*d^3-5670*b^8/e^10/(e*x+d)*B*a^2*d^4+2520*b^9
/e^11/(e*x+d)*B*a*d^5+210*b^6/e^7*ln(e*x+d)*A*a^4+210*b^10/e^11*ln(e*x+d)*A*d^4+252*b^5/e^7*ln(e*x+d)*B*a^5-46
2*b^10/e^12*ln(e*x+d)*B*d^5-2/e^2/(e*x+d)^5*A*a^9*b+2/e^11/(e*x+d)^5*A*b^10*d^9-11/5/e^12/(e*x+d)^5*b^10*B*d^1
0
________________________________________________________________________________________
Maxima [B] time = 1.9689, size = 2523, normalized size = 5.64 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^7,x, algorithm="maxima")
[Out]
-1/60*(20417*B*b^10*d^11 + 10*A*a^10*e^11 - 10655*(10*B*a*b^9 + A*b^10)*d^10*e + 25090*(9*B*a^2*b^8 + 2*A*a*b^
9)*d^9*e^2 - 30690*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 20070*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 6174*(6*B
*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4
*e^7 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 2*(B*a^10 + 10*A*a^9*b)*
d*e^10 + 2520*(11*B*b^10*d^6*e^5 - 6*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 20
*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 - 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*
d*e^10 + (5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 900*(143*B*b^10*d^7*e^4 - 77*(10*B*a*b^9 + A*b^10)*d^6*e^5 +
189*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 245*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 175*(7*B*a^4*b^6 + 4*A*a^3*b
^7)*d^3*e^8 - 63*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + (4*B*a^7*b^3 + 7
*A*a^6*b^4)*e^11)*x^4 + 300*(803*B*b^10*d^8*e^3 - 428*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 1036*(9*B*a^2*b^8 + 2*A*
a*b^9)*d^6*e^5 - 1316*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 910*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 308*(6*B
*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 4*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^1
0 + (3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 75*(3025*B*b^10*d^9*e^2 - 1599*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 382
8*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 4788*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 3234*(7*B*a^4*b^6 + 4*A*a^3*b
^7)*d^5*e^6 - 1050*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 12*(4*B*a^7*
b^3 + 7*A*a^6*b^4)*d^2*e^9 + 3*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + (2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 6*(1
7897*B*b^10*d^10*e - 9395*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 22290*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 27540*(8*B
*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 18270*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 5754*(6*B*a^5*b^5 + 5*A*a^4*b^6)
*d^5*e^6 + 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 15*(3*B*a^8*b^2
+ 8*A*a^7*b^3)*d^2*e^9 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 2*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^18*x^6 + 6*d
*e^17*x^5 + 15*d^2*e^16*x^4 + 20*d^3*e^15*x^3 + 15*d^4*e^14*x^2 + 6*d^5*e^13*x + d^6*e^12) + 1/60*(12*B*b^10*e
^4*x^5 - 15*(7*B*b^10*d*e^3 - (10*B*a*b^9 + A*b^10)*e^4)*x^4 + 20*(28*B*b^10*d^2*e^2 - 7*(10*B*a*b^9 + A*b^10)
*d*e^3 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^4)*x^3 - 30*(84*B*b^10*d^3*e - 28*(10*B*a*b^9 + A*b^10)*d^2*e^2 + 35*(9
*B*a^2*b^8 + 2*A*a*b^9)*d*e^3 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^4)*x^2 + 60*(210*B*b^10*d^4 - 84*(10*B*a*b^9
+ A*b^10)*d^3*e + 140*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^2 - 105*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^3 + 30*(7*B*a^4*
b^6 + 4*A*a^3*b^7)*e^4)*x)/e^11 - 42*(11*B*b^10*d^5 - 5*(10*B*a*b^9 + A*b^10)*d^4*e + 10*(9*B*a^2*b^8 + 2*A*a*
b^9)*d^3*e^2 - 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^3 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^4 - (6*B*a^5*b^5 + 5
*A*a^4*b^6)*e^5)*log(e*x + d)/e^12
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Fricas [B] time = 2.24831, size = 6174, normalized size = 13.81 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^7,x, algorithm="fricas")
[Out]
1/60*(12*B*b^10*e^11*x^11 - 20417*B*b^10*d^11 - 10*A*a^10*e^11 + 10655*(10*B*a*b^9 + A*b^10)*d^10*e - 25090*(9
*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 30690*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 - 20070*(7*B*a^4*b^6 + 4*A*a^3*b^7
)*d^7*e^4 + 6174*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 60*(4*B*a^7*b
^3 + 7*A*a^6*b^4)*d^4*e^7 - 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 2*(
B*a^10 + 10*A*a^9*b)*d*e^10 - 3*(11*B*b^10*d*e^10 - 5*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 10*(11*B*b^10*d^2*e^9
- 5*(10*B*a*b^9 + A*b^10)*d*e^10 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 45*(11*B*b^10*d^3*e^8 - 5*(10*B*a
*b^9 + A*b^10)*d^2*e^9 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 360*
(11*B*b^10*d^4*e^7 - 5*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 10*(8*B*a^3*b^7
+ 3*A*a^2*b^8)*d*e^10 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + (47497*B*b^10*d^5*e^6 - 20215*(10*B*a*b^9 +
A*b^10)*d^4*e^7 + 36650*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 31050*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 10800*
(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10)*x^6 + 6*(19777*B*b^10*d^6*e^5 - 7615*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 11450
*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 5850*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 - 1800*(7*B*a^4*b^6 + 4*A*a^3*b^
7)*d^2*e^9 + 2520*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 15*(5917*B*
b^10*d^7*e^4 - 1315*(10*B*a*b^9 + A*b^10)*d^6*e^5 - 1150*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 + 6750*(8*B*a^3*b^7
+ 3*A*a^2*b^8)*d^4*e^7 - 8100*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 3780*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9
- 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 - 20*(3323*B*b^10*d^8*e^3
- 2885*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 9550*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 15150*(8*B*a^3*b^7 + 3*A*a^2*b
^8)*d^5*e^6 + 12300*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 4620*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 420*(5*B*
a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 + 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*
x^3 - 15*(10253*B*b^10*d^9*e^2 - 6035*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 15850*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4
- 21450*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 15450*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 5250*(6*B*a^5*b^5 +
5*A*a^4*b^6)*d^4*e^7 + 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 + 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 15*(
3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 - 6*(15797*B*b^10*d^10*e - 8555*(10*
B*a*b^9 + A*b^10)*d^9*e^2 + 20890*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 26490*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^
4 + 17970*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 5754*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 420*(5*B*a^6*b^4 +
6*A*a^5*b^5)*d^4*e^7 + 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 5*(2*
B*a^9*b + 9*A*a^8*b^2)*d*e^10 + 2*(B*a^10 + 10*A*a^9*b)*e^11)*x - 2520*(11*B*b^10*d^11 - 5*(10*B*a*b^9 + A*b^1
0)*d^10*e + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 5*(7*B*a^4*b^6 + 4
*A*a^3*b^7)*d^7*e^4 - (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + (11*B*b^10*d^5*e^6 - 5*(10*B*a*b^9 + A*b^10)*d^4*e
^7 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 5*(7*B*a^4*b^6 + 4*A*a^3*
b^7)*d*e^10 - (6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 6*(11*B*b^10*d^6*e^5 - 5*(10*B*a*b^9 + A*b^10)*d^5*e^6 +
10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)
*d^2*e^9 - (6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10)*x^5 + 15*(11*B*b^10*d^7*e^4 - 5*(10*B*a*b^9 + A*b^10)*d^6*e^5 +
10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7)
*d^3*e^8 - (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^9)*x^4 + 20*(11*B*b^10*d^8*e^3 - 5*(10*B*a*b^9 + A*b^10)*d^7*e^4
+ 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7
)*d^4*e^7 - (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8)*x^3 + 15*(11*B*b^10*d^9*e^2 - 5*(10*B*a*b^9 + A*b^10)*d^8*e^3
+ 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^
7)*d^5*e^6 - (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^4*e^7)*x^2 + 6*(11*B*b^10*d^10*e - 5*(10*B*a*b^9 + A*b^10)*d^9*e^2
+ 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 10*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 5*(7*B*a^4*b^6 + 4*A*a^3*b^7
)*d^6*e^5 - (6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6)*x)*log(e*x + d))/(e^18*x^6 + 6*d*e^17*x^5 + 15*d^2*e^16*x^4 +
20*d^3*e^15*x^3 + 15*d^4*e^14*x^2 + 6*d^5*e^13*x + d^6*e^12)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**10*(B*x+A)/(e*x+d)**7,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 2.895, size = 2534, normalized size = 5.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^7,x, algorithm="giac")
[Out]
-42*(11*B*b^10*d^5 - 50*B*a*b^9*d^4*e - 5*A*b^10*d^4*e + 90*B*a^2*b^8*d^3*e^2 + 20*A*a*b^9*d^3*e^2 - 80*B*a^3*
b^7*d^2*e^3 - 30*A*a^2*b^8*d^2*e^3 + 35*B*a^4*b^6*d*e^4 + 20*A*a^3*b^7*d*e^4 - 6*B*a^5*b^5*e^5 - 5*A*a^4*b^6*e
^5)*e^(-12)*log(abs(x*e + d)) + 1/60*(12*B*b^10*x^5*e^28 - 105*B*b^10*d*x^4*e^27 + 560*B*b^10*d^2*x^3*e^26 - 2
520*B*b^10*d^3*x^2*e^25 + 12600*B*b^10*d^4*x*e^24 + 150*B*a*b^9*x^4*e^28 + 15*A*b^10*x^4*e^28 - 1400*B*a*b^9*d
*x^3*e^27 - 140*A*b^10*d*x^3*e^27 + 8400*B*a*b^9*d^2*x^2*e^26 + 840*A*b^10*d^2*x^2*e^26 - 50400*B*a*b^9*d^3*x*
e^25 - 5040*A*b^10*d^3*x*e^25 + 900*B*a^2*b^8*x^3*e^28 + 200*A*a*b^9*x^3*e^28 - 9450*B*a^2*b^8*d*x^2*e^27 - 21
00*A*a*b^9*d*x^2*e^27 + 75600*B*a^2*b^8*d^2*x*e^26 + 16800*A*a*b^9*d^2*x*e^26 + 3600*B*a^3*b^7*x^2*e^28 + 1350
*A*a^2*b^8*x^2*e^28 - 50400*B*a^3*b^7*d*x*e^27 - 18900*A*a^2*b^8*d*x*e^27 + 12600*B*a^4*b^6*x*e^28 + 7200*A*a^
3*b^7*x*e^28)*e^(-35) - 1/60*(20417*B*b^10*d^11 - 106550*B*a*b^9*d^10*e - 10655*A*b^10*d^10*e + 225810*B*a^2*b
^8*d^9*e^2 + 50180*A*a*b^9*d^9*e^2 - 245520*B*a^3*b^7*d^8*e^3 - 92070*A*a^2*b^8*d^8*e^3 + 140490*B*a^4*b^6*d^7
*e^4 + 80280*A*a^3*b^7*d^7*e^4 - 37044*B*a^5*b^5*d^6*e^5 - 30870*A*a^4*b^6*d^6*e^5 + 2100*B*a^6*b^4*d^5*e^6 +
2520*A*a^5*b^5*d^5*e^6 + 240*B*a^7*b^3*d^4*e^7 + 420*A*a^6*b^4*d^4*e^7 + 45*B*a^8*b^2*d^3*e^8 + 120*A*a^7*b^3*
d^3*e^8 + 10*B*a^9*b*d^2*e^9 + 45*A*a^8*b^2*d^2*e^9 + 2*B*a^10*d*e^10 + 20*A*a^9*b*d*e^10 + 10*A*a^10*e^11 + 2
520*(11*B*b^10*d^6*e^5 - 60*B*a*b^9*d^5*e^6 - 6*A*b^10*d^5*e^6 + 135*B*a^2*b^8*d^4*e^7 + 30*A*a*b^9*d^4*e^7 -
160*B*a^3*b^7*d^3*e^8 - 60*A*a^2*b^8*d^3*e^8 + 105*B*a^4*b^6*d^2*e^9 + 60*A*a^3*b^7*d^2*e^9 - 36*B*a^5*b^5*d*e
^10 - 30*A*a^4*b^6*d*e^10 + 5*B*a^6*b^4*e^11 + 6*A*a^5*b^5*e^11)*x^5 + 900*(143*B*b^10*d^7*e^4 - 770*B*a*b^9*d
^6*e^5 - 77*A*b^10*d^6*e^5 + 1701*B*a^2*b^8*d^5*e^6 + 378*A*a*b^9*d^5*e^6 - 1960*B*a^3*b^7*d^4*e^7 - 735*A*a^2
*b^8*d^4*e^7 + 1225*B*a^4*b^6*d^3*e^8 + 700*A*a^3*b^7*d^3*e^8 - 378*B*a^5*b^5*d^2*e^9 - 315*A*a^4*b^6*d^2*e^9
+ 35*B*a^6*b^4*d*e^10 + 42*A*a^5*b^5*d*e^10 + 4*B*a^7*b^3*e^11 + 7*A*a^6*b^4*e^11)*x^4 + 300*(803*B*b^10*d^8*e
^3 - 4280*B*a*b^9*d^7*e^4 - 428*A*b^10*d^7*e^4 + 9324*B*a^2*b^8*d^6*e^5 + 2072*A*a*b^9*d^6*e^5 - 10528*B*a^3*b
^7*d^5*e^6 - 3948*A*a^2*b^8*d^5*e^6 + 6370*B*a^4*b^6*d^4*e^7 + 3640*A*a^3*b^7*d^4*e^7 - 1848*B*a^5*b^5*d^3*e^8
- 1540*A*a^4*b^6*d^3*e^8 + 140*B*a^6*b^4*d^2*e^9 + 168*A*a^5*b^5*d^2*e^9 + 16*B*a^7*b^3*d*e^10 + 28*A*a^6*b^4
*d*e^10 + 3*B*a^8*b^2*e^11 + 8*A*a^7*b^3*e^11)*x^3 + 75*(3025*B*b^10*d^9*e^2 - 15990*B*a*b^9*d^8*e^3 - 1599*A*
b^10*d^8*e^3 + 34452*B*a^2*b^8*d^7*e^4 + 7656*A*a*b^9*d^7*e^4 - 38304*B*a^3*b^7*d^6*e^5 - 14364*A*a^2*b^8*d^6*
e^5 + 22638*B*a^4*b^6*d^5*e^6 + 12936*A*a^3*b^7*d^5*e^6 - 6300*B*a^5*b^5*d^4*e^7 - 5250*A*a^4*b^6*d^4*e^7 + 42
0*B*a^6*b^4*d^3*e^8 + 504*A*a^5*b^5*d^3*e^8 + 48*B*a^7*b^3*d^2*e^9 + 84*A*a^6*b^4*d^2*e^9 + 9*B*a^8*b^2*d*e^10
+ 24*A*a^7*b^3*d*e^10 + 2*B*a^9*b*e^11 + 9*A*a^8*b^2*e^11)*x^2 + 6*(17897*B*b^10*d^10*e - 93950*B*a*b^9*d^9*e
^2 - 9395*A*b^10*d^9*e^2 + 200610*B*a^2*b^8*d^8*e^3 + 44580*A*a*b^9*d^8*e^3 - 220320*B*a^3*b^7*d^7*e^4 - 82620
*A*a^2*b^8*d^7*e^4 + 127890*B*a^4*b^6*d^6*e^5 + 73080*A*a^3*b^7*d^6*e^5 - 34524*B*a^5*b^5*d^5*e^6 - 28770*A*a^
4*b^6*d^5*e^6 + 2100*B*a^6*b^4*d^4*e^7 + 2520*A*a^5*b^5*d^4*e^7 + 240*B*a^7*b^3*d^3*e^8 + 420*A*a^6*b^4*d^3*e^
8 + 45*B*a^8*b^2*d^2*e^9 + 120*A*a^7*b^3*d^2*e^9 + 10*B*a^9*b*d*e^10 + 45*A*a^8*b^2*d*e^10 + 2*B*a^10*e^11 + 2
0*A*a^9*b*e^11)*x)*e^(-12)/(x*e + d)^6