3.1104 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{16}} \, dx\)

Optimal. Leaf size=235 \[ \frac{b^3 (a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{60060 e (d+e x)^{11} (b d-a e)^5}+\frac{b^2 (a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{5460 e (d+e x)^{12} (b d-a e)^4}+\frac{b (a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{910 e (d+e x)^{13} (b d-a e)^3}+\frac{(a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{210 e (d+e x)^{14} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{15 e (d+e x)^{15} (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(15*e*(b*d - a*e)*(d + e*x)^15) + ((11*b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(
210*e*(b*d - a*e)^2*(d + e*x)^14) + (b*(11*b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(910*e*(b*d - a*e)^3*(d +
e*x)^13) + (b^2*(11*b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(5460*e*(b*d - a*e)^4*(d + e*x)^12) + (b^3*(11*
b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(60060*e*(b*d - a*e)^5*(d + e*x)^11)

________________________________________________________________________________________

Rubi [A]  time = 0.10911, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ \frac{b^3 (a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{60060 e (d+e x)^{11} (b d-a e)^5}+\frac{b^2 (a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{5460 e (d+e x)^{12} (b d-a e)^4}+\frac{b (a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{910 e (d+e x)^{13} (b d-a e)^3}+\frac{(a+b x)^{11} (-15 a B e+4 A b e+11 b B d)}{210 e (d+e x)^{14} (b d-a e)^2}-\frac{(a+b x)^{11} (B d-A e)}{15 e (d+e x)^{15} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^16,x]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(15*e*(b*d - a*e)*(d + e*x)^15) + ((11*b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(
210*e*(b*d - a*e)^2*(d + e*x)^14) + (b*(11*b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(910*e*(b*d - a*e)^3*(d +
e*x)^13) + (b^2*(11*b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(5460*e*(b*d - a*e)^4*(d + e*x)^12) + (b^3*(11*
b*B*d + 4*A*b*e - 15*a*B*e)*(a + b*x)^11)/(60060*e*(b*d - a*e)^5*(d + e*x)^11)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(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] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 45

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] - Dist[(d*Simplify[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] && NeQ[b*c - a*d, 0] && 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] ||  !SumSimplerQ[n, 1])

Rule 37

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] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{16}} \, dx &=-\frac{(B d-A e) (a+b x)^{11}}{15 e (b d-a e) (d+e x)^{15}}+\frac{(11 b B d+4 A b e-15 a B e) \int \frac{(a+b x)^{10}}{(d+e x)^{15}} \, dx}{15 e (b d-a e)}\\ &=-\frac{(B d-A e) (a+b x)^{11}}{15 e (b d-a e) (d+e x)^{15}}+\frac{(11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{210 e (b d-a e)^2 (d+e x)^{14}}+\frac{(b (11 b B d+4 A b e-15 a B e)) \int \frac{(a+b x)^{10}}{(d+e x)^{14}} \, dx}{70 e (b d-a e)^2}\\ &=-\frac{(B d-A e) (a+b x)^{11}}{15 e (b d-a e) (d+e x)^{15}}+\frac{(11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{210 e (b d-a e)^2 (d+e x)^{14}}+\frac{b (11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{910 e (b d-a e)^3 (d+e x)^{13}}+\frac{\left (b^2 (11 b B d+4 A b e-15 a B e)\right ) \int \frac{(a+b x)^{10}}{(d+e x)^{13}} \, dx}{455 e (b d-a e)^3}\\ &=-\frac{(B d-A e) (a+b x)^{11}}{15 e (b d-a e) (d+e x)^{15}}+\frac{(11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{210 e (b d-a e)^2 (d+e x)^{14}}+\frac{b (11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{910 e (b d-a e)^3 (d+e x)^{13}}+\frac{b^2 (11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{5460 e (b d-a e)^4 (d+e x)^{12}}+\frac{\left (b^3 (11 b B d+4 A b e-15 a B e)\right ) \int \frac{(a+b x)^{10}}{(d+e x)^{12}} \, dx}{5460 e (b d-a e)^4}\\ &=-\frac{(B d-A e) (a+b x)^{11}}{15 e (b d-a e) (d+e x)^{15}}+\frac{(11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{210 e (b d-a e)^2 (d+e x)^{14}}+\frac{b (11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{910 e (b d-a e)^3 (d+e x)^{13}}+\frac{b^2 (11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{5460 e (b d-a e)^4 (d+e x)^{12}}+\frac{b^3 (11 b B d+4 A b e-15 a B e) (a+b x)^{11}}{60060 e (b d-a e)^5 (d+e x)^{11}}\\ \end{align*}

Mathematica [B]  time = 0.825559, size = 1430, normalized size = 6.09 \[ -\frac{\left (4 A e \left (d^{10}+15 e x d^9+105 e^2 x^2 d^8+455 e^3 x^3 d^7+1365 e^4 x^4 d^6+3003 e^5 x^5 d^5+5005 e^6 x^6 d^4+6435 e^7 x^7 d^3+6435 e^8 x^8 d^2+5005 e^9 x^9 d+3003 e^{10} x^{10}\right )+11 B \left (d^{11}+15 e x d^{10}+105 e^2 x^2 d^9+455 e^3 x^3 d^8+1365 e^4 x^4 d^7+3003 e^5 x^5 d^6+5005 e^6 x^6 d^5+6435 e^7 x^7 d^4+6435 e^8 x^8 d^3+5005 e^9 x^9 d^2+3003 e^{10} x^{10} d+1365 e^{11} x^{11}\right )\right ) b^{10}+20 a e \left (A e \left (d^9+15 e x d^8+105 e^2 x^2 d^7+455 e^3 x^3 d^6+1365 e^4 x^4 d^5+3003 e^5 x^5 d^4+5005 e^6 x^6 d^3+6435 e^7 x^7 d^2+6435 e^8 x^8 d+5005 e^9 x^9\right )+2 B \left (d^{10}+15 e x d^9+105 e^2 x^2 d^8+455 e^3 x^3 d^7+1365 e^4 x^4 d^6+3003 e^5 x^5 d^5+5005 e^6 x^6 d^4+6435 e^7 x^7 d^3+6435 e^8 x^8 d^2+5005 e^9 x^9 d+3003 e^{10} x^{10}\right )\right ) b^9+30 a^2 e^2 \left (2 A e \left (d^8+15 e x d^7+105 e^2 x^2 d^6+455 e^3 x^3 d^5+1365 e^4 x^4 d^4+3003 e^5 x^5 d^3+5005 e^6 x^6 d^2+6435 e^7 x^7 d+6435 e^8 x^8\right )+3 B \left (d^9+15 e x d^8+105 e^2 x^2 d^7+455 e^3 x^3 d^6+1365 e^4 x^4 d^5+3003 e^5 x^5 d^4+5005 e^6 x^6 d^3+6435 e^7 x^7 d^2+6435 e^8 x^8 d+5005 e^9 x^9\right )\right ) b^8+20 a^3 e^3 \left (7 A e \left (d^7+15 e x d^6+105 e^2 x^2 d^5+455 e^3 x^3 d^4+1365 e^4 x^4 d^3+3003 e^5 x^5 d^2+5005 e^6 x^6 d+6435 e^7 x^7\right )+8 B \left (d^8+15 e x d^7+105 e^2 x^2 d^6+455 e^3 x^3 d^5+1365 e^4 x^4 d^4+3003 e^5 x^5 d^3+5005 e^6 x^6 d^2+6435 e^7 x^7 d+6435 e^8 x^8\right )\right ) b^7+35 a^4 e^4 \left (8 A e \left (d^6+15 e x d^5+105 e^2 x^2 d^4+455 e^3 x^3 d^3+1365 e^4 x^4 d^2+3003 e^5 x^5 d+5005 e^6 x^6\right )+7 B \left (d^7+15 e x d^6+105 e^2 x^2 d^5+455 e^3 x^3 d^4+1365 e^4 x^4 d^3+3003 e^5 x^5 d^2+5005 e^6 x^6 d+6435 e^7 x^7\right )\right ) b^6+168 a^5 e^5 \left (3 A e \left (d^5+15 e x d^4+105 e^2 x^2 d^3+455 e^3 x^3 d^2+1365 e^4 x^4 d+3003 e^5 x^5\right )+2 B \left (d^6+15 e x d^5+105 e^2 x^2 d^4+455 e^3 x^3 d^3+1365 e^4 x^4 d^2+3003 e^5 x^5 d+5005 e^6 x^6\right )\right ) b^5+420 a^6 e^6 \left (2 A e \left (d^4+15 e x d^3+105 e^2 x^2 d^2+455 e^3 x^3 d+1365 e^4 x^4\right )+B \left (d^5+15 e x d^4+105 e^2 x^2 d^3+455 e^3 x^3 d^2+1365 e^4 x^4 d+3003 e^5 x^5\right )\right ) b^4+120 a^7 e^7 \left (11 A e \left (d^3+15 e x d^2+105 e^2 x^2 d+455 e^3 x^3\right )+4 B \left (d^4+15 e x d^3+105 e^2 x^2 d^2+455 e^3 x^3 d+1365 e^4 x^4\right )\right ) b^3+495 a^8 e^8 \left (4 A e \left (d^2+15 e x d+105 e^2 x^2\right )+B \left (d^3+15 e x d^2+105 e^2 x^2 d+455 e^3 x^3\right )\right ) b^2+220 a^9 e^9 \left (13 A e (d+15 e x)+2 B \left (d^2+15 e x d+105 e^2 x^2\right )\right ) b+286 a^{10} e^{10} (14 A e+B (d+15 e x))}{60060 e^{12} (d+e x)^{15}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^16,x]

[Out]

-(286*a^10*e^10*(14*A*e + B*(d + 15*e*x)) + 220*a^9*b*e^9*(13*A*e*(d + 15*e*x) + 2*B*(d^2 + 15*d*e*x + 105*e^2
*x^2)) + 495*a^8*b^2*e^8*(4*A*e*(d^2 + 15*d*e*x + 105*e^2*x^2) + B*(d^3 + 15*d^2*e*x + 105*d*e^2*x^2 + 455*e^3
*x^3)) + 120*a^7*b^3*e^7*(11*A*e*(d^3 + 15*d^2*e*x + 105*d*e^2*x^2 + 455*e^3*x^3) + 4*B*(d^4 + 15*d^3*e*x + 10
5*d^2*e^2*x^2 + 455*d*e^3*x^3 + 1365*e^4*x^4)) + 420*a^6*b^4*e^6*(2*A*e*(d^4 + 15*d^3*e*x + 105*d^2*e^2*x^2 +
455*d*e^3*x^3 + 1365*e^4*x^4) + B*(d^5 + 15*d^4*e*x + 105*d^3*e^2*x^2 + 455*d^2*e^3*x^3 + 1365*d*e^4*x^4 + 300
3*e^5*x^5)) + 168*a^5*b^5*e^5*(3*A*e*(d^5 + 15*d^4*e*x + 105*d^3*e^2*x^2 + 455*d^2*e^3*x^3 + 1365*d*e^4*x^4 +
3003*e^5*x^5) + 2*B*(d^6 + 15*d^5*e*x + 105*d^4*e^2*x^2 + 455*d^3*e^3*x^3 + 1365*d^2*e^4*x^4 + 3003*d*e^5*x^5
+ 5005*e^6*x^6)) + 35*a^4*b^6*e^4*(8*A*e*(d^6 + 15*d^5*e*x + 105*d^4*e^2*x^2 + 455*d^3*e^3*x^3 + 1365*d^2*e^4*
x^4 + 3003*d*e^5*x^5 + 5005*e^6*x^6) + 7*B*(d^7 + 15*d^6*e*x + 105*d^5*e^2*x^2 + 455*d^4*e^3*x^3 + 1365*d^3*e^
4*x^4 + 3003*d^2*e^5*x^5 + 5005*d*e^6*x^6 + 6435*e^7*x^7)) + 20*a^3*b^7*e^3*(7*A*e*(d^7 + 15*d^6*e*x + 105*d^5
*e^2*x^2 + 455*d^4*e^3*x^3 + 1365*d^3*e^4*x^4 + 3003*d^2*e^5*x^5 + 5005*d*e^6*x^6 + 6435*e^7*x^7) + 8*B*(d^8 +
15*d^7*e*x + 105*d^6*e^2*x^2 + 455*d^5*e^3*x^3 + 1365*d^4*e^4*x^4 + 3003*d^3*e^5*x^5 + 5005*d^2*e^6*x^6 + 643
5*d*e^7*x^7 + 6435*e^8*x^8)) + 30*a^2*b^8*e^2*(2*A*e*(d^8 + 15*d^7*e*x + 105*d^6*e^2*x^2 + 455*d^5*e^3*x^3 + 1
365*d^4*e^4*x^4 + 3003*d^3*e^5*x^5 + 5005*d^2*e^6*x^6 + 6435*d*e^7*x^7 + 6435*e^8*x^8) + 3*B*(d^9 + 15*d^8*e*x
+ 105*d^7*e^2*x^2 + 455*d^6*e^3*x^3 + 1365*d^5*e^4*x^4 + 3003*d^4*e^5*x^5 + 5005*d^3*e^6*x^6 + 6435*d^2*e^7*x
^7 + 6435*d*e^8*x^8 + 5005*e^9*x^9)) + 20*a*b^9*e*(A*e*(d^9 + 15*d^8*e*x + 105*d^7*e^2*x^2 + 455*d^6*e^3*x^3 +
1365*d^5*e^4*x^4 + 3003*d^4*e^5*x^5 + 5005*d^3*e^6*x^6 + 6435*d^2*e^7*x^7 + 6435*d*e^8*x^8 + 5005*e^9*x^9) +
2*B*(d^10 + 15*d^9*e*x + 105*d^8*e^2*x^2 + 455*d^7*e^3*x^3 + 1365*d^6*e^4*x^4 + 3003*d^5*e^5*x^5 + 5005*d^4*e^
6*x^6 + 6435*d^3*e^7*x^7 + 6435*d^2*e^8*x^8 + 5005*d*e^9*x^9 + 3003*e^10*x^10)) + b^10*(4*A*e*(d^10 + 15*d^9*e
*x + 105*d^8*e^2*x^2 + 455*d^7*e^3*x^3 + 1365*d^6*e^4*x^4 + 3003*d^5*e^5*x^5 + 5005*d^4*e^6*x^6 + 6435*d^3*e^7
*x^7 + 6435*d^2*e^8*x^8 + 5005*d*e^9*x^9 + 3003*e^10*x^10) + 11*B*(d^11 + 15*d^10*e*x + 105*d^9*e^2*x^2 + 455*
d^8*e^3*x^3 + 1365*d^7*e^4*x^4 + 3003*d^6*e^5*x^5 + 5005*d^5*e^6*x^6 + 6435*d^4*e^7*x^7 + 6435*d^3*e^8*x^8 + 5
005*d^2*e^9*x^9 + 3003*d*e^10*x^10 + 1365*e^11*x^11)))/(60060*e^12*(d + e*x)^15)

________________________________________________________________________________________

Maple [B]  time = 0.013, size = 1942, normalized size = 8.3 \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)^16,x)

[Out]

-5/6*b^8*(2*A*a*b*e^2-2*A*b^2*d*e+9*B*a^2*e^2-20*B*a*b*d*e+11*B*b^2*d^2)/e^12/(e*x+d)^6-5/13*b*(9*A*a^8*b*e^9-
72*A*a^7*b^2*d*e^8+252*A*a^6*b^3*d^2*e^7-504*A*a^5*b^4*d^3*e^6+630*A*a^4*b^5*d^4*e^5-504*A*a^3*b^6*d^5*e^4+252
*A*a^2*b^7*d^6*e^3-72*A*a*b^8*d^7*e^2+9*A*b^9*d^8*e+2*B*a^9*e^9-27*B*a^8*b*d*e^8+144*B*a^7*b^2*d^2*e^7-420*B*a
^6*b^3*d^3*e^6+756*B*a^5*b^4*d^4*e^5-882*B*a^4*b^5*d^5*e^4+672*B*a^3*b^6*d^6*e^3-324*B*a^2*b^7*d^7*e^2+90*B*a*
b^8*d^8*e-11*B*b^9*d^9)/e^12/(e*x+d)^13-15/7*b^7*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+8*B*a^3*e^3-27*B
*a^2*b*d*e^2+30*B*a*b^2*d^2*e-11*B*b^3*d^3)/e^12/(e*x+d)^7-14/3*b^5*(5*A*a^4*b*e^5-20*A*a^3*b^2*d*e^4+30*A*a^2
*b^3*d^2*e^3-20*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e+6*B*a^5*e^5-35*B*a^4*b*d*e^4+80*B*a^3*b^2*d^2*e^3-90*B*a^2*b^3*d
^3*e^2+50*B*a*b^4*d^4*e-11*B*b^5*d^5)/e^12/(e*x+d)^9-1/14*(10*A*a^9*b*e^10-90*A*a^8*b^2*d*e^9+360*A*a^7*b^3*d^
2*e^8-840*A*a^6*b^4*d^3*e^7+1260*A*a^5*b^5*d^4*e^6-1260*A*a^4*b^6*d^5*e^5+840*A*a^3*b^7*d^6*e^4-360*A*a^2*b^8*
d^7*e^3+90*A*a*b^9*d^8*e^2-10*A*b^10*d^9*e+B*a^10*e^10-20*B*a^9*b*d*e^9+135*B*a^8*b^2*d^2*e^8-480*B*a^7*b^3*d^
3*e^7+1050*B*a^6*b^4*d^4*e^6-1512*B*a^5*b^5*d^5*e^5+1470*B*a^4*b^6*d^6*e^4-960*B*a^3*b^7*d^7*e^3+405*B*a^2*b^8
*d^8*e^2-100*B*a*b^9*d^9*e+11*B*b^10*d^10)/e^12/(e*x+d)^14-5/4*b^2*(8*A*a^7*b*e^8-56*A*a^6*b^2*d*e^7+168*A*a^5
*b^3*d^2*e^6-280*A*a^4*b^4*d^3*e^5+280*A*a^3*b^5*d^4*e^4-168*A*a^2*b^6*d^5*e^3+56*A*a*b^7*d^6*e^2-8*A*b^8*d^7*
e+3*B*a^8*e^8-32*B*a^7*b*d*e^7+140*B*a^6*b^2*d^2*e^6-336*B*a^5*b^3*d^3*e^5+490*B*a^4*b^4*d^4*e^4-448*B*a^3*b^5
*d^5*e^3+252*B*a^2*b^6*d^6*e^2-80*B*a*b^7*d^7*e+11*B*b^8*d^8)/e^12/(e*x+d)^12-1/5*b^9*(A*b*e+10*B*a*e-11*B*b*d
)/e^12/(e*x+d)^5-15/4*b^6*(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+7*B*a^4*e^4-32*B*
a^3*b*d*e^3+54*B*a^2*b^2*d^2*e^2-40*B*a*b^3*d^3*e+11*B*b^4*d^4)/e^12/(e*x+d)^8-21/5*b^4*(6*A*a^5*b*e^6-30*A*a^
4*b^2*d*e^5+60*A*a^3*b^3*d^2*e^4-60*A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2-6*A*b^6*d^5*e+5*B*a^6*e^6-36*B*a^5*b*
d*e^5+105*B*a^4*b^2*d^2*e^4-160*B*a^3*b^3*d^3*e^3+135*B*a^2*b^4*d^4*e^2-60*B*a*b^5*d^5*e+11*B*b^6*d^6)/e^12/(e
*x+d)^10-1/4*B*b^10/e^12/(e*x+d)^4-30/11*b^3*(7*A*a^6*b*e^7-42*A*a^5*b^2*d*e^6+105*A*a^4*b^3*d^2*e^5-140*A*a^3
*b^4*d^3*e^4+105*A*a^2*b^5*d^4*e^3-42*A*a*b^6*d^5*e^2+7*A*b^7*d^6*e+4*B*a^7*e^7-35*B*a^6*b*d*e^6+126*B*a^5*b^2
*d^2*e^5-245*B*a^4*b^3*d^3*e^4+280*B*a^3*b^4*d^4*e^3-189*B*a^2*b^5*d^5*e^2+70*B*a*b^6*d^6*e-11*B*b^7*d^7)/e^12
/(e*x+d)^11-1/15*(A*a^10*e^11-10*A*a^9*b*d*e^10+45*A*a^8*b^2*d^2*e^9-120*A*a^7*b^3*d^3*e^8+210*A*a^6*b^4*d^4*e
^7-252*A*a^5*b^5*d^5*e^6+210*A*a^4*b^6*d^6*e^5-120*A*a^3*b^7*d^7*e^4+45*A*a^2*b^8*d^8*e^3-10*A*a*b^9*d^9*e^2+A
*b^10*d^10*e-B*a^10*d*e^10+10*B*a^9*b*d^2*e^9-45*B*a^8*b^2*d^3*e^8+120*B*a^7*b^3*d^4*e^7-210*B*a^6*b^4*d^5*e^6
+252*B*a^5*b^5*d^6*e^5-210*B*a^4*b^6*d^7*e^4+120*B*a^3*b^7*d^8*e^3-45*B*a^2*b^8*d^9*e^2+10*B*a*b^9*d^10*e-B*b^
10*d^11)/e^12/(e*x+d)^15

________________________________________________________________________________________

Maxima [B]  time = 2.55169, size = 2664, normalized size = 11.34 \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)^16,x, algorithm="maxima")

[Out]

-1/60060*(15015*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 4004*A*a^10*e^11 + 4*(10*B*a*b^9 + A*b^10)*d^10*e + 10*(9*
B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e
^4 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 120*(4*B*a^7*b^3 + 7*A*
a^6*b^4)*d^4*e^7 + 165*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 220*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 286*(B*a^
10 + 10*A*a^9*b)*d*e^10 + 3003*(11*B*b^10*d*e^10 + 4*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 5005*(11*B*b^10*d^2*e^
9 + 4*(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 + 6435*(11*B*b^10*d^3*e^8 + 4*(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 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 6
435*(11*B*b^10*d^4*e^7 + 4*(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 + 20*(8*B*a^3*
b^7 + 3*A*a^2*b^8)*d*e^10 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 5005*(11*B*b^10*d^5*e^6 + 4*(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 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 35*(7*B*a
^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 3003*(11*B*b^10*d^6*e^5 + 4*(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 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 35*(7*
B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11
)*x^5 + 1365*(11*B*b^10*d^7*e^4 + 4*(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 + 20*
(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*
d^2*e^9 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 455*(11*B*b^10*d
^8*e^3 + 4*(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 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^
8)*d^5*e^6 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 84*(5*B*a^6*b^4
+ 6*A*a^5*b^5)*d^2*e^9 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 165*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 +
105*(11*B*b^10*d^9*e^2 + 4*(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 + 20*(8*B*a^3
*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 56*(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 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 165*(3*B*a^8*b^2 + 8*A*a^
7*b^3)*d*e^10 + 220*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 15*(11*B*b^10*d^10*e + 4*(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 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 35*(7*B*a^4*b^6 + 4*A*a^3
*b^7)*d^6*e^5 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 120*(4*B*a^7
*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 165*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 220*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 +
286*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^27*x^15 + 15*d*e^26*x^14 + 105*d^2*e^25*x^13 + 455*d^3*e^24*x^12 + 1365
*d^4*e^23*x^11 + 3003*d^5*e^22*x^10 + 5005*d^6*e^21*x^9 + 6435*d^7*e^20*x^8 + 6435*d^8*e^19*x^7 + 5005*d^9*e^1
8*x^6 + 3003*d^10*e^17*x^5 + 1365*d^11*e^16*x^4 + 455*d^12*e^15*x^3 + 105*d^13*e^14*x^2 + 15*d^14*e^13*x + d^1
5*e^12)

________________________________________________________________________________________

Fricas [B]  time = 2.29132, size = 4319, normalized size = 18.38 \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)^16,x, algorithm="fricas")

[Out]

-1/60060*(15015*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 4004*A*a^10*e^11 + 4*(10*B*a*b^9 + A*b^10)*d^10*e + 10*(9*
B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e
^4 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 120*(4*B*a^7*b^3 + 7*A*
a^6*b^4)*d^4*e^7 + 165*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 220*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 + 286*(B*a^
10 + 10*A*a^9*b)*d*e^10 + 3003*(11*B*b^10*d*e^10 + 4*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 5005*(11*B*b^10*d^2*e^
9 + 4*(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 + 6435*(11*B*b^10*d^3*e^8 + 4*(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 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 6
435*(11*B*b^10*d^4*e^7 + 4*(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 + 20*(8*B*a^3*
b^7 + 3*A*a^2*b^8)*d*e^10 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 5005*(11*B*b^10*d^5*e^6 + 4*(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 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + 35*(7*B*a
^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 3003*(11*B*b^10*d^6*e^5 + 4*(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 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 35*(7*
B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11
)*x^5 + 1365*(11*B*b^10*d^7*e^4 + 4*(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 + 20*
(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*
d^2*e^9 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 455*(11*B*b^10*d
^8*e^3 + 4*(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 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^
8)*d^5*e^6 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 + 84*(5*B*a^6*b^4
+ 6*A*a^5*b^5)*d^2*e^9 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 + 165*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 +
105*(11*B*b^10*d^9*e^2 + 4*(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 + 20*(8*B*a^3
*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 35*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 + 56*(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 + 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 + 165*(3*B*a^8*b^2 + 8*A*a^
7*b^3)*d*e^10 + 220*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 15*(11*B*b^10*d^10*e + 4*(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 + 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 35*(7*B*a^4*b^6 + 4*A*a^3
*b^7)*d^6*e^5 + 56*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 + 84*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^4*e^7 + 120*(4*B*a^7
*b^3 + 7*A*a^6*b^4)*d^3*e^8 + 165*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 + 220*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^10 +
286*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^27*x^15 + 15*d*e^26*x^14 + 105*d^2*e^25*x^13 + 455*d^3*e^24*x^12 + 1365
*d^4*e^23*x^11 + 3003*d^5*e^22*x^10 + 5005*d^6*e^21*x^9 + 6435*d^7*e^20*x^8 + 6435*d^8*e^19*x^7 + 5005*d^9*e^1
8*x^6 + 3003*d^10*e^17*x^5 + 1365*d^11*e^16*x^4 + 455*d^12*e^15*x^3 + 105*d^13*e^14*x^2 + 15*d^14*e^13*x + d^1
5*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)**16,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.31801, size = 2830, normalized size = 12.04 \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)^16,x, algorithm="giac")

[Out]

-1/60060*(15015*B*b^10*x^11*e^11 + 33033*B*b^10*d*x^10*e^10 + 55055*B*b^10*d^2*x^9*e^9 + 70785*B*b^10*d^3*x^8*
e^8 + 70785*B*b^10*d^4*x^7*e^7 + 55055*B*b^10*d^5*x^6*e^6 + 33033*B*b^10*d^6*x^5*e^5 + 15015*B*b^10*d^7*x^4*e^
4 + 5005*B*b^10*d^8*x^3*e^3 + 1155*B*b^10*d^9*x^2*e^2 + 165*B*b^10*d^10*x*e + 11*B*b^10*d^11 + 120120*B*a*b^9*
x^10*e^11 + 12012*A*b^10*x^10*e^11 + 200200*B*a*b^9*d*x^9*e^10 + 20020*A*b^10*d*x^9*e^10 + 257400*B*a*b^9*d^2*
x^8*e^9 + 25740*A*b^10*d^2*x^8*e^9 + 257400*B*a*b^9*d^3*x^7*e^8 + 25740*A*b^10*d^3*x^7*e^8 + 200200*B*a*b^9*d^
4*x^6*e^7 + 20020*A*b^10*d^4*x^6*e^7 + 120120*B*a*b^9*d^5*x^5*e^6 + 12012*A*b^10*d^5*x^5*e^6 + 54600*B*a*b^9*d
^6*x^4*e^5 + 5460*A*b^10*d^6*x^4*e^5 + 18200*B*a*b^9*d^7*x^3*e^4 + 1820*A*b^10*d^7*x^3*e^4 + 4200*B*a*b^9*d^8*
x^2*e^3 + 420*A*b^10*d^8*x^2*e^3 + 600*B*a*b^9*d^9*x*e^2 + 60*A*b^10*d^9*x*e^2 + 40*B*a*b^9*d^10*e + 4*A*b^10*
d^10*e + 450450*B*a^2*b^8*x^9*e^11 + 100100*A*a*b^9*x^9*e^11 + 579150*B*a^2*b^8*d*x^8*e^10 + 128700*A*a*b^9*d*
x^8*e^10 + 579150*B*a^2*b^8*d^2*x^7*e^9 + 128700*A*a*b^9*d^2*x^7*e^9 + 450450*B*a^2*b^8*d^3*x^6*e^8 + 100100*A
*a*b^9*d^3*x^6*e^8 + 270270*B*a^2*b^8*d^4*x^5*e^7 + 60060*A*a*b^9*d^4*x^5*e^7 + 122850*B*a^2*b^8*d^5*x^4*e^6 +
27300*A*a*b^9*d^5*x^4*e^6 + 40950*B*a^2*b^8*d^6*x^3*e^5 + 9100*A*a*b^9*d^6*x^3*e^5 + 9450*B*a^2*b^8*d^7*x^2*e
^4 + 2100*A*a*b^9*d^7*x^2*e^4 + 1350*B*a^2*b^8*d^8*x*e^3 + 300*A*a*b^9*d^8*x*e^3 + 90*B*a^2*b^8*d^9*e^2 + 20*A
*a*b^9*d^9*e^2 + 1029600*B*a^3*b^7*x^8*e^11 + 386100*A*a^2*b^8*x^8*e^11 + 1029600*B*a^3*b^7*d*x^7*e^10 + 38610
0*A*a^2*b^8*d*x^7*e^10 + 800800*B*a^3*b^7*d^2*x^6*e^9 + 300300*A*a^2*b^8*d^2*x^6*e^9 + 480480*B*a^3*b^7*d^3*x^
5*e^8 + 180180*A*a^2*b^8*d^3*x^5*e^8 + 218400*B*a^3*b^7*d^4*x^4*e^7 + 81900*A*a^2*b^8*d^4*x^4*e^7 + 72800*B*a^
3*b^7*d^5*x^3*e^6 + 27300*A*a^2*b^8*d^5*x^3*e^6 + 16800*B*a^3*b^7*d^6*x^2*e^5 + 6300*A*a^2*b^8*d^6*x^2*e^5 + 2
400*B*a^3*b^7*d^7*x*e^4 + 900*A*a^2*b^8*d^7*x*e^4 + 160*B*a^3*b^7*d^8*e^3 + 60*A*a^2*b^8*d^8*e^3 + 1576575*B*a
^4*b^6*x^7*e^11 + 900900*A*a^3*b^7*x^7*e^11 + 1226225*B*a^4*b^6*d*x^6*e^10 + 700700*A*a^3*b^7*d*x^6*e^10 + 735
735*B*a^4*b^6*d^2*x^5*e^9 + 420420*A*a^3*b^7*d^2*x^5*e^9 + 334425*B*a^4*b^6*d^3*x^4*e^8 + 191100*A*a^3*b^7*d^3
*x^4*e^8 + 111475*B*a^4*b^6*d^4*x^3*e^7 + 63700*A*a^3*b^7*d^4*x^3*e^7 + 25725*B*a^4*b^6*d^5*x^2*e^6 + 14700*A*
a^3*b^7*d^5*x^2*e^6 + 3675*B*a^4*b^6*d^6*x*e^5 + 2100*A*a^3*b^7*d^6*x*e^5 + 245*B*a^4*b^6*d^7*e^4 + 140*A*a^3*
b^7*d^7*e^4 + 1681680*B*a^5*b^5*x^6*e^11 + 1401400*A*a^4*b^6*x^6*e^11 + 1009008*B*a^5*b^5*d*x^5*e^10 + 840840*
A*a^4*b^6*d*x^5*e^10 + 458640*B*a^5*b^5*d^2*x^4*e^9 + 382200*A*a^4*b^6*d^2*x^4*e^9 + 152880*B*a^5*b^5*d^3*x^3*
e^8 + 127400*A*a^4*b^6*d^3*x^3*e^8 + 35280*B*a^5*b^5*d^4*x^2*e^7 + 29400*A*a^4*b^6*d^4*x^2*e^7 + 5040*B*a^5*b^
5*d^5*x*e^6 + 4200*A*a^4*b^6*d^5*x*e^6 + 336*B*a^5*b^5*d^6*e^5 + 280*A*a^4*b^6*d^6*e^5 + 1261260*B*a^6*b^4*x^5
*e^11 + 1513512*A*a^5*b^5*x^5*e^11 + 573300*B*a^6*b^4*d*x^4*e^10 + 687960*A*a^5*b^5*d*x^4*e^10 + 191100*B*a^6*
b^4*d^2*x^3*e^9 + 229320*A*a^5*b^5*d^2*x^3*e^9 + 44100*B*a^6*b^4*d^3*x^2*e^8 + 52920*A*a^5*b^5*d^3*x^2*e^8 + 6
300*B*a^6*b^4*d^4*x*e^7 + 7560*A*a^5*b^5*d^4*x*e^7 + 420*B*a^6*b^4*d^5*e^6 + 504*A*a^5*b^5*d^5*e^6 + 655200*B*
a^7*b^3*x^4*e^11 + 1146600*A*a^6*b^4*x^4*e^11 + 218400*B*a^7*b^3*d*x^3*e^10 + 382200*A*a^6*b^4*d*x^3*e^10 + 50
400*B*a^7*b^3*d^2*x^2*e^9 + 88200*A*a^6*b^4*d^2*x^2*e^9 + 7200*B*a^7*b^3*d^3*x*e^8 + 12600*A*a^6*b^4*d^3*x*e^8
+ 480*B*a^7*b^3*d^4*e^7 + 840*A*a^6*b^4*d^4*e^7 + 225225*B*a^8*b^2*x^3*e^11 + 600600*A*a^7*b^3*x^3*e^11 + 519
75*B*a^8*b^2*d*x^2*e^10 + 138600*A*a^7*b^3*d*x^2*e^10 + 7425*B*a^8*b^2*d^2*x*e^9 + 19800*A*a^7*b^3*d^2*x*e^9 +
495*B*a^8*b^2*d^3*e^8 + 1320*A*a^7*b^3*d^3*e^8 + 46200*B*a^9*b*x^2*e^11 + 207900*A*a^8*b^2*x^2*e^11 + 6600*B*
a^9*b*d*x*e^10 + 29700*A*a^8*b^2*d*x*e^10 + 440*B*a^9*b*d^2*e^9 + 1980*A*a^8*b^2*d^2*e^9 + 4290*B*a^10*x*e^11
+ 42900*A*a^9*b*x*e^11 + 286*B*a^10*d*e^10 + 2860*A*a^9*b*d*e^10 + 4004*A*a^10*e^11)*e^(-12)/(x*e + d)^15