∫ (a+bx)10(A+Bx)___________

3.1096 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^8} \, dx\)

Optimal. Leaf size=444 \[ -\frac{b^9 (d+e x)^3 (-10 a B e-A b e+11 b B d)}{3 e^{12}}+\frac{5 b^8 (d+e x)^2 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{2 e^{12}}-\frac{15 b^7 x (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{11}}+\frac{42 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)}-\frac{21 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)^2}+\frac{10 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^3}-\frac{15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{4 e^{12} (d+e x)^4}+\frac{30 b^6 (b d-a e)^3 \log (d+e x) (-7 a B e-4 A b e+11 b B d)}{e^{12}}+\frac{b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{12} (d+e x)^5}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{6 e^{12} (d+e x)^6}+\frac{(b d-a e)^{10} (B d-A e)}{7 e^{12} (d+e x)^7}+\frac{b^{10} B (d+e x)^4}{4 e^{12}} \]

[Out]

(-15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(7*e^12*(d + e*x)

^7) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(6*e^12*(d + e*x)^6) + (b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*

e - 2*a*B*e))/(e^12*(d + e*x)^5) - (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(4*e^12*(d + e*x)^4)

+ (10*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(e^12*(d + e*x)^3) - (21*b^4*(b*d - a*e)^5*(11*b*B*d -

6*A*b*e - 5*a*B*e))/(e^12*(d + e*x)^2) + (42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(e^12*(d + e*x

)) + (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^2)/(2*e^12) - (b^9*(11*b*B*d - A*b*e - 10*a*B

*e)*(d + e*x)^3)/(3*e^12) + (b^10*B*(d + e*x)^4)/(4*e^12) + (30*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*

e)*Log[d + e*x])/e^12

________________________________________________________________________________________

Rubi [A] time = 1.09414, antiderivative size = 444, 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} \[ -\frac{b^9 (d+e x)^3 (-10 a B e-A b e+11 b B d)}{3 e^{12}}+\frac{5 b^8 (d+e x)^2 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{2 e^{12}}-\frac{15 b^7 x (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{e^{11}}+\frac{42 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{e^{12} (d+e x)}-\frac{21 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{e^{12} (d+e x)^2}+\frac{10 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{e^{12} (d+e x)^3}-\frac{15 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{4 e^{12} (d+e x)^4}+\frac{30 b^6 (b d-a e)^3 \log (d+e x) (-7 a B e-4 A b e+11 b B d)}{e^{12}}+\frac{b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{e^{12} (d+e x)^5}-\frac{(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{6 e^{12} (d+e x)^6}+\frac{(b d-a e)^{10} (B d-A e)}{7 e^{12} (d+e x)^7}+\frac{b^{10} B (d+e x)^4}{4 e^{12}} \]

Antiderivative was successfully veriﬁed.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^8,x]

[Out]

(-15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e)*x)/e^11 + ((b*d - a*e)^10*(B*d - A*e))/(7*e^12*(d + e*x)

^7) - ((b*d - a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(6*e^12*(d + e*x)^6) + (b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*

e - 2*a*B*e))/(e^12*(d + e*x)^5) - (15*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(4*e^12*(d + e*x)^4)

+ (10*b^3*(b*d - a*e)^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(e^12*(d + e*x)^3) - (21*b^4*(b*d - a*e)^5*(11*b*B*d -

6*A*b*e - 5*a*B*e))/(e^12*(d + e*x)^2) + (42*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(e^12*(d + e*x

)) + (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e)*(d + e*x)^2)/(2*e^12) - (b^9*(11*b*B*d - A*b*e - 10*a*B

*e)*(d + e*x)^3)/(3*e^12) + (b^10*B*(d + e*x)^4)/(4*e^12) + (30*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*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)^8} \, dx &=\int \left (\frac{15 b^7 (b d-a e)^2 (-11 b B d+3 A b e+8 a B e)}{e^{11}}+\frac{(-b d+a e)^{10} (-B d+A e)}{e^{11} (d+e x)^8}+\frac{(-b d+a e)^9 (-11 b B d+10 A b e+a B e)}{e^{11} (d+e x)^7}+\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)^6}-\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)^5}+\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)^4}-\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)^3}+\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)^2}-\frac{30 b^6 (b d-a e)^3 (-11 b B d+4 A b e+7 a B e)}{e^{11} (d+e x)}-\frac{5 b^8 (b d-a e) (-11 b B d+2 A b e+9 a B e) (d+e x)}{e^{11}}+\frac{b^9 (-11 b B d+A b e+10 a B e) (d+e x)^2}{e^{11}}+\frac{b^{10} B (d+e x)^3}{e^{11}}\right ) \, dx\\ &=-\frac{15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e) x}{e^{11}}+\frac{(b d-a e)^{10} (B d-A e)}{7 e^{12} (d+e x)^7}-\frac{(b d-a e)^9 (11 b B d-10 A b e-a B e)}{6 e^{12} (d+e x)^6}+\frac{b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{e^{12} (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)}{4 e^{12} (d+e x)^4}+\frac{10 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{e^{12} (d+e x)^3}-\frac{21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{e^{12} (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^{12} (d+e x)}+\frac{5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e) (d+e x)^2}{2 e^{12}}-\frac{b^9 (11 b B d-A b e-10 a B e) (d+e x)^3}{3 e^{12}}+\frac{b^{10} B (d+e x)^4}{4 e^{12}}+\frac{30 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e) \log (d+e x)}{e^{12}}\\ \end{align*}

Mathematica [A] time = 0.571836, size = 450, normalized size = 1.01 \[ \frac{84 b^7 e x \left (45 a^2 b e^2 (A e-8 B d)+120 a^3 B e^3+40 a b^2 d e (9 B d-2 A e)+12 b^3 d^2 (3 A e-10 B d)\right )-42 b^8 e^2 x^2 \left (-45 a^2 B e^2-10 a b e (A e-8 B d)+4 b^2 d (2 A e-9 B d)\right )+28 b^9 e^3 x^3 (10 a B e+A b e-8 b B d)+\frac{3528 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{d+e x}-\frac{1764 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{(d+e x)^2}+\frac{840 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{(d+e x)^3}-\frac{315 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{(d+e x)^4}+2520 b^6 (b d-a e)^3 \log (d+e x) (-7 a B e-4 A b e+11 b B d)+\frac{84 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{(d+e x)^5}-\frac{14 (b d-a e)^9 (-a B e-10 A b e+11 b B d)}{(d+e x)^6}+\frac{12 (b d-a e)^{10} (B d-A e)}{(d+e x)^7}+21 b^{10} B e^4 x^4}{84 e^{12}} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^8,x]

[Out]

(84*b^7*e*(120*a^3*B*e^3 + 40*a*b^2*d*e*(9*B*d - 2*A*e) + 45*a^2*b*e^2*(-8*B*d + A*e) + 12*b^3*d^2*(-10*B*d +

3*A*e))*x - 42*b^8*e^2*(-45*a^2*B*e^2 - 10*a*b*e*(-8*B*d + A*e) + 4*b^2*d*(-9*B*d + 2*A*e))*x^2 + 28*b^9*e^3*(

-8*b*B*d + A*b*e + 10*a*B*e)*x^3 + 21*b^10*B*e^4*x^4 + (12*(b*d - a*e)^10*(B*d - A*e))/(d + e*x)^7 - (14*(b*d

- a*e)^9*(11*b*B*d - 10*A*b*e - a*B*e))/(d + e*x)^6 + (84*b*(b*d - a*e)^8*(11*b*B*d - 9*A*b*e - 2*a*B*e))/(d +

e*x)^5 - (315*b^2*(b*d - a*e)^7*(11*b*B*d - 8*A*b*e - 3*a*B*e))/(d + e*x)^4 + (840*b^3*(b*d - a*e)^6*(11*b*B*

d - 7*A*b*e - 4*a*B*e))/(d + e*x)^3 - (1764*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(d + e*x)^2 + (3

528*b^5*(b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(d + e*x) + 2520*b^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e -

7*a*B*e)*Log[d + e*x])/(84*e^12)

________________________________________________________________________________________

Maple [B] time = 0.034, size = 2823, normalized size = 6.4 \begin{align*} \text{output 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)^8,x)

[Out]

-2835*b^8/e^10/(e*x+d)^2*B*a^2*d^4+1260*b^9/e^11/(e*x+d)^2*B*a*d^5+840*b^7/e^8/(e*x+d)*A*a^3*d-1260*b^8/e^9/(e

*x+d)*A*a^2*d^2+840*b^9/e^10/(e*x+d)*A*a*d^3+1470*b^6/e^8/(e*x+d)*B*a^4*d-3360*b^7/e^9/(e*x+d)*B*a^3*d^2+3780*

b^8/e^10/(e*x+d)*B*a^2*d^3-2100*b^9/e^11/(e*x+d)*B*a*d^4-360*b^8/e^9*ln(e*x+d)*A*a^2*d+360*b^9/e^10*ln(e*x+d)*

A*a*d^2-960*b^7/e^9*ln(e*x+d)*B*a^3*d+1620*b^8/e^10*ln(e*x+d)*B*a^2*d^2-1200*b^9/e^11*ln(e*x+d)*B*a*d^3+72*b^3

/e^4/(e*x+d)^5*A*a^7*d-252*b^4/e^5/(e*x+d)^5*A*a^6*d^2+504*b^5/e^6/(e*x+d)^5*A*a^5*d^3-630*b^6/e^7/(e*x+d)^5*A

*a^4*d^4+504*b^7/e^8/(e*x+d)^5*A*a^3*d^5-252*b^8/e^9/(e*x+d)^5*A*a^2*d^6+72*b^9/e^10/(e*x+d)^5*A*a*d^7+27*b^2/

e^4/(e*x+d)^5*B*a^8*d-144*b^3/e^5/(e*x+d)^5*B*a^7*d^2+420*b^4/e^6/(e*x+d)^5*B*a^6*d^3-756*b^5/e^7/(e*x+d)^5*B*

a^5*d^4+882*b^6/e^8/(e*x+d)^5*B*a^4*d^5-672*b^7/e^9/(e*x+d)^5*B*a^3*d^6+324*b^8/e^10/(e*x+d)^5*B*a^2*d^7-90*b^

9/e^11/(e*x+d)^5*B*a*d^8+210*b^4/e^5/(e*x+d)^4*A*a^6*d-630*b^5/e^6/(e*x+d)^4*A*a^5*d^2+1050*b^6/e^7/(e*x+d)^4*

A*a^4*d^3-1050*b^7/e^8/(e*x+d)^4*A*a^3*d^4+630*b^8/e^9/(e*x+d)^4*A*a^2*d^5-210*b^9/e^10/(e*x+d)^4*A*a*d^6+120*

b^3/e^5/(e*x+d)^4*B*a^7*d-525*b^4/e^6/(e*x+d)^4*B*a^6*d^2+1260*b^5/e^7/(e*x+d)^4*B*a^5*d^3-3675/2*b^6/e^8/(e*x

+d)^4*B*a^4*d^4+5*b^9/e^8*A*x^2*a-4*b^10/e^9*A*x^2*d+1680*b^7/e^9/(e*x+d)^4*B*a^3*d^5-945*b^8/e^10/(e*x+d)^4*B

*a^2*d^6+300*b^9/e^11/(e*x+d)^4*B*a*d^7-40*b^9/e^9*B*x^2*a*d-80*b^9/e^9*A*a*d*x-360*b^8/e^9*B*a^2*d*x+360*b^9/

e^10*B*a*d^2*x+15/e^3/(e*x+d)^6*A*a^8*b^2*d-60/e^4/(e*x+d)^6*A*a^7*b^3*d^2+140/e^5/(e*x+d)^6*A*a^6*b^4*d^3-210

/e^6/(e*x+d)^6*A*a^5*b^5*d^4+210/e^7/(e*x+d)^6*A*a^4*b^6*d^5-140/e^8/(e*x+d)^6*A*a^3*b^7*d^6+60/e^9/(e*x+d)^6*

A*a^2*b^8*d^7-15/e^10/(e*x+d)^6*A*a*b^9*d^8+10/3/e^3/(e*x+d)^6*B*a^9*b*d-45/2/e^4/(e*x+d)^6*B*a^8*b^2*d^2+80/e

^5/(e*x+d)^6*B*a^7*b^3*d^3-175/e^6/(e*x+d)^6*B*a^6*b^4*d^4+252/e^7/(e*x+d)^6*B*a^5*b^5*d^5-245/e^8/(e*x+d)^6*B

*a^4*b^6*d^6+160/e^9/(e*x+d)^6*B*a^3*b^7*d^7-135/2/e^10/(e*x+d)^6*B*a^2*b^8*d^8+50/3/e^11/(e*x+d)^6*B*a*b^9*d^

9+420*b^5/e^6/(e*x+d)^3*A*a^5*d+1/4*b^10/e^8*B*x^4+1/3*b^10/e^8*A*x^3-1/6/e^2/(e*x+d)^6*B*a^10-1/7/e/(e*x+d)^7

*a^10*A-1050*b^6/e^7/(e*x+d)^3*A*a^4*d^2+1400*b^7/e^8/(e*x+d)^3*A*a^3*d^3-1050*b^8/e^9/(e*x+d)^3*A*a^2*d^4+420

*b^9/e^10/(e*x+d)^3*A*a*d^5+350*b^4/e^6/(e*x+d)^3*B*a^6*d-1260*b^5/e^7/(e*x+d)^3*B*a^5*d^2+2450*b^6/e^8/(e*x+d

)^3*B*a^4*d^3-2800*b^7/e^9/(e*x+d)^3*B*a^3*d^4+1890*b^8/e^10/(e*x+d)^3*B*a^2*d^5-700*b^9/e^11/(e*x+d)^3*B*a*d^

6+10/7/e^2/(e*x+d)^7*A*d*a^9*b-45/7/e^3/(e*x+d)^7*A*d^2*a^8*b^2+120/7/e^4/(e*x+d)^7*A*d^3*a^7*b^3-30/e^5/(e*x+

d)^7*A*d^4*a^6*b^4+36/e^6/(e*x+d)^7*A*d^5*a^5*b^5-30/e^7/(e*x+d)^7*A*d^6*a^4*b^6+120/7/e^8/(e*x+d)^7*A*d^7*a^3

*b^7-45/7/e^9/(e*x+d)^7*A*a^2*b^8*d^8+10/7/e^10/(e*x+d)^7*A*a*b^9*d^9-10/7/e^3/(e*x+d)^7*B*d^2*a^9*b+45/7/e^4/

(e*x+d)^7*B*d^3*a^8*b^2-120/7/e^5/(e*x+d)^7*B*d^4*a^7*b^3+30/e^6/(e*x+d)^7*B*d^5*a^6*b^4-36/e^7/(e*x+d)^7*B*d^

6*a^5*b^5+30/e^8/(e*x+d)^7*B*d^7*a^4*b^6-120/7/e^9/(e*x+d)^7*B*a^3*b^7*d^8+45/7/e^10/(e*x+d)^7*B*a^2*b^8*d^9-1

0/7/e^11/(e*x+d)^7*B*a*b^9*d^10+630*b^6/e^7/(e*x+d)^2*A*a^4*d-1260*b^7/e^8/(e*x+d)^2*A*a^3*d^2+1260*b^8/e^9/(e

*x+d)^2*A*a^2*d^3-630*b^9/e^10/(e*x+d)^2*A*a*d^4+756*b^5/e^7/(e*x+d)^2*B*a^5*d-2205*b^6/e^8/(e*x+d)^2*B*a^4*d^

2+3360*b^7/e^9/(e*x+d)^2*B*a^3*d^3+45/2*b^8/e^8*B*x^2*a^2+18*b^10/e^10*B*x^2*d^2+45*b^8/e^8*A*a^2*x+36*b^10/e^

10*A*d^2*x+120*b^7/e^8*a^3*B*x-120*b^10/e^11*B*d^3*x-5/3/e^2/(e*x+d)^6*A*a^9*b+5/3/e^11/(e*x+d)^6*A*b^10*d^9-1

1/6/e^12/(e*x+d)^6*b^10*B*d^10+120*b^7/e^8*ln(e*x+d)*A*a^3-120*b^10/e^11*ln(e*x+d)*A*d^3+210*b^6/e^8*ln(e*x+d)

*B*a^4+330*b^10/e^12*ln(e*x+d)*B*d^4+10/3*b^9/e^8*B*x^3*a-8/3*b^10/e^9*B*x^3*d-70*b^4/e^5/(e*x+d)^3*A*a^6-70*b

^10/e^11/(e*x+d)^3*A*d^6-40*b^3/e^5/(e*x+d)^3*B*a^7+110*b^10/e^12/(e*x+d)^3*B*d^7-1/7/e^11/(e*x+d)^7*A*b^10*d^

10+1/7/e^2/(e*x+d)^7*B*d*a^10+1/7/e^12/(e*x+d)^7*b^10*B*d^11-126*b^5/e^6/(e*x+d)^2*A*a^5+126*b^10/e^11/(e*x+d)

^2*A*d^5-105*b^4/e^6/(e*x+d)^2*B*a^6-231*b^10/e^12/(e*x+d)^2*B*d^6-210*b^6/e^7/(e*x+d)*A*a^4-210*b^10/e^11/(e*

x+d)*A*d^4-252*b^5/e^7/(e*x+d)*B*a^5+462*b^10/e^12/(e*x+d)*B*d^5-9*b^2/e^3/(e*x+d)^5*A*a^8-9*b^10/e^11/(e*x+d)

^5*A*d^8-2*b/e^3/(e*x+d)^5*B*a^9+11*b^10/e^12/(e*x+d)^5*B*d^9-30*b^3/e^4/(e*x+d)^4*A*a^7+30*b^10/e^11/(e*x+d)^

4*A*d^7-45/4*b^2/e^4/(e*x+d)^4*B*a^8-165/4*b^10/e^12/(e*x+d)^4*B*d^8

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Maxima [B] time = 2.3705, size = 2542, normalized size = 5.73 \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)^8,x, algorithm="maxima")

[Out]

1/84*(25961*B*b^10*d^11 - 12*A*a^10*e^11 - 11044*(10*B*a*b^9 + A*b^10)*d^10*e + 20094*(9*B*a^2*b^8 + 2*A*a*b^9

)*d^9*e^2 - 17316*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 6534*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 504*(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 - 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7

- 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 4*(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^1

0 + 3528*(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

+ 1764*(121*B*b^10*d^6*e^5 - 54*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 105*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 100*(

8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + 45*(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 + 420*(1177*B*b^10*d^7*e^4 - 518*(10*B*a*b^9 + A*b^10)*d^6*e^5 +

987*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 910*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + 385*(7*B*a^4*b^6 + 4*A*a^3*b

^7)*d^3*e^8 - 42*(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 - 2*(4*B*a^7*b^3 +

7*A*a^6*b^4)*e^11)*x^4 + 105*(5863*B*b^10*d^8*e^3 - 2552*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 4788*(9*B*a^2*b^8 +

2*A*a*b^9)*d^6*e^5 - 4312*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 1750*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 168

*(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 - 8*(4*B*a^7*b^3 + 7*A*a^6*b^4)*

d*e^10 - 3*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 21*(20669*B*b^10*d^9*e^2 - 8916*(10*B*a*b^9 + A*b^10)*d^8*e

^3 + 16524*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 14616*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 + 5754*(7*B*a^4*b^6 +

4*A*a^3*b^7)*d^5*e^6 - 504*(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 - 24*

(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 - 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 - 4*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)

*x^2 + 7*(23441*B*b^10*d^10*e - 10036*(10*B*a*b^9 + A*b^10)*d^9*e^2 + 18414*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3

- 16056*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 6174*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 504*(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 - 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 - 9*(3*B*

a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 - 4*(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^19*x

^7 + 7*d*e^18*x^6 + 21*d^2*e^17*x^5 + 35*d^3*e^16*x^4 + 35*d^4*e^15*x^3 + 21*d^5*e^14*x^2 + 7*d^6*e^13*x + d^7

*e^12) + 1/12*(3*B*b^10*e^3*x^4 - 4*(8*B*b^10*d*e^2 - (10*B*a*b^9 + A*b^10)*e^3)*x^3 + 6*(36*B*b^10*d^2*e - 8*

(10*B*a*b^9 + A*b^10)*d*e^2 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^3)*x^2 - 12*(120*B*b^10*d^3 - 36*(10*B*a*b^9 + A*b

^10)*d^2*e + 40*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^2 - 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^3)*x)/e^11 + 30*(11*B*b^10*

d^4 - 4*(10*B*a*b^9 + A*b^10)*d^3*e + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^2 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^

3 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*e^4)*log(e*x + d)/e^12

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Fricas [B] time = 2.13078, size = 6003, normalized size = 13.52 \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)^8,x, algorithm="fricas")

[Out]

1/84*(21*B*b^10*e^11*x^11 + 25961*B*b^10*d^11 - 12*A*a^10*e^11 - 11044*(10*B*a*b^9 + A*b^10)*d^10*e + 20094*(9

*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^2 - 17316*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 6534*(7*B*a^4*b^6 + 4*A*a^3*b^7)

*d^7*e^4 - 504*(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 - 24*(4*B*a^7*b^3

+ 7*A*a^6*b^4)*d^4*e^7 - 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 4*(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 - 7*(11*B*b^10*d*e^10 - 4*(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 35*(11*B*b^10*d^2*e^9 - 4

*(10*B*a*b^9 + A*b^10)*d*e^10 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 - 315*(11*B*b^10*d^3*e^8 - 4*(10*B*a*b^9

+ A*b^10)*d^2*e^9 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 - 49*(937*B*

b^10*d^4*e^7 - 308*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 390*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 180*(8*B*a^3*b^7 +

3*A*a^2*b^8)*d*e^10)*x^7 - 49*(2599*B*b^10*d^5*e^6 - 716*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 570*(9*B*a^2*b^8 + 2*

A*a*b^9)*d^3*e^8 + 180*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 - 360*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 72*(6*B*

a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 - 147*(619*B*b^10*d^6*e^5 + 4*(10*B*a*b^9 + A*b^10)*d^5*e^6 - 510*(9*B*a^2*b^

8 + 2*A*a*b^9)*d^4*e^7 + 900*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 - 540*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 7

2*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 12*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 35*(4907*B*b^10*d^7*e^4 - 33

88*(10*B*a*b^9 + A*b^10)*d^6*e^5 + 8610*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 9660*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d

^4*e^7 + 4620*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 504*(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 - 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 35*(11837*B*b^10*d^8*e^3 - 5908*(10*B*a*b^9

+ A*b^10)*d^7*e^4 + 12390*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 12180*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + 525

0*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 504*(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 - 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 - 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^11)*x^3 + 21*(17381*B*b^1

0*d^9*e^2 - 7924*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 15414*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 14196*(8*B*a^3*b^7

+ 3*A*a^2*b^8)*d^6*e^5 + 5754*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 504*(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 - 24*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 - 9*(3*B*a^8*b^2 + 8*A*a^7*b^3

)*d*e^10 - 4*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 7*(22001*B*b^10*d^10*e - 9604*(10*B*a*b^9 + A*b^10)*d^9*e^2

+ 17934*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 15876*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + 6174*(7*B*a^4*b^6 + 4

*A*a^3*b^7)*d^6*e^5 - 504*(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 - 24*(4

*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 - 9*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 - 4*(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 - 4*(10*B*a*b^9 + A*b^10)*d^10*e + 6*(9*B*a^2*b^8 +

2*A*a*b^9)*d^9*e^2 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 + (11*B*b^10*

d^4*e^7 - 4*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8

)*d*e^10 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 7*(11*B*b^10*d^5*e^6 - 4*(10*B*a*b^9 + A*b^10)*d^4*e^7 + 6*

(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10

)*x^6 + 21*(11*B*b^10*d^6*e^5 - 4*(10*B*a*b^9 + A*b^10)*d^5*e^6 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 4*(8*B

*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9)*x^5 + 35*(11*B*b^10*d^7*e^4 - 4*(10*B*a

*b^9 + A*b^10)*d^6*e^5 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 + (7*B*a^

4*b^6 + 4*A*a^3*b^7)*d^3*e^8)*x^4 + 35*(11*B*b^10*d^8*e^3 - 4*(10*B*a*b^9 + A*b^10)*d^7*e^4 + 6*(9*B*a^2*b^8 +

2*A*a*b^9)*d^6*e^5 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^6 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7)*x^3 + 21*(1

1*B*b^10*d^9*e^2 - 4*(10*B*a*b^9 + A*b^10)*d^8*e^3 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^4 - 4*(8*B*a^3*b^7 + 3*

A*a^2*b^8)*d^6*e^5 + (7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6)*x^2 + 7*(11*B*b^10*d^10*e - 4*(10*B*a*b^9 + A*b^10)*

d^9*e^2 + 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4 + (7*B*a^4*b^6 + 4*A*a^3

*b^7)*d^6*e^5)*x)*log(e*x + d))/(e^19*x^7 + 7*d*e^18*x^6 + 21*d^2*e^17*x^5 + 35*d^3*e^16*x^4 + 35*d^4*e^15*x^3

+ 21*d^5*e^14*x^2 + 7*d^6*e^13*x + d^7*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)**8,x)

[Out]

Timed out

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Giac [B] time = 1.86448, size = 2511, normalized size = 5.66 \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)^8,x, algorithm="giac")

[Out]

30*(11*B*b^10*d^4 - 40*B*a*b^9*d^3*e - 4*A*b^10*d^3*e + 54*B*a^2*b^8*d^2*e^2 + 12*A*a*b^9*d^2*e^2 - 32*B*a^3*b

^7*d*e^3 - 12*A*a^2*b^8*d*e^3 + 7*B*a^4*b^6*e^4 + 4*A*a^3*b^7*e^4)*e^(-12)*log(abs(x*e + d)) + 1/12*(3*B*b^10*

x^4*e^24 - 32*B*b^10*d*x^3*e^23 + 216*B*b^10*d^2*x^2*e^22 - 1440*B*b^10*d^3*x*e^21 + 40*B*a*b^9*x^3*e^24 + 4*A

*b^10*x^3*e^24 - 480*B*a*b^9*d*x^2*e^23 - 48*A*b^10*d*x^2*e^23 + 4320*B*a*b^9*d^2*x*e^22 + 432*A*b^10*d^2*x*e^

22 + 270*B*a^2*b^8*x^2*e^24 + 60*A*a*b^9*x^2*e^24 - 4320*B*a^2*b^8*d*x*e^23 - 960*A*a*b^9*d*x*e^23 + 1440*B*a^

3*b^7*x*e^24 + 540*A*a^2*b^8*x*e^24)*e^(-32) + 1/84*(25961*B*b^10*d^11 - 110440*B*a*b^9*d^10*e - 11044*A*b^10*

d^10*e + 180846*B*a^2*b^8*d^9*e^2 + 40188*A*a*b^9*d^9*e^2 - 138528*B*a^3*b^7*d^8*e^3 - 51948*A*a^2*b^8*d^8*e^3

+ 45738*B*a^4*b^6*d^7*e^4 + 26136*A*a^3*b^7*d^7*e^4 - 3024*B*a^5*b^5*d^6*e^5 - 2520*A*a^4*b^6*d^6*e^5 - 420*B

*a^6*b^4*d^5*e^6 - 504*A*a^5*b^5*d^5*e^6 - 96*B*a^7*b^3*d^4*e^7 - 168*A*a^6*b^4*d^4*e^7 - 27*B*a^8*b^2*d^3*e^8

- 72*A*a^7*b^3*d^3*e^8 - 8*B*a^9*b*d^2*e^9 - 36*A*a^8*b^2*d^2*e^9 - 2*B*a^10*d*e^10 - 20*A*a^9*b*d*e^10 - 12*

A*a^10*e^11 + 3528*(11*B*b^10*d^5*e^6 - 50*B*a*b^9*d^4*e^7 - 5*A*b^10*d^4*e^7 + 90*B*a^2*b^8*d^3*e^8 + 20*A*a*

b^9*d^3*e^8 - 80*B*a^3*b^7*d^2*e^9 - 30*A*a^2*b^8*d^2*e^9 + 35*B*a^4*b^6*d*e^10 + 20*A*a^3*b^7*d*e^10 - 6*B*a^

5*b^5*e^11 - 5*A*a^4*b^6*e^11)*x^6 + 1764*(121*B*b^10*d^6*e^5 - 540*B*a*b^9*d^5*e^6 - 54*A*b^10*d^5*e^6 + 945*

B*a^2*b^8*d^4*e^7 + 210*A*a*b^9*d^4*e^7 - 800*B*a^3*b^7*d^3*e^8 - 300*A*a^2*b^8*d^3*e^8 + 315*B*a^4*b^6*d^2*e^

9 + 180*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 + 420*(1177*B*b^10*d^7*e^4 - 5180*B*a*b^9*d^6*e^5 - 518*A*b^10*d^6*e^5 + 8883*B*a^2*b^8*d^5*e^6 + 1974*A*a*

b^9*d^5*e^6 - 7280*B*a^3*b^7*d^4*e^7 - 2730*A*a^2*b^8*d^4*e^7 + 2695*B*a^4*b^6*d^3*e^8 + 1540*A*a^3*b^7*d^3*e^

8 - 252*B*a^5*b^5*d^2*e^9 - 210*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 - 8*B*a^7*b^3*e^

11 - 14*A*a^6*b^4*e^11)*x^4 + 105*(5863*B*b^10*d^8*e^3 - 25520*B*a*b^9*d^7*e^4 - 2552*A*b^10*d^7*e^4 + 43092*B

*a^2*b^8*d^6*e^5 + 9576*A*a*b^9*d^6*e^5 - 34496*B*a^3*b^7*d^5*e^6 - 12936*A*a^2*b^8*d^5*e^6 + 12250*B*a^4*b^6*

d^4*e^7 + 7000*A*a^3*b^7*d^4*e^7 - 1008*B*a^5*b^5*d^3*e^8 - 840*A*a^4*b^6*d^3*e^8 - 140*B*a^6*b^4*d^2*e^9 - 16

8*A*a^5*b^5*d^2*e^9 - 32*B*a^7*b^3*d*e^10 - 56*A*a^6*b^4*d*e^10 - 9*B*a^8*b^2*e^11 - 24*A*a^7*b^3*e^11)*x^3 +

21*(20669*B*b^10*d^9*e^2 - 89160*B*a*b^9*d^8*e^3 - 8916*A*b^10*d^8*e^3 + 148716*B*a^2*b^8*d^7*e^4 + 33048*A*a*

b^9*d^7*e^4 - 116928*B*a^3*b^7*d^6*e^5 - 43848*A*a^2*b^8*d^6*e^5 + 40278*B*a^4*b^6*d^5*e^6 + 23016*A*a^3*b^7*d

^5*e^6 - 3024*B*a^5*b^5*d^4*e^7 - 2520*A*a^4*b^6*d^4*e^7 - 420*B*a^6*b^4*d^3*e^8 - 504*A*a^5*b^5*d^3*e^8 - 96*

B*a^7*b^3*d^2*e^9 - 168*A*a^6*b^4*d^2*e^9 - 27*B*a^8*b^2*d*e^10 - 72*A*a^7*b^3*d*e^10 - 8*B*a^9*b*e^11 - 36*A*

a^8*b^2*e^11)*x^2 + 7*(23441*B*b^10*d^10*e - 100360*B*a*b^9*d^9*e^2 - 10036*A*b^10*d^9*e^2 + 165726*B*a^2*b^8*

d^8*e^3 + 36828*A*a*b^9*d^8*e^3 - 128448*B*a^3*b^7*d^7*e^4 - 48168*A*a^2*b^8*d^7*e^4 + 43218*B*a^4*b^6*d^6*e^5

+ 24696*A*a^3*b^7*d^6*e^5 - 3024*B*a^5*b^5*d^5*e^6 - 2520*A*a^4*b^6*d^5*e^6 - 420*B*a^6*b^4*d^4*e^7 - 504*A*a

^5*b^5*d^4*e^7 - 96*B*a^7*b^3*d^3*e^8 - 168*A*a^6*b^4*d^3*e^8 - 27*B*a^8*b^2*d^2*e^9 - 72*A*a^7*b^3*d^2*e^9 -

8*B*a^9*b*d*e^10 - 36*A*a^8*b^2*d*e^10 - 2*B*a^10*e^11 - 20*A*a^9*b*e^11)*x)*e^(-12)/(x*e + d)^7

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