3.1804 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^3}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=306 \[ -\frac{2 b^5 (d+e x)^{13/2} (-6 a B e-A b e+7 b B d)}{13 e^8}+\frac{6 b^4 (d+e x)^{11/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{11 e^8}-\frac{10 b^3 (d+e x)^{9/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{9 e^8}+\frac{10 b^2 (d+e x)^{7/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{7 e^8}-\frac{6 b (d+e x)^{5/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{5 e^8}+\frac{2 (d+e x)^{3/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{3 e^8}-\frac{2 \sqrt{d+e x} (b d-a e)^6 (B d-A e)}{e^8}+\frac{2 b^6 B (d+e x)^{15/2}}{15 e^8} \]

[Out]

(-2*(b*d - a*e)^6*(B*d - A*e)*Sqrt[d + e*x])/e^8 + (2*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x)^(3/2
))/(3*e^8) - (6*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^(5/2))/(5*e^8) + (10*b^2*(b*d - a*e)^3
*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^(7/2))/(7*e^8) - (10*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*
(d + e*x)^(9/2))/(9*e^8) + (6*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(11/2))/(11*e^8) - (2*b^
5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^(13/2))/(13*e^8) + (2*b^6*B*(d + e*x)^(15/2))/(15*e^8)

________________________________________________________________________________________

Rubi [A]  time = 0.145515, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {27, 77} \[ -\frac{2 b^5 (d+e x)^{13/2} (-6 a B e-A b e+7 b B d)}{13 e^8}+\frac{6 b^4 (d+e x)^{11/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{11 e^8}-\frac{10 b^3 (d+e x)^{9/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{9 e^8}+\frac{10 b^2 (d+e x)^{7/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{7 e^8}-\frac{6 b (d+e x)^{5/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{5 e^8}+\frac{2 (d+e x)^{3/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{3 e^8}-\frac{2 \sqrt{d+e x} (b d-a e)^6 (B d-A e)}{e^8}+\frac{2 b^6 B (d+e x)^{15/2}}{15 e^8} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/Sqrt[d + e*x],x]

[Out]

(-2*(b*d - a*e)^6*(B*d - A*e)*Sqrt[d + e*x])/e^8 + (2*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x)^(3/2
))/(3*e^8) - (6*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^(5/2))/(5*e^8) + (10*b^2*(b*d - a*e)^3
*(7*b*B*d - 4*A*b*e - 3*a*B*e)*(d + e*x)^(7/2))/(7*e^8) - (10*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*
(d + e*x)^(9/2))/(9*e^8) + (6*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^(11/2))/(11*e^8) - (2*b^
5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^(13/2))/(13*e^8) + (2*b^6*B*(d + e*x)^(15/2))/(15*e^8)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt{d+e x}} \, dx &=\int \frac{(a+b x)^6 (A+B x)}{\sqrt{d+e x}} \, dx\\ &=\int \left (\frac{(-b d+a e)^6 (-B d+A e)}{e^7 \sqrt{d+e x}}+\frac{(-b d+a e)^5 (-7 b B d+6 A b e+a B e) \sqrt{d+e x}}{e^7}+\frac{3 b (b d-a e)^4 (-7 b B d+5 A b e+2 a B e) (d+e x)^{3/2}}{e^7}-\frac{5 b^2 (b d-a e)^3 (-7 b B d+4 A b e+3 a B e) (d+e x)^{5/2}}{e^7}+\frac{5 b^3 (b d-a e)^2 (-7 b B d+3 A b e+4 a B e) (d+e x)^{7/2}}{e^7}-\frac{3 b^4 (b d-a e) (-7 b B d+2 A b e+5 a B e) (d+e x)^{9/2}}{e^7}+\frac{b^5 (-7 b B d+A b e+6 a B e) (d+e x)^{11/2}}{e^7}+\frac{b^6 B (d+e x)^{13/2}}{e^7}\right ) \, dx\\ &=-\frac{2 (b d-a e)^6 (B d-A e) \sqrt{d+e x}}{e^8}+\frac{2 (b d-a e)^5 (7 b B d-6 A b e-a B e) (d+e x)^{3/2}}{3 e^8}-\frac{6 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) (d+e x)^{5/2}}{5 e^8}+\frac{10 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) (d+e x)^{7/2}}{7 e^8}-\frac{10 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^{9/2}}{9 e^8}+\frac{6 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^{11/2}}{11 e^8}-\frac{2 b^5 (7 b B d-A b e-6 a B e) (d+e x)^{13/2}}{13 e^8}+\frac{2 b^6 B (d+e x)^{15/2}}{15 e^8}\\ \end{align*}

Mathematica [A]  time = 0.216495, size = 259, normalized size = 0.85 \[ \frac{2 \sqrt{d+e x} \left (-3465 b^5 (d+e x)^6 (-6 a B e-A b e+7 b B d)+12285 b^4 (d+e x)^5 (b d-a e) (-5 a B e-2 A b e+7 b B d)-25025 b^3 (d+e x)^4 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)+32175 b^2 (d+e x)^3 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)-27027 b (d+e x)^2 (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)+15015 (d+e x) (b d-a e)^5 (-a B e-6 A b e+7 b B d)-45045 (b d-a e)^6 (B d-A e)+3003 b^6 B (d+e x)^7\right )}{45045 e^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(-45045*(b*d - a*e)^6*(B*d - A*e) + 15015*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*(d + e*x)
- 27027*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*(d + e*x)^2 + 32175*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*
e - 3*a*B*e)*(d + e*x)^3 - 25025*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^4 + 12285*b^4*(b*d
- a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^5 - 3465*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^6 + 3003*b^6
*B*(d + e*x)^7))/(45045*e^8)

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 913, normalized size = 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)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x)

[Out]

2/45045*(3003*B*b^6*e^7*x^7+3465*A*b^6*e^7*x^6+20790*B*a*b^5*e^7*x^6-3234*B*b^6*d*e^6*x^6+24570*A*a*b^5*e^7*x^
5-3780*A*b^6*d*e^6*x^5+61425*B*a^2*b^4*e^7*x^5-22680*B*a*b^5*d*e^6*x^5+3528*B*b^6*d^2*e^5*x^5+75075*A*a^2*b^4*
e^7*x^4-27300*A*a*b^5*d*e^6*x^4+4200*A*b^6*d^2*e^5*x^4+100100*B*a^3*b^3*e^7*x^4-68250*B*a^2*b^4*d*e^6*x^4+2520
0*B*a*b^5*d^2*e^5*x^4-3920*B*b^6*d^3*e^4*x^4+128700*A*a^3*b^3*e^7*x^3-85800*A*a^2*b^4*d*e^6*x^3+31200*A*a*b^5*
d^2*e^5*x^3-4800*A*b^6*d^3*e^4*x^3+96525*B*a^4*b^2*e^7*x^3-114400*B*a^3*b^3*d*e^6*x^3+78000*B*a^2*b^4*d^2*e^5*
x^3-28800*B*a*b^5*d^3*e^4*x^3+4480*B*b^6*d^4*e^3*x^3+135135*A*a^4*b^2*e^7*x^2-154440*A*a^3*b^3*d*e^6*x^2+10296
0*A*a^2*b^4*d^2*e^5*x^2-37440*A*a*b^5*d^3*e^4*x^2+5760*A*b^6*d^4*e^3*x^2+54054*B*a^5*b*e^7*x^2-115830*B*a^4*b^
2*d*e^6*x^2+137280*B*a^3*b^3*d^2*e^5*x^2-93600*B*a^2*b^4*d^3*e^4*x^2+34560*B*a*b^5*d^4*e^3*x^2-5376*B*b^6*d^5*
e^2*x^2+90090*A*a^5*b*e^7*x-180180*A*a^4*b^2*d*e^6*x+205920*A*a^3*b^3*d^2*e^5*x-137280*A*a^2*b^4*d^3*e^4*x+499
20*A*a*b^5*d^4*e^3*x-7680*A*b^6*d^5*e^2*x+15015*B*a^6*e^7*x-72072*B*a^5*b*d*e^6*x+154440*B*a^4*b^2*d^2*e^5*x-1
83040*B*a^3*b^3*d^3*e^4*x+124800*B*a^2*b^4*d^4*e^3*x-46080*B*a*b^5*d^5*e^2*x+7168*B*b^6*d^6*e*x+45045*A*a^6*e^
7-180180*A*a^5*b*d*e^6+360360*A*a^4*b^2*d^2*e^5-411840*A*a^3*b^3*d^3*e^4+274560*A*a^2*b^4*d^4*e^3-99840*A*a*b^
5*d^5*e^2+15360*A*b^6*d^6*e-30030*B*a^6*d*e^6+144144*B*a^5*b*d^2*e^5-308880*B*a^4*b^2*d^3*e^4+366080*B*a^3*b^3
*d^4*e^3-249600*B*a^2*b^4*d^5*e^2+92160*B*a*b^5*d^6*e-14336*B*b^6*d^7)*(e*x+d)^(1/2)/e^8

________________________________________________________________________________________

Maxima [B]  time = 1.00933, size = 1035, normalized size = 3.38 \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)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*B*b^6 - 3465*(7*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*(e*x + d)^(13/2) + 12285*(7*B*
b^6*d^2 - 2*(6*B*a*b^5 + A*b^6)*d*e + (5*B*a^2*b^4 + 2*A*a*b^5)*e^2)*(e*x + d)^(11/2) - 25025*(7*B*b^6*d^3 - 3
*(6*B*a*b^5 + A*b^6)*d^2*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^2 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^3)*(e*x + d)^(9
/2) + 32175*(7*B*b^6*d^4 - 4*(6*B*a*b^5 + A*b^6)*d^3*e + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^2 - 4*(4*B*a^3*b^3
+ 3*A*a^2*b^4)*d*e^3 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^4)*(e*x + d)^(7/2) - 27027*(7*B*b^6*d^5 - 5*(6*B*a*b^5 +
A*b^6)*d^4*e + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^2 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^3 + 5*(3*B*a^4*b^2
+ 4*A*a^3*b^3)*d*e^4 - (2*B*a^5*b + 5*A*a^4*b^2)*e^5)*(e*x + d)^(5/2) + 15015*(7*B*b^6*d^6 - 6*(6*B*a*b^5 + A*
b^6)*d^5*e + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^2 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^3 + 15*(3*B*a^4*b^2 +
4*A*a^3*b^3)*d^2*e^4 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)*e^6)*(e*x + d)^(3/2) - 45045*(
B*b^6*d^7 - A*a^6*e^7 - (6*B*a*b^5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 5*(4*B*a^3*b^3 + 3*A
*a^2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + (B*a^6 + 6*A
*a^5*b)*d*e^6)*sqrt(e*x + d))/e^8

________________________________________________________________________________________

Fricas [B]  time = 1.3269, size = 1736, 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)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3003*B*b^6*e^7*x^7 - 14336*B*b^6*d^7 + 45045*A*a^6*e^7 + 15360*(6*B*a*b^5 + A*b^6)*d^6*e - 49920*(5*B
*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 91520*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 - 102960*(3*B*a^4*b^2 + 4*A*a^3*b^3)
*d^3*e^4 + 72072*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 - 30030*(B*a^6 + 6*A*a^5*b)*d*e^6 - 231*(14*B*b^6*d*e^6 - 1
5*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 63*(56*B*b^6*d^2*e^5 - 60*(6*B*a*b^5 + A*b^6)*d*e^6 + 195*(5*B*a^2*b^4 + 2*A*
a*b^5)*e^7)*x^5 - 35*(112*B*b^6*d^3*e^4 - 120*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 390*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^
6 - 715*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 5*(896*B*b^6*d^4*e^3 - 960*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 3120*(
5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 5720*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 6435*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e
^7)*x^3 - 3*(1792*B*b^6*d^5*e^2 - 1920*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 6240*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 -
11440*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 12870*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - 9009*(2*B*a^5*b + 5*A*a^
4*b^2)*e^7)*x^2 + (7168*B*b^6*d^6*e - 7680*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 24960*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e
^3 - 45760*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 51480*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 36036*(2*B*a^5*b
+ 5*A*a^4*b^2)*d*e^6 + 15015*(B*a^6 + 6*A*a^5*b)*e^7)*x)*sqrt(e*x + d)/e^8

________________________________________________________________________________________

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)*(b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.1671, size = 1208, normalized size = 3.95 \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)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^6*e^(-1) + 90090*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d
)*A*a^5*b*e^(-1) + 18018*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^5*b*e^(-2) + 45
045*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^4*b^2*e^(-2) + 19305*(5*(x*e + d)^(7
/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a^4*b^2*e^(-3) + 25740*(5*(x*e +
d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a^3*b^3*e^(-3) + 2860*(35*
(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d
)*d^4)*B*a^3*b^3*e^(-4) + 2145*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*
e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*a^2*b^4*e^(-4) + 975*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d
+ 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a^2
*b^4*e^(-5) + 390*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2
)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*a*b^5*e^(-5) + 90*(231*(x*e + d)^(13/2) - 1638*(x*
e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e +
d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*a*b^5*e^(-6) + 15*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d +
5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 30
03*sqrt(x*e + d)*d^6)*A*b^6*e^(-6) + 7*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2
)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3
/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*B*b^6*e^(-7) + 45045*sqrt(x*e + d)*A*a^6)*e^(-1)