3.265 \(\int \frac{\sqrt{a+b x+c x^2} (A+B x+C x^2)}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=1904 \[ \text{result too large to display} \]

[Out]

(2*(2*c^3*d^3*(8*C*d^2 + e*(4*B*d + 5*A*e)) + 3*c^2*d*e*(2*a*e*(9*C*d^2 + 7*B*d*e - 9*A*e^2) - b*d*(16*C*d^2 +
7*B*d*e + 5*A*e^2)) + 3*c*e^2*(2*a^2*e^2*(17*C*d - 5*B*e) - a*b*e*(41*C*d^2 + 5*B*d*e - 9*A*e^2) + b^2*d*(15*
C*d^2 + 3*B*d*e + 7*A*e^2)) - b*e^3*(21*a^2*C*e^2 - 6*a*b*e*(3*C*d + 2*B*e) + b^2*(5*C*d^2 + 4*B*d*e + 8*A*e^2
)))*Sqrt[a + b*x + c*x^2])/(315*e^3*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(3/2)) + (2*(2*c^4*d^4*(8*C*d^2 + e*(4
*B*d + 5*A*e)) + 2*b^2*e^4*(21*a^2*C*e^2 - 6*a*b*e*(3*C*d + 2*B*e) + b^2*(5*C*d^2 + 4*B*d*e + 8*A*e^2)) - 6*c^
2*e^2*(a*b*d*e*(30*C*d^2 - 5*B*d*e - 34*A*e^2) - a^2*e^2*(30*C*d^2 - 36*B*d*e + 7*A*e^2) - b^2*d^2*(11*C*d^2 +
3*B*d*e + 11*A*e^2)) - c*e^3*(126*a^3*C*e^3 - 3*a^2*b*e^2*(12*C*d + 29*B*e) - 6*a*b^2*e*(5*C*d^2 + 7*B*d*e -
12*A*e^2) + b^3*d*(20*C*d^2 + 25*B*d*e + 56*A*e^2)) + c^3*d^2*e*(6*a*e*(11*C*d^2 + 8*B*d*e - 34*A*e^2) - b*d*(
56*C*d^2 + 5*e*(5*B*d + 4*A*e))))*Sqrt[a + b*x + c*x^2])/(315*e^3*(c*d^2 - b*d*e + a*e^2)^4*Sqrt[d + e*x]) - (
2*(c^2*d^3*(8*C*d^2 + e*(4*B*d + 5*A*e)) - e^2*(3*a^2*e^2*(3*C*d - 5*B*e) - a*b*e*(2*C*d^2 - 17*B*d*e - 10*A*e
^2) - b^2*d*(5*C*d^2 + 4*B*d*e + 8*A*e^2)) - c*d*e*(3*b*d*(5*C*d^2 + 2*B*d*e + 5*A*e^2) - a*e*(7*C*d^2 + 11*B*
d*e + 13*A*e^2)) + e*(3*c^2*d^2*(6*C*d^2 + e*(3*B*d - 5*A*e)) + c*e*(a*e*(47*C*d^2 + B*d*e - 7*A*e^2) - 3*b*d*
(15*C*d^2 + 2*B*d*e - 5*A*e^2)) + e^2*(21*a^2*C*e^2 - 3*a*b*e*(16*C*d - B*e) + b^2*(25*C*d^2 - e*(B*d + 2*A*e)
)))*x)*Sqrt[a + b*x + c*x^2])/(105*e^3*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(7/2)) - (2*(C*d^2 - e*(B*d - A*e))
*(a + b*x + c*x^2)^(3/2))/(9*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(9/2)) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c^4*d^
4*(8*C*d^2 + e*(4*B*d + 5*A*e)) + 2*b^2*e^4*(21*a^2*C*e^2 - 6*a*b*e*(3*C*d + 2*B*e) + b^2*(5*C*d^2 + 4*B*d*e +
8*A*e^2)) - 6*c^2*e^2*(a*b*d*e*(30*C*d^2 - 5*B*d*e - 34*A*e^2) - a^2*e^2*(30*C*d^2 - 36*B*d*e + 7*A*e^2) - b^
2*d^2*(11*C*d^2 + 3*B*d*e + 11*A*e^2)) - c*e^3*(126*a^3*C*e^3 - 3*a^2*b*e^2*(12*C*d + 29*B*e) - 6*a*b^2*e*(5*C
*d^2 + 7*B*d*e - 12*A*e^2) + b^3*d*(20*C*d^2 + 25*B*d*e + 56*A*e^2)) + c^3*d^2*e*(6*a*e*(11*C*d^2 + 8*B*d*e -
34*A*e^2) - b*d*(56*C*d^2 + 5*e*(5*B*d + 4*A*e))))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*
EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2
*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(315*e^4*(c*d^2 - b*d*e + a*e^2)^4*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b
^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c^3*d^3*(8*C*d^2 + e*(4*B*d + 5*A*e)
) + 3*c^2*d*e*(2*a*e*(9*C*d^2 + 7*B*d*e - 9*A*e^2) - b*d*(16*C*d^2 + 7*B*d*e + 5*A*e^2)) + 3*c*e^2*(2*a^2*e^2*
(17*C*d - 5*B*e) - a*b*e*(41*C*d^2 + 5*B*d*e - 9*A*e^2) + b^2*d*(15*C*d^2 + 3*B*d*e + 7*A*e^2)) - b*e^3*(21*a^
2*C*e^2 - 6*a*b*e*(3*C*d + 2*B*e) + b^2*(5*C*d^2 + 4*B*d*e + 8*A*e^2)))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[
b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] +
2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(315*e^4*(c
*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 6.24313, antiderivative size = 1904, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.206, Rules used = {1650, 810, 834, 843, 718, 424, 419} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(11/2),x]

[Out]

(2*(2*c^3*(8*C*d^5 + d^3*e*(4*B*d + 5*A*e)) + 3*c^2*d*e*(2*a*e*(9*C*d^2 + 7*B*d*e - 9*A*e^2) - b*d*(16*C*d^2 +
7*B*d*e + 5*A*e^2)) + 3*c*e^2*(2*a^2*e^2*(17*C*d - 5*B*e) - a*b*e*(41*C*d^2 + 5*B*d*e - 9*A*e^2) + b^2*d*(15*
C*d^2 + 3*B*d*e + 7*A*e^2)) - b*e^3*(21*a^2*C*e^2 - 6*a*b*e*(3*C*d + 2*B*e) + b^2*(5*C*d^2 + 4*B*d*e + 8*A*e^2
)))*Sqrt[a + b*x + c*x^2])/(315*e^3*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(3/2)) + (2*(2*c^4*(8*C*d^6 + d^4*e*(4
*B*d + 5*A*e)) - c^3*d^2*e*(56*b*C*d^3 + 5*b*d*e*(5*B*d + 4*A*e) - 6*a*e*(11*C*d^2 + 8*B*d*e - 34*A*e^2)) + 2*
b^2*e^4*(21*a^2*C*e^2 - 6*a*b*e*(3*C*d + 2*B*e) + b^2*(5*C*d^2 + 4*B*d*e + 8*A*e^2)) - 6*c^2*e^2*(a*b*d*e*(30*
C*d^2 - 5*B*d*e - 34*A*e^2) - a^2*e^2*(30*C*d^2 - 36*B*d*e + 7*A*e^2) - b^2*d^2*(11*C*d^2 + 3*B*d*e + 11*A*e^2
)) - c*e^3*(126*a^3*C*e^3 - 3*a^2*b*e^2*(12*C*d + 29*B*e) - 6*a*b^2*e*(5*C*d^2 + 7*B*d*e - 12*A*e^2) + b^3*d*(
20*C*d^2 + 25*B*d*e + 56*A*e^2)))*Sqrt[a + b*x + c*x^2])/(315*e^3*(c*d^2 - b*d*e + a*e^2)^4*Sqrt[d + e*x]) - (
2*(c^2*(8*C*d^5 + d^3*e*(4*B*d + 5*A*e)) - e^2*(3*a^2*e^2*(3*C*d - 5*B*e) - a*b*e*(2*C*d^2 - 17*B*d*e - 10*A*e
^2) - b^2*d*(5*C*d^2 + 4*B*d*e + 8*A*e^2)) - c*d*e*(3*b*d*(5*C*d^2 + 2*B*d*e + 5*A*e^2) - a*e*(7*C*d^2 + 11*B*
d*e + 13*A*e^2)) + e^2*((3*c^2*(6*C*d^4 + d^2*e*(3*B*d - 5*A*e)))/e + c*(a*e*(47*C*d^2 + e*(B*d - 7*A*e)) - 3*
b*(15*C*d^3 + d*e*(2*B*d - 5*A*e))) + e*(21*a^2*C*e^2 - 3*a*b*e*(16*C*d - B*e) + b^2*(25*C*d^2 - e*(B*d + 2*A*
e))))*x)*Sqrt[a + b*x + c*x^2])/(105*e^3*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(7/2)) - (2*(C*d^2 - e*(B*d - A*e
))*(a + b*x + c*x^2)^(3/2))/(9*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(9/2)) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c^4*
(8*C*d^6 + d^4*e*(4*B*d + 5*A*e)) - c^3*d^2*e*(56*b*C*d^3 + 5*b*d*e*(5*B*d + 4*A*e) - 6*a*e*(11*C*d^2 + 8*B*d*
e - 34*A*e^2)) + 2*b^2*e^4*(21*a^2*C*e^2 - 6*a*b*e*(3*C*d + 2*B*e) + b^2*(5*C*d^2 + 4*B*d*e + 8*A*e^2)) - 6*c^
2*e^2*(a*b*d*e*(30*C*d^2 - 5*B*d*e - 34*A*e^2) - a^2*e^2*(30*C*d^2 - 36*B*d*e + 7*A*e^2) - b^2*d^2*(11*C*d^2 +
3*B*d*e + 11*A*e^2)) - c*e^3*(126*a^3*C*e^3 - 3*a^2*b*e^2*(12*C*d + 29*B*e) - 6*a*b^2*e*(5*C*d^2 + 7*B*d*e -
12*A*e^2) + b^3*d*(20*C*d^2 + 25*B*d*e + 56*A*e^2)))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))
]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/
(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(315*e^4*(c*d^2 - b*d*e + a*e^2)^4*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt
[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c^3*(8*C*d^5 + d^3*e*(4*B*d + 5*A*
e)) + 3*c^2*d*e*(2*a*e*(9*C*d^2 + 7*B*d*e - 9*A*e^2) - b*d*(16*C*d^2 + 7*B*d*e + 5*A*e^2)) + 3*c*e^2*(2*a^2*e^
2*(17*C*d - 5*B*e) - a*b*e*(41*C*d^2 + 5*B*d*e - 9*A*e^2) + b^2*d*(15*C*d^2 + 3*B*d*e + 7*A*e^2)) - b*e^3*(21*
a^2*C*e^2 - 6*a*b*e*(3*C*d + 2*B*e) + b^2*(5*C*d^2 + 4*B*d*e + 8*A*e^2)))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqr
t[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c]
+ 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(315*e^4*
(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

Rule 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*
x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^
(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

Rule 810

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*
c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 718

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*Rt[b^2 - 4*a*c, 2]
*(d + e*x)^m*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))])/(c*Sqrt[a + b*x + c*x^2]*((2*c*(d + e*x))/(2*c*d -
b*e - e*Rt[b^2 - 4*a*c, 2]))^m), Subst[Int[(1 + (2*e*Rt[b^2 - 4*a*c, 2]*x^2)/(2*c*d - b*e - e*Rt[b^2 - 4*a*c,
2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{11/2}} \, dx &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{9 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}-\frac{2 \int \frac{\left (-\frac{3 \left (b C d^2-b e (B d+2 A e)+3 e (A c d-a C d+a B e)\right )}{2 e}-\frac{3}{2} \left (B c d-3 b C d+\frac{2 c C d^2}{e}-A c e+3 a C e\right ) x\right ) \sqrt{a+b x+c x^2}}{(d+e x)^{9/2}} \, dx}{9 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{2 \left (c^2 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )-e^2 \left (3 a^2 e^2 (3 C d-5 B e)-a b e \left (2 C d^2-17 B d e-10 A e^2\right )-b^2 d \left (5 C d^2+4 B d e+8 A e^2\right )\right )-c d e \left (3 b d \left (5 C d^2+2 B d e+5 A e^2\right )-a e \left (7 C d^2+11 B d e+13 A e^2\right )\right )+e^2 \left (\frac{3 c^2 \left (6 C d^4+d^2 e (3 B d-5 A e)\right )}{e}+c \left (a e \left (47 C d^2+e (B d-7 A e)\right )-3 b \left (15 C d^3+d e (2 B d-5 A e)\right )\right )+e \left (21 a^2 C e^2-3 a b e (16 C d-B e)+b^2 \left (25 C d^2-e (B d+2 A e)\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{9 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}+\frac{4 \int \frac{\frac{3 \left (b^3 e^2 \left (5 C d^2+4 e (B d+2 A e)\right )-10 a c e \left (3 a e^2 (2 C d-B e)+c d \left (2 C d^2+B d e-4 A e^2\right )\right )-3 b^2 \left (2 a e^3 (3 C d+2 B e)+c d e \left (5 C d^2+2 B d e+5 A e^2\right )\right )+b \left (21 a^2 C e^4+3 a c e^2 \left (19 C d^2+2 B d e-9 A e^2\right )+c^2 d^2 \left (8 C d^2+4 B d e+5 A e^2\right )\right )\right )}{4 e}+\frac{3 c \left (2 c^2 \left (8 C d^4+d^2 e (4 B d+5 A e)\right )+3 e^2 \left (14 a^2 C e^2-a b e (22 C d+3 B e)+b^2 \left (10 C d^2+B d e+2 A e^2\right )\right )+c e \left (2 a e \left (17 C d^2+16 B d e-7 A e^2\right )-b d \left (40 C d^2+17 B d e+10 A e^2\right )\right )\right ) x}{4 e}}{(d+e x)^{5/2} \sqrt{a+b x+c x^2}} \, dx}{315 e^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{2 \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}-\frac{2 \left (c^2 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )-e^2 \left (3 a^2 e^2 (3 C d-5 B e)-a b e \left (2 C d^2-17 B d e-10 A e^2\right )-b^2 d \left (5 C d^2+4 B d e+8 A e^2\right )\right )-c d e \left (3 b d \left (5 C d^2+2 B d e+5 A e^2\right )-a e \left (7 C d^2+11 B d e+13 A e^2\right )\right )+e^2 \left (\frac{3 c^2 \left (6 C d^4+d^2 e (3 B d-5 A e)\right )}{e}+c \left (a e \left (47 C d^2+e (B d-7 A e)\right )-3 b \left (15 C d^3+d e (2 B d-5 A e)\right )\right )+e \left (21 a^2 C e^2-3 a b e (16 C d-B e)+b^2 \left (25 C d^2-e (B d+2 A e)\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{9 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}-\frac{8 \int \frac{\frac{3 \left (2 b^4 e^3 \left (5 C d^2+4 e (B d+2 A e)\right )+b c \left (3 a^2 e^4 (19 C d+29 B e)-3 a c d e^2 \left (19 C d^2-15 B d e-59 A e^2\right )-c^2 d^3 \left (8 C d^2+4 B d e+5 A e^2\right )\right )-6 a c e \left (21 a^2 C e^4-c^2 d^2 \left (2 C d^2+B d e-25 A e^2\right )-a c e^2 \left (13 C d^2-31 B d e+7 A e^2\right )\right )-3 b^3 \left (4 a e^4 (3 C d+2 B e)+c d e^2 \left (5 C d^2+7 B d e+16 A e^2\right )\right )+3 b^2 \left (14 a^2 C e^5+2 a c e^3 \left (2 C d^2+e (5 B d-12 A e)\right )+c^2 d^2 e \left (7 C d^2+3 e (B d+5 A e)\right )\right )\right )}{8 e}-\frac{3 c \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right ) x}{8 e}}{(d+e x)^{3/2} \sqrt{a+b x+c x^2}} \, dx}{945 e^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{2 \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}+\frac{2 \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^4 \sqrt{d+e x}}-\frac{2 \left (c^2 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )-e^2 \left (3 a^2 e^2 (3 C d-5 B e)-a b e \left (2 C d^2-17 B d e-10 A e^2\right )-b^2 d \left (5 C d^2+4 B d e+8 A e^2\right )\right )-c d e \left (3 b d \left (5 C d^2+2 B d e+5 A e^2\right )-a e \left (7 C d^2+11 B d e+13 A e^2\right )\right )+e^2 \left (\frac{3 c^2 \left (6 C d^4+d^2 e (3 B d-5 A e)\right )}{e}+c \left (a e \left (47 C d^2+e (B d-7 A e)\right )-3 b \left (15 C d^3+d e (2 B d-5 A e)\right )\right )+e \left (21 a^2 C e^2-3 a b e (16 C d-B e)+b^2 \left (25 C d^2-e (B d+2 A e)\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{9 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}+\frac{16 \int \frac{-\frac{3 c \left (b^4 d e^3 \left (5 C d^2+4 e (B d+2 A e)\right )+3 b^2 \left (a^2 e^5 (C d-4 B e)-c^2 d^3 e \left (9 C d^2+4 B d e-10 A e^2\right )-2 a c d e^3 \left (26 C d^2-B d e+11 A e^2\right )\right )+b \left (21 a^3 C e^6+3 a^2 c e^4 \left (94 C d^2+24 B d e-9 A e^2\right )+c^3 d^4 \left (8 C d^2+4 B d e+5 A e^2\right )+3 a c^2 d^2 e^2 \left (15 C d^2+36 B d e+46 A e^2\right )\right )-2 a c e \left (3 a^2 e^4 (38 C d-5 B e)-6 a c d e^2 \left (2 C d^2-19 B d e+8 A e^2\right )+c^2 d^3 \left (2 C d^2+B d e+80 A e^2\right )\right )+b^3 e^2 \left (30 c C d^4-3 c d^2 e (4 B d+9 A e)-a e^2 \left (13 C d^2+8 e (B d-A e)\right )\right )\right )}{16 e}-\frac{3 c \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\right )\right )\right ) x}{16 e}}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{945 e^2 \left (c d^2-b d e+a e^2\right )^4}\\ &=\frac{2 \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}+\frac{2 \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^4 \sqrt{d+e x}}-\frac{2 \left (c^2 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )-e^2 \left (3 a^2 e^2 (3 C d-5 B e)-a b e \left (2 C d^2-17 B d e-10 A e^2\right )-b^2 d \left (5 C d^2+4 B d e+8 A e^2\right )\right )-c d e \left (3 b d \left (5 C d^2+2 B d e+5 A e^2\right )-a e \left (7 C d^2+11 B d e+13 A e^2\right )\right )+e^2 \left (\frac{3 c^2 \left (6 C d^4+d^2 e (3 B d-5 A e)\right )}{e}+c \left (a e \left (47 C d^2+e (B d-7 A e)\right )-3 b \left (15 C d^3+d e (2 B d-5 A e)\right )\right )+e \left (21 a^2 C e^2-3 a b e (16 C d-B e)+b^2 \left (25 C d^2-e (B d+2 A e)\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{9 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}+\frac{\left (c \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{315 e^4 \left (c d^2-b d e+a e^2\right )^3}-\frac{\left (c \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\right )\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{315 e^4 \left (c d^2-b d e+a e^2\right )^4}\\ &=\frac{2 \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}+\frac{2 \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^4 \sqrt{d+e x}}-\frac{2 \left (c^2 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )-e^2 \left (3 a^2 e^2 (3 C d-5 B e)-a b e \left (2 C d^2-17 B d e-10 A e^2\right )-b^2 d \left (5 C d^2+4 B d e+8 A e^2\right )\right )-c d e \left (3 b d \left (5 C d^2+2 B d e+5 A e^2\right )-a e \left (7 C d^2+11 B d e+13 A e^2\right )\right )+e^2 \left (\frac{3 c^2 \left (6 C d^4+d^2 e (3 B d-5 A e)\right )}{e}+c \left (a e \left (47 C d^2+e (B d-7 A e)\right )-3 b \left (15 C d^3+d e (2 B d-5 A e)\right )\right )+e \left (21 a^2 C e^2-3 a b e (16 C d-B e)+b^2 \left (25 C d^2-e (B d+2 A e)\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{9 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\right )\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{315 e^4 \left (c d^2-b d e+a e^2\right )^4 \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{a+b x+c x^2}}+\frac{\left (2 \sqrt{2} \sqrt{b^2-4 a c} \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{315 e^4 \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}+\frac{2 \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{315 e^3 \left (c d^2-b d e+a e^2\right )^4 \sqrt{d+e x}}-\frac{2 \left (c^2 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )-e^2 \left (3 a^2 e^2 (3 C d-5 B e)-a b e \left (2 C d^2-17 B d e-10 A e^2\right )-b^2 d \left (5 C d^2+4 B d e+8 A e^2\right )\right )-c d e \left (3 b d \left (5 C d^2+2 B d e+5 A e^2\right )-a e \left (7 C d^2+11 B d e+13 A e^2\right )\right )+e^2 \left (\frac{3 c^2 \left (6 C d^4+d^2 e (3 B d-5 A e)\right )}{e}+c \left (a e \left (47 C d^2+e (B d-7 A e)\right )-3 b \left (15 C d^3+d e (2 B d-5 A e)\right )\right )+e \left (21 a^2 C e^2-3 a b e (16 C d-B e)+b^2 \left (25 C d^2-e (B d+2 A e)\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{9 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (2 c^4 \left (8 C d^6+d^4 e (4 B d+5 A e)\right )-c^3 d^2 e \left (56 b C d^3+5 b d e (5 B d+4 A e)-6 a e \left (11 C d^2+8 B d e-34 A e^2\right )\right )+2 b^2 e^4 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )-6 c^2 e^2 \left (a b d e \left (30 C d^2-5 B d e-34 A e^2\right )-a^2 e^2 \left (30 C d^2-36 B d e+7 A e^2\right )-b^2 d^2 \left (11 C d^2+3 B d e+11 A e^2\right )\right )-c e^3 \left (126 a^3 C e^3-3 a^2 b e^2 (12 C d+29 B e)-6 a b^2 e \left (5 C d^2+7 B d e-12 A e^2\right )+b^3 d \left (20 C d^2+25 B d e+56 A e^2\right )\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 e^4 \left (c d^2-b d e+a e^2\right )^4 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (2 c^3 \left (8 C d^5+d^3 e (4 B d+5 A e)\right )+3 c^2 d e \left (2 a e \left (9 C d^2+7 B d e-9 A e^2\right )-b d \left (16 C d^2+7 B d e+5 A e^2\right )\right )+3 c e^2 \left (2 a^2 e^2 (17 C d-5 B e)-a b e \left (41 C d^2+5 B d e-9 A e^2\right )+b^2 d \left (15 C d^2+3 B d e+7 A e^2\right )\right )-b e^3 \left (21 a^2 C e^2-6 a b e (3 C d+2 B e)+b^2 \left (5 C d^2+4 B d e+8 A e^2\right )\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 e^4 \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 19.2101, size = 29140, normalized size = 15.3 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(11/2),x]

[Out]

Result too large to show

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Maple [B]  time = 1.017, size = 153623, normalized size = 80.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(11/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C x^{2} + B x + A\right )} \sqrt{c x^{2} + b x + a}}{{\left (e x + d\right )}^{\frac{11}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(11/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)/(e*x + d)^(11/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C x^{2} + B x + A\right )} \sqrt{c x^{2} + b x + a} \sqrt{e x + d}}{e^{6} x^{6} + 6 \, d e^{5} x^{5} + 15 \, d^{2} e^{4} x^{4} + 20 \, d^{3} e^{3} x^{3} + 15 \, d^{4} e^{2} x^{2} + 6 \, d^{5} e x + d^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

integral((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)/(e^6*x^6 + 6*d*e^5*x^5 + 15*d^2*e^4*x^4 + 20*d^
3*e^3*x^3 + 15*d^4*e^2*x^2 + 6*d^5*e*x + d^6), x)

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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((C*x**2+B*x+A)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**(11/2),x)

[Out]

Timed out

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Giac [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((C*x^2+B*x+A)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

Timed out