3.264 \(\int \frac{\sqrt{a+b x+c x^2} (A+B x+C x^2)}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=1363 \[ \text{result too large to display} \]

[Out]

(2*(2*c^3*d^3*(24*C*d^2 + e*(4*B*d + 3*A*e)) - b*e^3*(35*a^2*C*e^2 - 14*a*b*e*(3*C*d + B*e) + b^2*(15*C*d^2 +
6*B*d*e + 8*A*e^2)) + c^2*d*e*(2*a*e*(69*C*d^2 + e*(15*B*d - 29*A*e)) - b*d*(128*C*d^2 + e*(19*B*d + 9*A*e)))
+ c*e^2*(14*a^2*e^2*(11*C*d - 3*B*e) - a*b*e*(237*C*d^2 + e*(B*d - 29*A*e)) + b^2*d*(103*C*d^2 + e*(9*B*d + 19
*A*e))))*Sqrt[a + b*x + c*x^2])/(105*e^3*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x]) - (2*(c^2*d^3*(24*C*d^2 + e*
(4*B*d + 3*A*e)) - e^2*(7*a^2*e^2*(C*d - 3*B*e) - b^2*d*(15*C*d^2 + 6*B*d*e + 8*A*e^2) + a*b*e*(12*C*d^2 + 23*
B*d*e + 12*A*e^2)) - c*d*e*(b*d*(43*C*d^2 + 6*B*d*e + 15*A*e^2) - a*e*(33*C*d^2 + 9*B*d*e + 19*A*e^2)) + e*(7*
c^2*d^2*(6*C*d^2 + e*(B*d - 3*A*e)) + e^2*(35*a^2*C*e^2 - 7*a*b*e*(12*C*d - B*e) + b^2*(45*C*d^2 - 3*B*d*e - 4
*A*e^2)) + c*e*(a*e*(93*C*d^2 - 9*B*d*e - 5*A*e^2) - b*(91*C*d^3 - 21*A*d*e^2)))*x)*Sqrt[a + b*x + c*x^2])/(10
5*e^3*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(5/2)) - (2*(C*d^2 - e*(B*d - A*e))*(a + b*x + c*x^2)^(3/2))/(7*e*(c
*d^2 - b*d*e + a*e^2)*(d + e*x)^(7/2)) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c^3*d^3*(24*C*d^2 + e*(4*B*d + 3*A*e))
- b*e^3*(35*a^2*C*e^2 - 14*a*b*e*(3*C*d + B*e) + b^2*(15*C*d^2 + 6*B*d*e + 8*A*e^2)) + c^2*d*e*(2*a*e*(69*C*d^
2 + e*(15*B*d - 29*A*e)) - b*d*(128*C*d^2 + e*(19*B*d + 9*A*e))) + c*e^2*(14*a^2*e^2*(11*C*d - 3*B*e) - a*b*e*
(237*C*d^2 + e*(B*d - 29*A*e)) + b^2*d*(103*C*d^2 + e*(9*B*d + 19*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)])/(105*e^4*(c*d^2 - b*d*e + a*e^2)^3*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^2*d^2*(24*C
*d^2 + e*(4*B*d + 3*A*e)) + c*e*(2*a*e*(51*C*d^2 + e*(12*B*d - 5*A*e)) - b*d*(104*C*d^2 + 3*e*(5*B*d + 2*A*e))
) + e^2*(70*a^2*C*e^2 - 7*a*b*e*(18*C*d + B*e) + b^2*(60*C*d^2 + e*(3*B*d + 4*A*e))))*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)
])/(105*e^4*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 4.19999, antiderivative size = 1363, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{2 \left (C d^2-e (B d-A e)\right ) \left (c x^2+b x+a\right )^{3/2}}{7 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}-\frac{2 \left (\left (24 C d^5+e (4 B d+3 A e) d^3\right ) c^2-d e \left (b d \left (43 C d^2+6 B e d+15 A e^2\right )-a e \left (33 C d^2+9 B e d+19 A e^2\right )\right ) c-e^2 \left (-d \left (15 C d^2+6 B e d+8 A e^2\right ) b^2+a e \left (12 C d^2+23 B e d+12 A e^2\right ) b+7 a^2 e^2 (C d-3 B e)\right )+e \left (7 \left (6 C d^4+e (B d-3 A e) d^2\right ) c^2+e \left (a e \left (93 C d^2-9 B e d-5 A e^2\right )-b \left (91 C d^3-21 A d e^2\right )\right ) c+e^2 \left (\left (45 C d^2-3 B e d-4 A e^2\right ) b^2-7 a e (12 C d-B e) b+35 a^2 C e^2\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{105 e^3 \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{5/2}}+\frac{2 \left (\left (48 C d^5+2 e (4 B d+3 A e) d^3\right ) c^3+d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+e (19 B d+9 A e) d\right )\right ) c^2+e^2 \left (\left (103 C d^3+e (9 B d+19 A e) d\right ) b^2-a e \left (237 C d^2+e (B d-29 A e)\right ) b+14 a^2 e^2 (11 C d-3 B e)\right ) c-b e^3 \left (\left (15 C d^2+6 B e d+8 A e^2\right ) b^2-14 a e (3 C d+B e) b+35 a^2 C e^2\right )\right ) \sqrt{c x^2+b x+a}}{105 e^3 \left (c d^2-b e d+a e^2\right )^3 \sqrt{d+e x}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (2 \left (24 C d^5+e (4 B d+3 A e) d^3\right ) c^3+d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+e (19 B d+9 A e) d\right )\right ) c^2+e^2 \left (\left (103 C d^3+e (9 B d+19 A e) d\right ) b^2-a e \left (237 C d^2+e (B d-29 A e)\right ) b+14 a^2 e^2 (11 C d-3 B e)\right ) c-b e^3 \left (\left (15 C d^2+6 B e d+8 A e^2\right ) b^2-14 a e (3 C d+B e) b+35 a^2 C e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\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 )}{105 e^4 \left (c d^2-b e d+a e^2\right )^3 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (\left (48 C d^4+2 e (4 B d+3 A e) d^2\right ) c^2+e \left (2 a e \left (51 C d^2+e (12 B d-5 A e)\right )-b \left (104 C d^3+3 e (5 B d+2 A e) d\right )\right ) c+e^2 \left (\left (60 C d^2+e (3 B d+4 A e)\right ) b^2-7 a e (18 C d+B e) b+70 a^2 C e^2\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 (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\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 )}{105 e^4 \left (c d^2-b e d+a e^2\right )^2 \sqrt{d+e x} \sqrt{c x^2+b x+a}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(9/2),x]

[Out]

(2*(c^3*(48*C*d^5 + 2*d^3*e*(4*B*d + 3*A*e)) - b*e^3*(35*a^2*C*e^2 - 14*a*b*e*(3*C*d + B*e) + b^2*(15*C*d^2 +
6*B*d*e + 8*A*e^2)) + c^2*d*e*(2*a*e*(69*C*d^2 + e*(15*B*d - 29*A*e)) - b*(128*C*d^3 + d*e*(19*B*d + 9*A*e)))
+ c*e^2*(14*a^2*e^2*(11*C*d - 3*B*e) - a*b*e*(237*C*d^2 + e*(B*d - 29*A*e)) + b^2*(103*C*d^3 + d*e*(9*B*d + 19
*A*e))))*Sqrt[a + b*x + c*x^2])/(105*e^3*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x]) - (2*(c^2*(24*C*d^5 + d^3*e*
(4*B*d + 3*A*e)) - e^2*(7*a^2*e^2*(C*d - 3*B*e) - b^2*d*(15*C*d^2 + 6*B*d*e + 8*A*e^2) + a*b*e*(12*C*d^2 + 23*
B*d*e + 12*A*e^2)) - c*d*e*(b*d*(43*C*d^2 + 6*B*d*e + 15*A*e^2) - a*e*(33*C*d^2 + 9*B*d*e + 19*A*e^2)) + e*(7*
c^2*(6*C*d^4 + d^2*e*(B*d - 3*A*e)) + e^2*(35*a^2*C*e^2 - 7*a*b*e*(12*C*d - B*e) + b^2*(45*C*d^2 - 3*B*d*e - 4
*A*e^2)) + c*e*(a*e*(93*C*d^2 - 9*B*d*e - 5*A*e^2) - b*(91*C*d^3 - 21*A*d*e^2)))*x)*Sqrt[a + b*x + c*x^2])/(10
5*e^3*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(5/2)) - (2*(C*d^2 - e*(B*d - A*e))*(a + b*x + c*x^2)^(3/2))/(7*e*(c
*d^2 - b*d*e + a*e^2)*(d + e*x)^(7/2)) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c^3*(24*C*d^5 + d^3*e*(4*B*d + 3*A*e))
- b*e^3*(35*a^2*C*e^2 - 14*a*b*e*(3*C*d + B*e) + b^2*(15*C*d^2 + 6*B*d*e + 8*A*e^2)) + c^2*d*e*(2*a*e*(69*C*d^
2 + e*(15*B*d - 29*A*e)) - b*(128*C*d^3 + d*e*(19*B*d + 9*A*e))) + c*e^2*(14*a^2*e^2*(11*C*d - 3*B*e) - a*b*e*
(237*C*d^2 + e*(B*d - 29*A*e)) + b^2*(103*C*d^3 + d*e*(9*B*d + 19*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)])/(105*e^4*(c*d^2 - b*d*e + a*e^2)^3*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]*(c^2*(48*C*d^4 +
2*d^2*e*(4*B*d + 3*A*e)) + c*e*(2*a*e*(51*C*d^2 + e*(12*B*d - 5*A*e)) - b*(104*C*d^3 + 3*d*e*(5*B*d + 2*A*e))
) + e^2*(70*a^2*C*e^2 - 7*a*b*e*(18*C*d + B*e) + b^2*(60*C*d^2 + e*(3*B*d + 4*A*e))))*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)
])/(105*e^4*(c*d^2 - b*d*e + a*e^2)^2*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)^{9/2}} \, dx &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{7 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac{2 \int \frac{\left (-\frac{3 b C d^2-b e (3 B d+4 A e)+7 e (A c d-a C d+a B e)}{2 e}-\frac{1}{2} \left (B c d-7 b C d+\frac{6 c C d^2}{e}-A c e+7 a C e\right ) x\right ) \sqrt{a+b x+c x^2}}{(d+e x)^{7/2}} \, dx}{7 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{2 \left (c^2 \left (24 C d^5+d^3 e (4 B d+3 A e)\right )-e^2 \left (7 a^2 e^2 (C d-3 B e)-b^2 d \left (15 C d^2+6 B d e+8 A e^2\right )+a b e \left (12 C d^2+23 B d e+12 A e^2\right )\right )-c d e \left (b d \left (43 C d^2+6 B d e+15 A e^2\right )-a e \left (33 C d^2+9 B d e+19 A e^2\right )\right )+e \left (7 c^2 \left (6 C d^4+d^2 e (B d-3 A e)\right )+e^2 \left (35 a^2 C e^2-7 a b e (12 C d-B e)+b^2 \left (45 C d^2-3 B d e-4 A e^2\right )\right )+c e \left (a e \left (93 C d^2-9 B d e-5 A e^2\right )-b \left (91 C d^3-21 A d e^2\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)^{5/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{7 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}+\frac{4 \int \frac{\frac{b^3 e^2 \left (15 C d^2+6 B d e+8 A e^2\right )-6 a c e \left (7 a e^2 (2 C d-B e)+c d \left (6 C d^2+B d e-8 A e^2\right )\right )+b \left (35 a^2 C e^4+a c e^2 \left (111 C d^2-6 B d e-29 A e^2\right )+c^2 d^2 \left (24 C d^2+4 B d e+3 A e^2\right )\right )-b^2 \left (14 a e^3 (3 C d+B e)+c d e \left (43 C d^2+6 B d e+15 A e^2\right )\right )}{4 e}+\frac{c \left (c^2 \left (48 C d^4+2 d^2 e (4 B d+3 A e)\right )+e^2 \left (70 a^2 C e^2-7 a b e (18 C d+B e)+b^2 \left (60 C d^2+3 B d e+4 A e^2\right )\right )+c e \left (2 a e \left (51 C d^2+12 B d e-5 A e^2\right )-b d \left (104 C d^2+15 B d e+6 A e^2\right )\right )\right ) x}{4 e}}{(d+e x)^{3/2} \sqrt{a+b x+c x^2}} \, dx}{105 e^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{2 \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+d e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 \left (103 C d^3+d e (9 B d+19 A e)\right )\right )\right ) \sqrt{a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}-\frac{2 \left (c^2 \left (24 C d^5+d^3 e (4 B d+3 A e)\right )-e^2 \left (7 a^2 e^2 (C d-3 B e)-b^2 d \left (15 C d^2+6 B d e+8 A e^2\right )+a b e \left (12 C d^2+23 B d e+12 A e^2\right )\right )-c d e \left (b d \left (43 C d^2+6 B d e+15 A e^2\right )-a e \left (33 C d^2+9 B d e+19 A e^2\right )\right )+e \left (7 c^2 \left (6 C d^4+d^2 e (B d-3 A e)\right )+e^2 \left (35 a^2 C e^2-7 a b e (12 C d-B e)+b^2 \left (45 C d^2-3 B d e-4 A e^2\right )\right )+c e \left (a e \left (93 C d^2-9 B d e-5 A e^2\right )-b \left (91 C d^3-21 A d e^2\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)^{5/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{7 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac{8 \int \frac{\frac{c \left (b^3 d e^2 \left (45 C d^2-e (3 B d+4 A e)\right )-b^2 \left (c d^2 e \left (61 C d^2+9 B d e-9 A e^2\right )+4 a e^3 \left (36 C d^2-B d e+A e^2\right )\right )+b \left (7 a^2 e^4 (23 C d+B e)+c^2 d^3 \left (24 C d^2+4 B d e+3 A e^2\right )+5 a c d e^2 \left (19 C d^2+9 B d e+5 A e^2\right )\right )-2 a e \left (35 a^2 C e^4+a c e^2 \left (9 C d^2+33 B d e-5 A e^2\right )+c^2 d^2 \left (6 C d^2+B d e+27 A e^2\right )\right )\right )}{8 e}+\frac{c \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+15 B d e-29 A e^2\right )-b d \left (128 C d^2+19 B d e+9 A e^2\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+B d e-29 A e^2\right )+b^2 d \left (103 C d^2+9 B d e+19 A e^2\right )\right )\right ) x}{8 e}}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{105 e^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{2 \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+d e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 \left (103 C d^3+d e (9 B d+19 A e)\right )\right )\right ) \sqrt{a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}-\frac{2 \left (c^2 \left (24 C d^5+d^3 e (4 B d+3 A e)\right )-e^2 \left (7 a^2 e^2 (C d-3 B e)-b^2 d \left (15 C d^2+6 B d e+8 A e^2\right )+a b e \left (12 C d^2+23 B d e+12 A e^2\right )\right )-c d e \left (b d \left (43 C d^2+6 B d e+15 A e^2\right )-a e \left (33 C d^2+9 B d e+19 A e^2\right )\right )+e \left (7 c^2 \left (6 C d^4+d^2 e (B d-3 A e)\right )+e^2 \left (35 a^2 C e^2-7 a b e (12 C d-B e)+b^2 \left (45 C d^2-3 B d e-4 A e^2\right )\right )+c e \left (a e \left (93 C d^2-9 B d e-5 A e^2\right )-b \left (91 C d^3-21 A d e^2\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)^{5/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{7 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac{\left (c \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+d e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 \left (103 C d^3+d e (9 B d+19 A e)\right )\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{105 e^4 \left (c d^2-b d e+a e^2\right )^3}-\frac{\left (8 \left (-\frac{c d \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+15 B d e-29 A e^2\right )-b d \left (128 C d^2+19 B d e+9 A e^2\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+B d e-29 A e^2\right )+b^2 d \left (103 C d^2+9 B d e+19 A e^2\right )\right )\right )}{8 e}+\frac{1}{8} c \left (b^3 d e^2 \left (45 C d^2-e (3 B d+4 A e)\right )-b^2 \left (c d^2 e \left (61 C d^2+9 B d e-9 A e^2\right )+4 a e^3 \left (36 C d^2-B d e+A e^2\right )\right )+b \left (7 a^2 e^4 (23 C d+B e)+c^2 d^3 \left (24 C d^2+4 B d e+3 A e^2\right )+5 a c d e^2 \left (19 C d^2+9 B d e+5 A e^2\right )\right )-2 a e \left (35 a^2 C e^4+a c e^2 \left (9 C d^2+33 B d e-5 A e^2\right )+c^2 d^2 \left (6 C d^2+B d e+27 A e^2\right )\right )\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{105 e^3 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{2 \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+d e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 \left (103 C d^3+d e (9 B d+19 A e)\right )\right )\right ) \sqrt{a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}-\frac{2 \left (c^2 \left (24 C d^5+d^3 e (4 B d+3 A e)\right )-e^2 \left (7 a^2 e^2 (C d-3 B e)-b^2 d \left (15 C d^2+6 B d e+8 A e^2\right )+a b e \left (12 C d^2+23 B d e+12 A e^2\right )\right )-c d e \left (b d \left (43 C d^2+6 B d e+15 A e^2\right )-a e \left (33 C d^2+9 B d e+19 A e^2\right )\right )+e \left (7 c^2 \left (6 C d^4+d^2 e (B d-3 A e)\right )+e^2 \left (35 a^2 C e^2-7 a b e (12 C d-B e)+b^2 \left (45 C d^2-3 B d e-4 A e^2\right )\right )+c e \left (a e \left (93 C d^2-9 B d e-5 A e^2\right )-b \left (91 C d^3-21 A d e^2\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)^{5/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{7 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+d e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 \left (103 C d^3+d e (9 B d+19 A e)\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 )}{105 e^4 \left (c d^2-b d e+a e^2\right )^3 \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 (16 \sqrt{2} \sqrt{b^2-4 a c} \left (-\frac{c d \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+15 B d e-29 A e^2\right )-b d \left (128 C d^2+19 B d e+9 A e^2\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+B d e-29 A e^2\right )+b^2 d \left (103 C d^2+9 B d e+19 A e^2\right )\right )\right )}{8 e}+\frac{1}{8} c \left (b^3 d e^2 \left (45 C d^2-e (3 B d+4 A e)\right )-b^2 \left (c d^2 e \left (61 C d^2+9 B d e-9 A e^2\right )+4 a e^3 \left (36 C d^2-B d e+A e^2\right )\right )+b \left (7 a^2 e^4 (23 C d+B e)+c^2 d^3 \left (24 C d^2+4 B d e+3 A e^2\right )+5 a c d e^2 \left (19 C d^2+9 B d e+5 A e^2\right )\right )-2 a e \left (35 a^2 C e^4+a c e^2 \left (9 C d^2+33 B d e-5 A e^2\right )+c^2 d^2 \left (6 C d^2+B d e+27 A e^2\right )\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 )}{105 c e^3 \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 (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+d e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 \left (103 C d^3+d e (9 B d+19 A e)\right )\right )\right ) \sqrt{a+b x+c x^2}}{105 e^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}-\frac{2 \left (c^2 \left (24 C d^5+d^3 e (4 B d+3 A e)\right )-e^2 \left (7 a^2 e^2 (C d-3 B e)-b^2 d \left (15 C d^2+6 B d e+8 A e^2\right )+a b e \left (12 C d^2+23 B d e+12 A e^2\right )\right )-c d e \left (b d \left (43 C d^2+6 B d e+15 A e^2\right )-a e \left (33 C d^2+9 B d e+19 A e^2\right )\right )+e \left (7 c^2 \left (6 C d^4+d^2 e (B d-3 A e)\right )+e^2 \left (35 a^2 C e^2-7 a b e (12 C d-B e)+b^2 \left (45 C d^2-3 B d e-4 A e^2\right )\right )+c e \left (a e \left (93 C d^2-9 B d e-5 A e^2\right )-b \left (91 C d^3-21 A d e^2\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)^{5/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{7 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (c^3 \left (48 C d^5+2 d^3 e (4 B d+3 A e)\right )-b e^3 \left (35 a^2 C e^2-14 a b e (3 C d+B e)+b^2 \left (15 C d^2+6 B d e+8 A e^2\right )\right )+c^2 d e \left (2 a e \left (69 C d^2+e (15 B d-29 A e)\right )-b \left (128 C d^3+d e (19 B d+9 A e)\right )\right )+c e^2 \left (14 a^2 e^2 (11 C d-3 B e)-a b e \left (237 C d^2+e (B d-29 A e)\right )+b^2 \left (103 C d^3+d e (9 B d+19 A e)\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 )}{105 e^4 \left (c d^2-b d e+a e^2\right )^3 \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 (c^2 \left (48 C d^4+2 d^2 e (4 B d+3 A e)\right )+e^2 \left (70 a^2 C e^2-7 a b e (18 C d+B e)+b^2 \left (60 C d^2+3 B d e+4 A e^2\right )\right )+c e \left (2 a e \left (51 C d^2+12 B d e-5 A e^2\right )-b d \left (104 C d^2+15 B d e+6 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 )}{105 e^4 \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 16.486, size = 19853, normalized size = 14.57 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(9/2),x]

[Out]

Result too large to show

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Maple [B]  time = 0.738, size = 88790, normalized size = 65.1 \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)^(9/2),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{9}{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)^(9/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)/(e*x + d)^(9/2), 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^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5}}, 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)^(9/2),x, algorithm="fricas")

[Out]

integral((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)/(e^5*x^5 + 5*d*e^4*x^4 + 10*d^2*e^3*x^3 + 10*d^
3*e^2*x^2 + 5*d^4*e*x + d^5), 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)**(9/2),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)^(9/2),x, algorithm="giac")

[Out]

Timed out