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

Optimal. Leaf size=906 \[ \frac{2 C (d+e x)^{3/2} \left (c x^2+b x+a\right )^{3/2}}{9 c e}-\frac{2 (2 c C d-3 B c e+2 b C e) \sqrt{d+e x} \left (c x^2+b x+a\right )^{3/2}}{21 c^2 e}+\frac{2 \sqrt{d+e x} \left (d \left (8 C d^2-3 e (4 B d-7 A e)\right ) c^3+3 e \left (a e (C d-5 B e)-b \left (C d^2-2 B e d-7 A e^2\right )\right ) c^2-3 b e^2 (b C d+4 b B e-a C e) c+3 e \left (-\left (2 C d^2-3 e (B d+7 A e)\right ) c^2-e (b C d+12 b B e+7 a C e) c+8 b^2 C e^2\right ) x c+8 b^3 C e^3\right ) \sqrt{c x^2+b x+a}}{315 c^3 e^3}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (2 \left (4 c^2 d^2-b^2 e^2-\frac{3}{2} c e (b d-2 a e)\right ) \left (-\left (2 C d^2-3 e (B d+7 A e)\right ) c^2-e (b C d+12 b B e+7 a C e) c+8 b^2 C e^2\right )-5 c e (2 c d-b e) \left (6 C d e b^2+\left (2 a C e^2-c d (C d+9 B e)\right ) b+c e (21 A c d-5 a C d-3 a B e)\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 c^4 e^4 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (-2 d \left (8 C d^2-3 e (4 B d-7 A e)\right ) c^3-3 e^2 (b B d+2 a C d-7 A b e-10 a B e) c^2+3 b e^2 (b C d-4 b B e-9 a C e) c+8 b^3 C e^3\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 c^4 e^4 \sqrt{d+e x} \sqrt{c x^2+b x+a}} \]

[Out]

(2*Sqrt[d + e*x]*(8*b^3*C*e^3 - 3*b*c*e^2*(b*C*d + 4*b*B*e - a*C*e) + c^3*d*(8*C*d^2 - 3*e*(4*B*d - 7*A*e)) +
3*c^2*e*(a*e*(C*d - 5*B*e) - b*(C*d^2 - 2*B*d*e - 7*A*e^2)) + 3*c*e*(8*b^2*C*e^2 - c*e*(b*C*d + 12*b*B*e + 7*a
*C*e) - c^2*(2*C*d^2 - 3*e*(B*d + 7*A*e)))*x)*Sqrt[a + b*x + c*x^2])/(315*c^3*e^3) - (2*(2*c*C*d - 3*B*c*e + 2
*b*C*e)*Sqrt[d + e*x]*(a + b*x + c*x^2)^(3/2))/(21*c^2*e) + (2*C*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(9*c
*e) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*(4*c^2*d^2 - b^2*e^2 - (3*c*e*(b*d - 2*a*e))/2)*(8*b^2*C*e^2 - c*e*(b*C*d
+ 12*b*B*e + 7*a*C*e) - c^2*(2*C*d^2 - 3*e*(B*d + 7*A*e))) - 5*c*e*(2*c*d - b*e)*(6*b^2*C*d*e + c*e*(21*A*c*d
- 5*a*C*d - 3*a*B*e) + b*(2*a*C*e^2 - c*d*(C*d + 9*B*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*c^4*e^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]*(c*d^2 - b*d*e + a*e^2)*(8*b^3*C*e^3 - 3*c^2*e^2*(b*B
*d + 2*a*C*d - 7*A*b*e - 10*a*B*e) + 3*b*c*e^2*(b*C*d - 4*b*B*e - 9*a*C*e) - 2*c^3*d*(8*C*d^2 - 3*e*(4*B*d - 7
*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))]*E
llipticF[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*c^4*e^4*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 2.70038, antiderivative size = 905, 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 = {1653, 832, 814, 843, 718, 424, 419} \[ \frac{2 C (d+e x)^{3/2} \left (c x^2+b x+a\right )^{3/2}}{9 c e}-\frac{2 (2 c C d-3 B c e+2 b C e) \sqrt{d+e x} \left (c x^2+b x+a\right )^{3/2}}{21 c^2 e}+\frac{2 \sqrt{d+e x} \left (\left (8 C d^3-3 d e (4 B d-7 A e)\right ) c^3-3 e \left (b C d^2-b e (2 B d+7 A e)-a e (C d-5 B e)\right ) c^2-3 b e^2 (b C d+4 b B e-a C e) c+3 e \left (-\left (2 C d^2-3 e (B d+7 A e)\right ) c^2-e (b C d+12 b B e+7 a C e) c+8 b^2 C e^2\right ) x c+8 b^3 C e^3\right ) \sqrt{c x^2+b x+a}}{315 c^3 e^3}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (5 c e (2 c d-b e) \left (6 C d e b^2+2 a C e^2 b-c d (C d+9 B e) b+c e (21 A c d-5 a C d-3 a B e)\right )-2 \left (4 c^2 d^2-b^2 e^2-\frac{3}{2} c e (b d-2 a e)\right ) \left (-\left (2 C d^2-3 e (B d+7 A e)\right ) c^2-e (b C d+12 b B e+7 a C e) c+8 b^2 C e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 c^4 e^4 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (-2 \left (8 C d^3-3 d e (4 B d-7 A e)\right ) c^3-3 e^2 (b B d+2 a C d-7 A b e-10 a B e) c^2+3 b e^2 (b C d-4 b B e-9 a C e) c+8 b^3 C e^3\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 c^4 e^4 \sqrt{d+e x} \sqrt{c x^2+b x+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(8*b^3*C*e^3 - 3*b*c*e^2*(b*C*d + 4*b*B*e - a*C*e) + c^3*(8*C*d^3 - 3*d*e*(4*B*d - 7*A*e)) -
3*c^2*e*(b*C*d^2 - b*e*(2*B*d + 7*A*e) - a*e*(C*d - 5*B*e)) + 3*c*e*(8*b^2*C*e^2 - c*e*(b*C*d + 12*b*B*e + 7*a
*C*e) - c^2*(2*C*d^2 - 3*e*(B*d + 7*A*e)))*x)*Sqrt[a + b*x + c*x^2])/(315*c^3*e^3) - (2*(2*c*C*d - 3*B*c*e + 2
*b*C*e)*Sqrt[d + e*x]*(a + b*x + c*x^2)^(3/2))/(21*c^2*e) + (2*C*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(9*c
*e) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(5*c*e*(2*c*d - b*e)*(6*b^2*C*d*e + 2*a*b*C*e^2 - b*c*d*(C*d + 9*B*e) + c*e*(
21*A*c*d - 5*a*C*d - 3*a*B*e)) - 2*(4*c^2*d^2 - b^2*e^2 - (3*c*e*(b*d - 2*a*e))/2)*(8*b^2*C*e^2 - c*e*(b*C*d +
12*b*B*e + 7*a*C*e) - c^2*(2*C*d^2 - 3*e*(B*d + 7*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*c^4*e^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]*(c*d^2 - b*d*e + a*e^2)*(8*b^3*C*e^3 - 3*c^2*e^2*(b*B
*d + 2*a*C*d - 7*A*b*e - 10*a*B*e) + 3*b*c*e^2*(b*C*d - 4*b*B*e - 9*a*C*e) - 2*c^3*(8*C*d^3 - 3*d*e*(4*B*d - 7
*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))]*E
llipticF[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*c^4*e^4*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

Rule 1653

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 832

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

Rule 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (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 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (A+B x+C x^2\right ) \, dx &=\frac{2 C (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c e}+\frac{2 \int \sqrt{d+e x} \left (-\frac{3}{2} e (b C d-3 A c e+a C e)-\frac{3}{2} e (2 c C d-3 B c e+2 b C e) x\right ) \sqrt{a+b x+c x^2} \, dx}{9 c e^2}\\ &=-\frac{2 (2 c C d-3 B c e+2 b C e) \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2 e}+\frac{2 C (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c e}+\frac{4 \int \frac{\left (\frac{3}{4} e \left (6 b^2 C d e+2 a b C e^2-b c d (C d+9 B e)+c e (21 A c d-5 a C d-3 a B e)\right )+\frac{3}{4} e \left (8 b^2 C e^2-c e (b C d+12 b B e+7 a C e)-c^2 \left (2 C d^2-3 e (B d+7 A e)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{\sqrt{d+e x}} \, dx}{63 c^2 e^2}\\ &=\frac{2 \sqrt{d+e x} \left (8 b^3 C e^3-3 b c e^2 (b C d+4 b B e-a C e)+c^3 \left (8 C d^3-3 d e (4 B d-7 A e)\right )-3 c^2 e \left (b C d^2-b e (2 B d+7 A e)-a e (C d-5 B e)\right )+3 c e \left (8 b^2 C e^2-c e (b C d+12 b B e+7 a C e)-c^2 \left (2 C d^2-3 e (B d+7 A e)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{315 c^3 e^3}-\frac{2 (2 c C d-3 B c e+2 b C e) \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2 e}+\frac{2 C (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c e}-\frac{8 \int \frac{\frac{3}{8} e \left (5 c e (b d-2 a e) \left (6 b^2 C d e+2 a b C e^2-b c d (C d+9 B e)+c e (21 A c d-5 a C d-3 a B e)\right )-2 \left (\frac{1}{2} b d (4 c d-b e)-a e \left (c d+\frac{b e}{2}\right )\right ) \left (8 b^2 C e^2-c e (b C d+12 b B e+7 a C e)-c^2 \left (2 C d^2-3 e (B d+7 A e)\right )\right )\right )+\frac{3}{8} e \left (5 c e (2 c d-b e) \left (6 b^2 C d e+2 a b C e^2-b c d (C d+9 B e)+c e (21 A c d-5 a C d-3 a B e)\right )-2 \left (4 c^2 d^2-b^2 e^2-\frac{3}{2} c e (b d-2 a e)\right ) \left (8 b^2 C e^2-c e (b C d+12 b B e+7 a C e)-c^2 \left (2 C d^2-3 e (B d+7 A e)\right )\right )\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{945 c^3 e^4}\\ &=\frac{2 \sqrt{d+e x} \left (8 b^3 C e^3-3 b c e^2 (b C d+4 b B e-a C e)+c^3 \left (8 C d^3-3 d e (4 B d-7 A e)\right )-3 c^2 e \left (b C d^2-b e (2 B d+7 A e)-a e (C d-5 B e)\right )+3 c e \left (8 b^2 C e^2-c e (b C d+12 b B e+7 a C e)-c^2 \left (2 C d^2-3 e (B d+7 A e)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{315 c^3 e^3}-\frac{2 (2 c C d-3 B c e+2 b C e) \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2 e}+\frac{2 C (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c e}-\frac{\left (\left (c d^2-b d e+a e^2\right ) \left (8 b^3 C e^3-3 c^2 e^2 (b B d+2 a C d-7 A b e-10 a B e)+3 b c e^2 (b C d-4 b B e-9 a C e)-2 c^3 \left (8 C d^3-3 d e (4 B d-7 A e)\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{315 c^3 e^4}-\frac{\left (5 c e (2 c d-b e) \left (6 b^2 C d e+2 a b C e^2-b c d (C d+9 B e)+c e (21 A c d-5 a C d-3 a B e)\right )-2 \left (4 c^2 d^2-b^2 e^2-\frac{3}{2} c e (b d-2 a e)\right ) \left (8 b^2 C e^2-c e (b C d+12 b B e+7 a C e)-c^2 \left (2 C d^2-3 e (B d+7 A e)\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{315 c^3 e^4}\\ &=\frac{2 \sqrt{d+e x} \left (8 b^3 C e^3-3 b c e^2 (b C d+4 b B e-a C e)+c^3 \left (8 C d^3-3 d e (4 B d-7 A e)\right )-3 c^2 e \left (b C d^2-b e (2 B d+7 A e)-a e (C d-5 B e)\right )+3 c e \left (8 b^2 C e^2-c e (b C d+12 b B e+7 a C e)-c^2 \left (2 C d^2-3 e (B d+7 A e)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{315 c^3 e^3}-\frac{2 (2 c C d-3 B c e+2 b C e) \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2 e}+\frac{2 C (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c e}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (5 c e (2 c d-b e) \left (6 b^2 C d e+2 a b C e^2-b c d (C d+9 B e)+c e (21 A c d-5 a C d-3 a B e)\right )-2 \left (4 c^2 d^2-b^2 e^2-\frac{3}{2} c e (b d-2 a e)\right ) \left (8 b^2 C e^2-c e (b C d+12 b B e+7 a C e)-c^2 \left (2 C d^2-3 e (B d+7 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 )}{315 c^4 e^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 (c d^2-b d e+a e^2\right ) \left (8 b^3 C e^3-3 c^2 e^2 (b B d+2 a C d-7 A b e-10 a B e)+3 b c e^2 (b C d-4 b B e-9 a C e)-2 c^3 \left (8 C d^3-3 d e (4 B d-7 A e)\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 c^4 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (8 b^3 C e^3-3 b c e^2 (b C d+4 b B e-a C e)+c^3 \left (8 C d^3-3 d e (4 B d-7 A e)\right )-3 c^2 e \left (b C d^2-b e (2 B d+7 A e)-a e (C d-5 B e)\right )+3 c e \left (8 b^2 C e^2-c e (b C d+12 b B e+7 a C e)-c^2 \left (2 C d^2-3 e (B d+7 A e)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{315 c^3 e^3}-\frac{2 (2 c C d-3 B c e+2 b C e) \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2 e}+\frac{2 C (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c e}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (5 c e (2 c d-b e) \left (6 b^2 C d e+2 a b C e^2-b c d (C d+9 B e)+c e (21 A c d-5 a C d-3 a B e)\right )-2 \left (4 c^2 d^2-b^2 e^2-\frac{3}{2} c e (b d-2 a e)\right ) \left (8 b^2 C e^2-c e (b C d+12 b B e+7 a C e)-c^2 \left (2 C d^2-3 e (B d+7 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 )}{315 c^4 e^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 (c d^2-b d e+a e^2\right ) \left (8 b^3 C e^3-3 c^2 e^2 (b B d+2 a C d-7 A b e-10 a B e)+3 b c e^2 (b C d-4 b B e-9 a C e)-2 c^3 \left (8 C d^3-3 d e (4 B d-7 A e)\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 c^4 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 15.2902, size = 15669, normalized size = 17.29 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.426, size = 19955, normalized size = 22. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(d + e*x)*(A + B*x + C*x**2)*sqrt(a + b*x + c*x**2), x)

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

[Out]

Timed out