### 3.263 $$\int \frac{\sqrt{a+b x+c x^2} (A+B x+C x^2)}{(d+e x)^{7/2}} \, dx$$

Optimal. Leaf size=992 $-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (c x^2+b x+a\right )^{3/2}}{5 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (c^2 \left (24 C d^2-e (4 B d+A e)\right ) d^3-c e \left (b d \left (41 C d^2-6 B e d+A e^2\right )-a e \left (37 C d^2-7 B e d+7 A e^2\right )\right ) d+e^2 \left (15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left (22 C d^2+3 B e d+2 A e^2\right )\right )+e \left (5 c^2 \left (6 C d^2-e (B d+A e)\right ) d^2+e^2 \left (\left (23 C d^2-3 B e d-2 A e^2\right ) b^2-5 a e (8 C d-B e) b+15 a^2 C e^2\right )-c e \left (5 b d \left (11 C d^2-2 B e d-A e^2\right )-a e \left (53 C d^2-13 B e d+3 A e^2\right )\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{15 e^3 \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{3/2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (2 c^2 \left (24 C d^2-e (4 B d+A e)\right ) d^2+e^2 \left (\left (38 C d^2-3 B e d-2 A e^2\right ) b^2-5 a e (14 C d-B e) b+30 a^2 C e^2\right )-c e \left (b d \left (88 C d^2-13 B e d-2 A e^2\right )-2 a e \left (43 C d^2-8 B e d+3 A e^2\right )\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 )}{15 e^4 \left (c d^2-b e d+a e^2\right )^2 \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 (2 d \left (24 C d^2-e (4 B d+A e)\right ) c^2+e \left (10 a e (5 C d-B e)-b \left (64 C d^2-9 B e d-A e^2\right )\right ) c+15 b C e^2 (b d-a e)\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 )}{15 c e^4 \left (c d^2-b e d+a e^2\right ) \sqrt{d+e x} \sqrt{c x^2+b x+a}}$

[Out]

(-2*(c^2*d^3*(24*C*d^2 - e*(4*B*d + A*e)) + e^2*(15*b^2*C*d^3 + 5*a^2*e^2*(C*d + B*e) - a*b*e*(22*C*d^2 + 3*B*
d*e + 2*A*e^2)) - c*d*e*(b*d*(41*C*d^2 - 6*B*d*e + A*e^2) - a*e*(37*C*d^2 - 7*B*d*e + 7*A*e^2)) + e*(5*c^2*d^2
*(6*C*d^2 - e*(B*d + A*e)) + e^2*(15*a^2*C*e^2 - 5*a*b*e*(8*C*d - B*e) + b^2*(23*C*d^2 - 3*B*d*e - 2*A*e^2)) -
c*e*(5*b*d*(11*C*d^2 - 2*B*d*e - A*e^2) - a*e*(53*C*d^2 - 13*B*d*e + 3*A*e^2)))*x)*Sqrt[a + b*x + c*x^2])/(15
*e^3*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2)) - (2*(C*d^2 - e*(B*d - A*e))*(a + b*x + c*x^2)^(3/2))/(5*e*(c*
d^2 - b*d*e + a*e^2)*(d + e*x)^(5/2)) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c^2*d^2*(24*C*d^2 - e*(4*B*d + A*e)) + e
^2*(30*a^2*C*e^2 - 5*a*b*e*(14*C*d - B*e) + b^2*(38*C*d^2 - 3*B*d*e - 2*A*e^2)) - c*e*(b*d*(88*C*d^2 - 13*B*d*
e - 2*A*e^2) - 2*a*e*(43*C*d^2 - 8*B*d*e + 3*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)])/(15*e^4*(c*d^2 - b*d*e + a*e^2)^2*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]*(15*b*C*e^2*(b*d - a*e) + 2*c^2*d*(24
*C*d^2 - e*(4*B*d + A*e)) + c*e*(10*a*e*(5*C*d - B*e) - b*(64*C*d^2 - 9*B*d*e - 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 + Sqr
t[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)])/(15*c*e^4*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 1.91633, antiderivative size = 989, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 34, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.176, Rules used = {1650, 810, 843, 718, 424, 419} $-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (c x^2+b x+a\right )^{3/2}}{5 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (\left (24 C d^5-d^3 e (4 B d+A e)\right ) c^2-d e \left (b d \left (41 C d^2-6 B e d+A e^2\right )-a e \left (37 C d^2-7 B e d+7 A e^2\right )\right ) c+e^2 \left (15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left (22 C d^2+3 B e d+2 A e^2\right )\right )+e^2 \left (\frac{30 c^2 C d^4}{e}-5 c^2 (B d+A e) d^2-5 b c \left (11 C d^2-e (2 B d+A e)\right ) d+15 a^2 C e^3-5 a b e^2 (8 C d-B e)+a c e \left (53 C d^2-e (13 B d-3 A e)\right )+b^2 e \left (23 C d^2-e (3 B d+2 A e)\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{15 e^3 \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{3/2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (\left (48 C d^4-2 d^2 e (4 B d+A e)\right ) c^2-e \left (b d \left (88 C d^2-13 B e d-2 A e^2\right )-2 a e \left (43 C d^2-8 B e d+3 A e^2\right )\right ) c+e^2 \left (\left (38 C d^2-3 B e d-2 A e^2\right ) b^2-5 a e (14 C d-B e) b+30 a^2 C e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^4 \left (c d^2-b e d+a e^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (\left (48 C d^3-2 d e (4 B d+A e)\right ) c^2-e \left (64 b C d^2-b e (9 B d+A e)-10 a e (5 C d-B e)\right ) c+15 b C e^2 (b d-a e)\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 c e^4 \left (c d^2-b e d+a e^2\right ) \sqrt{d+e x} \sqrt{c x^2+b x+a}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(7/2),x]

[Out]

(-2*(c^2*(24*C*d^5 - d^3*e*(4*B*d + A*e)) + e^2*(15*b^2*C*d^3 + 5*a^2*e^2*(C*d + B*e) - a*b*e*(22*C*d^2 + 3*B*
d*e + 2*A*e^2)) - c*d*e*(b*d*(41*C*d^2 - 6*B*d*e + A*e^2) - a*e*(37*C*d^2 - 7*B*d*e + 7*A*e^2)) + e^2*((30*c^2
*C*d^4)/e + 15*a^2*C*e^3 - 5*c^2*d^2*(B*d + A*e) - 5*a*b*e^2*(8*C*d - B*e) + a*c*e*(53*C*d^2 - e*(13*B*d - 3*A
*e)) - 5*b*c*d*(11*C*d^2 - e*(2*B*d + A*e)) + b^2*e*(23*C*d^2 - e*(3*B*d + 2*A*e)))*x)*Sqrt[a + b*x + c*x^2])/
(15*e^3*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2)) - (2*(C*d^2 - e*(B*d - A*e))*(a + b*x + c*x^2)^(3/2))/(5*e*
(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(5/2)) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c^2*(48*C*d^4 - 2*d^2*e*(4*B*d + A*e))
+ e^2*(30*a^2*C*e^2 - 5*a*b*e*(14*C*d - B*e) + b^2*(38*C*d^2 - 3*B*d*e - 2*A*e^2)) - c*e*(b*d*(88*C*d^2 - 13*B
*d*e - 2*A*e^2) - 2*a*e*(43*C*d^2 - 8*B*d*e + 3*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)])/(15*e^4*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[(c*(d + e*x))/(2*c*d - (b + S
qrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(15*b*C*e^2*(b*d - a*e) + c^2*(48*
C*d^3 - 2*d*e*(4*B*d + A*e)) - c*e*(64*b*C*d^2 - b*e*(9*B*d + A*e) - 10*a*e*(5*C*d - B*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)])/(15*c*e^4*(c*d^2 - b*d*e + a*e^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 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)^{7/2}} \, dx &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \int \frac{\left (-\frac{3 b C d^2-b e (3 B d+2 A e)+5 e (A c d-a C d+a B e)}{2 e}+\frac{1}{2} \left (B c d+5 b C d-\frac{6 c C d^2}{e}-A c e-5 a C e\right ) x\right ) \sqrt{a+b x+c x^2}}{(d+e x)^{5/2}} \, dx}{5 \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+A e)\right )+e^2 \left (15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left (22 C d^2+3 B d e+2 A e^2\right )\right )-c d e \left (b d \left (41 C d^2-6 B d e+A e^2\right )-a e \left (37 C d^2-7 B d e+7 A e^2\right )\right )+e^2 \left (\frac{30 c^2 C d^4}{e}+15 a^2 C e^3-5 c^2 d^2 (B d+A e)-5 a b e^2 (8 C d-B e)+a c e \left (53 C d^2-e (13 B d-3 A e)\right )-5 b c d \left (11 C d^2-e (2 B d+A e)\right )+b^2 e \left (23 C d^2-e (3 B d+2 A e)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{15 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{4 \int \frac{\frac{15 b^3 C d^2 e^2-b^2 d e \left (41 c C d^2+30 a C e^2-c e (6 B d-A e)\right )-2 a c e \left (6 c C d^3-c d e (B d+4 A e)+5 a e^2 (2 C d-B e)\right )+b \left (15 a^2 C e^4+c^2 \left (24 C d^4-d^2 e (4 B d+A e)\right )+a c e^2 \left (59 C d^2-e (14 B d+A e)\right )\right )}{4 e}+\frac{c \left (c^2 \left (48 C d^4-2 d^2 e (4 B d+A e)\right )+e^2 \left (30 a^2 C e^2-5 a b e (14 C d-B e)+b^2 \left (38 C d^2-3 B d e-2 A e^2\right )\right )-c e \left (b d \left (88 C d^2-13 B d e-2 A e^2\right )-2 a e \left (43 C d^2-8 B d e+3 A e^2\right )\right )\right ) x}{4 e}}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{15 e^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (c^2 \left (24 C d^5-d^3 e (4 B d+A e)\right )+e^2 \left (15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left (22 C d^2+3 B d e+2 A e^2\right )\right )-c d e \left (b d \left (41 C d^2-6 B d e+A e^2\right )-a e \left (37 C d^2-7 B d e+7 A e^2\right )\right )+e^2 \left (\frac{30 c^2 C d^4}{e}+15 a^2 C e^3-5 c^2 d^2 (B d+A e)-5 a b e^2 (8 C d-B e)+a c e \left (53 C d^2-e (13 B d-3 A e)\right )-5 b c d \left (11 C d^2-e (2 B d+A e)\right )+b^2 e \left (23 C d^2-e (3 B d+2 A e)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{15 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac{\left (15 b C e^2 (b d-a e)+c^2 \left (48 C d^3-2 d e (4 B d+A e)\right )-c e \left (64 b C d^2-b e (9 B d+A e)-10 a e (5 C d-B e)\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{15 e^4 \left (c d^2-b d e+a e^2\right )}+\frac{\left (c \left (c^2 \left (48 C d^4-2 d^2 e (4 B d+A e)\right )+e^2 \left (30 a^2 C e^2-5 a b e (14 C d-B e)+b^2 \left (38 C d^2-3 B d e-2 A e^2\right )\right )-c e \left (b d \left (88 C d^2-13 B d e-2 A e^2\right )-2 a e \left (43 C d^2-8 B d e+3 A e^2\right )\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{15 e^4 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (c^2 \left (24 C d^5-d^3 e (4 B d+A e)\right )+e^2 \left (15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left (22 C d^2+3 B d e+2 A e^2\right )\right )-c d e \left (b d \left (41 C d^2-6 B d e+A e^2\right )-a e \left (37 C d^2-7 B d e+7 A e^2\right )\right )+e^2 \left (\frac{30 c^2 C d^4}{e}+15 a^2 C e^3-5 c^2 d^2 (B d+A e)-5 a b e^2 (8 C d-B e)+a c e \left (53 C d^2-e (13 B d-3 A e)\right )-5 b c d \left (11 C d^2-e (2 B d+A e)\right )+b^2 e \left (23 C d^2-e (3 B d+2 A e)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{15 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (c^2 \left (48 C d^4-2 d^2 e (4 B d+A e)\right )+e^2 \left (30 a^2 C e^2-5 a b e (14 C d-B e)+b^2 \left (38 C d^2-3 B d e-2 A e^2\right )\right )-c e \left (b d \left (88 C d^2-13 B d e-2 A e^2\right )-2 a e \left (43 C d^2-8 B d e+3 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 )}{15 e^4 \left (c d^2-b d e+a e^2\right )^2 \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 (15 b C e^2 (b d-a e)+c^2 \left (48 C d^3-2 d e (4 B d+A e)\right )-c e \left (64 b C d^2-b e (9 B d+A e)-10 a e (5 C d-B 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 )}{15 c e^4 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \left (c^2 \left (24 C d^5-d^3 e (4 B d+A e)\right )+e^2 \left (15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left (22 C d^2+3 B d e+2 A e^2\right )\right )-c d e \left (b d \left (41 C d^2-6 B d e+A e^2\right )-a e \left (37 C d^2-7 B d e+7 A e^2\right )\right )+e^2 \left (\frac{30 c^2 C d^4}{e}+15 a^2 C e^3-5 c^2 d^2 (B d+A e)-5 a b e^2 (8 C d-B e)+a c e \left (53 C d^2-e (13 B d-3 A e)\right )-5 b c d \left (11 C d^2-e (2 B d+A e)\right )+b^2 e \left (23 C d^2-e (3 B d+2 A e)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{15 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (c^2 \left (48 C d^4-2 d^2 e (4 B d+A e)\right )+e^2 \left (30 a^2 C e^2-5 a b e (14 C d-B e)+b^2 \left (38 C d^2-3 B d e-2 A e^2\right )\right )-c e \left (b d \left (88 C d^2-13 B d e-2 A e^2\right )-2 a e \left (43 C d^2-8 B d e+3 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 )}{15 e^4 \left (c d^2-b d e+a e^2\right )^2 \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 (15 b C e^2 (b d-a e)+c^2 \left (48 C d^3-2 d e (4 B d+A e)\right )-c e \left (64 b C d^2-b e (9 B d+A e)-10 a e (5 C d-B 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 )}{15 c e^4 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 14.864, size = 12997, normalized size = 13.1 $\text{Result too large to show}$

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(7/2),x]

[Out]

Result too large to show

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Maple [B]  time = 0.566, size = 48427, normalized size = 48.8 \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)^(7/2),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{7}{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)^(7/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)/(e*x + d)^(7/2), 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^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, 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)^(7/2),x, algorithm="fricas")

[Out]

integral((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)/(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*
e*x + d^4), 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)**(7/2),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)^(7/2),x, algorithm="giac")

[Out]

Timed out