\(\int \frac {\sqrt {a+b x+c x^2} (A+B x+C x^2)}{(d+e x)^{9/2}} \, dx\) [264]

Optimal result

Integrand size = 34, antiderivative size = 1363 \[ \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 (2 c^3 d^3 \left (24 C d^2+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 d \left (128 C d^2+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 d \left (103 C d^2+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 d^3 \left (24 C d^2+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 d^2 \left (6 C 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 (2 c^3 d^3 \left (24 C d^2+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 d \left (128 C d^2+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 d \left (103 C d^2+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 (\arcsin \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 (2 c^2 d^2 \left (24 C d^2+e (4 B d+3 A e)\right )+c e \left (2 a e \left (51 C d^2+e (12 B d-5 A e)\right )-b d \left (104 C d^2+3 e (5 B d+2 A e)\right )\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+e (3 B d+4 A e)\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}} \operatorname {EllipticF}\left (\arcsin \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}} \]

[Out]

-2/7*(C*d^2-e*(-A*e+B*d))*(c*x^2+b*x+a)^(3/2)/e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(7/2)-2/105*(c^2*d^3*(24*C*d^2+e*(
3*A*e+4*B*d))-e^2*(7*a^2*e^2*(-3*B*e+C*d)-b^2*d*(8*A*e^2+6*B*d*e+15*C*d^2)+a*b*e*(12*A*e^2+23*B*d*e+12*C*d^2))
-c*d*e*(b*d*(15*A*e^2+6*B*d*e+43*C*d^2)-a*e*(19*A*e^2+9*B*d*e+33*C*d^2))+e*(7*c^2*d^2*(6*C*d^2+e*(-3*A*e+B*d))
+e^2*(35*a^2*C*e^2-7*a*b*e*(-B*e+12*C*d)+b^2*(-4*A*e^2-3*B*d*e+45*C*d^2))+c*e*(a*e*(-5*A*e^2-9*B*d*e+93*C*d^2)
-b*(-21*A*d*e^2+91*C*d^3)))*x)*(c*x^2+b*x+a)^(1/2)/e^3/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^(5/2)+2/105*(2*c^3*d^3*(2
4*C*d^2+e*(3*A*e+4*B*d))-b*e^3*(35*a^2*C*e^2-14*a*b*e*(B*e+3*C*d)+b^2*(8*A*e^2+6*B*d*e+15*C*d^2))+c^2*d*e*(2*a
*e*(69*C*d^2+e*(-29*A*e+15*B*d))-b*d*(128*C*d^2+e*(9*A*e+19*B*d)))+c*e^2*(14*a^2*e^2*(-3*B*e+11*C*d)-a*b*e*(23
7*C*d^2+e*(-29*A*e+B*d))+b^2*d*(103*C*d^2+e*(19*A*e+9*B*d))))*(c*x^2+b*x+a)^(1/2)/e^3/(a*e^2-b*d*e+c*d^2)^3/(e
*x+d)^(1/2)-1/105*(2*c^3*d^3*(24*C*d^2+e*(3*A*e+4*B*d))-b*e^3*(35*a^2*C*e^2-14*a*b*e*(B*e+3*C*d)+b^2*(8*A*e^2+
6*B*d*e+15*C*d^2))+c^2*d*e*(2*a*e*(69*C*d^2+e*(-29*A*e+15*B*d))-b*d*(128*C*d^2+e*(9*A*e+19*B*d)))+c*e^2*(14*a^
2*e^2*(-3*B*e+11*C*d)-a*b*e*(237*C*d^2+e*(-29*A*e+B*d))+b^2*d*(103*C*d^2+e*(19*A*e+9*B*d))))*EllipticE(1/2*((b
+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)
^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/e^4/(a*e^2-b*d
*e+c*d^2)^3/(c*x^2+b*x+a)^(1/2)/(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)+2/105*(2*c^2*d^2*(24*C*d^2+
e*(3*A*e+4*B*d))+c*e*(2*a*e*(51*C*d^2+e*(-5*A*e+12*B*d))-b*d*(104*C*d^2+3*e*(2*A*e+5*B*d)))+e^2*(70*a^2*C*e^2-
7*a*b*e*(B*e+18*C*d)+b^2*(60*C*d^2+e*(4*A*e+3*B*d))))*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)
^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(
1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/e^4/(a*e^2-b*d*e
+c*d^2)^2/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 2.43 (sec) , antiderivative size = 1363, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {1664, 824, 848, 857, 732, 435, 430} \[ \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 (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 (\arcsin \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}} \operatorname {EllipticF}\left (\arcsin \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}} \]

[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 430

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

Rule 435

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

Rule 732

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 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((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), 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 848

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 857

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 1664

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]

Rubi steps \begin{align*} \text {integral}& = -\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} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 36.64 (sec) , antiderivative size = 19853, normalized size of antiderivative = 14.57 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\text {Result too large to show} \]

[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

Maple [A] (verified)

Time = 4.21 (sec) , antiderivative size = 2484, normalized size of antiderivative = 1.82

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2484\)
default	\(\text {Expression too large to display}\)	\(88790\)

[In]

int((C*x^2+B*x+A)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(9/2),x,method=_RETURNVERBOSE)

[Out]

((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2/7*(A*e^2-B*d*e+C*d^2)/e^7*(c*e*x^3+b*e*x^2
+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e)^4-2/35*(A*b*e^3-2*A*c*d*e^2+7*B*a*e^3-8*B*b*d*e^2+9*B*c*d^2*e-14*C*a*d
*e^2+15*C*b*d^2*e-16*C*c*d^3)/(a*e^2-b*d*e+c*d^2)/e^6*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e)^
3-2/105*(10*A*a*c*e^4-4*A*b^2*e^4+6*A*b*c*d*e^3-6*A*c^2*d^2*e^2+7*B*a*b*e^4-24*B*a*c*d*e^3-3*B*b^2*d*e^3+15*B*
b*c*d^2*e^2-8*B*c^2*d^3*e+35*C*a^2*e^4-84*C*a*b*d*e^3+108*C*a*c*d^2*e^2+45*C*b^2*d^2*e^2-106*C*b*c*d^3*e+57*C*
c^2*d^4)/e^5/(a*e^2-b*d*e+c*d^2)^2*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e)^2+2/105*(c*e*x^2+b*
e*x+a*e)/e^4/(a*e^2-b*d*e+c*d^2)^3*(29*A*a*b*c*e^5-58*A*a*c^2*d*e^4-8*A*b^3*e^5+19*A*b^2*c*d*e^4-9*A*b*c^2*d^2
*e^3+6*A*c^3*d^3*e^2-42*B*a^2*c*e^5+14*B*a*b^2*e^5-B*a*b*c*d*e^4+30*B*a*c^2*d^2*e^3-6*B*b^3*d*e^4+9*B*b^2*c*d^
2*e^3-19*B*b*c^2*d^3*e^2+8*B*c^3*d^4*e-35*C*a^2*b*e^5+154*C*a^2*c*d*e^4+42*C*a*b^2*d*e^4-237*C*a*b*c*d^2*e^3+1
38*C*a*c^2*d^3*e^2-15*C*b^3*d^2*e^3+103*C*b^2*c*d^3*e^2-128*C*b*c^2*d^4*e+48*C*c^3*d^5)/((x+d/e)*(c*e*x^2+b*e*
x+a*e))^(1/2)+2*(c*C/e^4-1/105*c*(10*A*a*c*e^4-4*A*b^2*e^4+6*A*b*c*d*e^3-6*A*c^2*d^2*e^2+7*B*a*b*e^4-24*B*a*c*
d*e^3-3*B*b^2*d*e^3+15*B*b*c*d^2*e^2-8*B*c^2*d^3*e+35*C*a^2*e^4-84*C*a*b*d*e^3+108*C*a*c*d^2*e^2+45*C*b^2*d^2*
e^2-106*C*b*c*d^3*e+57*C*c^2*d^4)/e^4/(a*e^2-b*d*e+c*d^2)^2+1/105/e^4*(b*e-c*d)*(29*A*a*b*c*e^5-58*A*a*c^2*d*e
^4-8*A*b^3*e^5+19*A*b^2*c*d*e^4-9*A*b*c^2*d^2*e^3+6*A*c^3*d^3*e^2-42*B*a^2*c*e^5+14*B*a*b^2*e^5-B*a*b*c*d*e^4+
30*B*a*c^2*d^2*e^3-6*B*b^3*d*e^4+9*B*b^2*c*d^2*e^3-19*B*b*c^2*d^3*e^2+8*B*c^3*d^4*e-35*C*a^2*b*e^5+154*C*a^2*c
*d*e^4+42*C*a*b^2*d*e^4-237*C*a*b*c*d^2*e^3+138*C*a*c^2*d^3*e^2-15*C*b^3*d^2*e^3+103*C*b^2*c*d^3*e^2-128*C*b*c
^2*d^4*e+48*C*c^3*d^5)/(a*e^2-b*d*e+c*d^2)^3-1/105*b/e^3/(a*e^2-b*d*e+c*d^2)^3*(29*A*a*b*c*e^5-58*A*a*c^2*d*e^
4-8*A*b^3*e^5+19*A*b^2*c*d*e^4-9*A*b*c^2*d^2*e^3+6*A*c^3*d^3*e^2-42*B*a^2*c*e^5+14*B*a*b^2*e^5-B*a*b*c*d*e^4+3
0*B*a*c^2*d^2*e^3-6*B*b^3*d*e^4+9*B*b^2*c*d^2*e^3-19*B*b*c^2*d^3*e^2+8*B*c^3*d^4*e-35*C*a^2*b*e^5+154*C*a^2*c*
d*e^4+42*C*a*b^2*d*e^4-237*C*a*b*c*d^2*e^3+138*C*a*c^2*d^3*e^2-15*C*b^3*d^2*e^3+103*C*b^2*c*d^3*e^2-128*C*b*c^
2*d^4*e+48*C*c^3*d^5))*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*(
(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c
)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*EllipticF(((x+d/e
)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^
(1/2))))^(1/2))-2/105*c/e^3*(29*A*a*b*c*e^5-58*A*a*c^2*d*e^4-8*A*b^3*e^5+19*A*b^2*c*d*e^4-9*A*b*c^2*d^2*e^3+6*
A*c^3*d^3*e^2-42*B*a^2*c*e^5+14*B*a*b^2*e^5-B*a*b*c*d*e^4+30*B*a*c^2*d^2*e^3-6*B*b^3*d*e^4+9*B*b^2*c*d^2*e^3-1
9*B*b*c^2*d^3*e^2+8*B*c^3*d^4*e-35*C*a^2*b*e^5+154*C*a^2*c*d*e^4+42*C*a*b^2*d*e^4-237*C*a*b*c*d^2*e^3+138*C*a*
c^2*d^3*e^2-15*C*b^3*d^2*e^3+103*C*b^2*c*d^3*e^2-128*C*b*c^2*d^4*e+48*C*c^3*d^5)/(a*e^2-b*d*e+c*d^2)^3*(d/e-1/
2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2
)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1
/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE
(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a
*c+b^2)^(1/2))))^(1/2))+1/2/c*(-b+(-4*a*c+b^2)^(1/2))*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(
1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.78 (sec) , antiderivative size = 4543, normalized size of antiderivative = 3.33 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\text {Too large to display} \]

[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]

2/315*((48*C*c^4*d^10 - 8*(19*C*b*c^3 - B*c^4)*d^9*e + (158*C*b^2*c^2 + 6*A*c^4 + (174*C*a - 23*B*b)*c^3)*d^8*
e^2 - (47*C*b^3*c - 12*(3*B*a - A*b)*c^3 + (384*C*a*b - 17*B*b^2)*c^2)*d^7*e^3 - (15*C*b^4 - 104*A*a*c^3 - (20
8*C*a^2 - 106*B*a*b - 17*A*b^2)*c^2 - 3*(79*C*a*b^2 + 4*B*b^3)*c)*d^6*e^4 + (42*C*a*b^3 - 6*B*b^4 + 52*(3*B*a^
2 - 2*A*a*b)*c^2 - (364*C*a^2*b - B*a*b^2 - 23*A*b^3)*c)*d^5*e^5 - (35*C*a^2*b^2 - 14*B*a*b^3 + 8*A*b^4 + 30*A
*a^2*c^2 - (210*C*a^3 - 63*B*a^2*b + 41*A*a*b^2)*c)*d^4*e^6 + (48*C*c^4*d^6*e^4 - 8*(19*C*b*c^3 - B*c^4)*d^5*e
^5 + (158*C*b^2*c^2 + 6*A*c^4 + (174*C*a - 23*B*b)*c^3)*d^4*e^6 - (47*C*b^3*c - 12*(3*B*a - A*b)*c^3 + (384*C*
a*b - 17*B*b^2)*c^2)*d^3*e^7 - (15*C*b^4 - 104*A*a*c^3 - (208*C*a^2 - 106*B*a*b - 17*A*b^2)*c^2 - 3*(79*C*a*b^
2 + 4*B*b^3)*c)*d^2*e^8 + (42*C*a*b^3 - 6*B*b^4 + 52*(3*B*a^2 - 2*A*a*b)*c^2 - (364*C*a^2*b - B*a*b^2 - 23*A*b
^3)*c)*d*e^9 - (35*C*a^2*b^2 - 14*B*a*b^3 + 8*A*b^4 + 30*A*a^2*c^2 - (210*C*a^3 - 63*B*a^2*b + 41*A*a*b^2)*c)*
e^10)*x^4 + 4*(48*C*c^4*d^7*e^3 - 8*(19*C*b*c^3 - B*c^4)*d^6*e^4 + (158*C*b^2*c^2 + 6*A*c^4 + (174*C*a - 23*B*
b)*c^3)*d^5*e^5 - (47*C*b^3*c - 12*(3*B*a - A*b)*c^3 + (384*C*a*b - 17*B*b^2)*c^2)*d^4*e^6 - (15*C*b^4 - 104*A
*a*c^3 - (208*C*a^2 - 106*B*a*b - 17*A*b^2)*c^2 - 3*(79*C*a*b^2 + 4*B*b^3)*c)*d^3*e^7 + (42*C*a*b^3 - 6*B*b^4
+ 52*(3*B*a^2 - 2*A*a*b)*c^2 - (364*C*a^2*b - B*a*b^2 - 23*A*b^3)*c)*d^2*e^8 - (35*C*a^2*b^2 - 14*B*a*b^3 + 8*
A*b^4 + 30*A*a^2*c^2 - (210*C*a^3 - 63*B*a^2*b + 41*A*a*b^2)*c)*d*e^9)*x^3 + 6*(48*C*c^4*d^8*e^2 - 8*(19*C*b*c
^3 - B*c^4)*d^7*e^3 + (158*C*b^2*c^2 + 6*A*c^4 + (174*C*a - 23*B*b)*c^3)*d^6*e^4 - (47*C*b^3*c - 12*(3*B*a - A
*b)*c^3 + (384*C*a*b - 17*B*b^2)*c^2)*d^5*e^5 - (15*C*b^4 - 104*A*a*c^3 - (208*C*a^2 - 106*B*a*b - 17*A*b^2)*c
^2 - 3*(79*C*a*b^2 + 4*B*b^3)*c)*d^4*e^6 + (42*C*a*b^3 - 6*B*b^4 + 52*(3*B*a^2 - 2*A*a*b)*c^2 - (364*C*a^2*b -
B*a*b^2 - 23*A*b^3)*c)*d^3*e^7 - (35*C*a^2*b^2 - 14*B*a*b^3 + 8*A*b^4 + 30*A*a^2*c^2 - (210*C*a^3 - 63*B*a^2*
b + 41*A*a*b^2)*c)*d^2*e^8)*x^2 + 4*(48*C*c^4*d^9*e - 8*(19*C*b*c^3 - B*c^4)*d^8*e^2 + (158*C*b^2*c^2 + 6*A*c^
4 + (174*C*a - 23*B*b)*c^3)*d^7*e^3 - (47*C*b^3*c - 12*(3*B*a - A*b)*c^3 + (384*C*a*b - 17*B*b^2)*c^2)*d^6*e^4
- (15*C*b^4 - 104*A*a*c^3 - (208*C*a^2 - 106*B*a*b - 17*A*b^2)*c^2 - 3*(79*C*a*b^2 + 4*B*b^3)*c)*d^5*e^5 + (4
2*C*a*b^3 - 6*B*b^4 + 52*(3*B*a^2 - 2*A*a*b)*c^2 - (364*C*a^2*b - B*a*b^2 - 23*A*b^3)*c)*d^4*e^6 - (35*C*a^2*b
^2 - 14*B*a*b^3 + 8*A*b^4 + 30*A*a^2*c^2 - (210*C*a^3 - 63*B*a^2*b + 41*A*a*b^2)*c)*d^3*e^7)*x)*sqrt(c*e)*weie
rstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b
^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 3*(48*C*c^4*d^9*e
- 8*(16*C*b*c^3 - B*c^4)*d^8*e^2 + (103*C*b^2*c^2 + 6*A*c^4 + (138*C*a - 19*B*b)*c^3)*d^7*e^3 - 3*(5*C*b^3*c
- (10*B*a - 3*A*b)*c^3 + (79*C*a*b - 3*B*b^2)*c^2)*d^6*e^4 - (58*A*a*c^3 - (154*C*a^2 - B*a*b + 19*A*b^2)*c^2
- 6*(7*C*a*b^2 - B*b^3)*c)*d^5*e^5 - ((42*B*a^2 - 29*A*a*b)*c^2 + (35*C*a^2*b - 14*B*a*b^2 + 8*A*b^3)*c)*d^4*e
^6 + (48*C*c^4*d^5*e^5 - 8*(16*C*b*c^3 - B*c^4)*d^4*e^6 + (103*C*b^2*c^2 + 6*A*c^4 + (138*C*a - 19*B*b)*c^3)*d
^3*e^7 - 3*(5*C*b^3*c - (10*B*a - 3*A*b)*c^3 + (79*C*a*b - 3*B*b^2)*c^2)*d^2*e^8 - (58*A*a*c^3 - (154*C*a^2 -
B*a*b + 19*A*b^2)*c^2 - 6*(7*C*a*b^2 - B*b^3)*c)*d*e^9 - ((42*B*a^2 - 29*A*a*b)*c^2 + (35*C*a^2*b - 14*B*a*b^2
+ 8*A*b^3)*c)*e^10)*x^4 + 4*(48*C*c^4*d^6*e^4 - 8*(16*C*b*c^3 - B*c^4)*d^5*e^5 + (103*C*b^2*c^2 + 6*A*c^4 + (
138*C*a - 19*B*b)*c^3)*d^4*e^6 - 3*(5*C*b^3*c - (10*B*a - 3*A*b)*c^3 + (79*C*a*b - 3*B*b^2)*c^2)*d^3*e^7 - (58
*A*a*c^3 - (154*C*a^2 - B*a*b + 19*A*b^2)*c^2 - 6*(7*C*a*b^2 - B*b^3)*c)*d^2*e^8 - ((42*B*a^2 - 29*A*a*b)*c^2
+ (35*C*a^2*b - 14*B*a*b^2 + 8*A*b^3)*c)*d*e^9)*x^3 + 6*(48*C*c^4*d^7*e^3 - 8*(16*C*b*c^3 - B*c^4)*d^6*e^4 + (
103*C*b^2*c^2 + 6*A*c^4 + (138*C*a - 19*B*b)*c^3)*d^5*e^5 - 3*(5*C*b^3*c - (10*B*a - 3*A*b)*c^3 + (79*C*a*b -
3*B*b^2)*c^2)*d^4*e^6 - (58*A*a*c^3 - (154*C*a^2 - B*a*b + 19*A*b^2)*c^2 - 6*(7*C*a*b^2 - B*b^3)*c)*d^3*e^7 -
((42*B*a^2 - 29*A*a*b)*c^2 + (35*C*a^2*b - 14*B*a*b^2 + 8*A*b^3)*c)*d^2*e^8)*x^2 + 4*(48*C*c^4*d^8*e^2 - 8*(16
*C*b*c^3 - B*c^4)*d^7*e^3 + (103*C*b^2*c^2 + 6*A*c^4 + (138*C*a - 19*B*b)*c^3)*d^6*e^4 - 3*(5*C*b^3*c - (10*B*
a - 3*A*b)*c^3 + (79*C*a*b - 3*B*b^2)*c^2)*d^5*e^5 - (58*A*a*c^3 - (154*C*a^2 - B*a*b + 19*A*b^2)*c^2 - 6*(7*C
*a*b^2 - B*b^3)*c)*d^4*e^6 - ((42*B*a^2 - 29*A*a*b)*c^2 + (35*C*a^2*b - 14*B*a*b^2 + 8*A*b^3)*c)*d^3*e^7)*x)*s
qrt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2
*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e
+ (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*
c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) + 3*(24*C*c^4*d^8*e^2 - 15*A*a^3*c*e^10 - 21*(2*B*a^2 - 5
*A*a*b)*c^2*d^3*e^7 - 6*(B*a^3 - 7*A*a^2*b)*c*d*e^9 - (61*C*b*c^3 - 4*B*c^4)*d^7*e^3 + 3*(15*C*b^2*c^2 + A*c^4
+ (22*C*a - 3*B*b)*c^3)*d^6*e^4 + ((32*B*a + 9*A*b)*c^3 - (119*C*a*b + 3*B*b^2)*c^2)*d^5*e^5 - (95*A*a*c^3 -
(98*C*a^2 + 7*B*a*b - 4*A*b^2)*c^2)*d^4*e^6 - (49*A*a^2*c^2 + (8*C*a^3 - 14*B*a^2*b + 35*A*a*b^2)*c)*d^2*e^8 +
(48*C*c^4*d^5*e^5 - 8*(16*C*b*c^3 - B*c^4)*d^4*e^6 + (103*C*b^2*c^2 + 6*A*c^4 + (138*C*a - 19*B*b)*c^3)*d^3*e
^7 - 3*(5*C*b^3*c - (10*B*a - 3*A*b)*c^3 + (79*C*a*b - 3*B*b^2)*c^2)*d^2*e^8 - (58*A*a*c^3 - (154*C*a^2 - B*a*
b + 19*A*b^2)*c^2 - 6*(7*C*a*b^2 - B*b^3)*c)*d*e^9 - ((42*B*a^2 - 29*A*a*b)*c^2 + (35*C*a^2*b - 14*B*a*b^2 + 8
*A*b^3)*c)*e^10)*x^3 + (87*C*c^4*d^6*e^4 - (221*C*b*c^3 - 32*B*c^4)*d^5*e^5 + (158*C*b^2*c^2 + 24*A*c^4 + (249
*C*a - 80*B*b)*c^3)*d^4*e^6 + ((122*B*a - 39*A*b)*c^3 - (413*C*a*b - 45*B*b^2)*c^2)*d^3*e^7 - (178*A*a*c^3 - (
319*C*a^2 - 49*B*a*b + 67*A*b^2)*c^2 + 3*(C*a*b^2 + 7*B*b^3)*c)*d^2*e^8 - ((102*B*a^2 - 91*A*a*b)*c^2 - 2*(7*C
*a^2*b + 26*B*a*b^2 - 14*A*b^3)*c)*d*e^9 - (10*A*a^2*c^2 + (35*C*a^3 + 7*B*a^2*b - 4*A*a*b^2)*c)*e^10)*x^2 + (
78*C*c^4*d^7*e^3 - 3*(7*B*a^3 + A*a^2*b)*c*e^10 - (199*C*b*c^3 - 13*B*c^4)*d^6*e^4 + (145*C*b^2*c^2 + 36*A*c^4
+ (222*C*a - 25*B*b)*c^3)*d^5*e^5 + ((79*B*a - 66*A*b)*c^3 - (385*C*a*b + 12*B*b^2)*c^2)*d^4*e^6 - (170*A*a*c
^3 - (308*C*a^2 + 49*B*a*b + 89*A*b^2)*c^2)*d^3*e^7 - (7*(21*B*a^2 - 11*A*a*b)*c^2 - (4*C*a^2*b - 7*B*a*b^2 -
35*A*b^3)*c)*d^2*e^8 - 2*(7*A*a^2*c^2 + (14*C*a^3 - 26*B*a^2*b - 7*A*a*b^2)*c)*d*e^9)*x)*sqrt(c*x^2 + b*x + a)
*sqrt(e*x + d))/(c^4*d^10*e^5 - 3*b*c^3*d^9*e^6 - 3*a^2*b*c*d^5*e^10 + a^3*c*d^4*e^11 + 3*(b^2*c^2 + a*c^3)*d^
8*e^7 - (b^3*c + 6*a*b*c^2)*d^7*e^8 + 3*(a*b^2*c + a^2*c^2)*d^6*e^9 + (c^4*d^6*e^9 - 3*b*c^3*d^5*e^10 - 3*a^2*
b*c*d*e^14 + a^3*c*e^15 + 3*(b^2*c^2 + a*c^3)*d^4*e^11 - (b^3*c + 6*a*b*c^2)*d^3*e^12 + 3*(a*b^2*c + a^2*c^2)*
d^2*e^13)*x^4 + 4*(c^4*d^7*e^8 - 3*b*c^3*d^6*e^9 - 3*a^2*b*c*d^2*e^13 + a^3*c*d*e^14 + 3*(b^2*c^2 + a*c^3)*d^5
*e^10 - (b^3*c + 6*a*b*c^2)*d^4*e^11 + 3*(a*b^2*c + a^2*c^2)*d^3*e^12)*x^3 + 6*(c^4*d^8*e^7 - 3*b*c^3*d^7*e^8
- 3*a^2*b*c*d^3*e^12 + a^3*c*d^2*e^13 + 3*(b^2*c^2 + a*c^3)*d^6*e^9 - (b^3*c + 6*a*b*c^2)*d^5*e^10 + 3*(a*b^2*
c + a^2*c^2)*d^4*e^11)*x^2 + 4*(c^4*d^9*e^6 - 3*b*c^3*d^8*e^7 - 3*a^2*b*c*d^4*e^11 + a^3*c*d^3*e^12 + 3*(b^2*c
^2 + a*c^3)*d^7*e^8 - (b^3*c + 6*a*b*c^2)*d^6*e^9 + 3*(a*b^2*c + a^2*c^2)*d^5*e^10)*x)

Sympy [F]

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

[In]

integrate((C*x**2+B*x+A)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**(9/2),x)

[Out]

Integral((A + B*x + C*x**2)*sqrt(a + b*x + c*x**2)/(d + e*x)**(9/2), x)

Maxima [F]

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

[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)

Giac [F]

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

[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]

integrate((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)/(e*x + d)^(9/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{9/2}} \, dx=\int \frac {\left (C\,x^2+B\,x+A\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]

[In]

int(((A + B*x + C*x^2)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(9/2),x)

[Out]

int(((A + B*x + C*x^2)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(9/2), x)