3.3.61 \(\int \frac {\sqrt {a+b x+c x^2} (A+B x+C x^2)}{(d+e x)^{3/2}} \, dx\) [261]

Optimal. Leaf size=749 \[ -\frac {2 \sqrt {d+e x} \left (b C e^2 (b d-a e)+c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )+c e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B d e+3 A e^2\right )\right )+3 c e^2 \left (5 B c d+b C d-\frac {6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{15 c e^3 \left (c d^2-b d e+a e^2\right )}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 b^2 C e^2+c e (8 b C d-5 b B e-6 a C e)-c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^2 e^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )-c e \left (2 a e (9 C d-5 B e)-b \left (32 C d^2-5 e (5 B d-3 A e)\right )\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^2 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

-2*(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)^(1/2)-2/15*(b*C*e^2*(-a*e+b*d)+c^2*d
*(24*C*d^2-5*e*(-3*A*e+4*B*d))+c*e*(a*e*(-5*B*e+9*C*d)-5*b*(3*A*e^2-4*B*d*e+5*C*d^2))+3*c*e^2*(5*B*c*d+b*C*d-6
*c*C*d^2/e-5*A*c*e-a*C*e)*x)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c/e^3/(a*e^2-b*d*e+c*d^2)-1/15*(2*b^2*C*e^2+c*e
*(-5*B*b*e-6*C*a*e+8*C*b*d)-c^2*(48*C*d^2-10*e*(-3*A*e+4*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)/c^2/e^4/(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)+2/15*(b*C*e^2*(-a*e+b*d)-2*c^2*d*(24*C*d^2-5*e*(-3*A*e+4*B*d))-c*e*(2*a*e*
(-5*B*e+9*C*d)-b*(32*C*d^2-5*e*(-3*A*e+5*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)/c^2/e^4/(e*x+d)^(1/2)
/(c*x^2+b*x+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.90, antiderivative size = 746, normalized size of antiderivative = 1.00, 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 = {1664, 828, 857, 732, 435, 430} \begin {gather*} \frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (c e \left (-2 a e (9 C d-5 B e)-5 b e (5 B d-3 A e)+32 b C d^2\right )+b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )\right ) F\left (\text {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 )}{15 c^2 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c e (-6 a C e-5 b B e+8 b C d)-\left (c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\right )+2 b^2 C e^2\right ) E\left (\text {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 )}{15 c^2 e^4 \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (3 c e^2 x \left (-a C e-5 A c e+b C d+5 B c d-\frac {6 c C d^2}{e}\right )+c e \left (a e (9 C d-5 B e)-5 b \left (3 A e^2-4 B d e+5 C d^2\right )\right )+b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )\right )}{15 c e^3 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{e \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(b*C*e^2*(b*d - a*e) + c^2*(24*C*d^3 - 5*d*e*(4*B*d - 3*A*e)) + c*e*(a*e*(9*C*d - 5*B*e) - 5
*b*(5*C*d^2 - 4*B*d*e + 3*A*e^2)) + 3*c*e^2*(5*B*c*d + b*C*d - (6*c*C*d^2)/e - 5*A*c*e - a*C*e)*x)*Sqrt[a + b*
x + c*x^2])/(15*c*e^3*(c*d^2 - b*d*e + a*e^2)) - (2*(C*d^2 - e*(B*d - A*e))*(a + b*x + c*x^2)^(3/2))/(e*(c*d^2
- b*d*e + a*e^2)*Sqrt[d + e*x]) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*b^2*C*e^2 + c*e*(8*b*C*d - 5*b*B*e - 6*a*C*e)
- c^2*(48*C*d^2 - 10*e*(4*B*d - 3*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)])/(15*c^2*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]*(b*C*e^2*(b*d - a*e) - 2*c^2*d*(24*C*d^2 - 5*e*(4*B*d - 3*A*e)) + c*e*(
32*b*C*d^2 - 5*b*e*(5*B*d - 3*A*e) - 2*a*e*(9*C*d - 5*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^2*e^4*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 828

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 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*} \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{3/2}} \, dx &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {2 \int \frac {\left (-\frac {3 b C d^2-b e (3 B d-2 A e)+e (A c d-a C d+a B e)}{2 e}+\frac {1}{2} \left (5 B c d+b C d-\frac {6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{\sqrt {d+e x}} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac {2 \sqrt {d+e x} \left (b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )+c e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B d e+3 A e^2\right )\right )+3 c e^2 \left (5 B c d+b C d-\frac {6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{15 c e^3 \left (c d^2-b d e+a e^2\right )}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {4 \int \frac {-\frac {\left (c d^2-b d e+a e^2\right ) \left (b^2 C d e+a b C e^2+2 a c e (6 C d-5 B e)-b c \left (24 C d^2-5 e (4 B d-3 A e)\right )\right )}{4 e}-\frac {\left (c d^2-b d e+a e^2\right ) \left (2 b^2 C e^2+c e (8 b C d-5 b B e-6 a C e)-c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\right ) x}{4 e}}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 c e^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \sqrt {d+e x} \left (b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )+c e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B d e+3 A e^2\right )\right )+3 c e^2 \left (5 B c d+b C d-\frac {6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{15 c e^3 \left (c d^2-b d e+a e^2\right )}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\left (2 b^2 C e^2+c e (8 b C d-5 b B e-6 a C e)-c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{15 c e^4}+\frac {\left (b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )+c e \left (32 b C d^2-5 b e (5 B d-3 A e)-2 a e (9 C d-5 B e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 c e^4}\\ &=-\frac {2 \sqrt {d+e x} \left (b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )+c e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B d e+3 A e^2\right )\right )+3 c e^2 \left (5 B c d+b C d-\frac {6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{15 c e^3 \left (c d^2-b d e+a e^2\right )}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (2 b^2 C e^2+c e (8 b C d-5 b B e-6 a C e)-c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {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 c^2 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 (b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )+c e \left (32 b C d^2-5 b e (5 B d-3 A e)-2 a e (9 C d-5 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 ) \text {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^2 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 \sqrt {d+e x} \left (b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )+c e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B d e+3 A e^2\right )\right )+3 c e^2 \left (5 B c d+b C d-\frac {6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{15 c e^3 \left (c d^2-b d e+a e^2\right )}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 b^2 C e^2+c e (8 b C d-5 b B e-6 a C e)-c^2 \left (48 C d^2-10 e (4 B d-3 A e)\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 c^2 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 (b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )+c e \left (32 b C d^2-5 b e (5 B d-3 A e)-2 a e (9 C d-5 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^2 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 32.90, size = 13240, normalized size = 17.68 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(3/2),x]

[Out]

Result too large to show

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(8220\) vs. \(2(687)=1374\).
time = 0.18, size = 8221, normalized size = 10.98

method	result	size
elliptic	\(\frac {\sqrt {\left (c \,x^{2}+b x +a \right ) \left (e x +d \right )}\, \left (-\frac {2 \left (c e \,x^{2}+b e x +a e \right ) \left (A \,e^{2}-B d e +C \,d^{2}\right )}{e^{4} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x +a e \right )}}+\frac {2 C x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}{5 e^{2}}+\frac {2 \left (\frac {c e B +C b e -c d C}{e^{2}}-\frac {2 \left (2 e b +2 c d \right ) C}{5 e^{2}}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}{3 c e}+\frac {2 \left (\frac {A b \,e^{3}-d \,e^{2} c A +B a \,e^{3}-B b d \,e^{2}+B c \,d^{2} e -C a d \,e^{2}+C b \,d^{2} e -C c \,d^{3}}{e^{4}}-\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \left (e b -c d \right )}{e^{4}}+\frac {b \left (A \,e^{2}-B d e +C \,d^{2}\right )}{e^{3}}-\frac {2 a d C}{5 e^{2}}-\frac {2 \left (\frac {c e B +C b e -c d C}{e^{2}}-\frac {2 \left (2 e b +2 c d \right ) C}{5 e^{2}}\right ) \left (\frac {a e}{2}+\frac {b d}{2}\right )}{3 c e}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}\right ) \sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {d}{e}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}, \sqrt {\frac {\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {d}{e}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}+\frac {2 \left (\frac {A c \,e^{2}+B b \,e^{2}-B c d e +a C \,e^{2}-C b d e +C c \,d^{2}}{e^{3}}+\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) c}{e^{3}}-\frac {2 C \left (\frac {3 a e}{2}+\frac {3 b d}{2}\right )}{5 e^{2}}-\frac {2 \left (\frac {c e B +C b e -c d C}{e^{2}}-\frac {2 \left (2 e b +2 c d \right ) C}{5 e^{2}}\right ) \left (e b +c d \right )}{3 c e}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}\right ) \sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {d}{e}}}\, \left (\left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}, \sqrt {\frac {\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {d}{e}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {d}{e}}}, \sqrt {\frac {\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {d}{e}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}\)	\(1215\)
risch	\(\text {Expression too large to display}\)	\(1864\)
default	\(\text {Expression too large to display}\)	\(8221\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)^(3/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)/(x*e + d)^(3/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.12, size = 790, normalized size = 1.05 \begin {gather*} -\frac {2 \, {\left ({\left (48 \, C c^{3} d^{4} - {\left (2 \, C b^{3} + 15 \, {\left (2 \, B a + A b\right )} c^{2} - {\left (9 \, C a b + 5 \, B b^{2}\right )} c\right )} x e^{4} - {\left ({\left (7 \, C b^{2} c - 30 \, A c^{3} - {\left (42 \, C a + 25 \, B b\right )} c^{2}\right )} d x + {\left (2 \, C b^{3} + 15 \, {\left (2 \, B a + A b\right )} c^{2} - {\left (9 \, C a b + 5 \, B b^{2}\right )} c\right )} d\right )} e^{3} - {\left (8 \, {\left (4 \, C b c^{2} + 5 \, B c^{3}\right )} d^{2} x + {\left (7 \, C b^{2} c - 30 \, A c^{3} - {\left (42 \, C a + 25 \, B b\right )} c^{2}\right )} d^{2}\right )} e^{2} + 8 \, {\left (6 \, C c^{3} d^{3} x - {\left (4 \, C b c^{2} + 5 \, B c^{3}\right )} d^{3}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 3 \, {\left (48 \, C c^{3} d^{3} e - {\left (2 \, C b^{2} c - 30 \, A c^{3} - {\left (6 \, C a + 5 \, B b\right )} c^{2}\right )} x e^{4} - {\left (8 \, {\left (C b c^{2} + 5 \, B c^{3}\right )} d x + {\left (2 \, C b^{2} c - 30 \, A c^{3} - {\left (6 \, C a + 5 \, B b\right )} c^{2}\right )} d\right )} e^{3} + 8 \, {\left (6 \, C c^{3} d^{2} x - {\left (C b c^{2} + 5 \, B c^{3}\right )} d^{2}\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (24 \, C c^{3} d^{2} e^{2} - {\left (3 \, C c^{3} x^{2} - 15 \, A c^{3} + {\left (C b c^{2} + 5 \, B c^{3}\right )} x\right )} e^{4} + {\left (6 \, C c^{3} d x - {\left (C b c^{2} + 20 \, B c^{3}\right )} d\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{45 \, {\left (c^{3} x e^{6} + c^{3} d e^{5}\right )}} \end {gather*}

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)^(3/2),x, algorithm="fricas")

[Out]

-2/45*((48*C*c^3*d^4 - (2*C*b^3 + 15*(2*B*a + A*b)*c^2 - (9*C*a*b + 5*B*b^2)*c)*x*e^4 - ((7*C*b^2*c - 30*A*c^3
- (42*C*a + 25*B*b)*c^2)*d*x + (2*C*b^3 + 15*(2*B*a + A*b)*c^2 - (9*C*a*b + 5*B*b^2)*c)*d)*e^3 - (8*(4*C*b*c^
2 + 5*B*c^3)*d^2*x + (7*C*b^2*c - 30*A*c^3 - (42*C*a + 25*B*b)*c^2)*d^2)*e^2 + 8*(6*C*c^3*d^3*x - (4*C*b*c^2 +
5*B*c^3)*d^3)*e)*sqrt(c)*e^(1/2)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/c^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)*e^(-3)/c^3, 1/3*(c*d + (
3*c*x + b)*e)*e^(-1)/c) + 3*(48*C*c^3*d^3*e - (2*C*b^2*c - 30*A*c^3 - (6*C*a + 5*B*b)*c^2)*x*e^4 - (8*(C*b*c^2
+ 5*B*c^3)*d*x + (2*C*b^2*c - 30*A*c^3 - (6*C*a + 5*B*b)*c^2)*d)*e^3 + 8*(6*C*c^3*d^2*x - (C*b*c^2 + 5*B*c^3)
*d^2)*e^2)*sqrt(c)*e^(1/2)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/c^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)*e^(-3)/c^3, weierstrassPInverse(4/3
*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/c^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)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c)) + 3*(24*C*c^3*d^2*e^2 - (3*C*c^3
*x^2 - 15*A*c^3 + (C*b*c^2 + 5*B*c^3)*x)*e^4 + (6*C*c^3*d*x - (C*b*c^2 + 20*B*c^3)*d)*e^3)*sqrt(c*x^2 + b*x +
a)*sqrt(x*e + d))/(c^3*x*e^6 + c^3*d*e^5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x + C x^{2}\right ) \sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

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)**(3/2),x)

[Out]

Integral((A + B*x + C*x**2)*sqrt(a + b*x + c*x**2)/(d + e*x)**(3/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)^(3/2),x, algorithm="giac")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)/(x*e + d)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (C\,x^2+B\,x+A\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x + C*x^2)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(3/2),x)

[Out]

int(((A + B*x + C*x^2)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(3/2), x)

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