\(\int \frac {\sqrt {c+d x} \sqrt {e+f x} (A+B x+C x^2)}{(a+b x)^{3/2}} \, dx\) [63]

Optimal result

Integrand size = 38, antiderivative size = 706 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2}} \, dx=\frac {2 \left (24 a^2 C d f^2-a b f (7 C d e+c C f+20 B d f)+b^2 (5 d f (B e+3 A f)-C e (2 d e-c f))\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{15 b^3 d f (b e-a f)}+\frac {2 \left (6 a^2 C d f+b^2 (c C e+5 A d f)-a b (C d e+c C f+5 B d f)\right ) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b^2 (b c-a d) f (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}+\frac {2 \sqrt {-b c+a d} \left (48 a^2 C d^2 f^2-8 a b d f (C d e+c C f+5 B d f)+b^2 \left (5 d f (B d e+B c f+6 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {-b c+a d} (d e-c f) \left (24 a^2 C d f^2-a b f (7 C d e+c C f+20 B d f)+b^2 (5 d f (B e+3 A f)-C e (2 d e-c f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt {c+d x} \sqrt {e+f x}} \]

[Out]

-2*(A*b^2-a*(B*b-C*a))*(d*x+c)^(3/2)*(f*x+e)^(3/2)/b/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^(1/2)+2/5*(6*a^2*C*d*f+b^2*
(5*A*d*f+C*c*e)-a*b*(5*B*d*f+C*c*f+C*d*e))*(f*x+e)^(3/2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^2/(-a*d+b*c)/f/(-a*f+b*
e)+2/15*(24*a^2*C*d*f^2-a*b*f*(20*B*d*f+C*c*f+7*C*d*e)+b^2*(5*d*f*(3*A*f+B*e)-C*e*(-c*f+2*d*e)))*(b*x+a)^(1/2)
*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^3/d/f/(-a*f+b*e)+2/15*(48*a^2*C*d^2*f^2-8*a*b*d*f*(5*B*d*f+C*c*f+C*d*e)+b^2*(5*
d*f*(6*A*d*f+B*c*f+B*d*e)-2*C*(c^2*f^2-c*d*e*f+d^2*e^2)))*EllipticE(d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a
*d+b*c)*f/d/(-a*f+b*e))^(1/2))*(a*d-b*c)^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(f*x+e)^(1/2)/b^4/d^(3/2)/f^2/(d*x
+c)^(1/2)/(b*(f*x+e)/(-a*f+b*e))^(1/2)-2/15*(-c*f+d*e)*(24*a^2*C*d*f^2-a*b*f*(20*B*d*f+C*c*f+7*C*d*e)+b^2*(5*d
*f*(3*A*f+B*e)-C*e*(-c*f+2*d*e)))*EllipticF(d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^
(1/2))*(a*d-b*c)^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(b*(f*x+e)/(-a*f+b*e))^(1/2)/b^4/d^(3/2)/f^2/(d*x+c)^(1/2)
/(f*x+e)^(1/2)

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {1628, 159, 164, 115, 114, 122, 121} \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2}} \, dx=\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left (5 d f (6 A d f+B c f+B d e)-2 C \left (c^2 f^2-c d e f+d^2 e^2\right )\right )\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {a d-b c} (d e-c f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt {c+d x} \sqrt {e+f x}}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} \left (6 a^2 C d f-a b (5 B d f+c C f+C d e)+b^2 (5 A d f+c C e)\right )}{5 b^2 f (b c-a d) (b e-a f)}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right )}{15 b^3 d f (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt {a+b x} (b c-a d) (b e-a f)} \]

[In]

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

[Out]

(2*(24*a^2*C*d*f^2 - a*b*f*(7*C*d*e + c*C*f + 20*B*d*f) + b^2*(5*d*f*(B*e + 3*A*f) - C*e*(2*d*e - c*f)))*Sqrt[
a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(15*b^3*d*f*(b*e - a*f)) + (2*(6*a^2*C*d*f + b^2*(c*C*e + 5*A*d*f) - a*b
*(C*d*e + c*C*f + 5*B*d*f))*Sqrt[a + b*x]*Sqrt[c + d*x]*(e + f*x)^(3/2))/(5*b^2*(b*c - a*d)*f*(b*e - a*f)) - (
2*(A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(b*(b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x]) + (2*Sqrt
[-(b*c) + a*d]*(48*a^2*C*d^2*f^2 - 8*a*b*d*f*(C*d*e + c*C*f + 5*B*d*f) + b^2*(5*d*f*(B*d*e + B*c*f + 6*A*d*f)
- 2*C*(d^2*e^2 - c*d*e*f + c^2*f^2)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*
Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(15*b^4*d^(3/2)*f^2*Sqrt[c + d*x]*Sqrt[(
b*(e + f*x))/(b*e - a*f)]) - (2*Sqrt[-(b*c) + a*d]*(d*e - c*f)*(24*a^2*C*d*f^2 - a*b*f*(7*C*d*e + c*C*f + 20*B
*d*f) + b^2*(5*d*f*(B*e + 3*A*f) - C*e*(2*d*e - c*f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e
- a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(15*b
^4*d^(3/2)*f^2*Sqrt[c + d*x]*Sqrt[e + f*x])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*Rt[-(b
*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /;
FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-(b*c - a*d)/d, 0] &&
!(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])

Rule 115

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[e + f*x
]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d*x]*Sqrt[b*((e + f*x)/(b*e - a*f))])), Int[Sqrt[b*(e/(b*e - a*f)
) + b*f*(x/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]

Rule 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)
/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && Si
mplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x] && (PosQ[-(b*c - a*d)/d] || NegQ[-(b*e - a*f)/f])

Rule 122

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[b*((c
+ d*x)/(b*c - a*d))]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 164

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rule 1628

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[{
Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[b*R*(a + b*x)^(m + 1)*
(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e
- a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f
*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x], x]] /; FreeQ[{a, b,
c, d, e, f, n, p}, x] && PolyQ[Px, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {2 \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (-\frac {3 a^2 C (d e+c f)-a b (c C e+3 B d e+3 B c f-A d f)+b^2 (B c e+2 A (d e+c f))}{2 b}+\frac {1}{2} \left (-\frac {6 a^2 C d f}{b}-b (c C e+5 A d f)+a (C d e+c C f+5 B d f)\right ) x\right )}{\sqrt {a+b x}} \, dx}{(b c-a d) (b e-a f)} \\ & = \frac {2 \left (6 a^2 C d f+b^2 (c C e+5 A d f)-a b (C d e+c C f+5 B d f)\right ) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b^2 (b c-a d) f (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {4 \int \frac {\sqrt {e+f x} \left (-\frac {(b c-a d) \left (6 a^2 C f (d e+3 c f)-b^2 \left (c C e^2-5 A d e f-5 c f (B e+2 A f)\right )-a b (5 B f (d e+3 c f)+C e (d e+7 c f))\right )}{4 b}-\frac {(b c-a d) \left (24 a^2 C d f^2-a b f (7 C d e+c C f+20 B d f)+b^2 (5 d f (B e+3 A f)-C e (2 d e-c f))\right ) x}{4 b}\right )}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{5 b (b c-a d) f (b e-a f)} \\ & = \frac {2 \left (24 a^2 C d f^2-a b f (7 C d e+c C f+20 B d f)+b^2 (5 d f (B e+3 A f)-C e (2 d e-c f))\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{15 b^3 d f (b e-a f)}+\frac {2 \left (6 a^2 C d f+b^2 (c C e+5 A d f)-a b (C d e+c C f+5 B d f)\right ) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b^2 (b c-a d) f (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {8 \int \frac {-\frac {(b c-a d) (b e-a f) \left (24 a^2 C d f (d e+c f)-a b \left (20 B d f (d e+c f)+C \left (d^2 e^2+14 c d e f+c^2 f^2\right )\right )-b^2 \left (c^2 C e f-15 A d^2 e f+c d \left (C e^2-5 f (2 B e+3 A f)\right )\right )\right )}{8 b}-\frac {(b c-a d) (b e-a f) \left (48 a^2 C d^2 f^2-8 a b d f (C d e+c C f+5 B d f)+b^2 \left (5 d f (B d e+B c f+6 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )\right ) x}{8 b}}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{15 b^2 d (b c-a d) f (b e-a f)} \\ & = \frac {2 \left (24 a^2 C d f^2-a b f (7 C d e+c C f+20 B d f)+b^2 (5 d f (B e+3 A f)-C e (2 d e-c f))\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{15 b^3 d f (b e-a f)}+\frac {2 \left (6 a^2 C d f+b^2 (c C e+5 A d f)-a b (C d e+c C f+5 B d f)\right ) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b^2 (b c-a d) f (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {\left ((d e-c f) \left (24 a^2 C d f^2-a b f (7 C d e+c C f+20 B d f)+b^2 (5 d f (B e+3 A f)-C e (2 d e-c f))\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{15 b^3 d f^2}+\frac {\left (48 a^2 C d^2 f^2-8 a b d f (C d e+c C f+5 B d f)+b^2 \left (5 d f (B d e+B c f+6 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{15 b^3 d f^2} \\ & = \frac {2 \left (24 a^2 C d f^2-a b f (7 C d e+c C f+20 B d f)+b^2 (5 d f (B e+3 A f)-C e (2 d e-c f))\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{15 b^3 d f (b e-a f)}+\frac {2 \left (6 a^2 C d f+b^2 (c C e+5 A d f)-a b (C d e+c C f+5 B d f)\right ) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b^2 (b c-a d) f (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {\left ((d e-c f) \left (24 a^2 C d f^2-a b f (7 C d e+c C f+20 B d f)+b^2 (5 d f (B e+3 A f)-C e (2 d e-c f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}} \, dx}{15 b^3 d f^2 \sqrt {c+d x}}+\frac {\left (\left (48 a^2 C d^2 f^2-8 a b d f (C d e+c C f+5 B d f)+b^2 \left (5 d f (B d e+B c f+6 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x}\right ) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}} \, dx}{15 b^3 d f^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}} \\ & = \frac {2 \left (24 a^2 C d f^2-a b f (7 C d e+c C f+20 B d f)+b^2 (5 d f (B e+3 A f)-C e (2 d e-c f))\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{15 b^3 d f (b e-a f)}+\frac {2 \left (6 a^2 C d f+b^2 (c C e+5 A d f)-a b (C d e+c C f+5 B d f)\right ) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b^2 (b c-a d) f (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}+\frac {2 \sqrt {-b c+a d} \left (48 a^2 C d^2 f^2-8 a b d f (C d e+c C f+5 B d f)+b^2 \left (5 d f (B d e+B c f+6 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {\left ((d e-c f) \left (24 a^2 C d f^2-a b f (7 C d e+c C f+20 B d f)+b^2 (5 d f (B e+3 A f)-C e (2 d e-c f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}} \, dx}{15 b^3 d f^2 \sqrt {c+d x} \sqrt {e+f x}} \\ & = \frac {2 \left (24 a^2 C d f^2-a b f (7 C d e+c C f+20 B d f)+b^2 (5 d f (B e+3 A f)-C e (2 d e-c f))\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{15 b^3 d f (b e-a f)}+\frac {2 \left (6 a^2 C d f+b^2 (c C e+5 A d f)-a b (C d e+c C f+5 B d f)\right ) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b^2 (b c-a d) f (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}+\frac {2 \sqrt {-b c+a d} \left (48 a^2 C d^2 f^2-8 a b d f (C d e+c C f+5 B d f)+b^2 \left (5 d f (B d e+B c f+6 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {-b c+a d} (d e-c f) \left (24 a^2 C d f^2-a b f (7 C d e+c C f+20 B d f)+b^2 (5 d f (B e+3 A f)-C e (2 d e-c f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt {c+d x} \sqrt {e+f x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.90 (sec) , antiderivative size = 633, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2}} \, dx=-\frac {2 \left (-b^2 \sqrt {-a+\frac {b c}{d}} \left (48 a^2 C d^2 f^2-8 a b d f (C d e+c C f+5 B d f)+b^2 \left (5 d f (B d e+B c f+6 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )\right ) (c+d x) (e+f x)+b^2 \sqrt {-a+\frac {b c}{d}} d f (c+d x) (e+f x) \left (15 \left (A b^2+a (-b B+a C)\right ) d f-(-9 a C d f+b (C d e+c C f+5 B d f)) (a+b x)-3 b C d f x (a+b x)\right )-i (b c-a d) f \left (48 a^2 C d^2 f^2-8 a b d f (C d e+c C f+5 B d f)+b^2 \left (5 d f (B d e+B c f+6 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )-i b f (d e-c f) \left (24 a^2 C d^2 f-a b d (C d e+7 c C f+20 B d f)+b^2 \left (-2 c^2 C f+15 A d^2 f+c d (C e+5 B f)\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )}{15 b^5 \sqrt {-a+\frac {b c}{d}} d^2 f^2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \]

[In]

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

[Out]

(-2*(-(b^2*Sqrt[-a + (b*c)/d]*(48*a^2*C*d^2*f^2 - 8*a*b*d*f*(C*d*e + c*C*f + 5*B*d*f) + b^2*(5*d*f*(B*d*e + B*
c*f + 6*A*d*f) - 2*C*(d^2*e^2 - c*d*e*f + c^2*f^2)))*(c + d*x)*(e + f*x)) + b^2*Sqrt[-a + (b*c)/d]*d*f*(c + d*
x)*(e + f*x)*(15*(A*b^2 + a*(-(b*B) + a*C))*d*f - (-9*a*C*d*f + b*(C*d*e + c*C*f + 5*B*d*f))*(a + b*x) - 3*b*C
*d*f*x*(a + b*x)) - I*(b*c - a*d)*f*(48*a^2*C*d^2*f^2 - 8*a*b*d*f*(C*d*e + c*C*f + 5*B*d*f) + b^2*(5*d*f*(B*d*
e + B*c*f + 6*A*d*f) - 2*C*(d^2*e^2 - c*d*e*f + c^2*f^2)))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*S
qrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f
- a*d*f)] - I*b*f*(d*e - c*f)*(24*a^2*C*d^2*f - a*b*d*(C*d*e + 7*c*C*f + 20*B*d*f) + b^2*(-2*c^2*C*f + 15*A*d
^2*f + c*d*(C*e + 5*B*f)))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]
*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)]))/(15*b^5*Sqrt[-a + (
b*c)/d]*d^2*f^2*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])

Maple [A] (verified)

Time = 2.28 (sec) , antiderivative size = 1163, normalized size of antiderivative = 1.65

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1163\)
default	\(\text {Expression too large to display}\)	\(5787\)

[In]

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

[Out]

((b*x+a)*(d*x+c)*(f*x+e))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)*(-2*(b*d*f*x^2+b*c*f*x+b*d*e*x+b*c*e
)*(A*b^2-B*a*b+C*a^2)/b^4/((x+a/b)*(b*d*f*x^2+b*c*f*x+b*d*e*x+b*c*e))^(1/2)+2/5*C/b^2*x*(b*d*f*x^3+a*d*f*x^2+b
*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)+2/3*(1/b^2*(B*b*d*f-C*a*d*f+C*b*c*f+C*b*d*e)-2/5*C/b^2
*(2*a*d*f+2*b*c*f+2*b*d*e))/b/d/f*(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2
)+2*(-(A*a*b^2*d*f-A*b^3*c*f-A*b^3*d*e-B*a^2*b*d*f+B*a*b^2*c*f+B*a*b^2*d*e-B*b^3*c*e+C*a^3*d*f-C*a^2*b*c*f-C*a
^2*b*d*e+C*a*b^2*c*e)/b^4+(A*b^2-B*a*b+C*a^2)/b^4*(a*d*f-b*c*f-b*d*e)+(b*c*f+b*d*e)*(A*b^2-B*a*b+C*a^2)/b^4-2/
5*C/b^2*a*c*e-2/3*(1/b^2*(B*b*d*f-C*a*d*f+C*b*c*f+C*b*d*e)-2/5*C/b^2*(2*a*d*f+2*b*c*f+2*b*d*e))/b/d/f*(1/2*a*c
*f+1/2*a*d*e+1/2*b*c*e))*(e/f-c/d)*((x+e/f)/(e/f-c/d))^(1/2)*((x+a/b)/(-e/f+a/b))^(1/2)*((x+c/d)/(-e/f+c/d))^(
1/2)/(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)*EllipticF(((x+e/f)/(e/f-c/d
))^(1/2),((-e/f+c/d)/(-e/f+a/b))^(1/2))+2*(1/b^3*(A*b^2*d*f-B*a*b*d*f+B*b^2*c*f+B*b^2*d*e+C*a^2*d*f-C*a*b*c*f-
C*a*b*d*e+C*b^2*c*e)+(A*b^2-B*a*b+C*a^2)/b^3*d*f-2/5*C/b^2*(3/2*a*c*f+3/2*a*d*e+3/2*b*c*e)-2/3*(1/b^2*(B*b*d*f
-C*a*d*f+C*b*c*f+C*b*d*e)-2/5*C/b^2*(2*a*d*f+2*b*c*f+2*b*d*e))/b/d/f*(a*d*f+b*c*f+b*d*e))*(e/f-c/d)*((x+e/f)/(
e/f-c/d))^(1/2)*((x+a/b)/(-e/f+a/b))^(1/2)*((x+c/d)/(-e/f+c/d))^(1/2)/(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2
+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)*((-e/f+a/b)*EllipticE(((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f+a/b))^
(1/2))-a/b*EllipticF(((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f+a/b))^(1/2))))

Fricas [C] (verification not implemented)

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

Time = 0.17 (sec) , antiderivative size = 1463, normalized size of antiderivative = 2.07 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

2/45*(3*(3*C*b^4*d^3*f^3*x^2 + C*a*b^3*d^3*e*f^2 + (C*a*b^3*c*d^2 - (24*C*a^2*b^2 - 20*B*a*b^3 + 15*A*b^4)*d^3
)*f^3 + (C*b^4*d^3*e*f^2 + (C*b^4*c*d^2 - (6*C*a*b^3 - 5*B*b^4)*d^3)*f^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(
f*x + e) + (2*C*a*b^3*d^3*e^3 - (3*C*a*b^3*c*d^2 - (7*C*a^2*b^2 - 5*B*a*b^3)*d^3)*e^2*f - (3*C*a*b^3*c^2*d + 4
*(7*C*a^2*b^2 - 5*B*a*b^3)*c*d^2 - (32*C*a^3*b - 25*B*a^2*b^2 + 15*A*a*b^3)*d^3)*e*f^2 + (2*C*a*b^3*c^3 + (7*C
*a^2*b^2 - 5*B*a*b^3)*c^2*d + (32*C*a^3*b - 25*B*a^2*b^2 + 15*A*a*b^3)*c*d^2 - 2*(24*C*a^4 - 20*B*a^3*b + 15*A
*a^2*b^2)*d^3)*f^3 + (2*C*b^4*d^3*e^3 - (3*C*b^4*c*d^2 - (7*C*a*b^3 - 5*B*b^4)*d^3)*e^2*f - (3*C*b^4*c^2*d + 4
*(7*C*a*b^3 - 5*B*b^4)*c*d^2 - (32*C*a^2*b^2 - 25*B*a*b^3 + 15*A*b^4)*d^3)*e*f^2 + (2*C*b^4*c^3 + (7*C*a*b^3 -
5*B*b^4)*c^2*d + (32*C*a^2*b^2 - 25*B*a*b^3 + 15*A*b^4)*c*d^2 - 2*(24*C*a^3*b - 20*B*a^2*b^2 + 15*A*a*b^3)*d^
3)*f^3)*x)*sqrt(b*d*f)*weierstrassPInverse(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c*d + a
^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*
d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), 1/3*(3*b*
d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f)) + 3*(2*C*a*b^3*d^3*e^2*f - (2*C*a*b^3*c*d^2 - (8*C*a^2*b^2 - 5*B*a*b^3
)*d^3)*e*f^2 + (2*C*a*b^3*c^2*d + (8*C*a^2*b^2 - 5*B*a*b^3)*c*d^2 - 2*(24*C*a^3*b - 20*B*a^2*b^2 + 15*A*a*b^3)
*d^3)*f^3 + (2*C*b^4*d^3*e^2*f - (2*C*b^4*c*d^2 - (8*C*a*b^3 - 5*B*b^4)*d^3)*e*f^2 + (2*C*b^4*c^2*d + (8*C*a*b
^3 - 5*B*b^4)*c*d^2 - 2*(24*C*a^2*b^2 - 20*B*a*b^3 + 15*A*b^4)*d^3)*f^3)*x)*sqrt(b*d*f)*weierstrassZeta(4/3*(b
^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3
- 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2
*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), weierstrassPInverse(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)
*e*f + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*
f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*
f^3)/(b^3*d^3*f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f))))/(b^6*d^3*f^3*x + a*b^5*d^3*f^3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2}} \, dx=\int \frac {\sqrt {e+f\,x}\,\sqrt {c+d\,x}\,\left (C\,x^2+B\,x+A\right )}{{\left (a+b\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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