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

Optimal. Leaf size=706 \[ -\frac{2 \sqrt{a d-b c} (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right ),\frac{f (b c-a d)}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt{c+d x} \sqrt{e+f x}}+\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left (5 d f (6 A d f+B c f+B d e)-2 C \left (c^2 f^2-c d e f+d^2 e^2\right )\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right )}{15 b^3 d f (b e-a f)}+\frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2} \left (6 a^2 C d f-a b (5 B d f+c C f+C d e)+b^2 (5 A d f+c C e)\right )}{5 b^2 f (b c-a d) (b e-a f)}-\frac{2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt{a+b x} (b c-a d) (b e-a f)} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 1.84465, antiderivative size = 706, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184, Rules used = {1614, 154, 158, 114, 113, 121, 120} \[ \frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left (5 d f (6 A d f+B c f+B d e)-2 C \left (c^2 f^2-c d e f+d^2 e^2\right )\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a d-b c} (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt{c+d x} \sqrt{e+f x}}+\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right )}{15 b^3 d f (b e-a f)}+\frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2} \left (6 a^2 C d f-a b (5 B d f+c C f+C d e)+b^2 (5 A d f+c C e)\right )}{5 b^2 f (b c-a d) (b e-a f)}-\frac{2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt{a+b x} (b c-a d) (b e-a f)} \]

Antiderivative was successfully verified.

[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 1614

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]

Rule 154

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 158

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 114

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 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*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))])/b, 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 121

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 120

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

Rubi steps

\begin{align*} \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 (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]  time = 8.11424, size = 633, normalized size = 0.9 \[ -\frac{2 \left (-i b f (a+b x)^{3/2} (d e-c f) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left (24 a^2 C d^2 f-a b d (20 B d f+7 c C f+C d e)+b^2 \left (15 A d^2 f+c d (5 B f+C e)-2 c^2 C f\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right ),\frac{b d e-a d f}{b c f-a d f}\right )+b^2 (c+d x) (e+f x) \left (-\sqrt{\frac{b c}{d}-a}\right ) \left (48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left (5 d f (6 A d f+B c f+B d e)-2 C \left (c^2 f^2-c d e f+d^2 e^2\right )\right )\right )-i f (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left (48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left (5 d f (6 A d f+B c f+B d e)-2 C \left (c^2 f^2-c d e f+d^2 e^2\right )\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )+b^2 d f (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} \left (15 d f \left (a (a C-b B)+A b^2\right )-(a+b x) (b (5 B d f+c C f+C d e)-9 a C d f)-3 b C d f x (a+b x)\right )\right )}{15 b^5 d^2 f^2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{\frac{b c}{d}-a}} \]

Antiderivative was successfully verified.

[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])

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Maple [B]  time = 0.051, size = 6257, normalized size = 8.9 \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)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^(3/2),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{d x + c} \sqrt{f x + e}}{{\left (b x + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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{b x + a} \sqrt{d x + c} \sqrt{f x + e}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)