3.64 \(\int \frac{\sqrt{c+d x} \sqrt{e+f x} (A+B x+C x^2)}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=687 \[ \frac{2 (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right ),\frac{f (b c-a d)}{d (b e-a f)}\right )}{3 b^4 \sqrt{d} f \sqrt{c+d x} \sqrt{e+f x} \sqrt{a d-b c}}+\frac{2 \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (-8 a^2 b d f (B d f+2 C (c f+d e))+16 a^3 C d^2 f^2+a b^2 \left (d f (2 A d f+7 B c f+7 B d e)+C \left (c^2 f^2+16 c d e f+d^2 e^2\right )\right )-b^3 \left (c d \left (A f^2+6 B e f+C e^2\right )+A d^2 e f+c^2 C e f\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^4 \sqrt{d} f \sqrt{c+d x} \sqrt{a d-b c} (b e-a f) \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right )}{3 b^3 (b c-a d) (b e-a f)}-\frac{2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}-\frac{2 \sqrt{c+d x} (e+f x)^{3/2} (b B-2 a C)}{b^2 \sqrt{a+b x} (b e-a f)} \]

[Out]

(2*(8*a^2*C*d*f + b^2*(c*C*e + 3*B*c*f + A*d*f) - a*b*(C*d*e + 7*c*C*f + 4*B*d*f))*Sqrt[a + b*x]*Sqrt[c + d*x]
*Sqrt[e + f*x])/(3*b^3*(b*c - a*d)*(b*e - a*f)) - (2*(b*B - 2*a*C)*Sqrt[c + d*x]*(e + f*x)^(3/2))/(b^2*(b*e -
a*f)*Sqrt[a + b*x]) - (2*(A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(3*b*(b*c - a*d)*(b*e - a*f)
*(a + b*x)^(3/2)) + (2*(16*a^3*C*d^2*f^2 - 8*a^2*b*d*f*(B*d*f + 2*C*(d*e + c*f)) - b^3*(c^2*C*e*f + A*d^2*e*f
+ c*d*(C*e^2 + 6*B*e*f + A*f^2)) + a*b^2*(d*f*(7*B*d*e + 7*B*c*f + 2*A*d*f) + C*(d^2*e^2 + 16*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))])/(3*b^4*Sqrt[d]*Sqrt[-(b*c) + a*d]*f*(b*e - a*f)*Sqrt[c + d*x]*Sqrt[(b*(e
+ f*x))/(b*e - a*f)]) + (2*(d*e - c*f)*(8*a^2*C*d*f + b^2*(c*C*e + 3*B*c*f + A*d*f) - a*b*(C*d*e + 7*c*C*f + 4
*B*d*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))])/(3*b^4*Sqrt[d]*Sqrt[-(b*c) + a*d]*f*Sqrt[c + d*x]*S
qrt[e + f*x])

________________________________________________________________________________________

Rubi [A]  time = 1.90276, antiderivative size = 687, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {1614, 150, 154, 158, 114, 113, 121, 120} \[ \frac{2 \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (-8 a^2 b d f (B d f+2 C (c f+d e))+16 a^3 C d^2 f^2+a b^2 \left (d f (2 A d f+7 B c f+7 B d e)+C \left (c^2 f^2+16 c d e f+d^2 e^2\right )\right )-b^3 \left (c d \left (A f^2+6 B e f+C e^2\right )+A d^2 e f+c^2 C e f\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^4 \sqrt{d} f \sqrt{c+d x} \sqrt{a d-b c} (b e-a f) \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right )}{3 b^3 (b c-a d) (b e-a f)}+\frac{2 (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^4 \sqrt{d} f \sqrt{c+d x} \sqrt{e+f x} \sqrt{a d-b c}}-\frac{2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}-\frac{2 \sqrt{c+d x} (e+f x)^{3/2} (b B-2 a C)}{b^2 \sqrt{a+b x} (b e-a f)} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^(5/2),x]

[Out]

(2*(8*a^2*C*d*f + b^2*(c*C*e + 3*B*c*f + A*d*f) - a*b*(C*d*e + 7*c*C*f + 4*B*d*f))*Sqrt[a + b*x]*Sqrt[c + d*x]
*Sqrt[e + f*x])/(3*b^3*(b*c - a*d)*(b*e - a*f)) - (2*(b*B - 2*a*C)*Sqrt[c + d*x]*(e + f*x)^(3/2))/(b^2*(b*e -
a*f)*Sqrt[a + b*x]) - (2*(A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(3*b*(b*c - a*d)*(b*e - a*f)
*(a + b*x)^(3/2)) + (2*(16*a^3*C*d^2*f^2 - 8*a^2*b*d*f*(B*d*f + 2*C*(d*e + c*f)) - b^3*(c^2*C*e*f + A*d^2*e*f
+ c*d*(C*e^2 + 6*B*e*f + A*f^2)) + a*b^2*(d*f*(7*B*d*e + 7*B*c*f + 2*A*d*f) + C*(d^2*e^2 + 16*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))])/(3*b^4*Sqrt[d]*Sqrt[-(b*c) + a*d]*f*(b*e - a*f)*Sqrt[c + d*x]*Sqrt[(b*(e
+ f*x))/(b*e - a*f)]) + (2*(d*e - c*f)*(8*a^2*C*d*f + b^2*(c*C*e + 3*B*c*f + A*d*f) - a*b*(C*d*e + 7*c*C*f + 4
*B*d*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))])/(3*b^4*Sqrt[d]*Sqrt[-(b*c) + a*d]*f*Sqrt[c + d*x]*S
qrt[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 150

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && 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)^{5/2}} \, dx &=-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}-\frac{2 \int \frac{\sqrt{c+d x} \sqrt{e+f x} \left (-\frac{3 \left (b^2 B c e+a^2 C (d e+c f)-a b (c C e+B d e+B c f-A d f)\right )}{2 b}-\frac{3}{2} \left (\frac{2 a^2 C d f}{b}+b (c C e+A d f)-a (C d e+c C f+B d f)\right ) x\right )}{(a+b x)^{3/2}} \, dx}{3 (b c-a d) (b e-a f)}\\ &=-\frac{2 (b B-2 a C) \sqrt{c+d x} (e+f x)^{3/2}}{b^2 (b e-a f) \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}-\frac{4 \int \frac{\sqrt{e+f x} \left (-\frac{3 (b e-a f) \left (2 a^2 C d (d e+3 c f)+b^2 c (c C e+B d e+2 B c f+A d f)-a b \left (B d^2 e+5 c^2 C f+3 c d (C e+B f)\right )\right )}{4 b}-\frac{3 d (b e-a f) \left (8 a^2 C d f+b^2 (c C e+3 B c f+A d f)-a b (C d e+7 c C f+4 B d f)\right ) x}{4 b}\right )}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{3 b (b c-a d) (b e-a f)^2}\\ &=\frac{2 \left (8 a^2 C d f+b^2 (c C e+3 B c f+A d f)-a b (C d e+7 c C f+4 B d f)\right ) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{3 b^3 (b c-a d) (b e-a f)}-\frac{2 (b B-2 a C) \sqrt{c+d x} (e+f x)^{3/2}}{b^2 (b e-a f) \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}-\frac{8 \int \frac{\frac{3 d (b e-a f) \left (8 a^3 C d f (d e+c f)-b^3 c e (2 c C e+3 B d e+3 B c f+2 A d f)+a b^2 \left (d^2 e (3 B e+A f)+3 c^2 f (3 C e+B f)+c d \left (9 C e^2+8 B e f+A f^2\right )\right )-a^2 b \left (4 B d f (d e+c f)+C \left (7 d^2 e^2+18 c d e f+7 c^2 f^2\right )\right )\right )}{8 b}+\frac{3 d (b e-a f) \left (16 a^3 C d^2 f^2-8 a^2 b d f (B d f+2 C (d e+c f))-b^3 \left (c^2 C e f+A d^2 e f+c d \left (C e^2+6 B e f+A f^2\right )\right )+a b^2 \left (d f (7 B d e+7 B c f+2 A d f)+C \left (d^2 e^2+16 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}{9 b^2 d (b c-a d) (b e-a f)^2}\\ &=\frac{2 \left (8 a^2 C d f+b^2 (c C e+3 B c f+A d f)-a b (C d e+7 c C f+4 B d f)\right ) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{3 b^3 (b c-a d) (b e-a f)}-\frac{2 (b B-2 a C) \sqrt{c+d x} (e+f x)^{3/2}}{b^2 (b e-a f) \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}-\frac{\left ((d e-c f) \left (8 a^2 C d f+b^2 (c C e+3 B c f+A d f)-a b (C d e+7 c C f+4 B d f)\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{3 b^3 (b c-a d) f}-\frac{\left (16 a^3 C d^2 f^2-8 a^2 b d f (B d f+2 C (d e+c f))-b^3 \left (c^2 C e f+A d^2 e f+c d \left (C e^2+6 B e f+A f^2\right )\right )+a b^2 \left (d f (7 B d e+7 B c f+2 A d f)+C \left (d^2 e^2+16 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}{3 b^3 (b c-a d) f (b e-a f)}\\ &=\frac{2 \left (8 a^2 C d f+b^2 (c C e+3 B c f+A d f)-a b (C d e+7 c C f+4 B d f)\right ) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{3 b^3 (b c-a d) (b e-a f)}-\frac{2 (b B-2 a C) \sqrt{c+d x} (e+f x)^{3/2}}{b^2 (b e-a f) \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}-\frac{\left ((d e-c f) \left (8 a^2 C d f+b^2 (c C e+3 B c f+A d f)-a b (C d e+7 c C f+4 B d 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}{3 b^3 (b c-a d) f \sqrt{c+d x}}-\frac{\left (\left (16 a^3 C d^2 f^2-8 a^2 b d f (B d f+2 C (d e+c f))-b^3 \left (c^2 C e f+A d^2 e f+c d \left (C e^2+6 B e f+A f^2\right )\right )+a b^2 \left (d f (7 B d e+7 B c f+2 A d f)+C \left (d^2 e^2+16 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}{3 b^3 (b c-a d) f (b e-a f) \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}\\ &=\frac{2 \left (8 a^2 C d f+b^2 (c C e+3 B c f+A d f)-a b (C d e+7 c C f+4 B d f)\right ) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{3 b^3 (b c-a d) (b e-a f)}-\frac{2 (b B-2 a C) \sqrt{c+d x} (e+f x)^{3/2}}{b^2 (b e-a f) \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac{2 \left (16 a^3 C d^2 f^2-8 a^2 b d f (B d f+2 C (d e+c f))-b^3 \left (c^2 C e f+A d^2 e f+c d \left (C e^2+6 B e f+A f^2\right )\right )+a b^2 \left (d f (7 B d e+7 B c f+2 A d f)+C \left (d^2 e^2+16 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 )}{3 b^4 \sqrt{d} \sqrt{-b c+a d} f (b e-a f) \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{\left ((d e-c f) \left (8 a^2 C d f+b^2 (c C e+3 B c f+A d f)-a b (C d e+7 c C f+4 B d 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}{3 b^3 (b c-a d) f \sqrt{c+d x} \sqrt{e+f x}}\\ &=\frac{2 \left (8 a^2 C d f+b^2 (c C e+3 B c f+A d f)-a b (C d e+7 c C f+4 B d f)\right ) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{3 b^3 (b c-a d) (b e-a f)}-\frac{2 (b B-2 a C) \sqrt{c+d x} (e+f x)^{3/2}}{b^2 (b e-a f) \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac{2 \left (16 a^3 C d^2 f^2-8 a^2 b d f (B d f+2 C (d e+c f))-b^3 \left (c^2 C e f+A d^2 e f+c d \left (C e^2+6 B e f+A f^2\right )\right )+a b^2 \left (d f (7 B d e+7 B c f+2 A d f)+C \left (d^2 e^2+16 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 )}{3 b^4 \sqrt{d} \sqrt{-b c+a d} f (b e-a f) \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 (d e-c f) \left (8 a^2 C d f+b^2 (c C e+3 B c f+A d f)-a b (C d e+7 c C f+4 B d 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 )}{3 b^4 \sqrt{d} \sqrt{-b c+a d} f \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}

Mathematica [C]  time = 13.4094, size = 938, normalized size = 1.37 \[ \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left (\frac{2 C}{3 b^3}-\frac{2 \left (-8 C d f a^3+7 b C d e a^2+7 b c C f a^2+5 b B d f a^2-6 b^2 c C e a-4 b^2 B d e a-4 b^2 B c f a-2 A b^2 d f a+3 b^3 B c e+A b^3 d e+A b^3 c f\right )}{3 b^3 (b c-a d) (b e-a f) (a+b x)}-\frac{2 \left (C a^2-b B a+A b^2\right )}{3 b^3 (a+b x)^2}\right )-\frac{2 (a+b x)^{3/2} \left (-\sqrt{\frac{b c}{d}-a} \left (-16 C d^2 f^2 a^3+8 b d f (B d f+2 C (d e+c f)) a^2-b^2 \left (d f (7 B d e+7 B c f+2 A d f)+C \left (d^2 e^2+16 c d f e+c^2 f^2\right )\right ) a+b^3 \left (C e f c^2+d \left (C e^2+6 B f e+A f^2\right ) c+A d^2 e f\right )\right ) \left (\frac{b c}{a+b x}+d-\frac{a d}{a+b x}\right ) \left (\frac{b e}{a+b x}+f-\frac{a f}{a+b x}\right )+\frac{i (a d-b c) f \left (-16 C d^2 f^2 a^3+8 b d f (B d f+2 C (d e+c f)) a^2-b^2 \left (d f (7 B d e+7 B c f+2 A d f)+C \left (d^2 e^2+16 c d f e+c^2 f^2\right )\right ) a+b^3 \left (C e f c^2+d \left (C e^2+6 B f e+A f^2\right ) c+A d^2 e f\right )\right ) \sqrt{-\frac{a}{a+b x}+\frac{b c}{d (a+b x)}+1} \sqrt{-\frac{a}{a+b x}+\frac{b e}{f (a+b x)}+1} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )}{\sqrt{a+b x}}+\frac{i b (a d-b c) f (d e-c f) \left (8 C d f a^2-b (7 C d e+c C f+4 B d f) a+b^2 (c C e+3 B d e+A d f)\right ) \sqrt{-\frac{a}{a+b x}+\frac{b c}{d (a+b x)}+1} \sqrt{-\frac{a}{a+b x}+\frac{b e}{f (a+b x)}+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right ),\frac{b d e-a d f}{b c f-a d f}\right )}{\sqrt{a+b x}}\right )}{3 b^5 \sqrt{\frac{b c}{d}-a} d (b c-a d) f (b e-a f) \sqrt{c+\frac{(a+b x) \left (d-\frac{a d}{a+b x}\right )}{b}} \sqrt{e+\frac{(a+b x) \left (f-\frac{a f}{a+b x}\right )}{b}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^(5/2),x]

[Out]

Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*((2*C)/(3*b^3) - (2*(A*b^2 - a*b*B + a^2*C))/(3*b^3*(a + b*x)^2) - (
2*(3*b^3*B*c*e - 6*a*b^2*c*C*e + A*b^3*d*e - 4*a*b^2*B*d*e + 7*a^2*b*C*d*e + A*b^3*c*f - 4*a*b^2*B*c*f + 7*a^2
*b*c*C*f - 2*a*A*b^2*d*f + 5*a^2*b*B*d*f - 8*a^3*C*d*f))/(3*b^3*(b*c - a*d)*(b*e - a*f)*(a + b*x))) - (2*(a +
b*x)^(3/2)*(-(Sqrt[-a + (b*c)/d]*(-16*a^3*C*d^2*f^2 + 8*a^2*b*d*f*(B*d*f + 2*C*(d*e + c*f)) + b^3*(c^2*C*e*f +
A*d^2*e*f + c*d*(C*e^2 + 6*B*e*f + A*f^2)) - a*b^2*(d*f*(7*B*d*e + 7*B*c*f + 2*A*d*f) + C*(d^2*e^2 + 16*c*d*e
*f + c^2*f^2)))*(d + (b*c)/(a + b*x) - (a*d)/(a + b*x))*(f + (b*e)/(a + b*x) - (a*f)/(a + b*x))) + (I*(-(b*c)
+ a*d)*f*(-16*a^3*C*d^2*f^2 + 8*a^2*b*d*f*(B*d*f + 2*C*(d*e + c*f)) + b^3*(c^2*C*e*f + A*d^2*e*f + c*d*(C*e^2
+ 6*B*e*f + A*f^2)) - a*b^2*(d*f*(7*B*d*e + 7*B*c*f + 2*A*d*f) + C*(d^2*e^2 + 16*c*d*e*f + c^2*f^2)))*Sqrt[1 -
a/(a + b*x) + (b*c)/(d*(a + b*x))]*Sqrt[1 - a/(a + b*x) + (b*e)/(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)])/Sqrt[a + b*x] + (I*b*(-(b*c) + a*d)*f*(d*e - c*f)*(
8*a^2*C*d*f + b^2*(c*C*e + 3*B*d*e + A*d*f) - a*b*(7*C*d*e + c*C*f + 4*B*d*f))*Sqrt[1 - a/(a + b*x) + (b*c)/(d
*(a + b*x))]*Sqrt[1 - a/(a + b*x) + (b*e)/(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)])/Sqrt[a + b*x]))/(3*b^5*Sqrt[-a + (b*c)/d]*d*(b*c - a*d)*f*(b*e - a*f)*Sqrt
[c + ((a + b*x)*(d - (a*d)/(a + b*x)))/b]*Sqrt[e + ((a + b*x)*(f - (a*f)/(a + b*x)))/b])

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Maple [B]  time = 0.096, size = 16177, normalized size = 23.6 \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)^(5/2),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{5}{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)^(5/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a)^(5/2), 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^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}, 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)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x**2+B*x+A)*(d*x+c)**(1/2)*(f*x+e)**(1/2)/(b*x+a)**(5/2),x)

[Out]

Timed out

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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{5}{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)^(5/2),x, algorithm="giac")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a)^(5/2), x)