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

Optimal. Leaf size=1716 \[ \text{result too large to display} \]

[Out]

(-2*(24*a^4*C*d^2*f^2 - a^3*b*d*f*(61*C*d*e + 43*c*C*f - 4*B*d*f) - 3*a*b^3*(d^2*e*(B*e - 3*A*f) + 2*c^2*f*(7*
C*e - B*f) + c*d*(28*C*e^2 - 5*B*e*f + 5*A*f^2)) - b^4*(4*A*d^2*e^2 - c*d*e*(7*B*e - A*f) - c^2*(35*C*e^2 - 14
*B*e*f + 8*A*f^2)) - 3*a^2*b^2*(d*f*(3*B*d*e + 2*B*c*f - A*d*f) - C*(15*d^2*e^2 + 37*c*d*e*f + 5*c^2*f^2)))*Sq
rt[c + d*x]*Sqrt[e + f*x])/(105*b^3*(b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)^(3/2)) + (2*(48*a^5*C*d^3*f^3 + 8*a^
4*b*d^2*f^2*(B*d*f - 16*C*(d*e + c*f)) - b^5*(8*A*d^3*e^3 - c*d^2*e^2*(14*B*e + 5*A*f) + c^2*d*e*(35*C*e^2 + 1
4*B*e*f - 5*A*f^2) + c^3*f*(35*C*e^2 - 14*B*e*f + 8*A*f^2)) - a*b^4*(d^3*e^2*(6*B*e - 19*A*f) - 6*c^3*f^2*(7*C
*e - B*f) - c^2*d*f*(238*C*e^2 - 19*f*(B*e - A*f)) - c*d^2*e*(42*C*e^2 - f*(19*B*e + 20*A*f))) + a^3*b^2*d*f*(
C*(103*d^2*e^2 + 344*c*d*e*f + 103*c^2*f^2) + d*f*(6*A*d*f - 19*B*(d*e + c*f))) - 3*a^2*b^3*(C*(5*d^3*e^3 + 94
*c*d^2*e^2*f + 94*c^2*d*e*f^2 + 5*c^3*f^3) + d*f*(3*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 16*c*d*e*f + 3*c^2*f^2)
)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(105*b^3*(b*c - a*d)^3*(b*e - a*f)^3*Sqrt[a + b*x]) + (2*(6*a^3*C*d*f + a*b^2
*(14*c*C*e + 3*B*d*e + 3*B*c*f - 8*A*d*f) - b^3*(7*B*c*e - 4*A*(d*e + c*f)) + a^2*b*(B*d*f - 10*C*(d*e + c*f))
)*Sqrt[c + d*x]*(e + f*x)^(3/2))/(35*b^2*(b*c - a*d)*(b*e - a*f)^2*(a + b*x)^(5/2)) - (2*(A*b^2 - a*(b*B - a*C
))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(7*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(7/2)) + (2*Sqrt[d]*(48*a^5*C*d^3*f
^3 + 8*a^4*b*d^2*f^2*(B*d*f - 16*C*(d*e + c*f)) - b^5*(8*A*d^3*e^3 - c*d^2*e^2*(14*B*e + 5*A*f) + c^2*d*e*(35*
C*e^2 + 14*B*e*f - 5*A*f^2) + c^3*f*(35*C*e^2 - 14*B*e*f + 8*A*f^2)) - a*b^4*(d^3*e^2*(6*B*e - 19*A*f) - 6*c^3
*f^2*(7*C*e - B*f) - c^2*d*f*(238*C*e^2 - 19*f*(B*e - A*f)) - c*d^2*e*(42*C*e^2 - f*(19*B*e + 20*A*f))) + a^3*
b^2*d*f*(C*(103*d^2*e^2 + 344*c*d*e*f + 103*c^2*f^2) + d*f*(6*A*d*f - 19*B*(d*e + c*f))) - 3*a^2*b^3*(C*(5*d^3
*e^3 + 94*c*d^2*e^2*f + 94*c^2*d*e*f^2 + 5*c^3*f^3) + d*f*(3*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 16*c*d*e*f + 3
*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))])/(105*b^4*(-(b*c) + a*d)^(5/2)*(b*e - a*f)^3*Sqrt[c + d*x]*Sqrt[(b
*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[d]*(d*e - c*f)*(24*a^4*C*d^2*f^2 - a^3*b*d*f*(43*C*d*e + 61*c*C*f - 4*B*d*
f) + b^4*(8*A*d^2*e^2 - c*d*e*(14*B*e + A*f) + c^2*(35*C*e^2 + 7*B*e*f - 4*A*f^2)) + 3*a*b^3*(d^2*e*(2*B*e - 5
*A*f) - c^2*f*(28*C*e + B*f) - c*d*(14*C*e^2 - 5*B*e*f - 3*A*f^2)) - 3*a^2*b^2*(d*f*(2*B*d*e + 3*B*c*f - A*d*f
) - C*(5*d^2*e^2 + 37*c*d*e*f + 15*c^2*f^2)))*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))])/(105*b^4*(-(b*
c) + a*d)^(5/2)*(b*e - a*f)^2*Sqrt[c + d*x]*Sqrt[e + f*x])

________________________________________________________________________________________

Rubi [A]  time = 7.04535, antiderivative size = 1716, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {1614, 150, 152, 158, 114, 113, 121, 120} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^(9/2),x]

[Out]

(-2*(24*a^4*C*d^2*f^2 - a^3*b*d*f*(61*C*d*e + 43*c*C*f - 4*B*d*f) - 3*a*b^3*(d^2*e*(B*e - 3*A*f) + 2*c^2*f*(7*
C*e - B*f) + c*d*(28*C*e^2 - 5*B*e*f + 5*A*f^2)) - b^4*(4*A*d^2*e^2 - c*d*e*(7*B*e - A*f) - c^2*(35*C*e^2 - 14
*B*e*f + 8*A*f^2)) - 3*a^2*b^2*(d*f*(3*B*d*e + 2*B*c*f - A*d*f) - C*(15*d^2*e^2 + 37*c*d*e*f + 5*c^2*f^2)))*Sq
rt[c + d*x]*Sqrt[e + f*x])/(105*b^3*(b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)^(3/2)) + (2*(48*a^5*C*d^3*f^3 + 8*a^
4*b*d^2*f^2*(B*d*f - 16*C*(d*e + c*f)) - b^5*(8*A*d^3*e^3 - c*d^2*e^2*(14*B*e + 5*A*f) + c^2*d*e*(35*C*e^2 + 1
4*B*e*f - 5*A*f^2) + c^3*f*(35*C*e^2 - 14*B*e*f + 8*A*f^2)) - a*b^4*(d^3*e^2*(6*B*e - 19*A*f) - 6*c^3*f^2*(7*C
*e - B*f) - c^2*d*f*(238*C*e^2 - 19*f*(B*e - A*f)) - c*d^2*e*(42*C*e^2 - f*(19*B*e + 20*A*f))) + a^3*b^2*d*f*(
C*(103*d^2*e^2 + 344*c*d*e*f + 103*c^2*f^2) + d*f*(6*A*d*f - 19*B*(d*e + c*f))) - 3*a^2*b^3*(C*(5*d^3*e^3 + 94
*c*d^2*e^2*f + 94*c^2*d*e*f^2 + 5*c^3*f^3) + d*f*(3*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 16*c*d*e*f + 3*c^2*f^2)
)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(105*b^3*(b*c - a*d)^3*(b*e - a*f)^3*Sqrt[a + b*x]) + (2*(6*a^3*C*d*f + a*b^2
*(14*c*C*e + 3*B*d*e + 3*B*c*f - 8*A*d*f) - b^3*(7*B*c*e - 4*A*(d*e + c*f)) + a^2*b*(B*d*f - 10*C*(d*e + c*f))
)*Sqrt[c + d*x]*(e + f*x)^(3/2))/(35*b^2*(b*c - a*d)*(b*e - a*f)^2*(a + b*x)^(5/2)) - (2*(A*b^2 - a*(b*B - a*C
))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(7*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(7/2)) + (2*Sqrt[d]*(48*a^5*C*d^3*f
^3 + 8*a^4*b*d^2*f^2*(B*d*f - 16*C*(d*e + c*f)) - b^5*(8*A*d^3*e^3 - c*d^2*e^2*(14*B*e + 5*A*f) + c^2*d*e*(35*
C*e^2 + 14*B*e*f - 5*A*f^2) + c^3*f*(35*C*e^2 - 14*B*e*f + 8*A*f^2)) - a*b^4*(d^3*e^2*(6*B*e - 19*A*f) - 6*c^3
*f^2*(7*C*e - B*f) - c^2*d*f*(238*C*e^2 - 19*f*(B*e - A*f)) - c*d^2*e*(42*C*e^2 - f*(19*B*e + 20*A*f))) + a^3*
b^2*d*f*(C*(103*d^2*e^2 + 344*c*d*e*f + 103*c^2*f^2) + d*f*(6*A*d*f - 19*B*(d*e + c*f))) - 3*a^2*b^3*(C*(5*d^3
*e^3 + 94*c*d^2*e^2*f + 94*c^2*d*e*f^2 + 5*c^3*f^3) + d*f*(3*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 16*c*d*e*f + 3
*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))])/(105*b^4*(-(b*c) + a*d)^(5/2)*(b*e - a*f)^3*Sqrt[c + d*x]*Sqrt[(b
*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[d]*(d*e - c*f)*(24*a^4*C*d^2*f^2 - a^3*b*d*f*(43*C*d*e + 61*c*C*f - 4*B*d*
f) + b^4*(8*A*d^2*e^2 - c*d*e*(14*B*e + A*f) + c^2*(35*C*e^2 + 7*B*e*f - 4*A*f^2)) + 3*a*b^3*(d^2*e*(2*B*e - 5
*A*f) - c^2*f*(28*C*e + B*f) - c*d*(14*C*e^2 - 5*B*e*f - 3*A*f^2)) - 3*a^2*b^2*(d*f*(2*B*d*e + 3*B*c*f - A*d*f
) - C*(5*d^2*e^2 + 37*c*d*e*f + 15*c^2*f^2)))*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))])/(105*b^4*(-(b*
c) + a*d)^(5/2)*(b*e - a*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 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 152

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 + 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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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)^{9/2}} \, dx &=-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}-\frac{2 \int \frac{\sqrt{c+d x} \sqrt{e+f x} \left (-\frac{3 a^2 C (d e+c f)-a b (7 c C e+3 B d e+3 B c f-7 A d f)+b^2 (7 B c e-4 A (d e+c f))}{2 b}+\frac{1}{2} \left (-7 b c C e+7 a C d e+7 a c C f+A b d f-a B d f-\frac{6 a^2 C d f}{b}\right ) x\right )}{(a+b x)^{7/2}} \, dx}{7 (b c-a d) (b e-a f)}\\ &=\frac{2 \left (6 a^3 C d f+a b^2 (14 c C e+3 B d e+3 B c f-8 A d f)-b^3 (7 B c e-4 A (d e+c f))+a^2 b (B d f-10 C (d e+c f))\right ) \sqrt{c+d x} (e+f x)^{3/2}}{35 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{5/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}-\frac{4 \int \frac{\sqrt{e+f x} \left (\frac{6 a^3 C d f (d e+3 c f)+b^3 \left (4 A d^2 e^2-c d e (7 B e-A f)-c^2 \left (35 C e^2-14 B e f+8 A f^2\right )\right )+a b^2 \left (d^2 e (3 B e-8 A f)+6 c^2 f (7 C e-B f)+c d \left (49 C e^2-8 B e f+11 A f^2\right )\right )+a^2 b \left (B d f (d e+3 c f)-5 C \left (2 d^2 e^2+11 c d e f+3 c^2 f^2\right )\right )}{4 b}+\frac{d \left (24 a^3 C d f^2-b^3 \left (35 c C e^2-A d e f-c f (7 B e-4 A f)\right )+a^2 b f (4 B d f-5 C (11 d e+5 c f))+a b^2 (7 C e (5 d e+8 c f)-f (8 B d e+3 B c f-3 A d f))\right ) x}{4 b}\right )}{(a+b x)^{5/2} \sqrt{c+d x}} \, dx}{35 b (b c-a d) (b e-a f)^2}\\ &=-\frac{2 \left (24 a^4 C d^2 f^2-a^3 b d f (61 C d e+43 c C f-4 B d f)-3 a b^3 \left (d^2 e (B e-3 A f)+2 c^2 f (7 C e-B f)+c d \left (28 C e^2-5 B e f+5 A f^2\right )\right )-b^4 \left (4 A d^2 e^2-c d e (7 B e-A f)-c^2 \left (35 C e^2-14 B e f+8 A f^2\right )\right )-3 a^2 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d e f+5 c^2 f^2\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \left (6 a^3 C d f+a b^2 (14 c C e+3 B d e+3 B c f-8 A d f)-b^3 (7 B c e-4 A (d e+c f))+a^2 b (B d f-10 C (d e+c f))\right ) \sqrt{c+d x} (e+f x)^{3/2}}{35 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{5/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}-\frac{8 \int \frac{-\frac{24 a^4 C d^2 f^2 (d e+c f)+3 a b^3 \left (d^3 e^2 (2 B e-5 A f)-2 c^3 f^2 (7 C e-B f)-2 c d^2 e \left (7 C e^2-2 B e f-3 A f^2\right )-c^2 d f \left (56 C e^2-4 B e f+5 A f^2\right )\right )+b^4 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )+a^3 b d f \left (4 B d f (d e+c f)-C \left (43 d^2 e^2+122 c d e f+43 c^2 f^2\right )\right )+3 a^2 b^2 \left (C \left (5 d^3 e^3+52 c d^2 e^2 f+52 c^2 d e f^2+5 c^3 f^3\right )+d f \left (A d f (d e+c f)-2 B \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right )\right )}{8 b}-\frac{d f \left (48 a^4 C d^2 f^2+8 a^3 b d f (B d f-13 C (d e+c f))+3 a b^3 \left (d^2 e (B e-2 A f)-c^2 f (42 C e-B f)-2 c d \left (21 C e^2-5 B e f+A f^2\right )\right )+b^4 \left (4 A d^2 e^2-c d e (7 B e+2 A f)+c^2 \left (70 C e^2-7 B e f+4 A f^2\right )\right )+3 a^2 b^2 \left (C \left (20 d^2 e^2+74 c d e f+20 c^2 f^2\right )+d f (2 A d f-5 B (d e+c f))\right )\right ) x}{8 b}}{(a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{105 b^2 (b c-a d)^2 (b e-a f)^2}\\ &=-\frac{2 \left (24 a^4 C d^2 f^2-a^3 b d f (61 C d e+43 c C f-4 B d f)-3 a b^3 \left (d^2 e (B e-3 A f)+2 c^2 f (7 C e-B f)+c d \left (28 C e^2-5 B e f+5 A f^2\right )\right )-b^4 \left (4 A d^2 e^2-c d e (7 B e-A f)-c^2 \left (35 C e^2-14 B e f+8 A f^2\right )\right )-3 a^2 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d e f+5 c^2 f^2\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 (b c-a d)^3 (b e-a f)^3 \sqrt{a+b x}}+\frac{2 \left (6 a^3 C d f+a b^2 (14 c C e+3 B d e+3 B c f-8 A d f)-b^3 (7 B c e-4 A (d e+c f))+a^2 b (B d f-10 C (d e+c f))\right ) \sqrt{c+d x} (e+f x)^{3/2}}{35 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{5/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}+\frac{16 \int \frac{-\frac{d f \left (24 a^5 C d^2 f^2 (d e+c f)-3 a^2 b^3 \left (d^3 e^2 (B e-3 A f)+c^3 f^2 (48 C e+B f)+3 c^2 d f \left (34 C e^2-4 B e f-A f^2\right )+12 c d^2 e \left (4 C e^2-B e f+A f^2\right )\right )-b^5 c e \left (4 A d^2 e^2-c d e (7 B e+2 A f)+c^2 \left (70 C e^2-7 B e f+4 A f^2\right )\right )-a b^4 \left (4 A d^3 e^3-c d^2 e^2 (4 B e+13 A f)-c^3 f \left (161 C e^2+4 B e f-4 A f^2\right )-c^2 d e \left (161 C e^2-58 B e f+13 A f^2\right )\right )+a^4 b d f \left (4 B d f (d e+c f)-C \left (61 d^2 e^2+134 c d e f+61 c^2 f^2\right )\right )+a^3 b^2 \left (5 C \left (9 d^3 e^3+46 c d^2 e^2 f+46 c^2 d e f^2+9 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (9 d^2 e^2+20 c d e f+9 c^2 f^2\right )\right )\right )\right )}{16 b}-\frac{d f \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right )\right )\right ) x}{16 b}}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{105 b^2 (b c-a d)^3 (b e-a f)^3}\\ &=-\frac{2 \left (24 a^4 C d^2 f^2-a^3 b d f (61 C d e+43 c C f-4 B d f)-3 a b^3 \left (d^2 e (B e-3 A f)+2 c^2 f (7 C e-B f)+c d \left (28 C e^2-5 B e f+5 A f^2\right )\right )-b^4 \left (4 A d^2 e^2-c d e (7 B e-A f)-c^2 \left (35 C e^2-14 B e f+8 A f^2\right )\right )-3 a^2 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d e f+5 c^2 f^2\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 (b c-a d)^3 (b e-a f)^3 \sqrt{a+b x}}+\frac{2 \left (6 a^3 C d f+a b^2 (14 c C e+3 B d e+3 B c f-8 A d f)-b^3 (7 B c e-4 A (d e+c f))+a^2 b (B d f-10 C (d e+c f))\right ) \sqrt{c+d x} (e+f x)^{3/2}}{35 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{5/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}-\frac{\left (d (d e-c f) \left (24 a^4 C d^2 f^2-a^3 b d f (43 C d e+61 c C f-4 B d f)+b^4 \left (8 A d^2 e^2-c d e (14 B e+A f)+c^2 \left (35 C e^2+7 B e f-4 A f^2\right )\right )+3 a b^3 \left (d^2 e (2 B e-5 A f)-c^2 f (28 C e+B f)-c d \left (14 C e^2-5 B e f-3 A f^2\right )\right )-3 a^2 b^2 \left (d f (2 B d e+3 B c f-A d f)-C \left (5 d^2 e^2+37 c d e f+15 c^2 f^2\right )\right )\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{105 b^3 (b c-a d)^3 (b e-a f)^2}-\frac{\left (d \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right )\right )\right )\right ) \int \frac{\sqrt{e+f x}}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{105 b^3 (b c-a d)^3 (b e-a f)^3}\\ &=-\frac{2 \left (24 a^4 C d^2 f^2-a^3 b d f (61 C d e+43 c C f-4 B d f)-3 a b^3 \left (d^2 e (B e-3 A f)+2 c^2 f (7 C e-B f)+c d \left (28 C e^2-5 B e f+5 A f^2\right )\right )-b^4 \left (4 A d^2 e^2-c d e (7 B e-A f)-c^2 \left (35 C e^2-14 B e f+8 A f^2\right )\right )-3 a^2 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d e f+5 c^2 f^2\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 (b c-a d)^3 (b e-a f)^3 \sqrt{a+b x}}+\frac{2 \left (6 a^3 C d f+a b^2 (14 c C e+3 B d e+3 B c f-8 A d f)-b^3 (7 B c e-4 A (d e+c f))+a^2 b (B d f-10 C (d e+c f))\right ) \sqrt{c+d x} (e+f x)^{3/2}}{35 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{5/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}-\frac{\left (d (d e-c f) \left (24 a^4 C d^2 f^2-a^3 b d f (43 C d e+61 c C f-4 B d f)+b^4 \left (8 A d^2 e^2-c d e (14 B e+A f)+c^2 \left (35 C e^2+7 B e f-4 A f^2\right )\right )+3 a b^3 \left (d^2 e (2 B e-5 A f)-c^2 f (28 C e+B f)-c d \left (14 C e^2-5 B e f-3 A f^2\right )\right )-3 a^2 b^2 \left (d f (2 B d e+3 B c f-A d f)-C \left (5 d^2 e^2+37 c d e f+15 c^2 f^2\right )\right )\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}{105 b^3 (b c-a d)^3 (b e-a f)^2 \sqrt{c+d x}}-\frac{\left (d \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\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}{105 b^3 (b c-a d)^3 (b e-a f)^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}\\ &=-\frac{2 \left (24 a^4 C d^2 f^2-a^3 b d f (61 C d e+43 c C f-4 B d f)-3 a b^3 \left (d^2 e (B e-3 A f)+2 c^2 f (7 C e-B f)+c d \left (28 C e^2-5 B e f+5 A f^2\right )\right )-b^4 \left (4 A d^2 e^2-c d e (7 B e-A f)-c^2 \left (35 C e^2-14 B e f+8 A f^2\right )\right )-3 a^2 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d e f+5 c^2 f^2\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 (b c-a d)^3 (b e-a f)^3 \sqrt{a+b x}}+\frac{2 \left (6 a^3 C d f+a b^2 (14 c C e+3 B d e+3 B c f-8 A d f)-b^3 (7 B c e-4 A (d e+c f))+a^2 b (B d f-10 C (d e+c f))\right ) \sqrt{c+d x} (e+f x)^{3/2}}{35 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{5/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}+\frac{2 \sqrt{d} \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\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 )}{105 b^4 (-b c+a d)^{5/2} (b e-a f)^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{\left (d (d e-c f) \left (24 a^4 C d^2 f^2-a^3 b d f (43 C d e+61 c C f-4 B d f)+b^4 \left (8 A d^2 e^2-c d e (14 B e+A f)+c^2 \left (35 C e^2+7 B e f-4 A f^2\right )\right )+3 a b^3 \left (d^2 e (2 B e-5 A f)-c^2 f (28 C e+B f)-c d \left (14 C e^2-5 B e f-3 A f^2\right )\right )-3 a^2 b^2 \left (d f (2 B d e+3 B c f-A d f)-C \left (5 d^2 e^2+37 c d e f+15 c^2 f^2\right )\right )\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}{105 b^3 (b c-a d)^3 (b e-a f)^2 \sqrt{c+d x} \sqrt{e+f x}}\\ &=-\frac{2 \left (24 a^4 C d^2 f^2-a^3 b d f (61 C d e+43 c C f-4 B d f)-3 a b^3 \left (d^2 e (B e-3 A f)+2 c^2 f (7 C e-B f)+c d \left (28 C e^2-5 B e f+5 A f^2\right )\right )-b^4 \left (4 A d^2 e^2-c d e (7 B e-A f)-c^2 \left (35 C e^2-14 B e f+8 A f^2\right )\right )-3 a^2 b^2 \left (d f (3 B d e+2 B c f-A d f)-C \left (15 d^2 e^2+37 c d e f+5 c^2 f^2\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 (b c-a d)^3 (b e-a f)^3 \sqrt{a+b x}}+\frac{2 \left (6 a^3 C d f+a b^2 (14 c C e+3 B d e+3 B c f-8 A d f)-b^3 (7 B c e-4 A (d e+c f))+a^2 b (B d f-10 C (d e+c f))\right ) \sqrt{c+d x} (e+f x)^{3/2}}{35 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{5/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{7 b (b c-a d) (b e-a f) (a+b x)^{7/2}}+\frac{2 \sqrt{d} \left (48 a^5 C d^3 f^3+8 a^4 b d^2 f^2 (B d f-16 C (d e+c f))-b^5 \left (8 A d^3 e^3-c d^2 e^2 (14 B e+5 A f)+c^2 d e \left (35 C e^2+14 B e f-5 A f^2\right )+c^3 f \left (35 C e^2-14 B e f+8 A f^2\right )\right )-a b^4 \left (d^3 e^2 (6 B e-19 A f)-6 c^3 f^2 (7 C e-B f)-c^2 d f \left (238 C e^2-19 f (B e-A f)\right )-c d^2 e \left (42 C e^2-f (19 B e+20 A f)\right )\right )+a^3 b^2 d f \left (C \left (103 d^2 e^2+344 c d e f+103 c^2 f^2\right )+d f (6 A d f-19 B (d e+c f))\right )-3 a^2 b^3 \left (C \left (5 d^3 e^3+94 c d^2 e^2 f+94 c^2 d e f^2+5 c^3 f^3\right )+d f \left (3 A d f (d e+c f)-B \left (3 d^2 e^2+16 c d e f+3 c^2 f^2\right )\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 )}{105 b^4 (-b c+a d)^{5/2} (b e-a f)^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 \sqrt{d} (d e-c f) \left (24 a^4 C d^2 f^2-a^3 b d f (43 C d e+61 c C f-4 B d f)+b^4 \left (8 A d^2 e^2-c d e (14 B e+A f)+c^2 \left (35 C e^2+7 B e f-4 A f^2\right )\right )+3 a b^3 \left (d^2 e (2 B e-5 A f)-c^2 f (28 C e+B f)-c d \left (14 C e^2-5 B e f-3 A f^2\right )\right )-3 a^2 b^2 \left (d f (2 B d e+3 B c f-A d f)-C \left (5 d^2 e^2+37 c d e f+15 c^2 f^2\right )\right )\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 )}{105 b^4 (-b c+a d)^{5/2} (b e-a f)^2 \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}

Mathematica [C]  time = 19.4264, size = 15719, normalized size = 9.16 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^(9/2),x]

[Out]

Result too large to show

________________________________________________________________________________________

Maple [B]  time = 0.36, size = 68345, normalized size = 39.8 \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)^(9/2),x)

[Out]

result too large to display

________________________________________________________________________________________

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{9}{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)^(9/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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^{5} x^{5} + 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5}}, 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)^(9/2),x, algorithm="fricas")

[Out]

integral((C*x^2 + B*x + A)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)/(b^5*x^5 + 5*a*b^4*x^4 + 10*a^2*b^3*x^3 +
10*a^3*b^2*x^2 + 5*a^4*b*x + a^5), x)

________________________________________________________________________________________

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)**(9/2),x)

[Out]

Timed out

________________________________________________________________________________________

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{9}{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)^(9/2),x, algorithm="giac")

[Out]

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