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

Optimal. Leaf size=774 \[ -\frac{2 \sqrt{a d-b c} (b e-a f) (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (24 a^2 C d^2 f^2+a b d f (-28 B d f-5 c C f+13 C d e)+b^2 \left (-\left (7 d f (-5 A d f-B c f+2 B d e)-C \left (-4 c^2 f^2-c d e f+8 d^2 e^2\right )\right )\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right ),\frac{f (b c-a d)}{d (b e-a f)}\right )}{105 b^4 d^{5/2} f^3 \sqrt{c+d x} \sqrt{e+f x}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))-(4 a d f-b c f+2 b d e) (6 a C d f-b (7 B d f-4 C (c f+d e))))}{105 b^3 d^2 f^2}-\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (3 b d f (5 b c f (3 a C (c f+d e)+b (c C e-7 A d f))-(3 a c f+a d e+b c e) (6 a C d f-b (7 B d f-4 C (c f+d e))))+2 \left (\frac{b d e}{2}-f (a d+b c)\right ) (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))-(4 a d f-b c f+2 b d e) (6 a C d f-b (7 B d f-4 C (c f+d e))))\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^4 d^{5/2} f^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2} (6 a C d f-b (7 B d f-4 C (c f+d e)))}{35 b^2 d f^2}+\frac{2 C \sqrt{a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f} \]

[Out]

(-2*(5*b*d*f*(3*a*C*(d*e + c*f) + b*(c*C*e - 7*A*d*f)) - (2*b*d*e - b*c*f + 4*a*d*f)*(6*a*C*d*f - b*(7*B*d*f -
4*C*(d*e + c*f))))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(105*b^3*d^2*f^2) - (2*(6*a*C*d*f - b*(7*B*d*f
- 4*C*(d*e + c*f)))*Sqrt[a + b*x]*Sqrt[c + d*x]*(e + f*x)^(3/2))/(35*b^2*d*f^2) + (2*C*Sqrt[a + b*x]*(c + d*x)
^(3/2)*(e + f*x)^(3/2))/(7*b*d*f) - (2*Sqrt[-(b*c) + a*d]*(3*b*d*f*(5*b*c*f*(3*a*C*(d*e + c*f) + b*(c*C*e - 7*
A*d*f)) - (b*c*e + a*d*e + 3*a*c*f)*(6*a*C*d*f - b*(7*B*d*f - 4*C*(d*e + c*f)))) + 2*((b*d*e)/2 - (b*c + a*d)*
f)*(5*b*d*f*(3*a*C*(d*e + c*f) + b*(c*C*e - 7*A*d*f)) - (2*b*d*e - b*c*f + 4*a*d*f)*(6*a*C*d*f - b*(7*B*d*f -
4*C*(d*e + c*f)))))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqr
t[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(105*b^4*d^(5/2)*f^3*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e
- a*f)]) - (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(d*e - c*f)*(24*a^2*C*d^2*f^2 + a*b*d*f*(13*C*d*e - 5*c*C*f - 28
*B*d*f) - b^2*(7*d*f*(2*B*d*e - B*c*f - 5*A*d*f) - C*(8*d^2*e^2 - c*d*e*f - 4*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*d^(5/2)*f^3*Sqrt[c + d*x]*Sqrt[e + f*x])

________________________________________________________________________________________

Rubi [A]  time = 2.23008, antiderivative size = 769, normalized size of antiderivative = 0.99, 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 = {1615, 154, 158, 114, 113, 121, 120} \[ -\frac{2 \sqrt{a d-b c} (b e-a f) (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (24 a^2 C d^2 f^2+a b d f (-28 B d f-5 c C f+13 C d e)+b^2 \left (-\left (7 d f (-5 A d f-B c f+2 B d e)-C \left (-4 c^2 f^2-c d e f+8 d^2 e^2\right )\right )\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^4 d^{5/2} f^3 \sqrt{c+d x} \sqrt{e+f x}}-\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (3 b d f (5 b c f (3 a C (c f+d e)+b (c C e-7 A d f))+(3 a c f+a d e+b c e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))+2 \left (\frac{b d e}{2}-f (a d+b c)\right ) (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))+(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^4 d^{5/2} f^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left (5 (3 a C (c f+d e)+b (c C e-7 A d f))+\frac{(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e))}{b d f}\right )}{105 b^2 d f}+\frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2} (-6 a C d f+7 b B d f-4 b C (c f+d e))}{35 b^2 d f^2}+\frac{2 C \sqrt{a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(((2*b*d*e - b*c*f + 4*a*d*f)*(7*b*B*d*f - 6*a*C*d*f - 4*b*C*(d*e + c*f)))/(b*d*f) + 5*(3*a*C*(d*e + c*f)
+ b*(c*C*e - 7*A*d*f)))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(105*b^2*d*f) + (2*(7*b*B*d*f - 6*a*C*d*f -
4*b*C*(d*e + c*f))*Sqrt[a + b*x]*Sqrt[c + d*x]*(e + f*x)^(3/2))/(35*b^2*d*f^2) + (2*C*Sqrt[a + b*x]*(c + d*x)
^(3/2)*(e + f*x)^(3/2))/(7*b*d*f) - (2*Sqrt[-(b*c) + a*d]*(3*b*d*f*((b*c*e + a*d*e + 3*a*c*f)*(7*b*B*d*f - 6*a
*C*d*f - 4*b*C*(d*e + c*f)) + 5*b*c*f*(3*a*C*(d*e + c*f) + b*(c*C*e - 7*A*d*f))) + 2*((b*d*e)/2 - (b*c + a*d)*
f)*((2*b*d*e - b*c*f + 4*a*d*f)*(7*b*B*d*f - 6*a*C*d*f - 4*b*C*(d*e + c*f)) + 5*b*d*f*(3*a*C*(d*e + c*f) + b*(
c*C*e - 7*A*d*f))))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqr
t[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(105*b^4*d^(5/2)*f^3*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e
- a*f)]) - (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(d*e - c*f)*(24*a^2*C*d^2*f^2 + a*b*d*f*(13*C*d*e - 5*c*C*f - 28
*B*d*f) - b^2*(7*d*f*(2*B*d*e - B*c*f - 5*A*d*f) - C*(8*d^2*e^2 - c*d*e*f - 4*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*d^(5/2)*f^3*Sqrt[c + d*x]*Sqrt[e + f*x])

Rule 1615

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[
{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[(k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*(e + f*x)^
(p + 1))/(d*f*b^(q - 1)*(m + n + p + q + 1)), x] + Dist[1/(d*f*b^q*(m + n + p + q + 1)), Int[(a + b*x)^m*(c +
d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a +
b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*
(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; F
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && 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 )}{\sqrt{a+b x}} \, dx &=\frac{2 C \sqrt{a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f}+\frac{2 \int \frac{\sqrt{c+d x} \sqrt{e+f x} \left (-\frac{1}{2} b (3 a C (d e+c f)+b (c C e-7 A d f))+\frac{1}{2} b (7 b B d f-6 a C d f-4 b C (d e+c f)) x\right )}{\sqrt{a+b x}} \, dx}{7 b^2 d f}\\ &=\frac{2 (7 b B d f-6 a C d f-4 b C (d e+c f)) \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2}}{35 b^2 d f^2}+\frac{2 C \sqrt{a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f}+\frac{4 \int \frac{\sqrt{e+f x} \left (-\frac{1}{4} b ((b c e+a d e+3 a c f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b c f (3 a C (d e+c f)+b (c C e-7 A d f)))-\frac{1}{4} b ((2 b d e-b c f+4 a d f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b d f (3 a C (d e+c f)+b (c C e-7 A d f))) x\right )}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{35 b^3 d f^2}\\ &=-\frac{2 ((2 b d e-b c f+4 a d f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b d f (3 a C (d e+c f)+b (c C e-7 A d f))) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 d^2 f^2}+\frac{2 (7 b B d f-6 a C d f-4 b C (d e+c f)) \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2}}{35 b^2 d f^2}+\frac{2 C \sqrt{a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f}+\frac{8 \int \frac{-\frac{1}{8} b (3 b d e ((b c e+a d e+3 a c f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b c f (3 a C (d e+c f)+b (c C e-7 A d f)))-(b c e+a d e+a c f) ((2 b d e-b c f+4 a d f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b d f (3 a C (d e+c f)+b (c C e-7 A d f))))-\frac{1}{8} b \left (3 b d f ((b c e+a d e+3 a c f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b c f (3 a C (d e+c f)+b (c C e-7 A d f)))+2 \left (\frac{b d e}{2}-(b c+a d) f\right ) ((2 b d e-b c f+4 a d f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b d f (3 a C (d e+c f)+b (c C e-7 A d f)))\right ) x}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{105 b^4 d^2 f^2}\\ &=-\frac{2 ((2 b d e-b c f+4 a d f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b d f (3 a C (d e+c f)+b (c C e-7 A d f))) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 d^2 f^2}+\frac{2 (7 b B d f-6 a C d f-4 b C (d e+c f)) \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2}}{35 b^2 d f^2}+\frac{2 C \sqrt{a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f}-\frac{\left ((b e-a f) (d e-c f) \left (24 a^2 C d^2 f^2+a b d f (13 C d e-5 c C f-28 B d f)-b^2 \left (7 d f (2 B d e-B c f-5 A d f)-C \left (8 d^2 e^2-c d e f-4 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 d^2 f^3}-\frac{\left (3 b d f ((b c e+a d e+3 a c f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b c f (3 a C (d e+c f)+b (c C e-7 A d f)))+2 \left (\frac{b d e}{2}-(b c+a d) f\right ) ((2 b d e-b c f+4 a d f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b d f (3 a C (d e+c f)+b (c C e-7 A d f)))\right ) \int \frac{\sqrt{e+f x}}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{105 b^3 d^2 f^3}\\ &=-\frac{2 ((2 b d e-b c f+4 a d f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b d f (3 a C (d e+c f)+b (c C e-7 A d f))) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 d^2 f^2}+\frac{2 (7 b B d f-6 a C d f-4 b C (d e+c f)) \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2}}{35 b^2 d f^2}+\frac{2 C \sqrt{a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f}-\frac{\left ((b e-a f) (d e-c f) \left (24 a^2 C d^2 f^2+a b d f (13 C d e-5 c C f-28 B d f)-b^2 \left (7 d f (2 B d e-B c f-5 A d f)-C \left (8 d^2 e^2-c d e f-4 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 d^2 f^3 \sqrt{c+d x}}-\frac{\left (\left (3 b d f ((b c e+a d e+3 a c f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b c f (3 a C (d e+c f)+b (c C e-7 A d f)))+2 \left (\frac{b d e}{2}-(b c+a d) f\right ) ((2 b d e-b c f+4 a d f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b d f (3 a C (d e+c f)+b (c C e-7 A d f)))\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 d^2 f^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}\\ &=-\frac{2 ((2 b d e-b c f+4 a d f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b d f (3 a C (d e+c f)+b (c C e-7 A d f))) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 d^2 f^2}+\frac{2 (7 b B d f-6 a C d f-4 b C (d e+c f)) \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2}}{35 b^2 d f^2}+\frac{2 C \sqrt{a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f}-\frac{2 \sqrt{-b c+a d} \left (3 b d f ((b c e+a d e+3 a c f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b c f (3 a C (d e+c f)+b (c C e-7 A d f)))+2 \left (\frac{b d e}{2}-(b c+a d) f\right ) ((2 b d e-b c f+4 a d f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b d f (3 a C (d e+c f)+b (c C e-7 A d f)))\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 d^{5/2} f^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{\left ((b e-a f) (d e-c f) \left (24 a^2 C d^2 f^2+a b d f (13 C d e-5 c C f-28 B d f)-b^2 \left (7 d f (2 B d e-B c f-5 A d f)-C \left (8 d^2 e^2-c d e f-4 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 d^2 f^3 \sqrt{c+d x} \sqrt{e+f x}}\\ &=-\frac{2 ((2 b d e-b c f+4 a d f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b d f (3 a C (d e+c f)+b (c C e-7 A d f))) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{105 b^3 d^2 f^2}+\frac{2 (7 b B d f-6 a C d f-4 b C (d e+c f)) \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2}}{35 b^2 d f^2}+\frac{2 C \sqrt{a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f}-\frac{2 \sqrt{-b c+a d} \left (3 b d f ((b c e+a d e+3 a c f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b c f (3 a C (d e+c f)+b (c C e-7 A d f)))+2 \left (\frac{b d e}{2}-(b c+a d) f\right ) ((2 b d e-b c f+4 a d f) (7 b B d f-6 a C d f-4 b C (d e+c f))+5 b d f (3 a C (d e+c f)+b (c C e-7 A d f)))\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 d^{5/2} f^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{-b c+a d} (b e-a f) (d e-c f) \left (24 a^2 C d^2 f^2+a b d f (13 C d e-5 c C f-28 B d f)-b^2 \left (7 d f (2 B d e-B c f-5 A d f)-C \left (8 d^2 e^2-c d e f-4 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 d^{5/2} f^3 \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}

Mathematica [C]  time = 13.3288, size = 917, normalized size = 1.18 \[ -\frac{2 \left (\sqrt{\frac{b c}{d}-a} \left (\left (C \left (-8 d^3 e^3+5 c d^2 f e^2+5 c^2 d f^2 e-8 c^3 f^3\right )-7 d f \left (5 A d f (d e+c f)-2 B \left (d^2 e^2-c d f e+c^2 f^2\right )\right )\right ) b^3+a d f \left (7 d f (3 B d e+3 B c f+10 A d f)+C \left (-9 d^2 e^2+8 c d f e-9 c^2 f^2\right )\right ) b^2-8 a^2 d^2 f^2 (7 B d f+2 C (d e+c f)) b+48 a^3 C d^3 f^3\right ) (c+d x) (e+f x) b^2+\sqrt{\frac{b c}{d}-a} d f (a+b x) (c+d x) (e+f x) \left (\left (C \left (\left (4 e^2-3 f x e-15 f^2 x^2\right ) d^2-c f (2 e+3 f x) d+4 c^2 f^2\right )-7 d f (B c f+5 A d f+B d (e+3 f x))\right ) b^2+a d f (28 B d f+C (5 d e+5 c f+18 d f x)) b-24 a^2 C d^2 f^2\right ) b^2-i (b c-a d) f (d e-c f) \left (\left (7 d f (B d e-2 B c f+5 A d f)-C \left (4 d^2 e^2+c d f e-8 c^2 f^2\right )\right ) b^2+a d f (-5 C d e+13 c C f-28 B d f) b+24 a^2 C d^2 f^2\right ) (a+b x)^{3/2} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right ),\frac{b d e-a d f}{b c f-a d f}\right ) b+i (b c-a d) f \left (\left (C \left (-8 d^3 e^3+5 c d^2 f e^2+5 c^2 d f^2 e-8 c^3 f^3\right )-7 d f \left (5 A d f (d e+c f)-2 B \left (d^2 e^2-c d f e+c^2 f^2\right )\right )\right ) b^3+a d f \left (7 d f (3 B d e+3 B c f+10 A d f)+C \left (-9 d^2 e^2+8 c d f e-9 c^2 f^2\right )\right ) b^2-8 a^2 d^2 f^2 (7 B d f+2 C (d e+c f)) b+48 a^3 C d^3 f^3\right ) (a+b x)^{3/2} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )\right )}{105 b^5 \sqrt{\frac{b c}{d}-a} d^3 f^3 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b^2*Sqrt[-a + (b*c)/d]*(48*a^3*C*d^3*f^3 - 8*a^2*b*d^2*f^2*(7*B*d*f + 2*C*(d*e + c*f)) + a*b^2*d*f*(7*d*f
*(3*B*d*e + 3*B*c*f + 10*A*d*f) + C*(-9*d^2*e^2 + 8*c*d*e*f - 9*c^2*f^2)) + b^3*(C*(-8*d^3*e^3 + 5*c*d^2*e^2*f
+ 5*c^2*d*e*f^2 - 8*c^3*f^3) - 7*d*f*(5*A*d*f*(d*e + c*f) - 2*B*(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*(a + b*x)*(c + d*x)*(e + f*x)*(-24*a^2*C*d^2*f^2 + a*b*d*f*(28*B*d*f + C*
(5*d*e + 5*c*f + 18*d*f*x)) + b^2*(-7*d*f*(B*c*f + 5*A*d*f + B*d*(e + 3*f*x)) + C*(4*c^2*f^2 - c*d*f*(2*e + 3*
f*x) + d^2*(4*e^2 - 3*e*f*x - 15*f^2*x^2)))) + I*(b*c - a*d)*f*(48*a^3*C*d^3*f^3 - 8*a^2*b*d^2*f^2*(7*B*d*f +
2*C*(d*e + c*f)) + a*b^2*d*f*(7*d*f*(3*B*d*e + 3*B*c*f + 10*A*d*f) + C*(-9*d^2*e^2 + 8*c*d*e*f - 9*c^2*f^2)) +
b^3*(C*(-8*d^3*e^3 + 5*c*d^2*e^2*f + 5*c^2*d*e*f^2 - 8*c^3*f^3) - 7*d*f*(5*A*d*f*(d*e + c*f) - 2*B*(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))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*Ell
ipticE[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*(b*c - a*d)*f*(d*e
- c*f)*(24*a^2*C*d^2*f^2 + a*b*d*f*(-5*C*d*e + 13*c*C*f - 28*B*d*f) + b^2*(7*d*f*(B*d*e - 2*B*c*f + 5*A*d*f) -
C*(4*d^2*e^2 + c*d*e*f - 8*c^2*f^2)))*(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)]))/(105*b^
5*Sqrt[-a + (b*c)/d]*d^3*f^3*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])

________________________________________________________________________________________

Maple [B]  time = 0.042, size = 10271, normalized size = 13.3 \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)^(1/2),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}}{\sqrt{b x + a}}\,{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)^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

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

________________________________________________________________________________________

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 )}{\sqrt{a + b x}}\, 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)**(1/2),x)

[Out]

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

________________________________________________________________________________________

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}}{\sqrt{b x + a}}\,{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)^(1/2),x, algorithm="giac")

[Out]

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