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

Optimal. Leaf size=766 \[ \frac{2 \sqrt{a d-b c} (b e-a f) (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (4 a^2 C d^2 f^2+a b d f (-7 B d f-2 c C f+8 C d e)+b^2 \left (-\left (7 d f (-10 A d f+B c f+8 B d e)-4 C \left (c^2 f^2+2 c d e f+12 d^2 e^2\right )\right )\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right ),\frac{f (b c-a d)}{d (b e-a f)}\right )}{105 b^3 d^{5/2} f^4 \sqrt{c+d x} \sqrt{e+f x}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))}{105 b^2 d^2 f^3}-\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (3 b d f (5 a d f (a c C f+3 a C d e-7 A b d f+3 b c C e)-(a c f+3 a d e+b c e) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))+2 \left (\frac{b c f}{2}-d (a f+b e)\right ) (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^3 d^{5/2} f^4 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a+b x} (c+d x)^{3/2} \sqrt{e+f x} (4 a C d f+b (-7 B d f+4 c C f+6 C d e))}{35 b d^2 f^2}+\frac{2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x}}{7 b d f} \]

[Out]

(-2*(5*b*d*f*(3*b*c*C*e + 3*a*C*d*e + a*c*C*f - 7*A*b*d*f) + (a*d*f - 2*b*(2*d*e + c*f))*(4*a*C*d*f + b*(6*C*d
*e + 4*c*C*f - 7*B*d*f)))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(105*b^2*d^2*f^3) - (2*(4*a*C*d*f + b*(6*
C*d*e + 4*c*C*f - 7*B*d*f))*Sqrt[a + b*x]*(c + d*x)^(3/2)*Sqrt[e + f*x])/(35*b*d^2*f^2) + (2*C*(a + b*x)^(3/2)
*(c + d*x)^(3/2)*Sqrt[e + f*x])/(7*b*d*f) - (2*Sqrt[-(b*c) + a*d]*(3*b*d*f*(5*a*d*f*(3*b*c*C*e + 3*a*C*d*e + a
*c*C*f - 7*A*b*d*f) - (b*c*e + 3*a*d*e + a*c*f)*(4*a*C*d*f + b*(6*C*d*e + 4*c*C*f - 7*B*d*f))) + 2*((b*c*f)/2
- d*(b*e + a*f))*(5*b*d*f*(3*b*c*C*e + 3*a*C*d*e + a*c*C*f - 7*A*b*d*f) + (a*d*f - 2*b*(2*d*e + c*f))*(4*a*C*d
*f + b*(6*C*d*e + 4*c*C*f - 7*B*d*f))))*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^3*d^(5/2)*f^4*Sqrt[c + d*x]*Sqr
t[(b*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(d*e - c*f)*(4*a^2*C*d^2*f^2 + a*b*d*f*(8*C*
d*e - 2*c*C*f - 7*B*d*f) - b^2*(7*d*f*(8*B*d*e + B*c*f - 10*A*d*f) - 4*C*(12*d^2*e^2 + 2*c*d*e*f + 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^3*d^(5/2)*f^4*Sqrt[c + d*x]*Sqrt[e + f*x])

________________________________________________________________________________________

Rubi [A]  time = 2.06141, antiderivative size = 766, 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 = {1615, 154, 158, 114, 113, 121, 120} \[ \frac{2 \sqrt{a d-b c} (b e-a f) (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (4 a^2 C d^2 f^2+a b d f (-7 B d f-2 c C f+8 C d e)+b^2 \left (-\left (7 d f (-10 A d f+B c f+8 B d e)-4 C \left (c^2 f^2+2 c d e f+12 d^2 e^2\right )\right )\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^3 d^{5/2} f^4 \sqrt{c+d x} \sqrt{e+f x}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))}{105 b^2 d^2 f^3}-\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (3 b d f (5 a d f (a c C f+3 a C d e-7 A b d f+3 b c C e)-(a c f+3 a d e+b c e) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))+2 \left (\frac{b c f}{2}-d (a f+b e)\right ) (5 b d f (a c C f+3 a C d e-7 A b d f+3 b c C e)+(a d f-2 b (c f+2 d e)) (4 a C d f+b (-7 B d f+4 c C f+6 C d e)))\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^3 d^{5/2} f^4 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a+b x} (c+d x)^{3/2} \sqrt{e+f x} (4 a C d f+b (-7 B d f+4 c C f+6 C d e))}{35 b d^2 f^2}+\frac{2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x}}{7 b d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(5*b*d*f*(3*b*c*C*e + 3*a*C*d*e + a*c*C*f - 7*A*b*d*f) + (a*d*f - 2*b*(2*d*e + c*f))*(4*a*C*d*f + b*(6*C*d
*e + 4*c*C*f - 7*B*d*f)))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(105*b^2*d^2*f^3) - (2*(4*a*C*d*f + b*(6*
C*d*e + 4*c*C*f - 7*B*d*f))*Sqrt[a + b*x]*(c + d*x)^(3/2)*Sqrt[e + f*x])/(35*b*d^2*f^2) + (2*C*(a + b*x)^(3/2)
*(c + d*x)^(3/2)*Sqrt[e + f*x])/(7*b*d*f) - (2*Sqrt[-(b*c) + a*d]*(3*b*d*f*(5*a*d*f*(3*b*c*C*e + 3*a*C*d*e + a
*c*C*f - 7*A*b*d*f) - (b*c*e + 3*a*d*e + a*c*f)*(4*a*C*d*f + b*(6*C*d*e + 4*c*C*f - 7*B*d*f))) + 2*((b*c*f)/2
- d*(b*e + a*f))*(5*b*d*f*(3*b*c*C*e + 3*a*C*d*e + a*c*C*f - 7*A*b*d*f) + (a*d*f - 2*b*(2*d*e + c*f))*(4*a*C*d
*f + b*(6*C*d*e + 4*c*C*f - 7*B*d*f))))*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^3*d^(5/2)*f^4*Sqrt[c + d*x]*Sqr
t[(b*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(d*e - c*f)*(4*a^2*C*d^2*f^2 + a*b*d*f*(8*C*
d*e - 2*c*C*f - 7*B*d*f) - b^2*(7*d*f*(8*B*d*e + B*c*f - 10*A*d*f) - 4*C*(12*d^2*e^2 + 2*c*d*e*f + 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^3*d^(5/2)*f^4*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{a+b x} \sqrt{c+d x} \left (A+B x+C x^2\right )}{\sqrt{e+f x}} \, dx &=\frac{2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x}}{7 b d f}+\frac{2 \int \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-\frac{1}{2} b (3 b c C e+3 a C d e+a c C f-7 A b d f)-\frac{1}{2} b (4 a C d f+b (6 C d e+4 c C f-7 B d f)) x\right )}{\sqrt{e+f x}} \, dx}{7 b^2 d f}\\ &=-\frac{2 (4 a C d f+b (6 C d e+4 c C f-7 B d f)) \sqrt{a+b x} (c+d x)^{3/2} \sqrt{e+f x}}{35 b d^2 f^2}+\frac{2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x}}{7 b d f}+\frac{4 \int \frac{\sqrt{c+d x} \left (-\frac{1}{4} b (5 a d f (3 b c C e+3 a C d e+a c C f-7 A b d f)-(b c e+3 a d e+a c f) (4 a C d f+b (6 C d e+4 c C f-7 B d f)))-\frac{1}{4} b (5 b d f (3 b c C e+3 a C d e+a c C f-7 A b d f)+(a d f-2 b (2 d e+c f)) (4 a C d f+b (6 C d e+4 c C f-7 B d f))) x\right )}{\sqrt{a+b x} \sqrt{e+f x}} \, dx}{35 b^2 d^2 f^2}\\ &=-\frac{2 (5 b d f (3 b c C e+3 a C d e+a c C f-7 A b d f)+(a d f-2 b (2 d e+c f)) (4 a C d f+b (6 C d e+4 c C f-7 B d f))) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{105 b^2 d^2 f^3}-\frac{2 (4 a C d f+b (6 C d e+4 c C f-7 B d f)) \sqrt{a+b x} (c+d x)^{3/2} \sqrt{e+f x}}{35 b d^2 f^2}+\frac{2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x}}{7 b d f}+\frac{8 \int \frac{-\frac{1}{8} b (3 b c f (5 a d f (3 b c C e+3 a C d e+a c C f-7 A b d f)-(b c e+3 a d e+a c f) (4 a C d f+b (6 C d e+4 c C f-7 B d f)))-(b c e+a d e+a c f) (5 b d f (3 b c C e+3 a C d e+a c C f-7 A b d f)+(a d f-2 b (2 d e+c f)) (4 a C d f+b (6 C d e+4 c C f-7 B d f))))-\frac{1}{8} b \left (3 b d f (5 a d f (3 b c C e+3 a C d e+a c C f-7 A b d f)-(b c e+3 a d e+a c f) (4 a C d f+b (6 C d e+4 c C f-7 B d f)))+2 \left (\frac{b c f}{2}-d (b e+a f)\right ) (5 b d f (3 b c C e+3 a C d e+a c C f-7 A b d f)+(a d f-2 b (2 d e+c f)) (4 a C d f+b (6 C d e+4 c C f-7 B d f)))\right ) x}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{105 b^3 d^2 f^3}\\ &=-\frac{2 (5 b d f (3 b c C e+3 a C d e+a c C f-7 A b d f)+(a d f-2 b (2 d e+c f)) (4 a C d f+b (6 C d e+4 c C f-7 B d f))) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{105 b^2 d^2 f^3}-\frac{2 (4 a C d f+b (6 C d e+4 c C f-7 B d f)) \sqrt{a+b x} (c+d x)^{3/2} \sqrt{e+f x}}{35 b d^2 f^2}+\frac{2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x}}{7 b d f}+\frac{\left ((b e-a f) (d e-c f) \left (4 a^2 C d^2 f^2+a b d f (8 C d e-2 c C f-7 B d f)-b^2 \left (7 d f (8 B d e+B c f-10 A d f)-4 C \left (12 d^2 e^2+2 c d e f+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^2 d^2 f^4}-\frac{\left (3 b d f (5 a d f (3 b c C e+3 a C d e+a c C f-7 A b d f)-(b c e+3 a d e+a c f) (4 a C d f+b (6 C d e+4 c C f-7 B d f)))+2 \left (\frac{b c f}{2}-d (b e+a f)\right ) (5 b d f (3 b c C e+3 a C d e+a c C f-7 A b d f)+(a d f-2 b (2 d e+c f)) (4 a C d f+b (6 C d e+4 c C f-7 B d f)))\right ) \int \frac{\sqrt{e+f x}}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{105 b^2 d^2 f^4}\\ &=-\frac{2 (5 b d f (3 b c C e+3 a C d e+a c C f-7 A b d f)+(a d f-2 b (2 d e+c f)) (4 a C d f+b (6 C d e+4 c C f-7 B d f))) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{105 b^2 d^2 f^3}-\frac{2 (4 a C d f+b (6 C d e+4 c C f-7 B d f)) \sqrt{a+b x} (c+d x)^{3/2} \sqrt{e+f x}}{35 b d^2 f^2}+\frac{2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x}}{7 b d f}+\frac{\left ((b e-a f) (d e-c f) \left (4 a^2 C d^2 f^2+a b d f (8 C d e-2 c C f-7 B d f)-b^2 \left (7 d f (8 B d e+B c f-10 A d f)-4 C \left (12 d^2 e^2+2 c d e f+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^2 d^2 f^4 \sqrt{c+d x}}-\frac{\left (\left (3 b d f (5 a d f (3 b c C e+3 a C d e+a c C f-7 A b d f)-(b c e+3 a d e+a c f) (4 a C d f+b (6 C d e+4 c C f-7 B d f)))+2 \left (\frac{b c f}{2}-d (b e+a f)\right ) (5 b d f (3 b c C e+3 a C d e+a c C f-7 A b d f)+(a d f-2 b (2 d e+c f)) (4 a C d f+b (6 C d e+4 c C f-7 B 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^2 d^2 f^4 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}\\ &=-\frac{2 (5 b d f (3 b c C e+3 a C d e+a c C f-7 A b d f)+(a d f-2 b (2 d e+c f)) (4 a C d f+b (6 C d e+4 c C f-7 B d f))) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{105 b^2 d^2 f^3}-\frac{2 (4 a C d f+b (6 C d e+4 c C f-7 B d f)) \sqrt{a+b x} (c+d x)^{3/2} \sqrt{e+f x}}{35 b d^2 f^2}+\frac{2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x}}{7 b d f}-\frac{2 \sqrt{-b c+a d} \left (3 b d f (5 a d f (3 b c C e+3 a C d e+a c C f-7 A b d f)-(b c e+3 a d e+a c f) (4 a C d f+b (6 C d e+4 c C f-7 B d f)))+2 \left (\frac{b c f}{2}-d (b e+a f)\right ) (5 b d f (3 b c C e+3 a C d e+a c C f-7 A b d f)+(a d f-2 b (2 d e+c f)) (4 a C d f+b (6 C d e+4 c C f-7 B 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^3 d^{5/2} f^4 \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 (4 a^2 C d^2 f^2+a b d f (8 C d e-2 c C f-7 B d f)-b^2 \left (7 d f (8 B d e+B c f-10 A d f)-4 C \left (12 d^2 e^2+2 c d e f+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^2 d^2 f^4 \sqrt{c+d x} \sqrt{e+f x}}\\ &=-\frac{2 (5 b d f (3 b c C e+3 a C d e+a c C f-7 A b d f)+(a d f-2 b (2 d e+c f)) (4 a C d f+b (6 C d e+4 c C f-7 B d f))) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{105 b^2 d^2 f^3}-\frac{2 (4 a C d f+b (6 C d e+4 c C f-7 B d f)) \sqrt{a+b x} (c+d x)^{3/2} \sqrt{e+f x}}{35 b d^2 f^2}+\frac{2 C (a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x}}{7 b d f}-\frac{2 \sqrt{-b c+a d} \left (3 b d f (5 a d f (3 b c C e+3 a C d e+a c C f-7 A b d f)-(b c e+3 a d e+a c f) (4 a C d f+b (6 C d e+4 c C f-7 B d f)))+2 \left (\frac{b c f}{2}-d (b e+a f)\right ) (5 b d f (3 b c C e+3 a C d e+a c C f-7 A b d f)+(a d f-2 b (2 d e+c f)) (4 a C d f+b (6 C d e+4 c C f-7 B 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^3 d^{5/2} f^4 \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 (4 a^2 C d^2 f^2+a b d f (8 C d e-2 c C f-7 B d f)-b^2 \left (7 d f (8 B d e+B c f-10 A d f)-4 C \left (12 d^2 e^2+2 c d e f+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^3 d^{5/2} f^4 \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}

Mathematica [C]  time = 13.0155, size = 922, normalized size = 1.2 \[ \frac{2 \left (\sqrt{\frac{b c}{d}-a} \left (\left (C \left (-48 d^3 e^3+16 c d^2 f e^2+9 c^2 d f^2 e+8 c^3 f^3\right )+7 d f \left (5 A d f (c f-2 d e)+B \left (8 d^2 e^2-3 c d f e-2 c^2 f^2\right )\right )\right ) b^3+a d f \left (7 d f (-3 B d e+2 B c f+5 A d f)+C \left (16 d^2 e^2-8 c d f e-5 c^2 f^2\right )\right ) b^2+a^2 d^2 f^2 (9 C d e-5 c C f-14 B d f) b+8 a^3 C d^3 f^3\right ) (c+d x) (e+f x) b^2+\sqrt{\frac{b c}{d}-a} d f (a+b x) (c+d x) (e+f x) \left (\left (7 d f (5 A d f+B (-4 d e+c f+3 d f x))+C \left (3 \left (8 e^2-6 f x e+5 f^2 x^2\right ) d^2+c f (3 f x-5 e) d-4 c^2 f^2\right )\right ) b^2+a d f (7 B d f+C (-5 d e+2 c f+3 d f x)) b-4 a^2 C d^2 f^2\right ) b^2-i (b c-a d) f (d e-c f) \left (-\left (7 d f (-4 B d e-2 B c f+5 A d f)+C \left (24 d^2 e^2+13 c d f e+8 c^2 f^2\right )\right ) b^2+a d f (5 C d e+c C f-7 B d f) b+4 a^2 C d^2 f^2\right ) (a+b x)^{3/2} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right ),\frac{b d e-a d f}{b c f-a d f}\right ) b+i (b c-a d) f \left (\left (C \left (-48 d^3 e^3+16 c d^2 f e^2+9 c^2 d f^2 e+8 c^3 f^3\right )+7 d f \left (5 A d f (c f-2 d e)+B \left (8 d^2 e^2-3 c d f e-2 c^2 f^2\right )\right )\right ) b^3+a d f \left (7 d f (-3 B d e+2 B c f+5 A d f)+C \left (16 d^2 e^2-8 c d f e-5 c^2 f^2\right )\right ) b^2+a^2 d^2 f^2 (9 C d e-5 c C f-14 B d f) b+8 a^3 C d^3 f^3\right ) (a+b x)^{3/2} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )\right )}{105 b^4 \sqrt{\frac{b c}{d}-a} d^3 f^4 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b^2*Sqrt[-a + (b*c)/d]*(8*a^3*C*d^3*f^3 + a^2*b*d^2*f^2*(9*C*d*e - 5*c*C*f - 14*B*d*f) + a*b^2*d*f*(7*d*f*
(-3*B*d*e + 2*B*c*f + 5*A*d*f) + C*(16*d^2*e^2 - 8*c*d*e*f - 5*c^2*f^2)) + b^3*(C*(-48*d^3*e^3 + 16*c*d^2*e^2*
f + 9*c^2*d*e*f^2 + 8*c^3*f^3) + 7*d*f*(5*A*d*f*(-2*d*e + c*f) + B*(8*d^2*e^2 - 3*c*d*e*f - 2*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)*(-4*a^2*C*d^2*f^2 + a*b*d*f*(7*B*d*
f + C*(-5*d*e + 2*c*f + 3*d*f*x)) + b^2*(7*d*f*(5*A*d*f + B*(-4*d*e + c*f + 3*d*f*x)) + C*(-4*c^2*f^2 + c*d*f*
(-5*e + 3*f*x) + 3*d^2*(8*e^2 - 6*e*f*x + 5*f^2*x^2)))) + I*(b*c - a*d)*f*(8*a^3*C*d^3*f^3 + a^2*b*d^2*f^2*(9*
C*d*e - 5*c*C*f - 14*B*d*f) + a*b^2*d*f*(7*d*f*(-3*B*d*e + 2*B*c*f + 5*A*d*f) + C*(16*d^2*e^2 - 8*c*d*e*f - 5*
c^2*f^2)) + b^3*(C*(-48*d^3*e^3 + 16*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 8*c^3*f^3) + 7*d*f*(5*A*d*f*(-2*d*e + c*f)
+ B*(8*d^2*e^2 - 3*c*d*e*f - 2*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))]*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
*(b*c - a*d)*f*(d*e - c*f)*(4*a^2*C*d^2*f^2 + a*b*d*f*(5*C*d*e + c*C*f - 7*B*d*f) - b^2*(7*d*f*(-4*B*d*e - 2*B
*c*f + 5*A*d*f) + C*(24*d^2*e^2 + 13*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^4*Sqrt[-a + (b*c)/d]*d^3*f^4*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])

________________________________________________________________________________________

Maple [B]  time = 0.043, size = 9544, normalized size = 12.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1/2)*(C*x^2+B*x+A)*(d*x+c)^(1/2)/(f*x+e)^(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{b x + a} \sqrt{d x + c}}{\sqrt{f x + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((C*x^2 + B*x + A)*sqrt(b*x + a)*sqrt(d*x + c)/sqrt(f*x + e), 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}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b x} \sqrt{c + d x} \left (A + B x + C x^{2}\right )}{\sqrt{e + f x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C x^{2} + B x + A\right )} \sqrt{b x + a} \sqrt{d x + c}}{\sqrt{f x + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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