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

Optimal. Leaf size=387 \[ \frac{2 \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} (a C f (d e-c f)-b (3 d f (B e-A f)-C e (c f+2 d e))) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right ),\frac{f (b c-a d)}{d (b e-a f)}\right )}{3 b^2 d^{3/2} f^2 \sqrt{c+d x} \sqrt{e+f x}}-\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} (2 a C d f-b (3 B d f-2 C (c f+d e))) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^2 d^{3/2} f^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 C \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{3 b d f} \]

[Out]

(2*C*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b*d*f) - (2*Sqrt[-(b*c) + a*d]*(2*a*C*d*f - b*(3*B*d*f - 2*
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])/Sqrt[-(
b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(3*b^2*d^(3/2)*f^2*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f
)]) + (2*Sqrt[-(b*c) + a*d]*(a*C*f*(d*e - c*f) - b*(3*d*f*(B*e - A*f) - C*e*(2*d*e + c*f)))*Sqrt[(b*(c + d*x))
/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((
b*c - a*d)*f)/(d*(b*e - a*f))])/(3*b^2*d^(3/2)*f^2*Sqrt[c + d*x]*Sqrt[e + f*x])

________________________________________________________________________________________

Rubi [A]  time = 0.505209, antiderivative size = 384, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1615, 158, 114, 113, 121, 120} \[ -\frac{2 \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} (-a C f (d e-c f)+3 b d f (B e-A f)-b C e (c f+2 d e)) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^2 d^{3/2} f^2 \sqrt{c+d x} \sqrt{e+f x}}+\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} (-2 a C d f+3 b B d f-2 b C (c f+d e)) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^2 d^{3/2} f^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 C \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{3 b d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*C*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b*d*f) + (2*Sqrt[-(b*c) + a*d]*(3*b*B*d*f - 2*a*C*d*f - 2*b
*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])/Sqrt[-(
b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(3*b^2*d^(3/2)*f^2*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f
)]) - (2*Sqrt[-(b*c) + a*d]*(3*b*d*f*(B*e - A*f) - a*C*f*(d*e - c*f) - b*C*e*(2*d*e + c*f))*Sqrt[(b*(c + d*x))
/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((
b*c - a*d)*f)/(d*(b*e - a*f))])/(3*b^2*d^(3/2)*f^2*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 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{A+B x+C x^2}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx &=\frac{2 C \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{3 b d f}+\frac{2 \int \frac{-\frac{1}{2} b (b c C e+a C d e+a c C f-3 A b d f)+\frac{1}{2} b (3 b B d f-2 a C d f-2 b C (d e+c f)) x}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{3 b^2 d f}\\ &=\frac{2 C \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{3 b d f}+\frac{(3 b B d f-2 a C d f-2 b C (d e+c f)) \int \frac{\sqrt{e+f x}}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{3 b d f^2}-\frac{(3 b d f (B e-A f)-a C f (d e-c f)-b C e (2 d e+c f)) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{3 b d f^2}\\ &=\frac{2 C \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{3 b d f}-\frac{\left ((3 b d f (B e-A f)-a C f (d e-c f)-b C e (2 d e+c f)) \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 d f^2 \sqrt{c+d x}}+\frac{\left ((3 b B d f-2 a C d f-2 b C (d e+c f)) \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 d f^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}\\ &=\frac{2 C \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{3 b d f}+\frac{2 \sqrt{-b c+a d} (3 b B d f-2 a C d f-2 b C (d e+c f)) \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^2 d^{3/2} f^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{\left ((3 b d f (B e-A f)-a C f (d e-c f)-b C e (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}}\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 d f^2 \sqrt{c+d x} \sqrt{e+f x}}\\ &=\frac{2 C \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{3 b d f}+\frac{2 \sqrt{-b c+a d} (3 b B d f-2 a C d f-2 b C (d e+c f)) \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^2 d^{3/2} f^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{-b c+a d} (3 b d f (B e-A f)-a C f (d e-c f)-b C e (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}} 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^2 d^{3/2} f^2 \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}

Mathematica [C]  time = 5.83552, size = 418, normalized size = 1.08 \[ \frac{\sqrt{a+b x} \left (\frac{2 i b f \sqrt{a+b x} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left (a C d (c f-d e)+b \left (3 A d^2 f+c d (C e-3 B f)+2 c^2 C f\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right ),\frac{b d e-a d f}{b c f-a d f}\right )}{\sqrt{\frac{b c}{d}-a}}-\frac{2 b^2 (c+d x) (e+f x) (2 a C d f-3 b B d f+2 b C (c f+d e))}{a+b x}+2 i d f \sqrt{a+b x} \sqrt{\frac{b c}{d}-a} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} (-2 a C d f+3 b B d f-2 b C (c f+d e)) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )+2 b^2 C d f (c+d x) (e+f x)\right )}{3 b^3 d^2 f^2 \sqrt{c+d x} \sqrt{e+f x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*(2*b^2*C*d*f*(c + d*x)*(e + f*x) - (2*b^2*(-3*b*B*d*f + 2*a*C*d*f + 2*b*C*(d*e + c*f))*(c + d*x
)*(e + f*x))/(a + b*x) + (2*I)*Sqrt[-a + (b*c)/d]*d*f*(3*b*B*d*f - 2*a*C*d*f - 2*b*C*(d*e + c*f))*Sqrt[a + b*x
]*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]/S
qrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] + ((2*I)*b*f*(a*C*d*(-(d*e) + c*f) + b*(2*c^2*C*f + 3*A*d^2*f
+ c*d*(C*e - 3*B*f)))*Sqrt[a + b*x]*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*Ellipt
icF[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*c)/d]))/(3*b^3
*d^2*f^2*Sqrt[c + d*x]*Sqrt[e + f*x])

________________________________________________________________________________________

Maple [B]  time = 0.028, size = 2497, normalized size = 6.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(C*a*b^2*c*d*e*f+2*C*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)
*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^3*d^2*f^2+2*C*(d*(b*x+a)/(a*d-b*c))^
(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*
c)*f/d/(a*f-b*e))^(1/2))*b^3*c*d*e^2+3*A*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/
(a*d-b*c))^(1/2)*EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^2*d^2*f^2-3*A*(d*(
b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)*EllipticF((d*(b*x+a)/(a*d-b*
c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*b^3*c*d*f^2+C*x^2*a*b^2*d^2*f^2+C*x^3*b^3*d^2*f^2+C*x*a*b^2*c*d*f^2
+C*x*a*b^2*d^2*e*f+C*x*b^3*c*d*e*f-3*B*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a
*d-b*c))^(1/2)*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*b^3*c*d*e*f+C*x^2*b^3*c*
d*f^2+C*x^2*b^3*d^2*e*f-3*B*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1
/2)*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b*d^2*f^2+C*(d*(b*x+a)/(a*d-b*c
))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)*EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d
-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^2*c^2*f^2+2*C*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+
c)*b/(a*d-b*c))^(1/2)*EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^2*d^2*e^2-C*(
d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)*EllipticF((d*(b*x+a)/(a*d
-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*b^3*c^2*e*f-2*C*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e
))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)*EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*b
^3*c*d*e^2-2*C*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)*EllipticE
((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^2*c^2*f^2-2*C*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-
(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(
a*f-b*e))^(1/2))*a*b^2*d^2*e^2+2*C*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b
*c))^(1/2)*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*b^3*c^2*e*f-2*C*(d*(b*x+a)/(
a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2
),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^2*c*d*e*f-C*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-
(d*x+c)*b/(a*d-b*c))^(1/2)*EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b*c*d*f^
2+C*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)*EllipticF((d*(b*x+a)
/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b*d^2*e*f-3*B*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(
a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)*EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(
1/2))*a*b^2*d^2*e*f+3*B*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)*
EllipticF((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*b^3*c*d*e*f+3*B*(d*(b*x+a)/(a*d-b*c))^(
1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c
)*f/d/(a*f-b*e))^(1/2))*a*b^2*c*d*f^2+3*B*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b
/(a*d-b*c))^(1/2)*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^2*d^2*e*f)*(b*x+a
)^(1/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/f^2/b^3/d^2/(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e
*x+a*c*e)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C x^{2} + B x + A}{\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((C*x^2+B*x+A)/(b*x+a)^(1/2)/(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)

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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 d f x^{3} + a c e +{\left (b d e +{\left (b c + a d\right )} f\right )} x^{2} +{\left (a c f +{\left (b c + a d\right )} e\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(b*x+a)^(1/2)/(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)/(b*d*f*x^3 + a*c*e + (b*d*e + (b*c + a*d)
*f)*x^2 + (a*c*f + (b*c + a*d)*e)*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C x^{2} + B x + A}{\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((C*x^2+B*x+A)/(b*x+a)^(1/2)/(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)