\(\int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [28]

Optimal result

Integrand size = 35, antiderivative size = 368 \[ \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {4 C \sqrt {-d e+c f} (d f g+d e h+c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {2 \sqrt {-d e+c f} \left (3 A d f h^2+C (c h (f g-e h)+d g (2 f g+e h))\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt {e+f x} \sqrt {g+h x}} \]

[Out]

2/3*C*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/d/f/h-4/3*C*(c*f*h+d*e*h+d*f*g)*EllipticE(f^(1/2)*(d*x+c)^(1/2
)/(c*f-d*e)^(1/2),((-c*f+d*e)*h/f/(-c*h+d*g))^(1/2))*(c*f-d*e)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(h*x+g)^(1/2
)/d^2/f^(3/2)/h^2/(f*x+e)^(1/2)/(d*(h*x+g)/(-c*h+d*g))^(1/2)+2/3*(3*A*d*f*h^2+C*(c*h*(-e*h+f*g)+d*g*(e*h+2*f*g
)))*EllipticF(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),((-c*f+d*e)*h/f/(-c*h+d*g))^(1/2))*(c*f-d*e)^(1/2)*(d*(f*x
+e)/(-c*f+d*e))^(1/2)*(d*(h*x+g)/(-c*h+d*g))^(1/2)/d^2/f^(3/2)/h^2/(f*x+e)^(1/2)/(h*x+g)^(1/2)

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1629, 164, 115, 114, 122, 121} \[ \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (3 A d f h^2+c C h (f g-e h)+C d g (e h+2 f g)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt {e+f x} \sqrt {g+h x}}-\frac {4 C \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} (c f h+d e h+d f g) E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h} \]

[In]

Int[(A + C*x^2)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(2*C*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*d*f*h) - (4*C*Sqrt[-(d*e) + c*f]*(d*f*g + d*e*h + c*f*h)*Sq
rt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*
e - c*f)*h)/(f*(d*g - c*h))])/(3*d^2*f^(3/2)*h^2*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) + (2*Sqrt[-(d*
e) + c*f]*(3*A*d*f*h^2 + c*C*h*(f*g - e*h) + C*d*g*(2*f*g + e*h))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g +
h*x))/(d*g - c*h)]*EllipticF[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*
h))])/(3*d^2*f^(3/2)*h^2*Sqrt[e + f*x]*Sqrt[g + h*x])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*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)))], 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 115

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 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*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)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && Si
mplerQ[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])

Rule 122

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 164

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 1629

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}+\frac {2 \int \frac {\frac {1}{2} d (3 A d f h-C (d e g+c f g+c e h))-C d (d f g+d e h+c f h) x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{3 d^2 f h} \\ & = \frac {2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {(2 C (d f g+d e h+c f h)) \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{3 d f h^2}+\frac {\left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{3 d f h^2} \\ & = \frac {2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}+\frac {\left (\left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right ) \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}} \, dx}{3 d f h^2 \sqrt {e+f x}}-\frac {\left (2 C (d f g+d e h+c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x}\right ) \int \frac {\sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}} \, dx}{3 d f h^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}} \\ & = \frac {2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {4 C \sqrt {-d e+c f} (d f g+d e h+c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {\left (\left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}} \, dx}{3 d f h^2 \sqrt {e+f x} \sqrt {g+h x}} \\ & = \frac {2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {4 C \sqrt {-d e+c f} (d f g+d e h+c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {2 \sqrt {-d e+c f} \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt {e+f x} \sqrt {g+h x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.77 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.06 \[ \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {\sqrt {c+d x} \left (2 C d^2 f h (e+f x) (g+h x)-\frac {4 C d^2 (d f g+d e h+c f h) (e+f x) (g+h x)}{c+d x}-4 i C \sqrt {-c+\frac {d e}{f}} f h (d f g+d e h+c f h) \sqrt {c+d x} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )+\frac {2 i d h \left (3 A d f^2 h+c C f (-f g+e h)+C d e (f g+2 e h)\right ) \sqrt {c+d x} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )}{\sqrt {-c+\frac {d e}{f}}}\right )}{3 d^3 f^2 h^2 \sqrt {e+f x} \sqrt {g+h x}} \]

[In]

Integrate[(A + C*x^2)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(Sqrt[c + d*x]*(2*C*d^2*f*h*(e + f*x)*(g + h*x) - (4*C*d^2*(d*f*g + d*e*h + c*f*h)*(e + f*x)*(g + h*x))/(c + d
*x) - (4*I)*C*Sqrt[-c + (d*e)/f]*f*h*(d*f*g + d*e*h + c*f*h)*Sqrt[c + d*x]*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*S
qrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h
- c*f*h)] + ((2*I)*d*h*(3*A*d*f^2*h + c*C*f*(-(f*g) + e*h) + C*d*e*(f*g + 2*e*h))*Sqrt[c + d*x]*Sqrt[(d*(e +
f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]],
(d*f*g - c*f*h)/(d*e*h - c*f*h)])/Sqrt[-c + (d*e)/f]))/(3*d^3*f^2*h^2*Sqrt[e + f*x]*Sqrt[g + h*x])

Maple [A] (verified)

Time = 2.32 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.66

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 C \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}{3 d f h}+\frac {2 \left (A -\frac {2 C \left (\frac {1}{2} c e h +\frac {1}{2} c f g +\frac {1}{2} d e g \right )}{3 d f h}\right ) \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}-\frac {4 C \left (c f h +d e h +d f g \right ) \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \left (\left (-\frac {g}{h}+\frac {c}{d}\right ) E\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )-\frac {c F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{d}\right )}{3 d f h \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\)	\(611\)
default	\(\text {Expression too large to display}\)	\(1812\)

[In]

int((C*x^2+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,method=_RETURNVERBOSE)

[Out]

((d*x+c)*(f*x+e)*(h*x+g))^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)*(2/3*C/d/f/h*(d*f*h*x^3+c*f*h*x^2+d*
e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)+2*(A-2/3*C/d/f/h*(1/2*c*e*h+1/2*c*f*g+1/2*d*e*g))*(g/h-
e/f)*((x+g/h)/(g/h-e/f))^(1/2)*((x+c/d)/(-g/h+c/d))^(1/2)*((x+e/f)/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d*e*
h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)*EllipticF(((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d
))^(1/2))-4/3*C/d/f/h*(c*f*h+d*e*h+d*f*g)*(g/h-e/f)*((x+g/h)/(g/h-e/f))^(1/2)*((x+c/d)/(-g/h+c/d))^(1/2)*((x+e
/f)/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)*((-g/h+c/d
)*EllipticE(((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2))-c/d*EllipticF(((x+g/h)/(g/h-e/f))^(1/2),(
(-g/h+e/f)/(-g/h+c/d))^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.15 (sec) , antiderivative size = 775, normalized size of antiderivative = 2.11 \[ \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g} C d^{2} f^{2} h^{2} + {\left (2 \, C d^{2} f^{2} g^{2} + {\left (C d^{2} e f + C c d f^{2}\right )} g h + {\left (2 \, C d^{2} e^{2} + C c d e f + {\left (2 \, C c^{2} + 9 \, A d^{2}\right )} f^{2}\right )} h^{2}\right )} \sqrt {d f h} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - {\left (d^{2} e f + c d f^{2}\right )} g h + {\left (d^{2} e^{2} - c d e f + c^{2} f^{2}\right )} h^{2}\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, {\left (d^{3} e f^{2} + c d^{2} f^{3}\right )} g^{2} h - 3 \, {\left (d^{3} e^{2} f - 4 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} g h^{2} + {\left (2 \, d^{3} e^{3} - 3 \, c d^{2} e^{2} f - 3 \, c^{2} d e f^{2} + 2 \, c^{3} f^{3}\right )} h^{3}\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + {\left (d e + c f\right )} h}{3 \, d f h}\right ) + 6 \, {\left (C d^{2} f^{2} g h + {\left (C d^{2} e f + C c d f^{2}\right )} h^{2}\right )} \sqrt {d f h} {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - {\left (d^{2} e f + c d f^{2}\right )} g h + {\left (d^{2} e^{2} - c d e f + c^{2} f^{2}\right )} h^{2}\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, {\left (d^{3} e f^{2} + c d^{2} f^{3}\right )} g^{2} h - 3 \, {\left (d^{3} e^{2} f - 4 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} g h^{2} + {\left (2 \, d^{3} e^{3} - 3 \, c d^{2} e^{2} f - 3 \, c^{2} d e f^{2} + 2 \, c^{3} f^{3}\right )} h^{3}\right )}}{27 \, d^{3} f^{3} h^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - {\left (d^{2} e f + c d f^{2}\right )} g h + {\left (d^{2} e^{2} - c d e f + c^{2} f^{2}\right )} h^{2}\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, {\left (d^{3} e f^{2} + c d^{2} f^{3}\right )} g^{2} h - 3 \, {\left (d^{3} e^{2} f - 4 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} g h^{2} + {\left (2 \, d^{3} e^{3} - 3 \, c d^{2} e^{2} f - 3 \, c^{2} d e f^{2} + 2 \, c^{3} f^{3}\right )} h^{3}\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + {\left (d e + c f\right )} h}{3 \, d f h}\right )\right )\right )}}{9 \, d^{3} f^{3} h^{3}} \]

[In]

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

[Out]

2/9*(3*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)*C*d^2*f^2*h^2 + (2*C*d^2*f^2*g^2 + (C*d^2*e*f + C*c*d*f^2)*g*
h + (2*C*d^2*e^2 + C*c*d*e*f + (2*C*c^2 + 9*A*d^2)*f^2)*h^2)*sqrt(d*f*h)*weierstrassPInverse(4/3*(d^2*f^2*g^2
- (d^2*e*f + c*d*f^2)*g*h + (d^2*e^2 - c*d*e*f + c^2*f^2)*h^2)/(d^2*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*(d^3*e*
f^2 + c*d^2*f^3)*g^2*h - 3*(d^3*e^2*f - 4*c*d^2*e*f^2 + c^2*d*f^3)*g*h^2 + (2*d^3*e^3 - 3*c*d^2*e^2*f - 3*c^2*
d*e*f^2 + 2*c^3*f^3)*h^3)/(d^3*f^3*h^3), 1/3*(3*d*f*h*x + d*f*g + (d*e + c*f)*h)/(d*f*h)) + 6*(C*d^2*f^2*g*h +
(C*d^2*e*f + C*c*d*f^2)*h^2)*sqrt(d*f*h)*weierstrassZeta(4/3*(d^2*f^2*g^2 - (d^2*e*f + c*d*f^2)*g*h + (d^2*e^
2 - c*d*e*f + c^2*f^2)*h^2)/(d^2*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*(d^3*e*f^2 + c*d^2*f^3)*g^2*h - 3*(d^3*e^2
*f - 4*c*d^2*e*f^2 + c^2*d*f^3)*g*h^2 + (2*d^3*e^3 - 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + 2*c^3*f^3)*h^3)/(d^3*f^3*
h^3), weierstrassPInverse(4/3*(d^2*f^2*g^2 - (d^2*e*f + c*d*f^2)*g*h + (d^2*e^2 - c*d*e*f + c^2*f^2)*h^2)/(d^2
*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*(d^3*e*f^2 + c*d^2*f^3)*g^2*h - 3*(d^3*e^2*f - 4*c*d^2*e*f^2 + c^2*d*f^3)*
g*h^2 + (2*d^3*e^3 - 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + 2*c^3*f^3)*h^3)/(d^3*f^3*h^3), 1/3*(3*d*f*h*x + d*f*g + (
d*e + c*f)*h)/(d*f*h))))/(d^3*f^3*h^3)

Sympy [F]

\[ \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {A + C x^{2}}{\sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]

[In]

integrate((C*x**2+A)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

Integral((A + C*x**2)/(sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*x)), x)

Maxima [F]

\[ \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C x^{2} + A}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]

[In]

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

[Out]

integrate((C*x^2 + A)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)

Giac [F]

\[ \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C x^{2} + A}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]

[In]

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

[Out]

integrate((C*x^2 + A)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {C\,x^2+A}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \]

[In]

int((A + C*x^2)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(c + d*x)^(1/2)),x)

[Out]

int((A + C*x^2)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(c + d*x)^(1/2)), x)