3.840 \(\int \frac{1}{\sqrt{x} \sqrt{a-b x} \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=75 \[ \frac{2 \sqrt{a} \sqrt{1-\frac{b x}{a}} \sqrt{\frac{b x}{a}+1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right |-1\right )}{\sqrt{b} \sqrt{a-b x} \sqrt{a+b x}} \]

[Out]

(2*Sqrt[a]*Sqrt[1 - (b*x)/a]*Sqrt[1 + (b*x)/a]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[x]
)/Sqrt[a]], -1])/(Sqrt[b]*Sqrt[a - b*x]*Sqrt[a + b*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.14423, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{2 \sqrt{a} \sqrt{1-\frac{b x}{a}} \sqrt{\frac{b x}{a}+1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right |-1\right )}{\sqrt{b} \sqrt{a-b x} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[x]*Sqrt[a - b*x]*Sqrt[a + b*x]),x]

[Out]

(2*Sqrt[a]*Sqrt[1 - (b*x)/a]*Sqrt[1 + (b*x)/a]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[x]
)/Sqrt[a]], -1])/(Sqrt[b]*Sqrt[a - b*x]*Sqrt[a + b*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.4533, size = 68, normalized size = 0.91 \[ \frac{2 \sqrt{a} \sqrt{1 - \frac{b x}{a}} \sqrt{1 + \frac{b x}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}\middle | -1\right )}{\sqrt{b} \sqrt{a - b x} \sqrt{a + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(1/2)/(-b*x+a)**(1/2)/(b*x+a)**(1/2),x)

[Out]

2*sqrt(a)*sqrt(1 - b*x/a)*sqrt(1 + b*x/a)*elliptic_f(asin(sqrt(b)*sqrt(x)/sqrt(a
)), -1)/(sqrt(b)*sqrt(a - b*x)*sqrt(a + b*x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.173253, size = 66, normalized size = 0.88 \[ -\frac{2 x \sqrt{1-\frac{a^2}{b^2 x^2}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{\frac{a}{b}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{a}{b}} \sqrt{a-b x} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[x]*Sqrt[a - b*x]*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[1 - a^2/(b^2*x^2)]*x*EllipticF[ArcSin[Sqrt[a/b]/Sqrt[x]], -1])/(Sqrt[a/
b]*Sqrt[a - b*x]*Sqrt[a + b*x])

_______________________________________________________________________________________

Maple [A]  time = 0.278, size = 91, normalized size = 1.2 \[ -{\frac{a}{b \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }\sqrt{-bx+a}\sqrt{bx+a}\sqrt{{\frac{bx+a}{a}}}\sqrt{-2\,{\frac{bx-a}{a}}}\sqrt{-{\frac{bx}{a}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+a}{a}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(1/2)/(-b*x+a)^(1/2)/(b*x+a)^(1/2),x)

[Out]

-1/x^(1/2)*(-b*x+a)^(1/2)*(b*x+a)^(1/2)*a*((b*x+a)/a)^(1/2)*(-2*(b*x-a)/a)^(1/2)
*(-b*x/a)^(1/2)*EllipticF(((b*x+a)/a)^(1/2),1/2*2^(1/2))/b/(b^2*x^2-a^2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + a} \sqrt{-b x + a} \sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*sqrt(-b*x + a)*sqrt(x)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x + a)*sqrt(-b*x + a)*sqrt(x)), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x + a} \sqrt{-b x + a} \sqrt{x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*sqrt(-b*x + a)*sqrt(x)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(b*x + a)*sqrt(-b*x + a)*sqrt(x)), x)

_______________________________________________________________________________________

Sympy [A]  time = 20.1512, size = 99, normalized size = 1.32 \[ \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{2}, 1, 1 & \frac{3}{4}, \frac{3}{4}, \frac{5}{4} \\\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4} & 0 \end{matrix} \middle |{\frac{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} \sqrt{a} \sqrt{b}} - \frac{i{G_{6, 6}^{3, 5}\left (\begin{matrix} - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4} & 1 \\0, \frac{1}{2}, 0 & - \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \end{matrix} \middle |{\frac{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} \sqrt{a} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(1/2)/(-b*x+a)**(1/2)/(b*x+a)**(1/2),x)

[Out]

I*meijerg(((1/2, 1, 1), (3/4, 3/4, 5/4)), ((1/4, 1/2, 3/4, 1, 5/4), (0,)), a**2/
(b**2*x**2))/(4*pi**(3/2)*sqrt(a)*sqrt(b)) - I*meijerg(((-1/4, 0, 1/4, 1/2, 3/4)
, (1,)), ((0, 1/2, 0), (-1/4, 1/4, 1/4)), a**2*exp_polar(-2*I*pi)/(b**2*x**2))/(
4*pi**(3/2)*sqrt(a)*sqrt(b))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + a} \sqrt{-b x + a} \sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*sqrt(-b*x + a)*sqrt(x)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x + a)*sqrt(-b*x + a)*sqrt(x)), x)