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

Optimal. Leaf size=77 \[ -\frac {2 \sqrt {a} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {-x}}{\sqrt {a}}\right ),-1\right )}{\sqrt {b} \sqrt {a-b x} \sqrt {a+b x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {117, 116} \[ -\frac {2 \sqrt {a} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {-x}}{\sqrt {a}}\right )\right |-1\right )}{\sqrt {b} \sqrt {a-b x} \sqrt {a+b x}} \]

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]*Sq
rt[a - b*x]*Sqrt[a + b*x])

Rule 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (PosQ[-(b/d)] || NegQ[-(b/f)])

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[1 + (d*x)/c]
*Sqrt[1 + (f*x)/e])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Int[1/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]*Sqrt[1 + (f*x)/e]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-x} \sqrt {a-b x} \sqrt {a+b x}} \, dx &=\frac {\left (\sqrt {1-\frac {b x}{a}} \sqrt {1+\frac {b x}{a}}\right ) \int \frac {1}{\sqrt {-x} \sqrt {1-\frac {b x}{a}} \sqrt {1+\frac {b x}{a}}} \, dx}{\sqrt {a-b x} \sqrt {a+b x}}\\ &=-\frac {2 \sqrt {a} \sqrt {1-\frac {b x}{a}} \sqrt {1+\frac {b x}{a}} 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}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.03, size = 66, normalized size = 0.86 \[ \frac {2 x \sqrt {1-\frac {b^2 x^2}{a^2}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {b^2 x^2}{a^2}\right )}{\sqrt {-x} \sqrt {a-b x} \sqrt {a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[1 - (b^2*x^2)/a^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (b^2*x^2)/a^2])/(Sqrt[-x]*Sqrt[a - b*x]*Sqrt[a +
b*x])

________________________________________________________________________________________

fricas [F]  time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} \sqrt {-b x + a} \sqrt {-x}}{b^{2} x^{3} - a^{2} x}, x\right ) \]

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, algorithm="fricas")

[Out]

integral(sqrt(b*x + a)*sqrt(-b*x + a)*sqrt(-x)/(b^2*x^3 - a^2*x), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x + a} \sqrt {-b x + a} \sqrt {-x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x)^(1/2)/(-b*x+a)^(1/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 93, normalized size = 1.21 \[ -\frac {\sqrt {-b x +a}\, \sqrt {b x +a}\, \sqrt {\frac {b x +a}{a}}\, \sqrt {-\frac {2 \left (b x -a \right )}{a}}\, \sqrt {-\frac {b x}{a}}\, a \EllipticF \left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-x}\, \left (b^{2} x^{2}-a^{2}\right ) b} \]

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]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x + a} \sqrt {-b x + a} \sqrt {-x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x)^(1/2)/(-b*x+a)^(1/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {-x}\,\sqrt {a+b\,x}\,\sqrt {a-b\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 6.75, size = 95, normalized size = 1.23 \[ \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {a} \sqrt {b}} - \frac {{G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {a} \sqrt {b}} \]

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]

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)) - 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))

________________________________________________________________________________________