3.857 \(\int \frac {1}{\sqrt {e x} \sqrt {2-b x} \sqrt {2+b x}} \, dx\)

Optimal. Leaf size=42 \[ \frac {\sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {2} \sqrt {e}}\right ),-1\right )}{\sqrt {b} \sqrt {e}} \]

[Out]

EllipticF(1/2*b^(1/2)*(e*x)^(1/2)*2^(1/2)/e^(1/2),I)*2^(1/2)/b^(1/2)/e^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {116} \[ \frac {\sqrt {2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {2} \sqrt {e}}\right )\right |-1\right )}{\sqrt {b} \sqrt {e}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[e*x]*Sqrt[2 - b*x]*Sqrt[2 + b*x]),x]

[Out]

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

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)])

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {e x} \sqrt {2-b x} \sqrt {2+b x}} \, dx &=\frac {\sqrt {2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {2} \sqrt {e}}\right )\right |-1\right )}{\sqrt {b} \sqrt {e}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 29, normalized size = 0.69 \[ \frac {x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {b^2 x^2}{4}\right )}{\sqrt {e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[e*x]*Sqrt[2 - b*x]*Sqrt[2 + b*x]),x]

[Out]

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

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fricas [F]  time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b x + 2} \sqrt {-b x + 2} \sqrt {e x}}{b^{2} e x^{3} - 4 \, e x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(b*x + 2)*sqrt(-b*x + 2)*sqrt(e*x)/(b^2*e*x^3 - 4*e*x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x + 2} \sqrt {-b x + 2} \sqrt {e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b*x + 2)*sqrt(-b*x + 2)*sqrt(e*x)), x)

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maple [A]  time = 0.05, size = 34, normalized size = 0.81 \[ \frac {\sqrt {-b x}\, \EllipticF \left (\frac {\sqrt {2}\, \sqrt {b x +2}}{2}, \frac {\sqrt {2}}{2}\right )}{\sqrt {e x}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x + 2} \sqrt {-b x + 2} \sqrt {e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b*x + 2)*sqrt(-b*x + 2)*sqrt(e*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {e\,x}\,\sqrt {2-b\,x}\,\sqrt {b\,x+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 6.52, size = 105, normalized size = 2.50 \[ \frac {\sqrt {2} i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {4}{b^{2} x^{2}}} \right )}}{8 \pi ^{\frac {3}{2}} \sqrt {b} \sqrt {e}} - \frac {\sqrt {2} i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {4 e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{8 \pi ^{\frac {3}{2}} \sqrt {b} \sqrt {e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x)**(1/2)/(-b*x+2)**(1/2)/(b*x+2)**(1/2),x)

[Out]

sqrt(2)*I*meijerg(((1/2, 1, 1), (3/4, 3/4, 5/4)), ((1/4, 1/2, 3/4, 1, 5/4), (0,)), 4/(b**2*x**2))/(8*pi**(3/2)
*sqrt(b)*sqrt(e)) - sqrt(2)*I*meijerg(((-1/4, 0, 1/4, 1/2, 3/4), (1,)), ((0, 1/2, 0), (-1/4, 1/4, 1/4)), 4*exp
_polar(-2*I*pi)/(b**2*x**2))/(8*pi**(3/2)*sqrt(b)*sqrt(e))

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