3.9.63 \(\int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+c x}} \, dx\) [863]

Optimal. Leaf size=33 \[ \frac {2 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {b} \sqrt {c}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 33, 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 = {117} \begin {gather*} \frac {2 F\left (\left .\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {b} \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2/(b*Sqrt[e]))*Rt
[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*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 {b x} \sqrt {1-c x} \sqrt {1+c x}} \, dx &=\frac {2 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {b} \sqrt {c}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.02, size = 27, normalized size = 0.82 \begin {gather*} \frac {2 x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};c^2 x^2\right )}{\sqrt {b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.10, size = 32, normalized size = 0.97

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {-c x}\, \EllipticF \left (\sqrt {c x +1}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {b x}}\)	\(32\)
elliptic	\(\frac {\sqrt {-b x \left (c^{2} x^{2}-1\right )}\, \sqrt {c \left (x +\frac {1}{c}\right )}\, \sqrt {-2 c \left (x -\frac {1}{c}\right )}\, \sqrt {-c x}\, \EllipticF \left (\sqrt {c \left (x +\frac {1}{c}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {b x}\, \sqrt {-c x +1}\, \sqrt {c x +1}\, c \sqrt {-b \,c^{2} x^{3}+b x}}\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x)^(1/2)/(-c*x+1)^(1/2)/(c*x+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 24, normalized size = 0.73 \begin {gather*} -\frac {2 \, \sqrt {-b c^{2}} {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )}{b c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-b*c^2)*weierstrassPInverse(4/c^2, 0, x)/(b*c^2)

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (29) = 58\).
time = 13.13, size = 94, normalized size = 2.85 \begin {gather*} \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 {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {b} \sqrt {c}} - \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 {e^{- 2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {b} \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x)**(1/2)/(-c*x+1)**(1/2)/(c*x+1)**(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,)), 1/(c**2*x**2))/(4*pi**(3/2)*sqrt(b)
*sqrt(c)) - I*meijerg(((-1/4, 0, 1/4, 1/2, 3/4), (1,)), ((0, 1/2, 0), (-1/4, 1/4, 1/4)), exp_polar(-2*I*pi)/(c
**2*x**2))/(4*pi**(3/2)*sqrt(b)*sqrt(c))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {b\,x}\,\sqrt {1-c\,x}\,\sqrt {c\,x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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