\(\int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx\) [852]

Optimal result

Integrand size = 25, antiderivative size = 75 \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\frac {2 \sqrt {a} \sqrt {1-\frac {b x}{a}} \sqrt {1+\frac {b x}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{\sqrt {b} \sqrt {a-b x} \sqrt {a+b x}} \]

[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] (verified)

Time = 0.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {118, 117} \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\frac {2 \sqrt {a} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{\sqrt {b} \sqrt {a-b x} \sqrt {a+b x}} \]

[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])

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])

Rule 118

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*} \text {integral}& = \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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.90 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\frac {2 \sqrt {x} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b^2 x^2}{a^2}\right )}{\sqrt {a-b x} \sqrt {a+b x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.49 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.20

method	result	size
default	\(\frac {\sqrt {-b x +a}\, \sqrt {b x +a}\, a \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, F\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x}\, b \left (-b^{2} x^{2}+a^{2}\right )}\)	\(90\)
elliptic	\(\frac {\sqrt {x \left (-b^{2} x^{2}+a^{2}\right )}\, a \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (-\frac {a}{b}+x \right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, F\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x}\, \sqrt {-b x +a}\, \sqrt {b x +a}\, b \sqrt {-b^{2} x^{3}+a^{2} x}}\)	\(115\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {-b x + a} \sqrt {x}} \,d x } \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 10.43 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {a} \sqrt {b}} - \frac {i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {a} \sqrt {b}} \]

[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))

Maxima [F]

\[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {-b x + a} \sqrt {x}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {-b x + a} \sqrt {x}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\int \frac {1}{\sqrt {x}\,\sqrt {a+b\,x}\,\sqrt {a-b\,x}} \,d x \]

[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)