Integrand size = 22, antiderivative size = 79 \[ \int \frac {\sqrt {a x}}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\frac {2 \sqrt {a} \sqrt {1-x} E\left (\left .\arcsin \left (\frac {\sqrt {a x}}{\sqrt {a}}\right )\right |-1\right )}{\sqrt {-1+x}}-\frac {2 \sqrt {a} \sqrt {1-x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a x}}{\sqrt {a}}\right ),-1\right )}{\sqrt {-1+x}} \] Output:
2*a^(1/2)*(1-x)^(1/2)*EllipticE((a*x)^(1/2)/a^(1/2),I)/(-1+x)^(1/2)-2*a^(1 /2)*(1-x)^(1/2)*EllipticF((a*x)^(1/2)/a^(1/2),I)/(-1+x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {a x}}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\frac {2 x \sqrt {a x} \sqrt {1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},x^2\right )}{3 \sqrt {-1+x} \sqrt {1+x}} \] Input:
Integrate[Sqrt[a*x]/(Sqrt[-1 + x]*Sqrt[1 + x]),x]
Output:
(2*x*Sqrt[a*x]*Sqrt[1 - x^2]*Hypergeometric2F1[1/2, 3/4, 7/4, x^2])/(3*Sqr t[-1 + x]*Sqrt[1 + x])
Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.61, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {124, 27, 123}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {a x}}{\sqrt {x-1} \sqrt {x+1}} \, dx\) |
\(\Big \downarrow \) 124 |
\(\displaystyle \frac {\sqrt {1-x} \sqrt {a x} \int \frac {\sqrt {2} \sqrt {-x}}{\sqrt {1-x} \sqrt {x+1}}dx}{\sqrt {2} \sqrt {x-1} \sqrt {-x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {1-x} \sqrt {a x} \int \frac {\sqrt {-x}}{\sqrt {1-x} \sqrt {x+1}}dx}{\sqrt {x-1} \sqrt {-x}}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle \frac {2 \sqrt {1-x} \sqrt {a x} E\left (\left .\arcsin \left (\frac {\sqrt {x+1}}{\sqrt {2}}\right )\right |2\right )}{\sqrt {x-1} \sqrt {-x}}\) |
Input:
Int[Sqrt[a*x]/(Sqrt[-1 + x]*Sqrt[1 + x]),x]
Output:
(2*Sqrt[1 - x]*Sqrt[a*x]*EllipticE[ArcSin[Sqrt[1 + x]/Sqrt[2]], 2])/(Sqrt[ -1 + x]*Sqrt[-x])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d *x]*Sqrt[b*((e + f*x)/(b*e - a*f))])) Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x /(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && Gt Q[b/(b*e - a*f), 0]) && !LtQ[-(b*c - a*d)/d, 0]
Time = 0.40 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70
method | result | size |
default | \(\frac {\left (-2 \operatorname {EllipticE}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {-x}\, \sqrt {1-x}\, \sqrt {2}\, \sqrt {a x}}{\sqrt {-1+x}\, x}\) | \(55\) |
elliptic | \(\frac {\sqrt {a x}\, \sqrt {a x \left (x^{2}-1\right )}\, \sqrt {2-2 x}\, \sqrt {-x}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-1+x}\, x \sqrt {a \,x^{3}-a x}}\) | \(74\) |
Input:
int((a*x)^(1/2)/(-1+x)^(1/2)/(1+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
(-2*EllipticE((1+x)^(1/2),1/2*2^(1/2))+EllipticF((1+x)^(1/2),1/2*2^(1/2))) *(-x)^(1/2)*(1-x)^(1/2)*2^(1/2)/(-1+x)^(1/2)*(a*x)^(1/2)/x
Time = 0.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.15 \[ \int \frac {\sqrt {a x}}{\sqrt {-1+x} \sqrt {1+x}} \, dx=-2 \, \sqrt {a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, x\right )\right ) \] Input:
integrate((a*x)^(1/2)/(-1+x)^(1/2)/(1+x)^(1/2),x, algorithm="fricas")
Output:
-2*sqrt(a)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, x))
Time = 103.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {a x}}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\frac {\sqrt {a} {G_{6, 6}^{6, 2}\left (\begin {matrix} 0, \frac {1}{2} & \frac {1}{4}, \frac {1}{4}, \frac {3}{4}, 1 \\- \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 0 & \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {\sqrt {a} {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4} & 1 \\- \frac {1}{2}, 0, 0 & - \frac {3}{4}, - \frac {1}{4}, - \frac {1}{4} \end {matrix} \middle | {\frac {e^{2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \] Input:
integrate((a*x)**(1/2)/(-1+x)**(1/2)/(1+x)**(1/2),x)
Output:
sqrt(a)*meijerg(((0, 1/2), (1/4, 1/4, 3/4, 1)), ((-1/4, 0, 1/4, 1/2, 3/4, 0), ()), x**(-2))/(4*pi**(3/2)) - sqrt(a)*meijerg(((-3/4, -1/2, -1/4, 0, 1 /4), (1,)), ((-1/2, 0, 0), (-3/4, -1/4, -1/4)), exp_polar(2*I*pi)/x**2)/(4 *pi**(3/2))
\[ \int \frac {\sqrt {a x}}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\int { \frac {\sqrt {a x}}{\sqrt {x + 1} \sqrt {x - 1}} \,d x } \] Input:
integrate((a*x)^(1/2)/(-1+x)^(1/2)/(1+x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(a*x)/(sqrt(x + 1)*sqrt(x - 1)), x)
\[ \int \frac {\sqrt {a x}}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\int { \frac {\sqrt {a x}}{\sqrt {x + 1} \sqrt {x - 1}} \,d x } \] Input:
integrate((a*x)^(1/2)/(-1+x)^(1/2)/(1+x)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(a*x)/(sqrt(x + 1)*sqrt(x - 1)), x)
Timed out. \[ \int \frac {\sqrt {a x}}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\int \frac {\sqrt {a\,x}}{\sqrt {x-1}\,\sqrt {x+1}} \,d x \] Input:
int((a*x)^(1/2)/((x - 1)^(1/2)*(x + 1)^(1/2)),x)
Output:
int((a*x)^(1/2)/((x - 1)^(1/2)*(x + 1)^(1/2)), x)
\[ \int \frac {\sqrt {a x}}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\sqrt {a}\, \left (\int \frac {\sqrt {x}\, \sqrt {x +1}\, \sqrt {x -1}}{x^{2}-1}d x \right ) \] Input:
int((a*x)^(1/2)/(-1+x)^(1/2)/(1+x)^(1/2),x)
Output:
sqrt(a)*int((sqrt(x)*sqrt(x + 1)*sqrt(x - 1))/(x**2 - 1),x)