Integrand size = 24, antiderivative size = 101 \[ \int \frac {\sqrt {a x}}{\sqrt {-1-x} \sqrt {-1+x}} \, dx=\frac {2 \sqrt {a} \sqrt {1-x^2} E\left (\left .\arcsin \left (\frac {\sqrt {a x}}{\sqrt {a}}\right )\right |-1\right )}{\sqrt {-1-x} \sqrt {-1+x}}-\frac {2 \sqrt {a} \sqrt {1-x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a x}}{\sqrt {a}}\right ),-1\right )}{\sqrt {-1-x} \sqrt {-1+x}} \] Output:
2*a^(1/2)*(-x^2+1)^(1/2)*EllipticE((a*x)^(1/2)/a^(1/2),I)/(-1-x)^(1/2)/(-1 +x)^(1/2)-2*a^(1/2)*(-x^2+1)^(1/2)*EllipticF((a*x)^(1/2)/a^(1/2),I)/(-1-x) ^(1/2)/(-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 = 52, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {a x}}{\sqrt {-1-x} \sqrt {-1+x}} \, dx=\frac {2 x \sqrt {a x} \sqrt {1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},x^2\right )}{3 \sqrt {-1-x} \sqrt {-1+x}} \] 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 = 0.17 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.58, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {136, 261, 259, 327}
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.
\(\displaystyle \int \frac {\sqrt {a x}}{\sqrt {-x-1} \sqrt {x-1}} \, dx\) |
\(\Big \downarrow \) 136 |
\(\displaystyle \frac {\sqrt {1-x^2} \int \frac {\sqrt {a x}}{\sqrt {1-x^2}}dx}{\sqrt {-x-1} \sqrt {x-1}}\) |
\(\Big \downarrow \) 261 |
\(\displaystyle \frac {\sqrt {1-x^2} \sqrt {a x} \int \frac {\sqrt {x}}{\sqrt {1-x^2}}dx}{\sqrt {-x-1} \sqrt {x-1} \sqrt {x}}\) |
\(\Big \downarrow \) 259 |
\(\displaystyle -\frac {2 \sqrt {1-x^2} \sqrt {a x} \int \frac {\sqrt {x}}{\sqrt {\frac {x-1}{2}+1}}d\frac {\sqrt {1-x}}{\sqrt {2}}}{\sqrt {-x-1} \sqrt {x-1} \sqrt {x}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle -\frac {2 \sqrt {1-x^2} \sqrt {a x} E\left (\left .\arcsin \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )\right |2\right )}{\sqrt {-x-1} \sqrt {x-1} \sqrt {x}}\) |
Input:
Int[Sqrt[a*x]/(Sqrt[-1 - x]*Sqrt[-1 + x]),x]
Output:
(-2*Sqrt[a*x]*Sqrt[1 - x^2]*EllipticE[ArcSin[Sqrt[1 - x]/Sqrt[2]], 2])/(Sq rt[-1 - x]*Sqrt[-1 + x]*Sqrt[x])
Int[((f_.)*(x_))^(p_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^Fr acPart[m]) Int[(a*c + b*d*x^2)^m*(f*x)^p, x], x] /; FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m]
Int[Sqrt[x_]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[-2/(Sqrt[a]*(-b/a )^(3/4)) Subst[Int[Sqrt[1 - 2*x^2]/Sqrt[1 - x^2], x], x, Sqrt[1 - Sqrt[-b /a]*x]/Sqrt[2]], x] /; FreeQ[{a, b}, x] && GtQ[-b/a, 0] && GtQ[a, 0]
Int[Sqrt[(c_)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sqrt[c*x]/ Sqrt[x] Int[Sqrt[x]/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ [-b/a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Time = 0.47 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {\sqrt {a x}\, \sqrt {-1-x}\, \sqrt {-1+x}\, \sqrt {1-x}\, \sqrt {2}\, \sqrt {1+x}\, \left (2 \operatorname {EllipticE}\left (\sqrt {1-x}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {1-x}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {x}\, \left (x^{2}-1\right )}\) | \(75\) |
elliptic | \(-\frac {\sqrt {a x}\, \sqrt {-a x \left (x^{2}-1\right )}\, \sqrt {1-x}\, \sqrt {2+2 x}\, \left (2 \operatorname {EllipticE}\left (\sqrt {1-x}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {1-x}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-1-x}\, \sqrt {-1+x}\, \sqrt {x}\, \sqrt {-a \,x^{3}+a x}}\) | \(91\) |
Input:
int((a*x)^(1/2)/(-1-x)^(1/2)/(-1+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
(a*x)^(1/2)*(-1-x)^(1/2)*(-1+x)^(1/2)*(1-x)^(1/2)*2^(1/2)*(1+x)^(1/2)/x^(1 /2)*(2*EllipticE((1-x)^(1/2),1/2*2^(1/2))-EllipticF((1-x)^(1/2),1/2*2^(1/2 )))/(x^2-1)
Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt {a x}}{\sqrt {-1-x} \sqrt {-1+x}} \, dx=2 \, \sqrt {-a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, x\right )\right ) \] 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 = 99.97 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a x}}{\sqrt {-1-x} \sqrt {-1+x}} \, dx=\frac {\sqrt {a} {G_{6, 6}^{6, 2}\left (\begin {matrix} 0, \frac {1}{2} & \frac {1}{4}, \frac {1}{4}, \frac {3}{4}, 1 \\- \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 0 & \end {matrix} \middle | {\frac {e^{2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {\sqrt {a} {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4} & 1 \\- \frac {1}{2}, 0, 0 & - \frac {3}{4}, - \frac {1}{4}, - \frac {1}{4} \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \] 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), ()), exp_polar(2*I*pi)/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)), 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)
\[ \int \frac {\sqrt {a x}}{\sqrt {-1-x} \sqrt {-1+x}} \, dx=-\sqrt {a}\, \left (\int \frac {\sqrt {x}\, \sqrt {x +1}\, \sqrt {x -1}}{x^{2}-1}d x \right ) i \] 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)*i