Integrand size = 25, antiderivative size = 67 \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\frac {2 \sqrt {a} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{\sqrt {b} \sqrt {a-b x} \sqrt {a+b x}} \] Output:
2*a^(1/2)*(1-b^2*x^2/a^2)^(1/2)*EllipticF(b^(1/2)*x^(1/2)/a^(1/2),I)/b^(1/ 2)/(-b*x+a)^(1/2)/(b*x+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.72 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\frac {2 \sqrt {x} \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b^2 x^2}{a^2}\right )}{\sqrt {a-b x} \sqrt {a+b x}} \] Input:
Integrate[1/(Sqrt[x]*Sqrt[a - b*x]*Sqrt[a + b*x]),x]
Output:
(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])
Time = 0.17 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {127, 126}
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.
\(\displaystyle \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx\) |
\(\Big \downarrow \) 127 |
\(\displaystyle \frac {\sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \int \frac {1}{\sqrt {x} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1}}dx}{\sqrt {a-b x} \sqrt {a+b x}}\) |
\(\Big \downarrow \) 126 |
\(\displaystyle \frac {2 \sqrt {a} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),-1\right )}{\sqrt {b} \sqrt {a-b x} \sqrt {a+b x}}\) |
Input:
Int[1/(Sqrt[x]*Sqrt[a - b*x]*Sqrt[a + b*x]),x]
Output:
(2*Sqrt[a]*Sqrt[1 - (b*x)/a]*Sqrt[1 + (b*x)/a]*EllipticF[ArcSin[(Sqrt[b]*S qrt[x])/Sqrt[a]], -1])/(Sqrt[b]*Sqrt[a - b*x]*Sqrt[a + b*x])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> 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])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[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] /; Free Q[{b, c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Time = 0.78 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.34
method | result | size |
default | \(\frac {\sqrt {-b x +a}\, \sqrt {b x +a}\, a \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x}\, b \left (-b^{2} x^{2}+a^{2}\right )}\) | \(90\) |
elliptic | \(\frac {\sqrt {x \left (-b^{2} x^{2}+a^{2}\right )}\, a \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (x -\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}\, b \sqrt {-b^{2} x^{3}+a^{2} x}}\) | \(115\) |
Input:
int(1/x^(1/2)/(-b*x+a)^(1/2)/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
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)
Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.34 \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=-\frac {2 \, \sqrt {-b^{2}} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )}{b^{2}} \] Input:
integrate(1/x^(1/2)/(-b*x+a)^(1/2)/(b*x+a)^(1/2),x, algorithm="fricas")
Output:
-2*sqrt(-b^2)*weierstrassPInverse(4*a^2/b^2, 0, x)/b^2
Time = 13.58 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.48 \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {a} \sqrt {b}} - \frac {i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {a} \sqrt {b}} \] Input:
integrate(1/x**(1/2)/(-b*x+a)**(1/2)/(b*x+a)**(1/2),x)
Output:
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))
\[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {-b x + a} \sqrt {x}} \,d x } \] Input:
integrate(1/x^(1/2)/(-b*x+a)^(1/2)/(b*x+a)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(b*x + a)*sqrt(-b*x + a)*sqrt(x)), x)
\[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {-b x + a} \sqrt {x}} \,d x } \] Input:
integrate(1/x^(1/2)/(-b*x+a)^(1/2)/(b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(b*x + a)*sqrt(-b*x + a)*sqrt(x)), x)
Timed out. \[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\int \frac {1}{\sqrt {x}\,\sqrt {a+b\,x}\,\sqrt {a-b\,x}} \,d x \] Input:
int(1/(x^(1/2)*(a + b*x)^(1/2)*(a - b*x)^(1/2)),x)
Output:
int(1/(x^(1/2)*(a + b*x)^(1/2)*(a - b*x)^(1/2)), x)
\[ \int \frac {1}{\sqrt {x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\int \frac {\sqrt {x}\, \sqrt {b x +a}\, \sqrt {-b x +a}}{-b^{2} x^{3}+a^{2} x}d x \] Input:
int(1/x^(1/2)/(-b*x+a)^(1/2)/(b*x+a)^(1/2),x)
Output:
int((sqrt(x)*sqrt(a + b*x)*sqrt(a - b*x))/(a**2*x - b**2*x**3),x)