Integrand size = 22, antiderivative size = 120 \[ \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}} \, dx=\frac {\sqrt [4]{6} \sqrt {c} \sqrt {3-2 x^2} E\left (\left .\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{\sqrt {3 a-2 a x^2}}-\frac {\sqrt [4]{6} \sqrt {c} \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{\sqrt {3 a-2 a x^2}} \] Output:
6^(1/4)*c^(1/2)*(-2*x^2+3)^(1/2)*EllipticE(1/3*2^(1/4)*3^(3/4)*(c*x)^(1/2) /c^(1/2),I)/(-2*a*x^2+3*a)^(1/2)-6^(1/4)*c^(1/2)*(-2*x^2+3)^(1/2)*Elliptic F(1/3*2^(1/4)*3^(3/4)*(c*x)^(1/2)/c^(1/2),I)/(-2*a*x^2+3*a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}} \, dx=\frac {2 x \sqrt {c x} \sqrt {3-2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {2 x^2}{3}\right )}{3 \sqrt {a \left (9-6 x^2\right )}} \] Input:
Integrate[Sqrt[c*x]/Sqrt[3*a - 2*a*x^2],x]
Output:
(2*x*Sqrt[c*x]*Sqrt[3 - 2*x^2]*Hypergeometric2F1[1/2, 3/4, 7/4, (2*x^2)/3] )/(3*Sqrt[a*(9 - 6*x^2)])
Time = 0.18 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.56, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {261, 260, 27, 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.
\(\displaystyle \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}} \, dx\) |
\(\Big \downarrow \) 261 |
\(\displaystyle \frac {\sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3 a-2 a x^2}}dx}{\sqrt {x}}\) |
\(\Big \downarrow \) 260 |
\(\displaystyle \frac {\sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {3} \sqrt {x}}{\sqrt {3-2 x^2}}dx}{\sqrt {3} \sqrt {x} \sqrt {3 a-2 a x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3-2 x^2}}dx}{\sqrt {x} \sqrt {3 a-2 a x^2}}\) |
\(\Big \downarrow \) 259 |
\(\displaystyle -\frac {\sqrt [4]{6} \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {\frac {1}{3} \left (\sqrt {6} x-3\right )+1}}{\sqrt {\frac {1}{6} \left (\sqrt {6} x-3\right )+1}}d\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}}{\sqrt {x} \sqrt {3 a-2 a x^2}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle -\frac {\sqrt [4]{6} \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\arcsin \left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{\sqrt {x} \sqrt {3 a-2 a x^2}}\) |
Input:
Int[Sqrt[c*x]/Sqrt[3*a - 2*a*x^2],x]
Output:
-((6^(1/4)*Sqrt[c*x]*Sqrt[3 - 2*x^2]*EllipticE[ArcSin[Sqrt[3 - Sqrt[6]*x]/ Sqrt[6]], 2])/(Sqrt[x]*Sqrt[3*a - 2*a*x^2]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[x_]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sqrt[1 + b*(x^2/a )]/Sqrt[a + b*x^2] Int[Sqrt[x]/Sqrt[1 + b*(x^2/a)], x], 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 = 0.39 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.27
method | result | size |
elliptic | \(\frac {\sqrt {c x}\, \sqrt {-c x a \left (2 x^{2}-3\right )}\, \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 \sqrt {6}\, x}\, \left (-\sqrt {6}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right )}{54 x \sqrt {-a \left (2 x^{2}-3\right )}\, \sqrt {-2 a c \,x^{3}+3 a c x}}\) | \(152\) |
default | \(\frac {\sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {\left (-2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {3}\, \sqrt {-\sqrt {3}\, \sqrt {2}\, x}\, \left (2 \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right )\right )}{12 x a \left (2 x^{2}-3\right )}\) | \(165\) |
Input:
int((c*x)^(1/2)/(-2*a*x^2+3*a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/54/x*(c*x)^(1/2)/(-a*(2*x^2-3))^(1/2)*(-c*x*a*(2*x^2-3))^(1/2)*6^(1/2)*3 ^(1/2)*((x+1/2*6^(1/2))*6^(1/2))^(1/2)*(-6*(x-1/2*6^(1/2))*6^(1/2))^(1/2)* (-3*6^(1/2)*x)^(1/2)/(-2*a*c*x^3+3*a*c*x)^(1/2)*(-6^(1/2)*EllipticE(1/3*3^ (1/2)*((x+1/2*6^(1/2))*6^(1/2))^(1/2),1/2*2^(1/2))+1/2*6^(1/2)*EllipticF(1 /3*3^(1/2)*((x+1/2*6^(1/2))*6^(1/2))^(1/2),1/2*2^(1/2)))
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}} \, dx=\frac {\sqrt {2} \sqrt {-a c} {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right )}{a} \] Input:
integrate((c*x)^(1/2)/(-2*a*x^2+3*a)^(1/2),x, algorithm="fricas")
Output:
sqrt(2)*sqrt(-a*c)*weierstrassZeta(6, 0, weierstrassPInverse(6, 0, x))/a
Time = 0.51 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}} \, dx=\frac {\sqrt {3} \sqrt {c} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{6 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((c*x)**(1/2)/(-2*a*x**2+3*a)**(1/2),x)
Output:
sqrt(3)*sqrt(c)*x**(3/2)*gamma(3/4)*hyper((1/2, 3/4), (7/4,), 2*x**2*exp_p olar(2*I*pi)/3)/(6*sqrt(a)*gamma(7/4))
\[ \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}} \, dx=\int { \frac {\sqrt {c x}}{\sqrt {-2 \, a x^{2} + 3 \, a}} \,d x } \] Input:
integrate((c*x)^(1/2)/(-2*a*x^2+3*a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(c*x)/sqrt(-2*a*x^2 + 3*a), x)
\[ \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}} \, dx=\int { \frac {\sqrt {c x}}{\sqrt {-2 \, a x^{2} + 3 \, a}} \,d x } \] Input:
integrate((c*x)^(1/2)/(-2*a*x^2+3*a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(c*x)/sqrt(-2*a*x^2 + 3*a), x)
Timed out. \[ \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}} \, dx=\int \frac {\sqrt {c\,x}}{\sqrt {3\,a-2\,a\,x^2}} \,d x \] Input:
int((c*x)^(1/2)/(3*a - 2*a*x^2)^(1/2),x)
Output:
int((c*x)^(1/2)/(3*a - 2*a*x^2)^(1/2), x)
\[ \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}} \, dx=-\frac {\sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{2}+3}}{2 x^{2}-3}d x \right )}{a} \] Input:
int((c*x)^(1/2)/(-2*a*x^2+3*a)^(1/2),x)
Output:
( - sqrt(c)*sqrt(a)*int((sqrt(x)*sqrt( - 2*x**2 + 3))/(2*x**2 - 3),x))/a