Integrand size = 22, antiderivative size = 155 \[ \int \sqrt {c x} \sqrt {3 a-2 a x^2} \, dx=\frac {2 (c x)^{3/2} \sqrt {3 a-2 a x^2}}{5 c}+\frac {6 \sqrt [4]{6} a \sqrt {c} \sqrt {3-2 x^2} E\left (\left .\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{5 \sqrt {3 a-2 a x^2}}-\frac {6 \sqrt [4]{6} a \sqrt {c} \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{5 \sqrt {3 a-2 a x^2}} \] Output:
2/5*(c*x)^(3/2)*(-2*a*x^2+3*a)^(1/2)/c+6/5*6^(1/4)*a*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)-6/5*6^(1/4)*a*c^(1/2)*(-2*x^2+3)^(1/2)*EllipticF(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 = 6.57 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.33 \[ \int \sqrt {c x} \sqrt {3 a-2 a x^2} \, dx=\frac {2 x \sqrt {c x} \sqrt {a \left (3-2 x^2\right )} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {2 x^2}{3}\right )}{\sqrt {9-6 x^2}} \] Input:
Integrate[Sqrt[c*x]*Sqrt[3*a - 2*a*x^2],x]
Output:
(2*x*Sqrt[c*x]*Sqrt[a*(3 - 2*x^2)]*Hypergeometric2F1[-1/2, 3/4, 7/4, (2*x^ 2)/3])/Sqrt[9 - 6*x^2]
Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.64, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {248, 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.
| \(\displaystyle \int \sqrt {3 a-2 a x^2} \sqrt {c x} \, dx\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {6}{5} a \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}}dx+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 261 |
| \(\displaystyle \frac {6 a \sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3 a-2 a x^2}}dx}{5 \sqrt {x}}+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 260 |
| \(\displaystyle \frac {2 \sqrt {3} a \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {3} \sqrt {x}}{\sqrt {3-2 x^2}}dx}{5 \sqrt {x} \sqrt {3 a-2 a x^2}}+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6 a \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3-2 x^2}}dx}{5 \sqrt {x} \sqrt {3 a-2 a x^2}}+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 259 |
| \(\displaystyle \frac {2 \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 c}-\frac {6 \sqrt [4]{6} a \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {\frac {1}{3} \left (\sqrt {6} x-3\right )+1}}{\sqrt {\frac {1}{6} \left (\sqrt {6} x-3\right )+1}}d\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}}{5 \sqrt {x} \sqrt {3 a-2 a x^2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 c}-\frac {6 \sqrt [4]{6} a \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\arcsin \left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{5 \sqrt {x} \sqrt {3 a-2 a x^2}}\) |
Input:
Int[Sqrt[c*x]*Sqrt[3*a - 2*a*x^2],x]
Output:
(2*(c*x)^(3/2)*Sqrt[3*a - 2*a*x^2])/(5*c) - (6*6^(1/4)*a*Sqrt[c*x]*Sqrt[3 - 2*x^2]*EllipticE[ArcSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6]], 2])/(5*Sqrt[x]*Sqr t[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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, 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 = 0.46 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(-\frac {2 x^{2} \left (2 x^{2}-3\right ) a c}{5 \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}+\frac {\sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 \sqrt {6}\, x}\, \left (-\sqrt {6}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right ) a c \sqrt {-c x a \left (2 x^{2}-3\right )}}{45 \sqrt {-2 a c \,x^{3}+3 a c x}\, \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}\) | \(183\) |
| elliptic | \(-\frac {\sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (\frac {2 x \sqrt {-2 a c \,x^{3}+3 a c x}}{5}+\frac {a c \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 \sqrt {6}\, x}\, \left (-\sqrt {6}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right )}{45 \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{c x a \left (2 x^{2}-3\right )}\) | \(190\) |
| default | \(\frac {\sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \left (2 \sqrt {\left (-2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {3}\, \sqrt {-\sqrt {3}\, \sqrt {2}\, x}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}-\sqrt {\left (-2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {3}\, \sqrt {-\sqrt {3}\, \sqrt {2}\, x}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}+8 x^{4}-12 x^{2}\right )}{10 x \left (2 x^{2}-3\right )}\) | \(229\) |
Input:
int((c*x)^(1/2)*(-2*a*x^2+3*a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/5*x^2*(2*x^2-3)*a*c/(c*x)^(1/2)/(-a*(2*x^2-3))^(1/2)+1/45*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)))*a*c*(-c*x*a*(2*x^2-3) )^(1/2)/(c*x)^(1/2)/(-a*(2*x^2-3))^(1/2)
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.25 \[ \int \sqrt {c x} \sqrt {3 a-2 a x^2} \, dx=\frac {2}{5} \, \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} x + \frac {6}{5} \, \sqrt {2} \sqrt {-a c} {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right ) \] Input:
integrate((c*x)^(1/2)*(-2*a*x^2+3*a)^(1/2),x, algorithm="fricas")
Output:
2/5*sqrt(-2*a*x^2 + 3*a)*sqrt(c*x)*x + 6/5*sqrt(2)*sqrt(-a*c)*weierstrassZ eta(6, 0, weierstrassPInverse(6, 0, x))
Time = 0.59 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.34 \[ \int \sqrt {c x} \sqrt {3 a-2 a x^2} \, dx=\frac {\sqrt {3} \sqrt {a} \sqrt {c} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((c*x)**(1/2)*(-2*a*x**2+3*a)**(1/2),x)
Output:
sqrt(3)*sqrt(a)*sqrt(c)*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), 2*x **2*exp_polar(2*I*pi)/3)/(2*gamma(7/4))
\[ \int \sqrt {c x} \sqrt {3 a-2 a x^2} \, dx=\int { \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} \,d x } \] Input:
integrate((c*x)^(1/2)*(-2*a*x^2+3*a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-2*a*x^2 + 3*a)*sqrt(c*x), x)
\[ \int \sqrt {c x} \sqrt {3 a-2 a x^2} \, dx=\int { \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} \,d x } \] Input:
integrate((c*x)^(1/2)*(-2*a*x^2+3*a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-2*a*x^2 + 3*a)*sqrt(c*x), x)
Timed out. \[ \int \sqrt {c x} \sqrt {3 a-2 a x^2} \, dx=\int \sqrt {c\,x}\,\sqrt {3\,a-2\,a\,x^2} \,d x \] 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)
\[ \int \sqrt {c x} \sqrt {3 a-2 a x^2} \, dx=\frac {2 \sqrt {c}\, \sqrt {a}\, \left (\sqrt {x}\, \sqrt {-2 x^{2}+3}\, x -3 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{2}+3}}{2 x^{2}-3}d x \right )\right )}{5} \] Input:
int((c*x)^(1/2)*(-2*a*x^2+3*a)^(1/2),x)
Output:
(2*sqrt(c)*sqrt(a)*(sqrt(x)*sqrt( - 2*x**2 + 3)*x - 3*int((sqrt(x)*sqrt( - 2*x**2 + 3))/(2*x**2 - 3),x)))/5