Integrand size = 22, antiderivative size = 162 \[ \int \frac {1}{(c x)^{3/2} \sqrt {3 a-2 a x^2}} \, dx=-\frac {2 \sqrt {3 a-2 a x^2}}{3 a c \sqrt {c x}}-\frac {2 \sqrt [4]{2} \sqrt {3-2 x^2} E\left (\left .\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{3^{3/4} c^{3/2} \sqrt {3 a-2 a x^2}}+\frac {2 \sqrt [4]{2} \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{3^{3/4} c^{3/2} \sqrt {3 a-2 a x^2}} \] Output:
-2/3*(-2*a*x^2+3*a)^(1/2)/a/c/(c*x)^(1/2)-2/3*2^(1/4)*(-2*x^2+3)^(1/2)*Ell ipticE(1/3*2^(1/4)*3^(3/4)*(c*x)^(1/2)/c^(1/2),I)*3^(1/4)/c^(3/2)/(-2*a*x^ 2+3*a)^(1/2)+2/3*2^(1/4)*(-2*x^2+3)^(1/2)*EllipticF(1/3*2^(1/4)*3^(3/4)*(c *x)^(1/2)/c^(1/2),I)*3^(1/4)/c^(3/2)/(-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 = 51, normalized size of antiderivative = 0.31 \[ \int \frac {1}{(c x)^{3/2} \sqrt {3 a-2 a x^2}} \, dx=-\frac {2 x \sqrt {3-2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\frac {2 x^2}{3}\right )}{(c x)^{3/2} \sqrt {a \left (9-6 x^2\right )}} \] Input:
Integrate[1/((c*x)^(3/2)*Sqrt[3*a - 2*a*x^2]),x]
Output:
(-2*x*Sqrt[3 - 2*x^2]*Hypergeometric2F1[-1/4, 1/2, 3/4, (2*x^2)/3])/((c*x) ^(3/2)*Sqrt[a*(9 - 6*x^2)])
Time = 0.21 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.66, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {264, 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.
| \(\displaystyle \int \frac {1}{\sqrt {3 a-2 a x^2} (c x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle -\frac {2 \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}}dx}{3 c^2}-\frac {2 \sqrt {3 a-2 a x^2}}{3 a c \sqrt {c x}}\) |
| \(\Big \downarrow \) 261 |
| \(\displaystyle -\frac {2 \sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3 a-2 a x^2}}dx}{3 c^2 \sqrt {x}}-\frac {2 \sqrt {3 a-2 a x^2}}{3 a c \sqrt {c x}}\) |
| \(\Big \downarrow \) 260 |
| \(\displaystyle -\frac {2 \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {3} \sqrt {x}}{\sqrt {3-2 x^2}}dx}{3 \sqrt {3} c^2 \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {2 \sqrt {3 a-2 a x^2}}{3 a c \sqrt {c x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3-2 x^2}}dx}{3 c^2 \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {2 \sqrt {3 a-2 a x^2}}{3 a c \sqrt {c x}}\) |
| \(\Big \downarrow \) 259 |
| \(\displaystyle \frac {2 \sqrt [4]{2} \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {\frac {1}{3} \left (\sqrt {6} x-3\right )+1}}{\sqrt {\frac {1}{6} \left (\sqrt {6} x-3\right )+1}}d\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}}{3^{3/4} c^2 \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {2 \sqrt {3 a-2 a x^2}}{3 a c \sqrt {c x}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 \sqrt [4]{2} \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\arcsin \left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{3^{3/4} c^2 \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {2 \sqrt {3 a-2 a x^2}}{3 a c \sqrt {c x}}\) |
Input:
Int[1/((c*x)^(3/2)*Sqrt[3*a - 2*a*x^2]),x]
Output:
(-2*Sqrt[3*a - 2*a*x^2])/(3*a*c*Sqrt[c*x]) + (2*2^(1/4)*Sqrt[c*x]*Sqrt[3 - 2*x^2]*EllipticE[ArcSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6]], 2])/(3^(3/4)*c^2*Sq rt[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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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 = 0.62 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.12
| method | result | size |
| risch | \(\frac {\frac {4 x^{2}}{3}-2}{c \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}-\frac {\sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 \sqrt {6}\, x}\, \left (-\sqrt {6}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right ) \sqrt {-c x a \left (2 x^{2}-3\right )}}{81 \sqrt {-2 a c \,x^{3}+3 a c x}\, c \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}\) | \(182\) |
| elliptic | \(\frac {\sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (-\frac {2 \left (-2 a c \,x^{2}+3 a c \right )}{3 a \,c^{2} \sqrt {x \left (-2 a c \,x^{2}+3 a c \right )}}-\frac {\sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 \sqrt {6}\, x}\, \left (-\sqrt {6}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right )}{81 c \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{\sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}\) | \(190\) |
| default | \(-\frac {\sqrt {-a \left (2 x^{2}-3\right )}\, \left (2 \sqrt {\left (-2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {3}\, \sqrt {-\sqrt {3}\, \sqrt {2}\, x}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}-\sqrt {\left (-2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {3}\, \sqrt {-\sqrt {3}\, \sqrt {2}\, x}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}+24 x^{2}-36\right )}{18 c \sqrt {c x}\, a \left (2 x^{2}-3\right )}\) | \(228\) |
Input:
int(1/(c*x)^(3/2)/(-2*a*x^2+3*a)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/3*(2*x^2-3)/c/(c*x)^(1/2)/(-a*(2*x^2-3))^(1/2)-1/81*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)))/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 = 48, normalized size of antiderivative = 0.30 \[ \int \frac {1}{(c x)^{3/2} \sqrt {3 a-2 a x^2}} \, dx=-\frac {2 \, {\left (\sqrt {2} \sqrt {-a c} x {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right ) + \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}\right )}}{3 \, a c^{2} x} \] Input:
integrate(1/(c*x)^(3/2)/(-2*a*x^2+3*a)^(1/2),x, algorithm="fricas")
Output:
-2/3*(sqrt(2)*sqrt(-a*c)*x*weierstrassZeta(6, 0, weierstrassPInverse(6, 0, x)) + sqrt(-2*a*x^2 + 3*a)*sqrt(c*x))/(a*c^2*x)
Time = 0.82 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.33 \[ \int \frac {1}{(c x)^{3/2} \sqrt {3 a-2 a x^2}} \, dx=\frac {\sqrt {3} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{6 \sqrt {a} c^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \] Input:
integrate(1/(c*x)**(3/2)/(-2*a*x**2+3*a)**(1/2),x)
Output:
sqrt(3)*gamma(-1/4)*hyper((-1/4, 1/2), (3/4,), 2*x**2*exp_polar(2*I*pi)/3) /(6*sqrt(a)*c**(3/2)*sqrt(x)*gamma(3/4))
\[ \int \frac {1}{(c x)^{3/2} \sqrt {3 a-2 a x^2}} \, dx=\int { \frac {1}{\sqrt {-2 \, a x^{2} + 3 \, a} \left (c x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(3/2)/(-2*a*x^2+3*a)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(-2*a*x^2 + 3*a)*(c*x)^(3/2)), x)
\[ \int \frac {1}{(c x)^{3/2} \sqrt {3 a-2 a x^2}} \, dx=\int { \frac {1}{\sqrt {-2 \, a x^{2} + 3 \, a} \left (c x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(3/2)/(-2*a*x^2+3*a)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(-2*a*x^2 + 3*a)*(c*x)^(3/2)), x)
Timed out. \[ \int \frac {1}{(c x)^{3/2} \sqrt {3 a-2 a x^2}} \, dx=\int \frac {1}{{\left (c\,x\right )}^{3/2}\,\sqrt {3\,a-2\,a\,x^2}} \,d x \] Input:
int(1/((c*x)^(3/2)*(3*a - 2*a*x^2)^(1/2)),x)
Output:
int(1/((c*x)^(3/2)*(3*a - 2*a*x^2)^(1/2)), x)
\[ \int \frac {1}{(c x)^{3/2} \sqrt {3 a-2 a x^2}} \, dx=-\frac {\sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{2}+3}}{2 x^{4}-3 x^{2}}d x \right )}{a \,c^{2}} \] Input:
int(1/(c*x)^(3/2)/(-2*a*x^2+3*a)^(1/2),x)
Output:
( - sqrt(c)*sqrt(a)*int((sqrt(x)*sqrt( - 2*x**2 + 3))/(2*x**4 - 3*x**2),x) )/(a*c**2)