Integrand size = 22, antiderivative size = 97 \[ \int \frac {\sqrt {3 a-2 a x^2}}{(c x)^{5/2}} \, dx=-\frac {2 \sqrt {3 a-2 a x^2}}{3 c (c x)^{3/2}}-\frac {4\ 2^{3/4} a \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{3 \sqrt [4]{3} c^{5/2} \sqrt {3 a-2 a x^2}} \] Output:
-2/3*(-2*a*x^2+3*a)^(1/2)/c/(c*x)^(3/2)-4/9*2^(3/4)*a*(-2*x^2+3)^(1/2)*Ell ipticF(1/3*2^(1/4)*3^(3/4)*(c*x)^(1/2)/c^(1/2),I)*3^(3/4)/c^(5/2)/(-2*a*x^ 2+3*a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {3 a-2 a x^2}}{(c x)^{5/2}} \, dx=-\frac {2 x \sqrt {a \left (3-2 x^2\right )} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {1}{2},\frac {1}{4},\frac {2 x^2}{3}\right )}{(c x)^{5/2} \sqrt {9-6 x^2}} \] Input:
Integrate[Sqrt[3*a - 2*a*x^2]/(c*x)^(5/2),x]
Output:
(-2*x*Sqrt[a*(3 - 2*x^2)]*Hypergeometric2F1[-3/4, -1/2, 1/4, (2*x^2)/3])/( (c*x)^(5/2)*Sqrt[9 - 6*x^2])
Time = 0.20 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {247, 266, 765, 762}
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.
\(\displaystyle \int \frac {\sqrt {3 a-2 a x^2}}{(c x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 247 |
\(\displaystyle -\frac {4 a \int \frac {1}{\sqrt {c x} \sqrt {3 a-2 a x^2}}dx}{3 c^2}-\frac {2 \sqrt {3 a-2 a x^2}}{3 c (c x)^{3/2}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle -\frac {8 a \int \frac {1}{\sqrt {3 a-2 a x^2}}d\sqrt {c x}}{3 c^3}-\frac {2 \sqrt {3 a-2 a x^2}}{3 c (c x)^{3/2}}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle -\frac {8 a \sqrt {3-2 x^2} \int \frac {1}{\sqrt {1-\frac {2 x^2}{3}}}d\sqrt {c x}}{3 \sqrt {3} c^3 \sqrt {3 a-2 a x^2}}-\frac {2 \sqrt {3 a-2 a x^2}}{3 c (c x)^{3/2}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle -\frac {4\ 2^{3/4} a \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{3 \sqrt [4]{3} c^{5/2} \sqrt {3 a-2 a x^2}}-\frac {2 \sqrt {3 a-2 a x^2}}{3 c (c x)^{3/2}}\) |
Input:
Int[Sqrt[3*a - 2*a*x^2]/(c*x)^(5/2),x]
Output:
(-2*Sqrt[3*a - 2*a*x^2])/(3*c*(c*x)^(3/2)) - (4*2^(3/4)*a*Sqrt[3 - 2*x^2]* EllipticF[ArcSin[((2/3)^(1/4)*Sqrt[c*x])/Sqrt[c]], -1])/(3*3^(1/4)*c^(5/2) *Sqrt[3*a - 2*a*x^2])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Time = 0.45 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.33
method | result | size |
default | \(\frac {2 \sqrt {-a \left (2 x^{2}-3\right )}\, \left (\sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {\left (-2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {-\sqrt {3}\, \sqrt {2}\, x}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right ) x -6 x^{2}+9\right )}{9 x \,c^{2} \sqrt {c x}\, \left (2 x^{2}-3\right )}\) | \(129\) |
risch | \(\frac {2 \left (2 x^{2}-3\right ) a}{3 x \,c^{2} \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}-\frac {2 \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 \sqrt {6}\, x}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right ) a \sqrt {-c x a \left (2 x^{2}-3\right )}}{81 \sqrt {-2 a c \,x^{3}+3 a c x}\, c^{2} \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}\) | \(152\) |
elliptic | \(-\frac {\sqrt {-a \left (2 x^{2}-3\right )}\, \sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (-\frac {2 \sqrt {-2 a c \,x^{3}+3 a c x}}{3 c^{3} x^{2}}-\frac {2 a \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 \sqrt {6}\, x}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{81 c^{2} \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{\sqrt {c x}\, a \left (2 x^{2}-3\right )}\) | \(156\) |
Input:
int((-2*a*x^2+3*a)^(1/2)/(c*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/9*(-a*(2*x^2-3))^(1/2)*(((2*x+3^(1/2)*2^(1/2))*3^(1/2)*2^(1/2))^(1/2)*(( -2*x+3^(1/2)*2^(1/2))*3^(1/2)*2^(1/2))^(1/2)*(-3^(1/2)*2^(1/2)*x)^(1/2)*El lipticF(1/6*3^(1/2)*2^(1/2)*((2*x+3^(1/2)*2^(1/2))*3^(1/2)*2^(1/2))^(1/2), 1/2*2^(1/2))*x-6*x^2+9)/x/c^2/(c*x)^(1/2)/(2*x^2-3)
Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {3 a-2 a x^2}}{(c x)^{5/2}} \, dx=\frac {2 \, {\left (2 \, \sqrt {2} \sqrt {-a c} x^{2} {\rm weierstrassPInverse}\left (6, 0, x\right ) - \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}\right )}}{3 \, c^{3} x^{2}} \] Input:
integrate((-2*a*x^2+3*a)^(1/2)/(c*x)^(5/2),x, algorithm="fricas")
Output:
2/3*(2*sqrt(2)*sqrt(-a*c)*x^2*weierstrassPInverse(6, 0, x) - sqrt(-2*a*x^2 + 3*a)*sqrt(c*x))/(c^3*x^2)
Time = 1.49 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {3 a-2 a x^2}}{(c x)^{5/2}} \, dx=\frac {\sqrt {2} i \sqrt {a} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {3}{2 x^{2}}} \right )}}{2 c^{\frac {5}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \] Input:
integrate((-2*a*x**2+3*a)**(1/2)/(c*x)**(5/2),x)
Output:
sqrt(2)*I*sqrt(a)*gamma(-1/4)*hyper((-1/2, 1/4), (5/4,), 3/(2*x**2))/(2*c* *(5/2)*sqrt(x)*gamma(3/4))
\[ \int \frac {\sqrt {3 a-2 a x^2}}{(c x)^{5/2}} \, dx=\int { \frac {\sqrt {-2 \, a x^{2} + 3 \, a}}{\left (c x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((-2*a*x^2+3*a)^(1/2)/(c*x)^(5/2),x, algorithm="maxima")
Output:
integrate(sqrt(-2*a*x^2 + 3*a)/(c*x)^(5/2), x)
\[ \int \frac {\sqrt {3 a-2 a x^2}}{(c x)^{5/2}} \, dx=\int { \frac {\sqrt {-2 \, a x^{2} + 3 \, a}}{\left (c x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((-2*a*x^2+3*a)^(1/2)/(c*x)^(5/2),x, algorithm="giac")
Output:
integrate(sqrt(-2*a*x^2 + 3*a)/(c*x)^(5/2), x)
Timed out. \[ \int \frac {\sqrt {3 a-2 a x^2}}{(c x)^{5/2}} \, dx=\int \frac {\sqrt {3\,a-2\,a\,x^2}}{{\left (c\,x\right )}^{5/2}} \,d x \] Input:
int((3*a - 2*a*x^2)^(1/2)/(c*x)^(5/2),x)
Output:
int((3*a - 2*a*x^2)^(1/2)/(c*x)^(5/2), x)
\[ \int \frac {\sqrt {3 a-2 a x^2}}{(c x)^{5/2}} \, dx=\frac {2 \sqrt {c}\, \sqrt {a}\, \left (-\sqrt {-2 x^{2}+3}+3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{2}+3}}{2 x^{5}-3 x^{3}}d x \right ) x \right )}{\sqrt {x}\, c^{3} x} \] Input:
int((-2*a*x^2+3*a)^(1/2)/(c*x)^(5/2),x)
Output:
(2*sqrt(c)*sqrt(a)*( - sqrt( - 2*x**2 + 3) + 3*sqrt(x)*int((sqrt(x)*sqrt( - 2*x**2 + 3))/(2*x**5 - 3*x**3),x)*x))/(sqrt(x)*c**3*x)