Integrand size = 22, antiderivative size = 170 \[ \int \frac {(c x)^{5/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}}-\frac {3 \sqrt [4]{3} c^{5/2} \sqrt {3-2 x^2} E\left (\left .\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{2\ 2^{3/4} a \sqrt {3 a-2 a x^2}}+\frac {3 \sqrt [4]{3} c^{5/2} \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{2\ 2^{3/4} a \sqrt {3 a-2 a x^2}} \] Output:
1/2*c*(c*x)^(3/2)/a/(-2*a*x^2+3*a)^(1/2)-3/4*3^(1/4)*c^(5/2)*(-2*x^2+3)^(1 /2)*EllipticE(1/3*2^(1/4)*3^(3/4)*(c*x)^(1/2)/c^(1/2),I)*2^(1/4)/a/(-2*a*x ^2+3*a)^(1/2)+3/4*3^(1/4)*c^(5/2)*(-2*x^2+3)^(1/2)*EllipticF(1/3*2^(1/4)*3 ^(3/4)*(c*x)^(1/2)/c^(1/2),I)*2^(1/4)/a/(-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 = 59, normalized size of antiderivative = 0.35 \[ \int \frac {(c x)^{5/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {c (c x)^{3/2} \left (-3+\sqrt {9-6 x^2} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {2 x^2}{3}\right )\right )}{3 a \sqrt {a \left (3-2 x^2\right )}} \] Input:
Integrate[(c*x)^(5/2)/(3*a - 2*a*x^2)^(3/2),x]
Output:
(c*(c*x)^(3/2)*(-3 + Sqrt[9 - 6*x^2]*Hypergeometric2F1[3/4, 3/2, 7/4, (2*x ^2)/3]))/(3*a*Sqrt[a*(3 - 2*x^2)])
Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.65, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {252, 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.
| \(\displaystyle \int \frac {(c x)^{5/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}}-\frac {3 c^2 \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}}dx}{4 a}\) |
| \(\Big \downarrow \) 261 |
| \(\displaystyle \frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}}-\frac {3 c^2 \sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3 a-2 a x^2}}dx}{4 a \sqrt {x}}\) |
| \(\Big \downarrow \) 260 |
| \(\displaystyle \frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}}-\frac {\sqrt {3} c^2 \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {3} \sqrt {x}}{\sqrt {3-2 x^2}}dx}{4 a \sqrt {x} \sqrt {3 a-2 a x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}}-\frac {3 c^2 \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3-2 x^2}}dx}{4 a \sqrt {x} \sqrt {3 a-2 a x^2}}\) |
| \(\Big \downarrow \) 259 |
| \(\displaystyle \frac {3 \sqrt [4]{3} c^2 \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {\frac {1}{3} \left (\sqrt {6} x-3\right )+1}}{\sqrt {\frac {1}{6} \left (\sqrt {6} x-3\right )+1}}d\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}}{2\ 2^{3/4} a \sqrt {x} \sqrt {3 a-2 a x^2}}+\frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {3 \sqrt [4]{3} c^2 \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\arcsin \left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{2\ 2^{3/4} a \sqrt {x} \sqrt {3 a-2 a x^2}}+\frac {c (c x)^{3/2}}{2 a \sqrt {3 a-2 a x^2}}\) |
Input:
Int[(c*x)^(5/2)/(3*a - 2*a*x^2)^(3/2),x]
Output:
(c*(c*x)^(3/2))/(2*a*Sqrt[3*a - 2*a*x^2]) + (3*3^(1/4)*c^2*Sqrt[c*x]*Sqrt[ 3 - 2*x^2]*EllipticE[ArcSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6]], 2])/(2*2^(3/4)*a *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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[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 = 0.54 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.09
| method | result | size |
| elliptic | \(\frac {\sqrt {c x}\, \sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (\frac {c^{3} x^{2}}{2 a \sqrt {-2 \left (x^{2}-\frac {3}{2}\right ) a c x}}-\frac {c^{3} \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 \sqrt {6}\, x}\, \left (-\sqrt {6}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right )}{72 a \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{c x \sqrt {-a \left (2 x^{2}-3\right )}}\) | \(186\) |
| default | \(-\frac {c^{2} \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \left (2 \sqrt {\left (-2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {3}\, \sqrt {-\sqrt {3}\, \sqrt {2}\, x}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}-\sqrt {\left (-2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {3}\, \sqrt {-\sqrt {3}\, \sqrt {2}\, x}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}+8 x^{2}\right )}{16 x \,a^{2} \left (2 x^{2}-3\right )}\) | \(230\) |
Input:
int((c*x)^(5/2)/(-2*a*x^2+3*a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/c/x*(c*x)^(1/2)/(-a*(2*x^2-3))^(1/2)*(-c*x*a*(2*x^2-3))^(1/2)*(1/2/a*c^3 *x^2/(-2*(x^2-3/2)*a*c*x)^(1/2)-1/72*c^3/a*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 = 74, normalized size of antiderivative = 0.44 \[ \int \frac {(c x)^{5/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} c^{2} x + 3 \, \sqrt {2} {\left (2 \, c^{2} x^{2} - 3 \, c^{2}\right )} \sqrt {-a c} {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right )}{4 \, {\left (2 \, a^{2} x^{2} - 3 \, a^{2}\right )}} \] Input:
integrate((c*x)^(5/2)/(-2*a*x^2+3*a)^(3/2),x, algorithm="fricas")
Output:
-1/4*(2*sqrt(-2*a*x^2 + 3*a)*sqrt(c*x)*c^2*x + 3*sqrt(2)*(2*c^2*x^2 - 3*c^ 2)*sqrt(-a*c)*weierstrassZeta(6, 0, weierstrassPInverse(6, 0, x)))/(2*a^2* x^2 - 3*a^2)
Time = 3.45 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.30 \[ \int \frac {(c x)^{5/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {\sqrt {3} c^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{18 a^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate((c*x)**(5/2)/(-2*a*x**2+3*a)**(3/2),x)
Output:
sqrt(3)*c**(5/2)*x**(7/2)*gamma(7/4)*hyper((3/2, 7/4), (11/4,), 2*x**2*exp _polar(2*I*pi)/3)/(18*a**(3/2)*gamma(11/4))
\[ \int \frac {(c x)^{5/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{\frac {5}{2}}}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c*x)^(5/2)/(-2*a*x^2+3*a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x)^(5/2)/(-2*a*x^2 + 3*a)^(3/2), x)
\[ \int \frac {(c x)^{5/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{\frac {5}{2}}}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c*x)^(5/2)/(-2*a*x^2+3*a)^(3/2),x, algorithm="giac")
Output:
integrate((c*x)^(5/2)/(-2*a*x^2 + 3*a)^(3/2), x)
Timed out. \[ \int \frac {(c x)^{5/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\int \frac {{\left (c\,x\right )}^{5/2}}{{\left (3\,a-2\,a\,x^2\right )}^{3/2}} \,d x \] Input:
int((c*x)^(5/2)/(3*a - 2*a*x^2)^(3/2),x)
Output:
int((c*x)^(5/2)/(3*a - 2*a*x^2)^(3/2), x)
\[ \int \frac {(c x)^{5/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, c^{2} \left (2 \sqrt {x}\, \sqrt {-2 x^{2}+3}\, x +18 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{2}+3}}{4 x^{4}-12 x^{2}+9}d x \right ) x^{2}-27 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{2}+3}}{4 x^{4}-12 x^{2}+9}d x \right )\right )}{2 a^{2} \left (2 x^{2}-3\right )} \] Input:
int((c*x)^(5/2)/(-2*a*x^2+3*a)^(3/2),x)
Output:
(sqrt(c)*sqrt(a)*c**2*(2*sqrt(x)*sqrt( - 2*x**2 + 3)*x + 18*int((sqrt(x)*s qrt( - 2*x**2 + 3))/(4*x**4 - 12*x**2 + 9),x)*x**2 - 27*int((sqrt(x)*sqrt( - 2*x**2 + 3))/(4*x**4 - 12*x**2 + 9),x)))/(2*a**2*(2*x**2 - 3))