Integrand size = 22, antiderivative size = 128 \[ \int (c x)^{5/2} \sqrt {3 a-2 a x^2} \, dx=-\frac {2}{15} c (c x)^{3/2} \sqrt {3 a-2 a x^2}+\frac {2 (c x)^{7/2} \sqrt {3 a-2 a x^2}}{9 c}-\frac {3 \sqrt [4]{6} a c^2 \sqrt {c x} \sqrt {3-2 x^2} E\left (\left .\arcsin \left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{5 \sqrt {x} \sqrt {3 a-2 a x^2}} \]
-3/5*6^(1/4)*a*c^2*EllipticE(1/6*(3-x*6^(1/2))^(1/2)*6^(1/2),2^(1/2))*(c*x )^(1/2)*(-2*x^2+3)^(1/2)/x^(1/2)/(-2*a*x^2+3*a)^(1/2)-2/15*c*(c*x)^(3/2)*( -2*a*x^2+3*a)^(1/2)+2/9*(c*x)^(7/2)*(-2*a*x^2+3*a)^(1/2)/c
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 9.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.58 \[ \int (c x)^{5/2} \sqrt {3 a-2 a x^2} \, dx=\frac {c (c x)^{3/2} \sqrt {a \left (3-2 x^2\right )} \left (-\left (3-2 x^2\right )^{3/2}+3 \sqrt {3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {2 x^2}{3}\right )\right )}{9 \sqrt {3-2 x^2}} \]
(c*(c*x)^(3/2)*Sqrt[a*(3 - 2*x^2)]*(-(3 - 2*x^2)^(3/2) + 3*Sqrt[3]*Hyperge ometric2F1[-1/2, 3/4, 7/4, (2*x^2)/3]))/(9*Sqrt[3 - 2*x^2])
Time = 0.25 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {248, 262, 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.
\(\displaystyle \int \sqrt {3 a-2 a x^2} (c x)^{5/2} \, dx\) |
\(\Big \downarrow \) 248 |
\(\displaystyle \frac {2}{3} a \int \frac {(c x)^{5/2}}{\sqrt {3 a-2 a x^2}}dx+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{7/2}}{9 c}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {2}{3} a \left (\frac {9}{10} c^2 \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}}dx-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\right )+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{7/2}}{9 c}\) |
\(\Big \downarrow \) 261 |
\(\displaystyle \frac {2}{3} a \left (\frac {9 c^2 \sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3 a-2 a x^2}}dx}{10 \sqrt {x}}-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\right )+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{7/2}}{9 c}\) |
\(\Big \downarrow \) 260 |
\(\displaystyle \frac {2}{3} a \left (\frac {3 \sqrt {3} c^2 \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {3} \sqrt {x}}{\sqrt {3-2 x^2}}dx}{10 \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\right )+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{7/2}}{9 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{3} a \left (\frac {9 c^2 \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3-2 x^2}}dx}{10 \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\right )+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{7/2}}{9 c}\) |
\(\Big \downarrow \) 259 |
\(\displaystyle \frac {2}{3} a \left (-\frac {9 \sqrt [4]{3} c^2 \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {\frac {1}{3} \left (\sqrt {6} x-3\right )+1}}{\sqrt {\frac {1}{6} \left (\sqrt {6} x-3\right )+1}}d\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}}{5\ 2^{3/4} \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\right )+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{7/2}}{9 c}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {2}{3} a \left (-\frac {9 \sqrt [4]{3} c^2 \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\arcsin \left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{5\ 2^{3/4} \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\right )+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{7/2}}{9 c}\) |
(2*(c*x)^(7/2)*Sqrt[3*a - 2*a*x^2])/(9*c) + (2*a*(-1/5*(c*(c*x)^(3/2)*Sqrt [3*a - 2*a*x^2])/a - (9*3^(1/4)*c^2*Sqrt[c*x]*Sqrt[3 - 2*x^2]*EllipticE[Ar cSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6]], 2])/(5*2^(3/4)*Sqrt[x]*Sqrt[3*a - 2*a*x ^2])))/3
3.7.7.3.1 Defintions of rubi rules used
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && 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 = 2.16 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.52
method | result | size |
risch | \(-\frac {2 x^{2} \left (5 x^{2}-3\right ) \left (2 x^{2}-3\right ) c^{3} a}{45 \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}+\frac {\sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 x \sqrt {6}}\, \left (-\sqrt {6}\, E\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, F\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right ) c^{3} a \sqrt {-c x a \left (2 x^{2}-3\right )}}{90 \sqrt {-2 a c \,x^{3}+3 a c x}\, \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}\) | \(194\) |
elliptic | \(-\frac {\sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (\frac {2 c^{2} x^{3} \sqrt {-2 a c \,x^{3}+3 a c x}}{9}-\frac {2 c^{2} x \sqrt {-2 a c \,x^{3}+3 a c x}}{15}+\frac {c^{3} a \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 x \sqrt {6}}\, \left (-\sqrt {6}\, E\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, F\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right )}{90 \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{c x a \left (2 x^{2}-3\right )}\) | \(218\) |
default | \(\frac {c^{2} \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \left (80 x^{6}+18 \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-x \sqrt {2}\, \sqrt {3}}\, E\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )-9 \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-x \sqrt {2}\, \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )-168 x^{4}+72 x^{2}\right )}{180 x \left (2 x^{2}-3\right )}\) | \(237\) |
-2/45*x^2*(5*x^2-3)*(2*x^2-3)*c^3*a/(c*x)^(1/2)/(-a*(2*x^2-3))^(1/2)+1/90* 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*x*6^(1/2))^(1/2)/(-2*a*c*x^3+3*a*c*x)^(1/2)*(-6^(1/2)*Ellipti cE(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)*El lipticF(1/3*3^(1/2)*((x+1/2*6^(1/2))*6^(1/2))^(1/2),1/2*2^(1/2)))*c^3*a*(- c*x*a*(2*x^2-3))^(1/2)/(c*x)^(1/2)/(-a*(2*x^2-3))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.44 \[ \int (c x)^{5/2} \sqrt {3 a-2 a x^2} \, dx=\frac {3}{5} \, \sqrt {2} \sqrt {-a c} c^{2} {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right ) + \frac {2}{45} \, {\left (5 \, c^{2} x^{3} - 3 \, c^{2} x\right )} \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} \]
3/5*sqrt(2)*sqrt(-a*c)*c^2*weierstrassZeta(6, 0, weierstrassPInverse(6, 0, x)) + 2/45*(5*c^2*x^3 - 3*c^2*x)*sqrt(-2*a*x^2 + 3*a)*sqrt(c*x)
Result contains complex when optimal does not.
Time = 4.72 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.41 \[ \int (c x)^{5/2} \sqrt {3 a-2 a x^2} \, dx=\frac {\sqrt {3} \sqrt {a} c^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} \]
sqrt(3)*sqrt(a)*c**(5/2)*x**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), 2 *x**2*exp_polar(2*I*pi)/3)/(2*gamma(11/4))
\[ \int (c x)^{5/2} \sqrt {3 a-2 a x^2} \, dx=\int { \sqrt {-2 \, a x^{2} + 3 \, a} \left (c x\right )^{\frac {5}{2}} \,d x } \]
\[ \int (c x)^{5/2} \sqrt {3 a-2 a x^2} \, dx=\int { \sqrt {-2 \, a x^{2} + 3 \, a} \left (c x\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (c x)^{5/2} \sqrt {3 a-2 a x^2} \, dx=\int {\left (c\,x\right )}^{5/2}\,\sqrt {3\,a-2\,a\,x^2} \,d x \]