Integrand size = 22, antiderivative size = 117 \[ \int (c x)^{3/2} \sqrt {3 a-2 a x^2} \, dx=-\frac {2}{7} c \sqrt {c x} \sqrt {3 a-2 a x^2}+\frac {2 (c x)^{5/2} \sqrt {3 a-2 a x^2}}{7 c}+\frac {6^{3/4} a c^{3/2} \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{7 \sqrt {a \left (3-2 x^2\right )}} \]
1/7*6^(3/4)*a*c^(3/2)*EllipticF(1/3*2^(1/4)*3^(3/4)*(c*x)^(1/2)/c^(1/2),I) *(-2*x^2+3)^(1/2)/(a*(-2*x^2+3))^(1/2)+2/7*(c*x)^(5/2)*(-2*a*x^2+3*a)^(1/2 )/c-2/7*c*(c*x)^(1/2)*(-2*a*x^2+3*a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 7.74 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.63 \[ \int (c x)^{3/2} \sqrt {3 a-2 a x^2} \, dx=\frac {c \sqrt {c x} \sqrt {a \left (3-2 x^2\right )} \left (-\left (3-2 x^2\right )^{3/2}+3 \sqrt {3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {2 x^2}{3}\right )\right )}{7 \sqrt {3-2 x^2}} \]
(c*Sqrt[c*x]*Sqrt[a*(3 - 2*x^2)]*(-(3 - 2*x^2)^(3/2) + 3*Sqrt[3]*Hypergeom etric2F1[-1/2, 1/4, 5/4, (2*x^2)/3]))/(7*Sqrt[3 - 2*x^2])
Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {248, 262, 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.
\(\displaystyle \int \sqrt {3 a-2 a x^2} (c x)^{3/2} \, dx\) |
\(\Big \downarrow \) 248 |
\(\displaystyle \frac {6}{7} a \int \frac {(c x)^{3/2}}{\sqrt {3 a-2 a x^2}}dx+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{5/2}}{7 c}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {6}{7} a \left (\frac {1}{2} c^2 \int \frac {1}{\sqrt {c x} \sqrt {3 a-2 a x^2}}dx-\frac {c \sqrt {3 a-2 a x^2} \sqrt {c x}}{3 a}\right )+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{5/2}}{7 c}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {6}{7} a \left (c \int \frac {1}{\sqrt {3 a-2 a x^2}}d\sqrt {c x}-\frac {c \sqrt {3 a-2 a x^2} \sqrt {c x}}{3 a}\right )+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{5/2}}{7 c}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle \frac {6}{7} a \left (\frac {c \sqrt {3-2 x^2} \int \frac {1}{\sqrt {1-\frac {2 x^2}{3}}}d\sqrt {c x}}{\sqrt {3} \sqrt {3 a-2 a x^2}}-\frac {c \sqrt {3 a-2 a x^2} \sqrt {c x}}{3 a}\right )+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{5/2}}{7 c}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {6}{7} a \left (\frac {c^{3/2} \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{\sqrt [4]{6} \sqrt {3 a-2 a x^2}}-\frac {c \sqrt {3 a-2 a x^2} \sqrt {c x}}{3 a}\right )+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{5/2}}{7 c}\) |
(2*(c*x)^(5/2)*Sqrt[3*a - 2*a*x^2])/(7*c) + (6*a*(-1/3*(c*Sqrt[c*x]*Sqrt[3 *a - 2*a*x^2])/a + (c^(3/2)*Sqrt[3 - 2*x^2]*EllipticF[ArcSin[((2/3)^(1/4)* Sqrt[c*x])/Sqrt[c]], -1])/(6^(1/4)*Sqrt[3*a - 2*a*x^2])))/7
3.7.8.3.1 Defintions of rubi rules used
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[((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[((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 = 2.00 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.14
method | result | size |
default | \(-\frac {c \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \left (-8 x^{5}+\sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {-x \sqrt {2}\, \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )+20 x^{3}-12 x \right )}{14 x \left (2 x^{2}-3\right )}\) | \(133\) |
risch | \(-\frac {2 \left (x^{2}-1\right ) x \left (2 x^{2}-3\right ) c^{2} a}{7 \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}+\frac {\sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 x \sqrt {6}}\, F\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right ) c^{2} a \sqrt {-c x a \left (2 x^{2}-3\right )}}{126 \sqrt {-2 a c \,x^{3}+3 a c x}\, \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}\) | \(155\) |
elliptic | \(-\frac {\sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (\frac {2 c \,x^{2} \sqrt {-2 a c \,x^{3}+3 a c x}}{7}-\frac {2 c \sqrt {-2 a c \,x^{3}+3 a c x}}{7}+\frac {c^{2} a \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 x \sqrt {6}}\, F\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{126 \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{c x a \left (2 x^{2}-3\right )}\) | \(178\) |
-1/14*c*(c*x)^(1/2)*(-a*(2*x^2-3))^(1/2)*(-8*x^5+((2*x+2^(1/2)*3^(1/2))*2^ (1/2)*3^(1/2))^(1/2)*((-2*x+2^(1/2)*3^(1/2))*2^(1/2)*3^(1/2))^(1/2)*(-x*2^ (1/2)*3^(1/2))^(1/2)*EllipticF(1/6*3^(1/2)*2^(1/2)*((2*x+2^(1/2)*3^(1/2))* 2^(1/2)*3^(1/2))^(1/2),1/2*2^(1/2))+20*x^3-12*x)/x/(2*x^2-3)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.38 \[ \int (c x)^{3/2} \sqrt {3 a-2 a x^2} \, dx=-\frac {3}{7} \, \sqrt {2} \sqrt {-a c} c {\rm weierstrassPInverse}\left (6, 0, x\right ) + \frac {2}{7} \, \sqrt {-2 \, a x^{2} + 3 \, a} {\left (c x^{2} - c\right )} \sqrt {c x} \]
-3/7*sqrt(2)*sqrt(-a*c)*c*weierstrassPInverse(6, 0, x) + 2/7*sqrt(-2*a*x^2 + 3*a)*(c*x^2 - c)*sqrt(c*x)
Time = 1.44 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.45 \[ \int (c x)^{3/2} \sqrt {3 a-2 a x^2} \, dx=\frac {\sqrt {3} \sqrt {a} c^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} \]
sqrt(3)*sqrt(a)*c**(3/2)*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), 2* x**2*exp_polar(2*I*pi)/3)/(2*gamma(9/4))
\[ \int (c x)^{3/2} \sqrt {3 a-2 a x^2} \, dx=\int { \sqrt {-2 \, a x^{2} + 3 \, a} \left (c x\right )^{\frac {3}{2}} \,d x } \]
\[ \int (c x)^{3/2} \sqrt {3 a-2 a x^2} \, dx=\int { \sqrt {-2 \, a x^{2} + 3 \, a} \left (c x\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (c x)^{3/2} \sqrt {3 a-2 a x^2} \, dx=\int {\left (c\,x\right )}^{3/2}\,\sqrt {3\,a-2\,a\,x^2} \,d x \]