Integrand size = 22, antiderivative size = 201 \[ \int \frac {(c x)^{9/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {c (c x)^{7/2}}{2 a \sqrt {3 a-2 a x^2}}+\frac {7 c^3 (c x)^{3/2} \sqrt {3 a-2 a x^2}}{20 a^2}-\frac {63 \sqrt [4]{3} c^{9/2} \sqrt {3-2 x^2} E\left (\left .\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{20\ 2^{3/4} a \sqrt {3 a-2 a x^2}}+\frac {63 \sqrt [4]{3} c^{9/2} \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{20\ 2^{3/4} a \sqrt {3 a-2 a x^2}} \] Output:
1/2*c*(c*x)^(7/2)/a/(-2*a*x^2+3*a)^(1/2)+7/20*c^3*(c*x)^(3/2)*(-2*a*x^2+3* a)^(1/2)/a^2-63/40*3^(1/4)*c^(9/2)*(-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)+63/40*3^(1/4 )*c^(9/2)*(-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 = 10.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.33 \[ \int \frac {(c x)^{9/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {c^3 (c x)^{3/2} \left (-21-2 x^2+7 \sqrt {9-6 x^2} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {2 x^2}{3}\right )\right )}{10 a \sqrt {a \left (3-2 x^2\right )}} \] Input:
Integrate[(c*x)^(9/2)/(3*a - 2*a*x^2)^(3/2),x]
Output:
(c^3*(c*x)^(3/2)*(-21 - 2*x^2 + 7*Sqrt[9 - 6*x^2]*Hypergeometric2F1[3/4, 3 /2, 7/4, (2*x^2)/3]))/(10*a*Sqrt[a*(3 - 2*x^2)])
Time = 0.23 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.73, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {252, 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.
| \(\displaystyle \int \frac {(c x)^{9/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {c (c x)^{7/2}}{2 a \sqrt {3 a-2 a x^2}}-\frac {7 c^2 \int \frac {(c x)^{5/2}}{\sqrt {3 a-2 a x^2}}dx}{4 a}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {c (c x)^{7/2}}{2 a \sqrt {3 a-2 a x^2}}-\frac {7 c^2 \left (\frac {9}{10} c^2 \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}}dx-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\right )}{4 a}\) |
| \(\Big \downarrow \) 261 |
| \(\displaystyle \frac {c (c x)^{7/2}}{2 a \sqrt {3 a-2 a x^2}}-\frac {7 c^2 \left (\frac {9 c^2 \sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3 a-2 a x^2}}dx}{10 \sqrt {x}}-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\right )}{4 a}\) |
| \(\Big \downarrow \) 260 |
| \(\displaystyle \frac {c (c x)^{7/2}}{2 a \sqrt {3 a-2 a x^2}}-\frac {7 c^2 \left (\frac {3 \sqrt {3} c^2 \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {3} \sqrt {x}}{\sqrt {3-2 x^2}}dx}{10 \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\right )}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c (c x)^{7/2}}{2 a \sqrt {3 a-2 a x^2}}-\frac {7 c^2 \left (\frac {9 c^2 \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3-2 x^2}}dx}{10 \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\right )}{4 a}\) |
| \(\Big \downarrow \) 259 |
| \(\displaystyle \frac {c (c x)^{7/2}}{2 a \sqrt {3 a-2 a x^2}}-\frac {7 c^2 \left (-\frac {9 \sqrt [4]{3} c^2 \sqrt {3-2 x^2} \sqrt {c x} \int \frac {\sqrt {\frac {1}{3} \left (\sqrt {6} x-3\right )+1}}{\sqrt {\frac {1}{6} \left (\sqrt {6} x-3\right )+1}}d\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}}{5\ 2^{3/4} \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\right )}{4 a}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {c (c x)^{7/2}}{2 a \sqrt {3 a-2 a x^2}}-\frac {7 c^2 \left (-\frac {9 \sqrt [4]{3} c^2 \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\arcsin \left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{5\ 2^{3/4} \sqrt {x} \sqrt {3 a-2 a x^2}}-\frac {c \sqrt {3 a-2 a x^2} (c x)^{3/2}}{5 a}\right )}{4 a}\) |
Input:
Int[(c*x)^(9/2)/(3*a - 2*a*x^2)^(3/2),x]
Output:
(c*(c*x)^(7/2))/(2*a*Sqrt[3*a - 2*a*x^2]) - (7*c^2*(-1/5*(c*(c*x)^(3/2)*Sq rt[3*a - 2*a*x^2])/a - (9*3^(1/4)*c^2*Sqrt[c*x]*Sqrt[3 - 2*x^2]*EllipticE[ ArcSin[Sqrt[3 - Sqrt[6]*x]/Sqrt[6]], 2])/(5*2^(3/4)*Sqrt[x]*Sqrt[3*a - 2*a *x^2])))/(4*a)
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[((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 = 0.68 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.04
| method | result | size |
| elliptic | \(\frac {\sqrt {c x}\, \sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (\frac {3 c^{5} x^{2}}{4 a \sqrt {-2 \left (x^{2}-\frac {3}{2}\right ) a c x}}+\frac {c^{4} x \sqrt {-2 a c \,x^{3}+3 a c x}}{10 a^{2}}-\frac {7 c^{5} \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 \sqrt {6}\, x}\, \left (-\sqrt {6}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )+\frac {\sqrt {6}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{2}\right )}{240 a \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{c x \sqrt {-a \left (2 x^{2}-3\right )}}\) | \(210\) |
| default | \(-\frac {c^{4} \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \left (42 \sqrt {\left (-2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {3}\, \sqrt {-\sqrt {3}\, \sqrt {2}\, x}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}-21 \sqrt {\left (-2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {3}\, \sqrt {-\sqrt {3}\, \sqrt {2}\, x}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}-32 x^{4}+168 x^{2}\right )}{160 x \,a^{2} \left (2 x^{2}-3\right )}\) | \(235\) |
Input:
int((c*x)^(9/2)/(-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)*(3/4/a*c^5 *x^2/(-2*(x^2-3/2)*a*c*x)^(1/2)+1/10/a^2*c^4*x*(-2*a*c*x^3+3*a*c*x)^(1/2)- 7/240*c^5/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 = 85, normalized size of antiderivative = 0.42 \[ \int \frac {(c x)^{9/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=-\frac {63 \, \sqrt {2} {\left (2 \, c^{4} x^{2} - 3 \, c^{4}\right )} \sqrt {-a c} {\rm weierstrassZeta}\left (6, 0, {\rm weierstrassPInverse}\left (6, 0, x\right )\right ) - 2 \, {\left (4 \, c^{4} x^{3} - 21 \, c^{4} x\right )} \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}}{40 \, {\left (2 \, a^{2} x^{2} - 3 \, a^{2}\right )}} \] Input:
integrate((c*x)^(9/2)/(-2*a*x^2+3*a)^(3/2),x, algorithm="fricas")
Output:
-1/40*(63*sqrt(2)*(2*c^4*x^2 - 3*c^4)*sqrt(-a*c)*weierstrassZeta(6, 0, wei erstrassPInverse(6, 0, x)) - 2*(4*c^4*x^3 - 21*c^4*x)*sqrt(-2*a*x^2 + 3*a) *sqrt(c*x))/(2*a^2*x^2 - 3*a^2)
Time = 36.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.25 \[ \int \frac {(c x)^{9/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {\sqrt {3} c^{\frac {9}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{18 a^{\frac {3}{2}} \Gamma \left (\frac {15}{4}\right )} \] Input:
integrate((c*x)**(9/2)/(-2*a*x**2+3*a)**(3/2),x)
Output:
sqrt(3)*c**(9/2)*x**(11/2)*gamma(11/4)*hyper((3/2, 11/4), (15/4,), 2*x**2* exp_polar(2*I*pi)/3)/(18*a**(3/2)*gamma(15/4))
\[ \int \frac {(c x)^{9/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{\frac {9}{2}}}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c*x)^(9/2)/(-2*a*x^2+3*a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x)^(9/2)/(-2*a*x^2 + 3*a)^(3/2), x)
\[ \int \frac {(c x)^{9/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{\frac {9}{2}}}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c*x)^(9/2)/(-2*a*x^2+3*a)^(3/2),x, algorithm="giac")
Output:
integrate((c*x)^(9/2)/(-2*a*x^2 + 3*a)^(3/2), x)
Timed out. \[ \int \frac {(c x)^{9/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\int \frac {{\left (c\,x\right )}^{9/2}}{{\left (3\,a-2\,a\,x^2\right )}^{3/2}} \,d x \] Input:
int((c*x)^(9/2)/(3*a - 2*a*x^2)^(3/2),x)
Output:
int((c*x)^(9/2)/(3*a - 2*a*x^2)^(3/2), x)
\[ \int \frac {(c x)^{9/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, c^{4} \left (4 \sqrt {x}\, \sqrt {-2 x^{2}+3}\, x^{3}+42 \sqrt {x}\, \sqrt {-2 x^{2}+3}\, x +378 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{2}+3}}{4 x^{4}-12 x^{2}+9}d x \right ) x^{2}-567 \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{2}+3}}{4 x^{4}-12 x^{2}+9}d x \right )\right )}{20 a^{2} \left (2 x^{2}-3\right )} \] Input:
int((c*x)^(9/2)/(-2*a*x^2+3*a)^(3/2),x)
Output:
(sqrt(c)*sqrt(a)*c**4*(4*sqrt(x)*sqrt( - 2*x**2 + 3)*x**3 + 42*sqrt(x)*sqr t( - 2*x**2 + 3)*x + 378*int((sqrt(x)*sqrt( - 2*x**2 + 3))/(4*x**4 - 12*x* *2 + 9),x)*x**2 - 567*int((sqrt(x)*sqrt( - 2*x**2 + 3))/(4*x**4 - 12*x**2 + 9),x)))/(20*a**2*(2*x**2 - 3))