Integrand size = 22, antiderivative size = 95 \[ \int \frac {(c x)^{3/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {c \sqrt {c x}}{2 a \sqrt {3 a-2 a x^2}}-\frac {c^{3/2} \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{2 \sqrt [4]{6} a \sqrt {3 a-2 a x^2}} \] Output:
1/2*c*(c*x)^(1/2)/a/(-2*a*x^2+3*a)^(1/2)-1/12*c^(3/2)*(-2*x^2+3)^(1/2)*Ell ipticF(1/3*2^(1/4)*3^(3/4)*(c*x)^(1/2)/c^(1/2),I)*6^(3/4)/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 = 9.85 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.62 \[ \int \frac {(c x)^{3/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=-\frac {c \sqrt {c x} \left (-3+\sqrt {9-6 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {2 x^2}{3}\right )\right )}{6 a \sqrt {a \left (3-2 x^2\right )}} \] Input:
Integrate[(c*x)^(3/2)/(3*a - 2*a*x^2)^(3/2),x]
Output:
-1/6*(c*Sqrt[c*x]*(-3 + Sqrt[9 - 6*x^2]*Hypergeometric2F1[1/4, 1/2, 5/4, ( 2*x^2)/3]))/(a*Sqrt[a*(3 - 2*x^2)])
Time = 0.20 (sec) , antiderivative size = 95, 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 = {252, 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.
\(\displaystyle \int \frac {(c x)^{3/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {c \sqrt {c x}}{2 a \sqrt {3 a-2 a x^2}}-\frac {c^2 \int \frac {1}{\sqrt {c x} \sqrt {3 a-2 a x^2}}dx}{4 a}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {c \sqrt {c x}}{2 a \sqrt {3 a-2 a x^2}}-\frac {c \int \frac {1}{\sqrt {3 a-2 a x^2}}d\sqrt {c x}}{2 a}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle \frac {c \sqrt {c x}}{2 a \sqrt {3 a-2 a x^2}}-\frac {c \sqrt {3-2 x^2} \int \frac {1}{\sqrt {1-\frac {2 x^2}{3}}}d\sqrt {c x}}{2 \sqrt {3} a \sqrt {3 a-2 a x^2}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {c \sqrt {c x}}{2 a \sqrt {3 a-2 a x^2}}-\frac {c^{3/2} \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{2 \sqrt [4]{6} a \sqrt {3 a-2 a x^2}}\) |
Input:
Int[(c*x)^(3/2)/(3*a - 2*a*x^2)^(3/2),x]
Output:
(c*Sqrt[c*x])/(2*a*Sqrt[3*a - 2*a*x^2]) - (c^(3/2)*Sqrt[3 - 2*x^2]*Ellipti cF[ArcSin[((2/3)^(1/4)*Sqrt[c*x])/Sqrt[c]], -1])/(2*6^(1/4)*a*Sqrt[3*a - 2 *a*x^2])
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[((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 = 0.46 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.33
method | result | size |
default | \(\frac {c \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \left (\sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {\left (-2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}\, \sqrt {-\sqrt {3}\, \sqrt {2}\, x}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\, \sqrt {2}\right ) \sqrt {3}\, \sqrt {2}}}{6}, \frac {\sqrt {2}}{2}\right )-12 x \right )}{24 x \,a^{2} \left (2 x^{2}-3\right )}\) | \(126\) |
elliptic | \(\frac {\sqrt {c x}\, \sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (\frac {c^{2} x}{2 a \sqrt {-2 \left (x^{2}-\frac {3}{2}\right ) a c x}}-\frac {c^{2} \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 \sqrt {6}\, x}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{216 a \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{c x \sqrt {-a \left (2 x^{2}-3\right )}}\) | \(149\) |
Input:
int((c*x)^(3/2)/(-2*a*x^2+3*a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/24*c*(c*x)^(1/2)*(-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)*EllipticF(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))-12*x)/x/a^2/(2*x^2-3)
Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int \frac {(c x)^{3/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {\sqrt {2} {\left (2 \, c x^{2} - 3 \, c\right )} \sqrt {-a c} {\rm weierstrassPInverse}\left (6, 0, x\right ) - 2 \, \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} c}{4 \, {\left (2 \, a^{2} x^{2} - 3 \, a^{2}\right )}} \] Input:
integrate((c*x)^(3/2)/(-2*a*x^2+3*a)^(3/2),x, algorithm="fricas")
Output:
1/4*(sqrt(2)*(2*c*x^2 - 3*c)*sqrt(-a*c)*weierstrassPInverse(6, 0, x) - 2*s qrt(-2*a*x^2 + 3*a)*sqrt(c*x)*c)/(2*a^2*x^2 - 3*a^2)
Time = 1.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.54 \[ \int \frac {(c x)^{3/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {\sqrt {3} c^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{18 a^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate((c*x)**(3/2)/(-2*a*x**2+3*a)**(3/2),x)
Output:
sqrt(3)*c**(3/2)*x**(5/2)*gamma(5/4)*hyper((5/4, 3/2), (9/4,), 2*x**2*exp_ polar(2*I*pi)/3)/(18*a**(3/2)*gamma(9/4))
\[ \int \frac {(c x)^{3/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{\frac {3}{2}}}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c*x)^(3/2)/(-2*a*x^2+3*a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x)^(3/2)/(-2*a*x^2 + 3*a)^(3/2), x)
\[ \int \frac {(c x)^{3/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{\frac {3}{2}}}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c*x)^(3/2)/(-2*a*x^2+3*a)^(3/2),x, algorithm="giac")
Output:
integrate((c*x)^(3/2)/(-2*a*x^2 + 3*a)^(3/2), x)
Timed out. \[ \int \frac {(c x)^{3/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\int \frac {{\left (c\,x\right )}^{3/2}}{{\left (3\,a-2\,a\,x^2\right )}^{3/2}} \,d x \] Input:
int((c*x)^(3/2)/(3*a - 2*a*x^2)^(3/2),x)
Output:
int((c*x)^(3/2)/(3*a - 2*a*x^2)^(3/2), x)
\[ \int \frac {(c x)^{3/2}}{\left (3 a-2 a x^2\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, c \left (\sqrt {-2 x^{2}+3}\, \left (\int \frac {\sqrt {x}\, \sqrt {-2 x^{2}+3}}{2 x^{3}-3 x}d x \right )+2 \sqrt {x}\right )}{4 \sqrt {-2 x^{2}+3}\, a^{2}} \] Input:
int((c*x)^(3/2)/(-2*a*x^2+3*a)^(3/2),x)
Output:
(sqrt(c)*sqrt(a)*c*(sqrt( - 2*x**2 + 3)*int((sqrt(x)*sqrt( - 2*x**2 + 3))/ (2*x**3 - 3*x),x) + 2*sqrt(x)))/(4*sqrt( - 2*x**2 + 3)*a**2)