Integrand size = 15, antiderivative size = 141 \[ \int \frac {1}{x^6 \left (-2-3 x^2\right )^{3/4}} \, dx=\frac {\sqrt [4]{-2-3 x^2}}{10 x^5}-\frac {9 \sqrt [4]{-2-3 x^2}}{40 x^3}+\frac {27 \sqrt [4]{-2-3 x^2}}{32 x}+\frac {27 \sqrt {3} \sqrt {-\frac {x^2}{\left (\sqrt {2}+\sqrt {-2-3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2-3 x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-2-3 x^2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{64 \sqrt [4]{2} x} \] Output:
1/10*(-3*x^2-2)^(1/4)/x^5-9/40*(-3*x^2-2)^(1/4)/x^3+27/32*(-3*x^2-2)^(1/4) /x+27/128*2^(3/4)*(-x^2/(2^(1/2)+(-3*x^2-2)^(1/2))^2)^(1/2)*(2^(1/2)+(-3*x ^2-2)^(1/2))*InverseJacobiAM(2*arctan(1/2*(-3*x^2-2)^(1/4)*2^(3/4)),1/2*2^ (1/2))*3^(1/2)/x
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.34 \[ \int \frac {1}{x^6 \left (-2-3 x^2\right )^{3/4}} \, dx=-\frac {\left (1+\frac {3 x^2}{2}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{4},-\frac {3}{2},-\frac {3 x^2}{2}\right )}{5 x^5 \left (-2-3 x^2\right )^{3/4}} \] Input:
Integrate[1/(x^6*(-2 - 3*x^2)^(3/4)),x]
Output:
-1/5*((1 + (3*x^2)/2)^(3/4)*Hypergeometric2F1[-5/2, 3/4, -3/2, (-3*x^2)/2] )/(x^5*(-2 - 3*x^2)^(3/4))
Time = 0.23 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {264, 264, 264, 232, 761}
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.
\(\displaystyle \int \frac {1}{x^6 \left (-3 x^2-2\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {\sqrt [4]{-3 x^2-2}}{10 x^5}-\frac {27}{20} \int \frac {1}{x^4 \left (-3 x^2-2\right )^{3/4}}dx\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {\sqrt [4]{-3 x^2-2}}{10 x^5}-\frac {27}{20} \left (\frac {\sqrt [4]{-3 x^2-2}}{6 x^3}-\frac {5}{4} \int \frac {1}{x^2 \left (-3 x^2-2\right )^{3/4}}dx\right )\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {\sqrt [4]{-3 x^2-2}}{10 x^5}-\frac {27}{20} \left (\frac {\sqrt [4]{-3 x^2-2}}{6 x^3}-\frac {5}{4} \left (\frac {\sqrt [4]{-3 x^2-2}}{2 x}-\frac {3}{4} \int \frac {1}{\left (-3 x^2-2\right )^{3/4}}dx\right )\right )\) |
\(\Big \downarrow \) 232 |
\(\displaystyle \frac {\sqrt [4]{-3 x^2-2}}{10 x^5}-\frac {27}{20} \left (\frac {\sqrt [4]{-3 x^2-2}}{6 x^3}-\frac {5}{4} \left (\frac {\sqrt {\frac {3}{2}} \sqrt {-x^2} \int \frac {1}{\sqrt {\frac {1}{2} \left (-3 x^2-2\right )+1}}d\sqrt [4]{-3 x^2-2}}{2 x}+\frac {\sqrt [4]{-3 x^2-2}}{2 x}\right )\right )\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {\sqrt [4]{-3 x^2-2}}{10 x^5}-\frac {27}{20} \left (\frac {\sqrt [4]{-3 x^2-2}}{6 x^3}-\frac {5}{4} \left (\frac {\sqrt {3} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} x}+\frac {\sqrt [4]{-3 x^2-2}}{2 x}\right )\right )\) |
Input:
Int[1/(x^6*(-2 - 3*x^2)^(3/4)),x]
Output:
(-2 - 3*x^2)^(1/4)/(10*x^5) - (27*((-2 - 3*x^2)^(1/4)/(6*x^3) - (5*((-2 - 3*x^2)^(1/4)/(2*x) + (Sqrt[3]*Sqrt[-(x^2/(Sqrt[2] + Sqrt[-2 - 3*x^2])^2)]* (Sqrt[2] + Sqrt[-2 - 3*x^2])*EllipticF[2*ArcTan[(-2 - 3*x^2)^(1/4)/2^(1/4) ], 1/2])/(4*2^(1/4)*x)))/4))/20
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[2*(Sqrt[(-b)*(x^2/a)]/( b*x)) Subst[Int[1/Sqrt[1 - x^4/a], x], x, (a + b*x^2)^(1/4)], x] /; FreeQ [{a, b}, x] && NegQ[a]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.16
method | result | size |
meijerg | \(\frac {\left (-1\right )^{\frac {1}{4}} 2^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {5}{2}, \frac {3}{4}\right ], \left [-\frac {3}{2}\right ], -\frac {3 x^{2}}{2}\right )}{10 x^{5}}\) | \(23\) |
Input:
int(1/x^6/(-3*x^2-2)^(3/4),x,method=_RETURNVERBOSE)
Output:
1/10*(-1)^(1/4)*2^(1/4)/x^5*hypergeom([-5/2,3/4],[-3/2],-3/2*x^2)
\[ \int \frac {1}{x^6 \left (-2-3 x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (-3 \, x^{2} - 2\right )}^{\frac {3}{4}} x^{6}} \,d x } \] Input:
integrate(1/x^6/(-3*x^2-2)^(3/4),x, algorithm="fricas")
Output:
1/160*(160*x^5*integral(81/64*(-3*x^2 - 2)^(1/4)/(3*x^2 + 2), x) + (135*x^ 4 - 36*x^2 + 16)*(-3*x^2 - 2)^(1/4))/x^5
Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^6 \left (-2-3 x^2\right )^{3/4}} \, dx=\frac {\sqrt [4]{2} e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {3}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{10 x^{5}} \] Input:
integrate(1/x**6/(-3*x**2-2)**(3/4),x)
Output:
2**(1/4)*exp(I*pi/4)*hyper((-5/2, 3/4), (-3/2,), 3*x**2*exp_polar(I*pi)/2) /(10*x**5)
\[ \int \frac {1}{x^6 \left (-2-3 x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (-3 \, x^{2} - 2\right )}^{\frac {3}{4}} x^{6}} \,d x } \] Input:
integrate(1/x^6/(-3*x^2-2)^(3/4),x, algorithm="maxima")
Output:
integrate(1/((-3*x^2 - 2)^(3/4)*x^6), x)
\[ \int \frac {1}{x^6 \left (-2-3 x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (-3 \, x^{2} - 2\right )}^{\frac {3}{4}} x^{6}} \,d x } \] Input:
integrate(1/x^6/(-3*x^2-2)^(3/4),x, algorithm="giac")
Output:
integrate(1/((-3*x^2 - 2)^(3/4)*x^6), x)
Timed out. \[ \int \frac {1}{x^6 \left (-2-3 x^2\right )^{3/4}} \, dx=\int \frac {1}{x^6\,{\left (-3\,x^2-2\right )}^{3/4}} \,d x \] Input:
int(1/(x^6*(- 3*x^2 - 2)^(3/4)),x)
Output:
int(1/(x^6*(- 3*x^2 - 2)^(3/4)), x)
\[ \int \frac {1}{x^6 \left (-2-3 x^2\right )^{3/4}} \, dx=\int \frac {1}{\left (-3 x^{2}-2\right )^{\frac {3}{4}} x^{6}}d x \] Input:
int(1/x^6/(-3*x^2-2)^(3/4),x)
Output:
int(1/(( - 3*x**2 - 2)**(3/4)*x**6),x)