Integrand size = 15, antiderivative size = 139 \[ \int \frac {x^6}{\left (-2-3 x^2\right )^{3/4}} \, dx=-\frac {160 x \sqrt [4]{-2-3 x^2}}{2079}+\frac {40}{693} x^3 \sqrt [4]{-2-3 x^2}-\frac {2}{33} x^5 \sqrt [4]{-2-3 x^2}+\frac {160\ 2^{3/4} \sqrt {-\frac {x^2}{\left (\sqrt {2}+\sqrt {-2-3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2-3 x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-2-3 x^2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{2079 \sqrt {3} x} \] Output:
-160/2079*x*(-3*x^2-2)^(1/4)+40/693*x^3*(-3*x^2-2)^(1/4)-2/33*x^5*(-3*x^2- 2)^(1/4)+160/6237*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 = 5.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.49 \[ \int \frac {x^6}{\left (-2-3 x^2\right )^{3/4}} \, dx=\frac {2 x \left (160+120 x^2-54 x^4+189 x^6-80 \sqrt [4]{2} \left (2+3 x^2\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\frac {3 x^2}{2}\right )\right )}{2079 \left (-2-3 x^2\right )^{3/4}} \] Input:
Integrate[x^6/(-2 - 3*x^2)^(3/4),x]
Output:
(2*x*(160 + 120*x^2 - 54*x^4 + 189*x^6 - 80*2^(1/4)*(2 + 3*x^2)^(3/4)*Hype rgeometric2F1[1/2, 3/4, 3/2, (-3*x^2)/2]))/(2079*(-2 - 3*x^2)^(3/4))
Time = 0.23 (sec) , antiderivative size = 149, 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 = {262, 262, 262, 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.
\(\displaystyle \int \frac {x^6}{\left (-3 x^2-2\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {20}{33} \int \frac {x^4}{\left (-3 x^2-2\right )^{3/4}}dx-\frac {2}{33} \sqrt [4]{-3 x^2-2} x^5\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {20}{33} \left (-\frac {4}{7} \int \frac {x^2}{\left (-3 x^2-2\right )^{3/4}}dx-\frac {2}{21} \sqrt [4]{-3 x^2-2} x^3\right )-\frac {2}{33} \sqrt [4]{-3 x^2-2} x^5\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {20}{33} \left (-\frac {4}{7} \left (-\frac {4}{9} \int \frac {1}{\left (-3 x^2-2\right )^{3/4}}dx-\frac {2}{9} \sqrt [4]{-3 x^2-2} x\right )-\frac {2}{21} \sqrt [4]{-3 x^2-2} x^3\right )-\frac {2}{33} \sqrt [4]{-3 x^2-2} x^5\) |
\(\Big \downarrow \) 232 |
\(\displaystyle -\frac {20}{33} \left (-\frac {4}{7} \left (\frac {4 \sqrt {\frac {2}{3}} \sqrt {-x^2} \int \frac {1}{\sqrt {\frac {1}{2} \left (-3 x^2-2\right )+1}}d\sqrt [4]{-3 x^2-2}}{9 x}-\frac {2}{9} x \sqrt [4]{-3 x^2-2}\right )-\frac {2}{21} \sqrt [4]{-3 x^2-2} x^3\right )-\frac {2}{33} \sqrt [4]{-3 x^2-2} x^5\) |
\(\Big \downarrow \) 761 |
\(\displaystyle -\frac {20}{33} \left (-\frac {4}{7} \left (\frac {2\ 2^{3/4} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{9 \sqrt {3} x}-\frac {2}{9} x \sqrt [4]{-3 x^2-2}\right )-\frac {2}{21} \sqrt [4]{-3 x^2-2} x^3\right )-\frac {2}{33} \sqrt [4]{-3 x^2-2} x^5\) |
Input:
Int[x^6/(-2 - 3*x^2)^(3/4),x]
Output:
(-2*x^5*(-2 - 3*x^2)^(1/4))/33 - (20*((-2*x^3*(-2 - 3*x^2)^(1/4))/21 - (4* ((-2*x*(-2 - 3*x^2)^(1/4))/9 + (2*2^(3/4)*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])/(9*Sqrt[3]*x)))/7))/33
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*(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[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 = 0.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.17
method | result | size |
meijerg | \(-\frac {\left (-1\right )^{\frac {1}{4}} 2^{\frac {1}{4}} x^{7} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {7}{2}\right ], \left [\frac {9}{2}\right ], -\frac {3 x^{2}}{2}\right )}{14}\) | \(23\) |
Input:
int(x^6/(-3*x^2-2)^(3/4),x,method=_RETURNVERBOSE)
Output:
-1/14*(-1)^(1/4)*2^(1/4)*x^7*hypergeom([3/4,7/2],[9/2],-3/2*x^2)
\[ \int \frac {x^6}{\left (-2-3 x^2\right )^{3/4}} \, dx=\int { \frac {x^{6}}{{\left (-3 \, x^{2} - 2\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^6/(-3*x^2-2)^(3/4),x, algorithm="fricas")
Output:
-2/2079*(63*x^5 - 60*x^3 + 80*x)*(-3*x^2 - 2)^(1/4) + integral(320/2079*(- 3*x^2 - 2)^(1/4)/(3*x^2 + 2), x)
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.26 \[ \int \frac {x^6}{\left (-2-3 x^2\right )^{3/4}} \, dx=\frac {\sqrt [4]{2} x^{7} e^{- \frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{14} \] Input:
integrate(x**6/(-3*x**2-2)**(3/4),x)
Output:
2**(1/4)*x**7*exp(-3*I*pi/4)*hyper((3/4, 7/2), (9/2,), 3*x**2*exp_polar(I* pi)/2)/14
\[ \int \frac {x^6}{\left (-2-3 x^2\right )^{3/4}} \, dx=\int { \frac {x^{6}}{{\left (-3 \, x^{2} - 2\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^6/(-3*x^2-2)^(3/4),x, algorithm="maxima")
Output:
integrate(x^6/(-3*x^2 - 2)^(3/4), x)
\[ \int \frac {x^6}{\left (-2-3 x^2\right )^{3/4}} \, dx=\int { \frac {x^{6}}{{\left (-3 \, x^{2} - 2\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^6/(-3*x^2-2)^(3/4),x, algorithm="giac")
Output:
integrate(x^6/(-3*x^2 - 2)^(3/4), x)
Timed out. \[ \int \frac {x^6}{\left (-2-3 x^2\right )^{3/4}} \, dx=\int \frac {x^6}{{\left (-3\,x^2-2\right )}^{3/4}} \,d x \] Input:
int(x^6/(- 3*x^2 - 2)^(3/4),x)
Output:
int(x^6/(- 3*x^2 - 2)^(3/4), x)
\[ \int \frac {x^6}{\left (-2-3 x^2\right )^{3/4}} \, dx=\int \frac {x^{6}}{\left (-3 x^{2}-2\right )^{\frac {3}{4}}}d x \] Input:
int(x^6/(-3*x^2-2)^(3/4),x)
Output:
int(x**6/( - 3*x**2 - 2)**(3/4),x)