3.467 \(\int \frac{x^5}{\left (-4+x^2\right )^{13/6}} \, dx\)

Optimal. Leaf size=38 \[ \frac{3}{5} \left (x^2-4\right )^{5/6}-\frac{24}{\sqrt [6]{x^2-4}}-\frac{48}{7 \left (x^2-4\right )^{7/6}} \]

[Out]

-48/(7*(-4 + x^2)^(7/6)) - 24/(-4 + x^2)^(1/6) + (3*(-4 + x^2)^(5/6))/5

_______________________________________________________________________________________

Rubi [A]  time = 0.0377135, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3}{5} \left (x^2-4\right )^{5/6}-\frac{24}{\sqrt [6]{x^2-4}}-\frac{48}{7 \left (x^2-4\right )^{7/6}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(-4 + x^2)^(13/6),x]

[Out]

-48/(7*(-4 + x^2)^(7/6)) - 24/(-4 + x^2)^(1/6) + (3*(-4 + x^2)^(5/6))/5

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.26905, size = 32, normalized size = 0.84 \[ \frac{3 \left (x^{2} - 4\right )^{\frac{5}{6}}}{5} - \frac{24}{\sqrt [6]{x^{2} - 4}} - \frac{48}{7 \left (x^{2} - 4\right )^{\frac{7}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(x**2-4)**(13/6),x)

[Out]

3*(x**2 - 4)**(5/6)/5 - 24/(x**2 - 4)**(1/6) - 48/(7*(x**2 - 4)**(7/6))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0147151, size = 25, normalized size = 0.66 \[ \frac{3 \left (7 x^4-336 x^2+1152\right )}{35 \left (x^2-4\right )^{7/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(-4 + x^2)^(13/6),x]

[Out]

(3*(1152 - 336*x^2 + 7*x^4))/(35*(-4 + x^2)^(7/6))

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 28, normalized size = 0.7 \[{\frac{ \left ( -6+3\,x \right ) \left ( 2+x \right ) \left ( 7\,{x}^{4}-336\,{x}^{2}+1152 \right ) }{35} \left ({x}^{2}-4 \right ) ^{-{\frac{13}{6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(x^2-4)^(13/6),x)

[Out]

3/35*(-2+x)*(2+x)*(7*x^4-336*x^2+1152)/(x^2-4)^(13/6)

_______________________________________________________________________________________

Maxima [A]  time = 1.34152, size = 38, normalized size = 1. \[ \frac{3}{5} \,{\left (x^{2} - 4\right )}^{\frac{5}{6}} - \frac{24}{{\left (x^{2} - 4\right )}^{\frac{1}{6}}} - \frac{48}{7 \,{\left (x^{2} - 4\right )}^{\frac{7}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^2 - 4)^(13/6),x, algorithm="maxima")

[Out]

3/5*(x^2 - 4)^(5/6) - 24/(x^2 - 4)^(1/6) - 48/7/(x^2 - 4)^(7/6)

_______________________________________________________________________________________

Fricas [A]  time = 0.202315, size = 28, normalized size = 0.74 \[ \frac{3 \,{\left (7 \, x^{4} - 336 \, x^{2} + 1152\right )}}{35 \,{\left (x^{2} - 4\right )}^{\frac{7}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^2 - 4)^(13/6),x, algorithm="fricas")

[Out]

3/35*(7*x^4 - 336*x^2 + 1152)/(x^2 - 4)^(7/6)

_______________________________________________________________________________________

Sympy [A]  time = 11.7506, size = 82, normalized size = 2.16 \[ \frac{21 x^{4}}{35 x^{2} \sqrt [6]{x^{2} - 4} - 140 \sqrt [6]{x^{2} - 4}} - \frac{1008 x^{2}}{35 x^{2} \sqrt [6]{x^{2} - 4} - 140 \sqrt [6]{x^{2} - 4}} + \frac{3456}{35 x^{2} \sqrt [6]{x^{2} - 4} - 140 \sqrt [6]{x^{2} - 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(x**2-4)**(13/6),x)

[Out]

21*x**4/(35*x**2*(x**2 - 4)**(1/6) - 140*(x**2 - 4)**(1/6)) - 1008*x**2/(35*x**2
*(x**2 - 4)**(1/6) - 140*(x**2 - 4)**(1/6)) + 3456/(35*x**2*(x**2 - 4)**(1/6) -
140*(x**2 - 4)**(1/6))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.200564, size = 35, normalized size = 0.92 \[ \frac{3}{5} \,{\left (x^{2} - 4\right )}^{\frac{5}{6}} - \frac{24 \,{\left (7 \, x^{2} - 26\right )}}{7 \,{\left (x^{2} - 4\right )}^{\frac{7}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^2 - 4)^(13/6),x, algorithm="giac")

[Out]

3/5*(x^2 - 4)^(5/6) - 24/7*(7*x^2 - 26)/(x^2 - 4)^(7/6)