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

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

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {3}{5} \left (x^2-4\right )^{5/6}-\frac {24}{\sqrt [6]{x^2-4}}-\frac {48}{7 \left (x^2-4\right )^{7/6}} \]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^5}{\left (-4+x^2\right )^{13/6}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{(-4+x)^{13/6}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {16}{(-4+x)^{13/6}}+\frac {8}{(-4+x)^{7/6}}+\frac {1}{\sqrt [6]{-4+x}}\right ) \, dx,x,x^2\right )\\ &=-\frac {48}{7 \left (-4+x^2\right )^{7/6}}-\frac {24}{\sqrt [6]{-4+x^2}}+\frac {3}{5} \left (-4+x^2\right )^{5/6}\\ \end {align*}

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Mathematica [A]  time = 0.01, 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))

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IntegrateAlgebraic [A]  time = 0.03, 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]

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

[Out]

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

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fricas [A]  time = 1.10, size = 33, normalized size = 0.87 \[ \frac {3 \, {\left (7 \, x^{4} - 336 \, x^{2} + 1152\right )} {\left (x^{2} - 4\right )}^{\frac {5}{6}}}{35 \, {\left (x^{4} - 8 \, x^{2} + 16\right )}} \]

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)^(5/6)/(x^4 - 8*x^2 + 16)

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giac [A]  time = 0.61, size = 26, normalized size = 0.68 \[ \frac {3}{5} \, {\left (x^{2} - 4\right )}^{\frac {5}{6}} - \frac {24 \, {\left (7 \, x^{2} - 26\right )}}{7 \, {\left (x^{2} - 4\right )}^{\frac {7}{6}}} \]

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)

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maple [A]  time = 0.33, size = 22, normalized size = 0.58

method	result	size
trager	\(\frac {\frac {3}{5} x^{4}-\frac {144}{5} x^{2}+\frac {3456}{35}}{\left (x^{2}-4\right )^{\frac {7}{6}}}\)	\(22\)
risch	\(\frac {\frac {3}{5} x^{4}-\frac {144}{5} x^{2}+\frac {3456}{35}}{\left (x^{2}-4\right )^{\frac {7}{6}}}\)	\(22\)
gosper	\(\frac {3 \left (-2+x \right ) \left (2+x \right ) \left (7 x^{4}-336 x^{2}+1152\right )}{35 \left (x^{2}-4\right )^{\frac {13}{6}}}\)	\(28\)
meijerg	\(\frac {2^{\frac {2}{3}} \left (-\mathrm {signum}\left (-1+\frac {x^{2}}{4}\right )\right )^{\frac {13}{6}} x^{6} \hypergeom \left (\left [\frac {13}{6}, 3\right ], \relax [4], \frac {x^{2}}{4}\right )}{192 \mathrm {signum}\left (-1+\frac {x^{2}}{4}\right )^{\frac {13}{6}}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^2-4)^(13/6),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.42, size = 28, normalized size = 0.74 \[ \frac {3}{5} \, {\left (x^{2} - 4\right )}^{\frac {5}{6}} - \frac {24}{{\left (x^{2} - 4\right )}^{\frac {1}{6}}} - \frac {48}{7 \, {\left (x^{2} - 4\right )}^{\frac {7}{6}}} \]

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)

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mupad [B]  time = 0.45, size = 21, normalized size = 0.55 \[ \frac {3\,\left (7\,x^4-336\,x^2+1152\right )}{35\,{\left (x^2-4\right )}^{7/6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^2 - 4)^(13/6),x)

[Out]

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

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sympy [B]  time = 5.06, 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))

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