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

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

[Out]

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

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Rubi [A]  time = 0.0154403, 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, 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 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]]

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])

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*}

Mathematica [A]  time = 0.0090649, 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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Maple [A]  time = 0.004, size = 28, normalized size = 0.7 \begin{align*}{\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}}}} \end{align*}

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)

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Maxima [A]  time = 0.976044, size = 38, normalized size = 1. \begin{align*} \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}}} \end{align*}

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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Fricas [A]  time = 2.02865, size = 89, normalized size = 2.34 \begin{align*} \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 )}} \end{align*}

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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Sympy [B]  time = 6.20819, size = 82, normalized size = 2.16 \begin{align*} \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}} \end{align*}

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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Giac [A]  time = 1.06423, size = 35, normalized size = 0.92 \begin{align*} \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}}} \end{align*}

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)