3.3.50 \(\int x^3 (-1+x^2)^{2/3} \, dx\)

Optimal. Leaf size=25 \[ \frac {3}{80} \left (x^2-1\right )^{2/3} \left (5 x^4-2 x^2-3\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.08, 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} \begin {gather*} \frac {3}{16} \left (x^2-1\right )^{8/3}+\frac {3}{10} \left (x^2-1\right )^{5/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(-1 + x^2)^(2/3),x]

[Out]

(3*(-1 + x^2)^(5/3))/10 + (3*(-1 + x^2)^(8/3))/16

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 x^3 \left (-1+x^2\right )^{2/3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (-1+x)^{2/3} x \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left ((-1+x)^{2/3}+(-1+x)^{5/3}\right ) \, dx,x,x^2\right )\\ &=\frac {3}{10} \left (-1+x^2\right )^{5/3}+\frac {3}{16} \left (-1+x^2\right )^{8/3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.80 \begin {gather*} \frac {3}{80} \left (x^2-1\right )^{5/3} \left (5 x^2+3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(-1 + x^2)^(2/3),x]

[Out]

(3*(-1 + x^2)^(5/3)*(3 + 5*x^2))/80

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IntegrateAlgebraic [A]  time = 0.02, size = 20, normalized size = 0.80 \begin {gather*} \frac {3}{80} \left (-1+x^2\right )^{5/3} \left (3+5 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^3*(-1 + x^2)^(2/3),x]

[Out]

(3*(-1 + x^2)^(5/3)*(3 + 5*x^2))/80

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fricas [A]  time = 0.45, size = 21, normalized size = 0.84 \begin {gather*} \frac {3}{80} \, {\left (5 \, x^{4} - 2 \, x^{2} - 3\right )} {\left (x^{2} - 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^2-1)^(2/3),x, algorithm="fricas")

[Out]

3/80*(5*x^4 - 2*x^2 - 3)*(x^2 - 1)^(2/3)

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giac [A]  time = 0.29, size = 19, normalized size = 0.76 \begin {gather*} \frac {3}{16} \, {\left (x^{2} - 1\right )}^{\frac {8}{3}} + \frac {3}{10} \, {\left (x^{2} - 1\right )}^{\frac {5}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^2-1)^(2/3),x, algorithm="giac")

[Out]

3/16*(x^2 - 1)^(8/3) + 3/10*(x^2 - 1)^(5/3)

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maple [A]  time = 0.08, size = 21, normalized size = 0.84

method	result	size
trager	\(\left (\frac {3}{16} x^{4}-\frac {3}{40} x^{2}-\frac {9}{80}\right ) \left (x^{2}-1\right )^{\frac {2}{3}}\)	\(21\)
risch	\(\frac {3 \left (x^{2}-1\right )^{\frac {2}{3}} \left (5 x^{4}-2 x^{2}-3\right )}{80}\)	\(22\)
gosper	\(\frac {3 \left (-1+x \right ) \left (1+x \right ) \left (5 x^{2}+3\right ) \left (x^{2}-1\right )^{\frac {2}{3}}}{80}\)	\(23\)
meijerg	\(\frac {\mathrm {signum}\left (x^{2}-1\right )^{\frac {2}{3}} \hypergeom \left (\left [-\frac {2}{3}, 2\right ], \relax [3], x^{2}\right ) x^{4}}{4 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {2}{3}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(x^2-1)^(2/3),x,method=_RETURNVERBOSE)

[Out]

(3/16*x^4-3/40*x^2-9/80)*(x^2-1)^(2/3)

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maxima [A]  time = 0.34, size = 19, normalized size = 0.76 \begin {gather*} \frac {3}{16} \, {\left (x^{2} - 1\right )}^{\frac {8}{3}} + \frac {3}{10} \, {\left (x^{2} - 1\right )}^{\frac {5}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^2-1)^(2/3),x, algorithm="maxima")

[Out]

3/16*(x^2 - 1)^(8/3) + 3/10*(x^2 - 1)^(5/3)

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mupad [B]  time = 0.21, size = 21, normalized size = 0.84 \begin {gather*} -{\left (x^2-1\right )}^{2/3}\,\left (-\frac {3\,x^4}{16}+\frac {3\,x^2}{40}+\frac {9}{80}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(x^2 - 1)^(2/3),x)

[Out]

-(x^2 - 1)^(2/3)*((3*x^2)/40 - (3*x^4)/16 + 9/80)

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sympy [A]  time = 0.37, size = 41, normalized size = 1.64 \begin {gather*} \frac {3 x^{4} \left (x^{2} - 1\right )^{\frac {2}{3}}}{16} - \frac {3 x^{2} \left (x^{2} - 1\right )^{\frac {2}{3}}}{40} - \frac {9 \left (x^{2} - 1\right )^{\frac {2}{3}}}{80} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(x**2-1)**(2/3),x)

[Out]

3*x**4*(x**2 - 1)**(2/3)/16 - 3*x**2*(x**2 - 1)**(2/3)/40 - 9*(x**2 - 1)**(2/3)/80

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