3.1.6 \(\int x \sqrt [4]{-1+x^2} \, dx\)

Optimal. Leaf size=13 \[ \frac {2}{5} \left (x^2-1\right )^{5/4} \]

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {261} \begin {gather*} \frac {2}{5} \left (x^2-1\right )^{5/4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(-1 + x^2)^(1/4),x]

[Out]

(2*(-1 + x^2)^(5/4))/5

Rule 261

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

Rubi steps

\begin {align*} \int x \sqrt [4]{-1+x^2} \, dx &=\frac {2}{5} \left (-1+x^2\right )^{5/4}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {2}{5} \left (x^2-1\right )^{5/4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(-1 + x^2)^(1/4),x]

[Out]

(2*(-1 + x^2)^(5/4))/5

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IntegrateAlgebraic [A]  time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {2}{5} \left (-1+x^2\right )^{5/4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x*(-1 + x^2)^(1/4),x]

[Out]

(2*(-1 + x^2)^(5/4))/5

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fricas [A]  time = 0.43, size = 9, normalized size = 0.69 \begin {gather*} \frac {2}{5} \, {\left (x^{2} - 1\right )}^{\frac {5}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2-1)^(1/4),x, algorithm="fricas")

[Out]

2/5*(x^2 - 1)^(5/4)

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giac [A]  time = 0.45, size = 9, normalized size = 0.69 \begin {gather*} \frac {2}{5} \, {\left (x^{2} - 1\right )}^{\frac {5}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2-1)^(1/4),x, algorithm="giac")

[Out]

2/5*(x^2 - 1)^(5/4)

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maple [A]  time = 0.07, size = 10, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {2 \left (x^{2}-1\right )^{\frac {5}{4}}}{5}\)	\(10\)
default	\(\frac {2 \left (x^{2}-1\right )^{\frac {5}{4}}}{5}\)	\(10\)
risch	\(\frac {2 \left (x^{2}-1\right )^{\frac {5}{4}}}{5}\)	\(10\)
gosper	\(\frac {2 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}-1\right )^{\frac {1}{4}}}{5}\)	\(16\)
trager	\(\left (\frac {2 x^{2}}{5}-\frac {2}{5}\right ) \left (x^{2}-1\right )^{\frac {1}{4}}\)	\(16\)
meijerg	\(\frac {\mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {1}{4}, 1\right ], \relax [2], x^{2}\right ) x^{2}}{2 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{4}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^2-1)^(1/4),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.49, size = 9, normalized size = 0.69 \begin {gather*} \frac {2}{5} \, {\left (x^{2} - 1\right )}^{\frac {5}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2-1)^(1/4),x, algorithm="maxima")

[Out]

2/5*(x^2 - 1)^(5/4)

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mupad [B]  time = 0.08, size = 9, normalized size = 0.69 \begin {gather*} \frac {2\,{\left (x^2-1\right )}^{5/4}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^2 - 1)^(1/4),x)

[Out]

(2*(x^2 - 1)^(5/4))/5

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sympy [B]  time = 0.17, size = 26, normalized size = 2.00 \begin {gather*} \frac {2 x^{2} \sqrt [4]{x^{2} - 1}}{5} - \frac {2 \sqrt [4]{x^{2} - 1}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x**2-1)**(1/4),x)

[Out]

2*x**2*(x**2 - 1)**(1/4)/5 - 2*(x**2 - 1)**(1/4)/5

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