3.8.1 \(\int x (-1+x^2)^{7/3} \, dx\) [701]

Optimal. Leaf size=13 \[ \frac {3}{20} \left (-1+x^2\right )^{10/3} \]

[Out]

3/20*(x^2-1)^(10/3)

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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 = {267} \begin {gather*} \frac {3}{20} \left (x^2-1\right )^{10/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(-1 + x^2)^(7/3),x]

[Out]

(3*(-1 + x^2)^(10/3))/20

Rule 267

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 \left (-1+x^2\right )^{7/3} \, dx &=\frac {3}{20} \left (-1+x^2\right )^{10/3}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {3}{20} \left (-1+x^2\right )^{10/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(-1 + x^2)^(7/3),x]

[Out]

(3*(-1 + x^2)^(10/3))/20

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Maple [A]
time = 0.10, size = 10, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {3 \left (x^{2}-1\right )^{\frac {10}{3}}}{20}\)	\(10\)
default	\(\frac {3 \left (x^{2}-1\right )^{\frac {10}{3}}}{20}\)	\(10\)
gosper	\(\frac {3 \left (x -1\right ) \left (x +1\right ) \left (x^{2}-1\right )^{\frac {7}{3}}}{20}\)	\(16\)
risch	\(\frac {3 \left (x^{2}-1\right )^{\frac {1}{3}} \left (x^{6}-3 x^{4}+3 x^{2}-1\right )}{20}\)	\(25\)
trager	\(\left (\frac {3}{20} x^{6}-\frac {9}{20} x^{4}+\frac {9}{20} x^{2}-\frac {3}{20}\right ) \left (x^{2}-1\right )^{\frac {1}{3}}\)	\(26\)
meijerg	\(\frac {\mathrm {signum}\left (x^{2}-1\right )^{\frac {7}{3}} x^{2} \hypergeom \left (\left [-\frac {7}{3}, 1\right ], \left [2\right ], x^{2}\right )}{2 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {7}{3}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^2-1)^(7/3),x,method=_RETURNVERBOSE)

[Out]

3/20*(x^2-1)^(10/3)

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Maxima [A]
time = 0.29, size = 9, normalized size = 0.69 \begin {gather*} \frac {3}{20} \, {\left (x^{2} - 1\right )}^{\frac {10}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2-1)^(7/3),x, algorithm="maxima")

[Out]

3/20*(x^2 - 1)^(10/3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (9) = 18\).
time = 1.03, size = 24, normalized size = 1.85 \begin {gather*} \frac {3}{20} \, {\left (x^{6} - 3 \, x^{4} + 3 \, x^{2} - 1\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2-1)^(7/3),x, algorithm="fricas")

[Out]

3/20*(x^6 - 3*x^4 + 3*x^2 - 1)*(x^2 - 1)^(1/3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (10) = 20\).
time = 0.36, size = 56, normalized size = 4.31 \begin {gather*} \frac {3 x^{6} \sqrt [3]{x^{2} - 1}}{20} - \frac {9 x^{4} \sqrt [3]{x^{2} - 1}}{20} + \frac {9 x^{2} \sqrt [3]{x^{2} - 1}}{20} - \frac {3 \sqrt [3]{x^{2} - 1}}{20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x**2-1)**(7/3),x)

[Out]

3*x**6*(x**2 - 1)**(1/3)/20 - 9*x**4*(x**2 - 1)**(1/3)/20 + 9*x**2*(x**2 - 1)**(1/3)/20 - 3*(x**2 - 1)**(1/3)/
20

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Giac [A]
time = 1.60, size = 9, normalized size = 0.69 \begin {gather*} \frac {3}{20} \, {\left (x^{2} - 1\right )}^{\frac {10}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2-1)^(7/3),x, algorithm="giac")

[Out]

3/20*(x^2 - 1)^(10/3)

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Mupad [B]
time = 5.13, size = 25, normalized size = 1.92 \begin {gather*} {\left (x^2-1\right )}^{1/3}\,\left (\frac {3\,x^6}{20}-\frac {9\,x^4}{20}+\frac {9\,x^2}{20}-\frac {3}{20}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^2 - 1)^(7/3),x)

[Out]

(x^2 - 1)^(1/3)*((9*x^2)/20 - (9*x^4)/20 + (3*x^6)/20 - 3/20)

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