3.48 \(\int x (-1+x^2)^9 \, dx\)

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

[Out]

(1 - x^2)^10/20

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Rubi [A]  time = 0.0014274, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {261} \[ \frac{1}{20} \left (1-x^2\right )^{10} \]

Antiderivative was successfully verified.

[In]

Int[x*(-1 + x^2)^9,x]

[Out]

(1 - x^2)^10/20

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 \left (-1+x^2\right )^9 \, dx &=\frac{1}{20} \left (1-x^2\right )^{10}\\ \end{align*}

Mathematica [A]  time = 0.0013549, size = 11, normalized size = 0.85 \[ \frac{1}{20} \left (x^2-1\right )^{10} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(-1 + x^2)^9,x]

[Out]

(-1 + x^2)^10/20

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Maple [B]  time = 0.001, size = 52, normalized size = 4. \begin{align*}{\frac{{x}^{20}}{20}}-{\frac{{x}^{18}}{2}}+{\frac{9\,{x}^{16}}{4}}-6\,{x}^{14}+{\frac{21\,{x}^{12}}{2}}-{\frac{63\,{x}^{10}}{5}}+{\frac{21\,{x}^{8}}{2}}-6\,{x}^{6}+{\frac{9\,{x}^{4}}{4}}-{\frac{{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^2-1)^9,x)

[Out]

1/20*x^20-1/2*x^18+9/4*x^16-6*x^14+21/2*x^12-63/5*x^10+21/2*x^8-6*x^6+9/4*x^4-1/2*x^2

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Maxima [A]  time = 0.955849, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{20} \,{\left (x^{2} - 1\right )}^{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2-1)^9,x, algorithm="maxima")

[Out]

1/20*(x^2 - 1)^10

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Fricas [B]  time = 0.358493, size = 142, normalized size = 10.92 \begin{align*} \frac{1}{20} x^{20} - \frac{1}{2} x^{18} + \frac{9}{4} x^{16} - 6 x^{14} + \frac{21}{2} x^{12} - \frac{63}{5} x^{10} + \frac{21}{2} x^{8} - 6 x^{6} + \frac{9}{4} x^{4} - \frac{1}{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2-1)^9,x, algorithm="fricas")

[Out]

1/20*x^20 - 1/2*x^18 + 9/4*x^16 - 6*x^14 + 21/2*x^12 - 63/5*x^10 + 21/2*x^8 - 6*x^6 + 9/4*x^4 - 1/2*x^2

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Sympy [B]  time = 0.056396, size = 58, normalized size = 4.46 \begin{align*} \frac{x^{20}}{20} - \frac{x^{18}}{2} + \frac{9 x^{16}}{4} - 6 x^{14} + \frac{21 x^{12}}{2} - \frac{63 x^{10}}{5} + \frac{21 x^{8}}{2} - 6 x^{6} + \frac{9 x^{4}}{4} - \frac{x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x**2-1)**9,x)

[Out]

x**20/20 - x**18/2 + 9*x**16/4 - 6*x**14 + 21*x**12/2 - 63*x**10/5 + 21*x**8/2 - 6*x**6 + 9*x**4/4 - x**2/2

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Giac [A]  time = 1.12956, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{20} \,{\left (x^{2} - 1\right )}^{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2-1)^9,x, algorithm="giac")

[Out]

1/20*(x^2 - 1)^10