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

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

[Out]

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

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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 = 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*}

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Mathematica [A]  time = 0.00, 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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fricas [B]  time = 0.35, size = 51, normalized size = 3.92 \[ \frac {1}{20} x^{20} - \frac {1}{2} x^{18} + \frac {9}{4} x^{16} - 6 x^{14} + \frac {21}{2} x^{12} - \frac {63}{5} x^{10} + \frac {21}{2} x^{8} - 6 x^{6} + \frac {9}{4} x^{4} - \frac {1}{2} x^{2} \]

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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giac [A]  time = 0.01, size = 9, normalized size = 0.69 \[ \frac {1}{20} \, {\left (x^{2} - 1\right )}^{10} \]

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

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maple [B]  time = 0.00, size = 52, normalized size = 4.00 \[ \frac {1}{20} x^{20}-\frac {1}{2} x^{18}+\frac {9}{4} x^{16}-6 x^{14}+\frac {21}{2} x^{12}-\frac {63}{5} x^{10}+\frac {21}{2} x^{8}-6 x^{6}+\frac {9}{4} x^{4}-\frac {1}{2} x^{2} \]

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 = 0.46, size = 9, normalized size = 0.69 \[ \frac {1}{20} \, {\left (x^{2} - 1\right )}^{10} \]

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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mupad [B]  time = 0.10, size = 9, normalized size = 0.69 \[ \frac {{\left (x^2-1\right )}^{10}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^2 - 1)^9,x)

[Out]

(x^2 - 1)^10/20

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sympy [B]  time = 0.06, size = 58, normalized size = 4.46 \[ \frac {x^{20}}{20} - \frac {x^{18}}{2} + \frac {9 x^{16}}{4} - 6 x^{14} + \frac {21 x^{12}}{2} - \frac {63 x^{10}}{5} + \frac {21 x^{8}}{2} - 6 x^{6} + \frac {9 x^{4}}{4} - \frac {x^{2}}{2} \]

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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