3.1.48 \(\int x (-1+x^2)^9 \, dx\) [48]

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

[Out]

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

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

Int[x*(-1 + x^2)^9,x]

[Out]

(1 - x^2)^10/20

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 11, normalized size = 0.85 \begin {gather*} \frac {1}{20} \left (-1+x^2\right )^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(-1 + x^2)^9,x]

[Out]

(-1 + x^2)^10/20

________________________________________________________________________________________

Maple [A]
time = 0.06, size = 10, normalized size = 0.77

method	result	size
default	\(\frac {\left (x^{2}-1\right )^{10}}{20}\)	\(10\)
gosper	\(\frac {x^{2} \left (x^{18}-10 x^{16}+45 x^{14}-120 x^{12}+210 x^{10}-252 x^{8}+210 x^{6}-120 x^{4}+45 x^{2}-10\right )}{20}\)	\(51\)
norman	\(-\frac {1}{2} x^{2}+\frac {9}{4} x^{4}-6 x^{6}+\frac {21}{2} x^{8}-\frac {63}{5} x^{10}+\frac {21}{2} x^{12}-6 x^{14}+\frac {9}{4} x^{16}-\frac {1}{2} x^{18}+\frac {1}{20} x^{20}\)	\(52\)
risch	\(\frac {1}{20} x^{20}-\frac {1}{2} x^{18}+\frac {9}{4} x^{16}-6 x^{14}+\frac {21}{2} x^{12}-\frac {63}{5} x^{10}+\frac {21}{2} x^{8}-6 x^{6}+\frac {9}{4} x^{4}-\frac {1}{2} x^{2}+\frac {1}{20}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^2-1)^9,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 2.56, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{20} \, {\left (x^{2} - 1\right )}^{10} \end {gather*}

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (9) = 18\).
time = 0.96, size = 51, normalized size = 3.92 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (7) = 14\).
time = 0.01, size = 58, normalized size = 4.46 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Giac [A]
time = 0.45, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{20} \, {\left (x^{2} - 1\right )}^{10} \end {gather*}

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^2 - 1)^9,x)

[Out]

(x^2 - 1)^10/20

________________________________________________________________________________________