3.341 \(\int x (5+x^2)^8 \, dx\)

Optimal. Leaf size=11 \[ \frac {1}{18} \left (x^2+5\right )^9 \]

[Out]

1/18*(x^2+5)^9

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Rubi [A]  time = 0.00, antiderivative size = 11, 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} \[ \frac {1}{18} \left (x^2+5\right )^9 \]

Antiderivative was successfully verified.

[In]

Int[x*(5 + x^2)^8,x]

[Out]

(5 + x^2)^9/18

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 (5+x^2\right )^8 \, dx &=\frac {1}{18} \left (5+x^2\right )^9\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \[ \frac {1}{18} \left (x^2+5\right )^9 \]

Antiderivative was successfully verified.

[In]

Integrate[x*(5 + x^2)^8,x]

[Out]

(5 + x^2)^9/18

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fricas [B]  time = 0.34, size = 46, normalized size = 4.18 \[ \frac {1}{18} x^{18} + \frac {5}{2} x^{16} + 50 x^{14} + \frac {1750}{3} x^{12} + 4375 x^{10} + 21875 x^{8} + \frac {218750}{3} x^{6} + 156250 x^{4} + \frac {390625}{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2+5)^8,x, algorithm="fricas")

[Out]

1/18*x^18 + 5/2*x^16 + 50*x^14 + 1750/3*x^12 + 4375*x^10 + 21875*x^8 + 218750/3*x^6 + 156250*x^4 + 390625/2*x^
2

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giac [A]  time = 0.98, size = 9, normalized size = 0.82 \[ \frac {1}{18} \, {\left (x^{2} + 5\right )}^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2+5)^8,x, algorithm="giac")

[Out]

1/18*(x^2 + 5)^9

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maple [B]  time = 0.00, size = 47, normalized size = 4.27 \[ \frac {1}{18} x^{18}+\frac {5}{2} x^{16}+50 x^{14}+\frac {1750}{3} x^{12}+4375 x^{10}+21875 x^{8}+\frac {218750}{3} x^{6}+156250 x^{4}+\frac {390625}{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^2+5)^8,x)

[Out]

1/18*x^18+5/2*x^16+50*x^14+1750/3*x^12+4375*x^10+21875*x^8+218750/3*x^6+156250*x^4+390625/2*x^2

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maxima [A]  time = 0.58, size = 9, normalized size = 0.82 \[ \frac {1}{18} \, {\left (x^{2} + 5\right )}^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x^2+5)^8,x, algorithm="maxima")

[Out]

1/18*(x^2 + 5)^9

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mupad [B]  time = 0.21, size = 9, normalized size = 0.82 \[ \frac {{\left (x^2+5\right )}^9}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^2 + 5)^8,x)

[Out]

(x^2 + 5)^9/18

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sympy [B]  time = 0.06, size = 51, normalized size = 4.64 \[ \frac {x^{18}}{18} + \frac {5 x^{16}}{2} + 50 x^{14} + \frac {1750 x^{12}}{3} + 4375 x^{10} + 21875 x^{8} + \frac {218750 x^{6}}{3} + 156250 x^{4} + \frac {390625 x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(x**2+5)**8,x)

[Out]

x**18/18 + 5*x**16/2 + 50*x**14 + 1750*x**12/3 + 4375*x**10 + 21875*x**8 + 218750*x**6/3 + 156250*x**4 + 39062
5*x**2/2

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