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

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

[Out]

(5 + x^2)^9/18

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Rubi [A]  time = 0.0014835, antiderivative size = 11, 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} $\frac{1}{18} \left (x^2+5\right )^9$

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*}

Mathematica [A]  time = 0.0019411, size = 11, normalized size = 1. $\frac{1}{18} \left (x^2+5\right )^9$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*(5 + x^2)^8,x]

[Out]

(5 + x^2)^9/18

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Maple [B]  time = 0., size = 47, normalized size = 4.3 \begin{align*}{\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}} \end{align*}

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 = 0.931201, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{18} \,{\left (x^{2} + 5\right )}^{9} \end{align*}

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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Fricas [B]  time = 1.57474, size = 153, normalized size = 13.91 \begin{align*} \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} \end{align*}

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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Sympy [B]  time = 0.055313, size = 51, normalized size = 4.64 \begin{align*} \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} \end{align*}

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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Giac [A]  time = 1.05683, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{18} \,{\left (x^{2} + 5\right )}^{9} \end{align*}

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