3.91 \(\int x (a+b x^2)^8 \, dx\)

Optimal. Leaf size=16 \[ \frac{\left (a+b x^2\right )^9}{18 b} \]

[Out]

(a + b*x^2)^9/(18*b)

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Rubi [A]  time = 0.0023727, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {261} \[ \frac{\left (a+b x^2\right )^9}{18 b} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^8,x]

[Out]

(a + b*x^2)^9/(18*b)

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

Mathematica [A]  time = 0.0018185, size = 16, normalized size = 1. \[ \frac{\left (a+b x^2\right )^9}{18 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^8,x]

[Out]

(a + b*x^2)^9/(18*b)

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Maple [B]  time = 0., size = 91, normalized size = 5.7 \begin{align*}{\frac{{b}^{8}{x}^{18}}{18}}+{\frac{a{b}^{7}{x}^{16}}{2}}+2\,{a}^{2}{b}^{6}{x}^{14}+{\frac{14\,{a}^{3}{b}^{5}{x}^{12}}{3}}+7\,{a}^{4}{b}^{4}{x}^{10}+7\,{a}^{5}{b}^{3}{x}^{8}+{\frac{14\,{a}^{6}{b}^{2}{x}^{6}}{3}}+2\,{a}^{7}b{x}^{4}+{\frac{{a}^{8}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^2+a)^8,x)

[Out]

1/18*b^8*x^18+1/2*a*b^7*x^16+2*a^2*b^6*x^14+14/3*a^3*b^5*x^12+7*a^4*b^4*x^10+7*a^5*b^3*x^8+14/3*a^6*b^2*x^6+2*
a^7*b*x^4+1/2*a^8*x^2

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Maxima [A]  time = 2.53358, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{9}}{18 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(b*x^2 + a)^9/b

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Fricas [B]  time = 1.32205, size = 201, normalized size = 12.56 \begin{align*} \frac{1}{18} x^{18} b^{8} + \frac{1}{2} x^{16} b^{7} a + 2 x^{14} b^{6} a^{2} + \frac{14}{3} x^{12} b^{5} a^{3} + 7 x^{10} b^{4} a^{4} + 7 x^{8} b^{3} a^{5} + \frac{14}{3} x^{6} b^{2} a^{6} + 2 x^{4} b a^{7} + \frac{1}{2} x^{2} a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*x^18*b^8 + 1/2*x^16*b^7*a + 2*x^14*b^6*a^2 + 14/3*x^12*b^5*a^3 + 7*x^10*b^4*a^4 + 7*x^8*b^3*a^5 + 14/3*x^
6*b^2*a^6 + 2*x^4*b*a^7 + 1/2*x^2*a^8

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Sympy [B]  time = 0.078476, size = 99, normalized size = 6.19 \begin{align*} \frac{a^{8} x^{2}}{2} + 2 a^{7} b x^{4} + \frac{14 a^{6} b^{2} x^{6}}{3} + 7 a^{5} b^{3} x^{8} + 7 a^{4} b^{4} x^{10} + \frac{14 a^{3} b^{5} x^{12}}{3} + 2 a^{2} b^{6} x^{14} + \frac{a b^{7} x^{16}}{2} + \frac{b^{8} x^{18}}{18} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**2+a)**8,x)

[Out]

a**8*x**2/2 + 2*a**7*b*x**4 + 14*a**6*b**2*x**6/3 + 7*a**5*b**3*x**8 + 7*a**4*b**4*x**10 + 14*a**3*b**5*x**12/
3 + 2*a**2*b**6*x**14 + a*b**7*x**16/2 + b**8*x**18/18

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Giac [A]  time = 1.67879, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{9}}{18 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(b*x^2 + a)^9/b