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

Optimal. Leaf size=16 \[ \frac{\left (a+b x^2\right )^3}{6 b} \]

[Out]

(a + b*x^2)^3/(6*b)

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Rubi [A]  time = 0.0022917, 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} \[ \frac{\left (a+b x^2\right )^3}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^2,x]

[Out]

(a + b*x^2)^3/(6*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 )^2 \, dx &=\frac{\left (a+b x^2\right )^3}{6 b}\\ \end{align*}

Mathematica [A]  time = 0.0017881, size = 16, normalized size = 1. \[ \frac{\left (a+b x^2\right )^3}{6 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^2,x]

[Out]

(a + b*x^2)^3/(6*b)

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Maple [A]  time = 0., size = 25, normalized size = 1.6 \begin{align*}{\frac{{b}^{2}{x}^{6}}{6}}+{\frac{ab{x}^{4}}{2}}+{\frac{{a}^{2}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*x^6+1/2*a*b*x^4+1/2*a^2*x^2

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Maxima [A]  time = 2.24358, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{3}}{6 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b*x^2 + a)^3/b

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Fricas [A]  time = 1.24651, size = 55, normalized size = 3.44 \begin{align*} \frac{1}{6} x^{6} b^{2} + \frac{1}{2} x^{4} b a + \frac{1}{2} x^{2} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*b^2 + 1/2*x^4*b*a + 1/2*x^2*a^2

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Sympy [B]  time = 0.059924, size = 24, normalized size = 1.5 \begin{align*} \frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**2/2 + a*b*x**4/2 + b**2*x**6/6

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Giac [A]  time = 2.27716, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{3}}{6 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b*x^2 + a)^3/b