3.411 \(\int x (a^2+2 a b x^2+b^2 x^4) \, dx\)

Optimal. Leaf size=30 \[ \frac{a^2 x^2}{2}+\frac{1}{2} a b x^4+\frac{b^2 x^6}{6} \]

[Out]

(a^2*x^2)/2 + (a*b*x^4)/2 + (b^2*x^6)/6

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Rubi [A]  time = 0.0094222, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {14} \[ \frac{a^2 x^2}{2}+\frac{1}{2} a b x^4+\frac{b^2 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Int[x*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(a^2*x^2)/2 + (a*b*x^4)/2 + (b^2*x^6)/6

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int x \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx &=\int \left (a^2 x+2 a b x^3+b^2 x^5\right ) \, dx\\ &=\frac{a^2 x^2}{2}+\frac{1}{2} a b x^4+\frac{b^2 x^6}{6}\\ \end{align*}

Mathematica [A]  time = 0.0018107, size = 16, normalized size = 0.53 \[ \frac{\left (a+b x^2\right )^3}{6 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(a + b*x^2)^3/(6*b)

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Maple [A]  time = 0.041, size = 25, normalized size = 0.8 \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^2*x^4+2*a*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 = 0.992762, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{6} \, b^{2} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.2158, size = 55, normalized size = 1.83 \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^2*x^4+2*a*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 [A]  time = 0.061395, size = 24, normalized size = 0.8 \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**2*x**4+2*a*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 = 1.18048, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{6} \, b^{2} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="giac")

[Out]

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