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

Optimal. Leaf size=25 \[ \frac{a x^2}{2}+\frac{b x^4}{4}+\frac{c x^6}{6} \]

[Out]

(a*x^2)/2 + (b*x^4)/4 + (c*x^6)/6

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Rubi [A]  time = 0.0071979, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {14} \[ \frac{a x^2}{2}+\frac{b x^4}{4}+\frac{c x^6}{6} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2 + c*x^4),x]

[Out]

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

Mathematica [A]  time = 0.0012317, size = 25, normalized size = 1. \[ \frac{a x^2}{2}+\frac{b x^4}{4}+\frac{c x^6}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2 + c*x^4),x]

[Out]

(a*x^2)/2 + (b*x^4)/4 + (c*x^6)/6

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Maple [A]  time = 0.041, size = 20, normalized size = 0.8 \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{b{x}^{4}}{4}}+{\frac{c{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c*x^4+b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 0.964554, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{6} \, c x^{6} + \frac{1}{4} \, b x^{4} + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c*x^6 + 1/4*b*x^4 + 1/2*a*x^2

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Fricas [A]  time = 1.27351, size = 47, normalized size = 1.88 \begin{align*} \frac{1}{6} x^{6} c + \frac{1}{4} x^{4} b + \frac{1}{2} x^{2} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*c + 1/4*x^4*b + 1/2*x^2*a

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Sympy [A]  time = 0.062213, size = 19, normalized size = 0.76 \begin{align*} \frac{a x^{2}}{2} + \frac{b x^{4}}{4} + \frac{c x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x**4+b*x**2+a),x)

[Out]

a*x**2/2 + b*x**4/4 + c*x**6/6

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Giac [A]  time = 1.22588, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{6} \, c x^{6} + \frac{1}{4} \, b x^{4} + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c*x^6 + 1/4*b*x^4 + 1/2*a*x^2