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

Optimal. Leaf size=17 \[ \frac{a x^2}{2}+\frac{b x^4}{4} \]

[Out]

(a*x^2)/2 + (b*x^4)/4

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Rubi [A]  time = 0.0050435, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {14} \[ \frac{a x^2}{2}+\frac{b x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2),x]

[Out]

(a*x^2)/2 + (b*x^4)/4

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

Mathematica [A]  time = 0.0009521, size = 17, normalized size = 1. \[ \frac{a x^2}{2}+\frac{b x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2),x]

[Out]

(a*x^2)/2 + (b*x^4)/4

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Maple [A]  time = 0.001, size = 14, normalized size = 0.8 \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{b{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.31886, size = 19, normalized size = 1.12 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{2}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b*x^2 + a)^2/b

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Fricas [A]  time = 1.24602, size = 31, normalized size = 1.82 \begin{align*} \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*(b*x^2+a),x, algorithm="fricas")

[Out]

1/4*x^4*b + 1/2*x^2*a

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Sympy [A]  time = 0.054487, size = 12, normalized size = 0.71 \begin{align*} \frac{a x^{2}}{2} + \frac{b x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**2/2 + b*x**4/4

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Giac [A]  time = 2.10797, size = 18, normalized size = 1.06 \begin{align*} \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*(b*x^2+a),x, algorithm="giac")

[Out]

1/4*b*x^4 + 1/2*a*x^2