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

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

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {a x^2}{2}+\frac {b x^4}{4} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {a x^2}{2}+\frac {b x^4}{4} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a+b x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x*(a + b*x^2),x]

[Out]

IntegrateAlgebraic[x*(a + b*x^2), x]

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fricas [A]  time = 0.99, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} x^{4} b + \frac {1}{2} x^{2} a \end {gather*}

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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giac [A]  time = 1.03, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, b x^{4} + \frac {1}{2} \, a x^{2} \end {gather*}

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

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \begin {gather*} \frac {1}{4} b \,x^{4}+\frac {1}{2} a \,x^{2} \end {gather*}

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 = 1.34, size = 14, normalized size = 0.82 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{2}}{4 \, b} \end {gather*}

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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mupad [B]  time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {b\,x^4}{4}+\frac {a\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^2),x)

[Out]

(a*x^2)/2 + (b*x^4)/4

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sympy [A]  time = 0.06, size = 12, normalized size = 0.71 \begin {gather*} \frac {a x^{2}}{2} + \frac {b x^{4}}{4} \end {gather*}

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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