3.1808 \(\int (a+\frac{b}{x^2}) x^2 \, dx\)

Optimal. Leaf size=12 \[ \frac{a x^3}{3}+b x \]

[Out]

b*x + (a*x^3)/3

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Rubi [A]  time = 0.0034542, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {14} \[ \frac{a x^3}{3}+b x \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)*x^2,x]

[Out]

b*x + (a*x^3)/3

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

Mathematica [A]  time = 0.0004023, size = 12, normalized size = 1. \[ \frac{a x^3}{3}+b x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)*x^2,x]

[Out]

b*x + (a*x^3)/3

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Maple [A]  time = 0.001, size = 11, normalized size = 0.9 \begin{align*} bx+{\frac{a{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x^2*b)*x^2,x)

[Out]

b*x+1/3*a*x^3

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Maxima [A]  time = 0.970803, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{3} \, a x^{3} + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3 + b*x

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Fricas [A]  time = 1.41487, size = 23, normalized size = 1.92 \begin{align*} \frac{1}{3} \, a x^{3} + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3 + b*x

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Sympy [A]  time = 0.052588, size = 8, normalized size = 0.67 \begin{align*} \frac{a x^{3}}{3} + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**3/3 + b*x

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Giac [A]  time = 1.16601, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{3} \, a x^{3} + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3 + b*x