3.1809 \(\int (a+\frac{b}{x^2}) x \, dx\)

Optimal. Leaf size=13 \[ \frac{a x^2}{2}+b \log (x) \]

[Out]

(a*x^2)/2 + b*Log[x]

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Rubi [A]  time = 0.0034573, antiderivative size = 13, 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} \[ \frac{a x^2}{2}+b \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)*x,x]

[Out]

(a*x^2)/2 + b*Log[x]

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

Mathematica [A]  time = 0.0026699, size = 13, normalized size = 1. \[ \frac{a x^2}{2}+b \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)*x,x]

[Out]

(a*x^2)/2 + b*Log[x]

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Maple [A]  time = 0.002, size = 12, normalized size = 0.9 \begin{align*}{\frac{a{x}^{2}}{2}}+b\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2+b*ln(x)

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Maxima [A]  time = 0.97708, size = 19, normalized size = 1.46 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{2} \, b \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2 + 1/2*b*log(x^2)

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Fricas [A]  time = 1.42743, size = 30, normalized size = 2.31 \begin{align*} \frac{1}{2} \, a x^{2} + b \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2 + b*log(x)

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Sympy [A]  time = 0.080173, size = 10, normalized size = 0.77 \begin{align*} \frac{a x^{2}}{2} + b \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**2/2 + b*log(x)

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Giac [A]  time = 1.10239, size = 16, normalized size = 1.23 \begin{align*} \frac{1}{2} \, a x^{2} + b \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2 + b*log(abs(x))