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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 13, 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} \[ \frac {a x^2}{2}+b \log (x) \]

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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ \frac {a x^2}{2}+b \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)*x,x]

[Out]

(a*x^2)/2 + b*Log[x]

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fricas [A]  time = 0.82, size = 11, normalized size = 0.85 \[ \frac {1}{2} \, a x^{2} + b \log \relax (x) \]

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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giac [A]  time = 0.17, size = 12, normalized size = 0.92 \[ \frac {1}{2} \, a x^{2} + b \log \left ({\left | x \right |}\right ) \]

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))

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maple [A]  time = 0.00, size = 12, normalized size = 0.92 \[ \frac {a \,x^{2}}{2}+b \ln \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.00, size = 14, normalized size = 1.08 \[ \frac {1}{2} \, a x^{2} + \frac {1}{2} \, b \log \left (x^{2}\right ) \]

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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mupad [B]  time = 0.02, size = 11, normalized size = 0.85 \[ \frac {a\,x^2}{2}+b\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b/x^2),x)

[Out]

(a*x^2)/2 + b*log(x)

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sympy [A]  time = 0.09, size = 10, normalized size = 0.77 \[ \frac {a x^{2}}{2} + b \log {\relax (x )} \]

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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