3.1551 \(\int (a+\frac{b}{x}) x \, dx\)

Optimal. Leaf size=12 \[ \frac{a x^2}{2}+b x \]

[Out]

b*x + (a*x^2)/2

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Rubi [A]  time = 0.0037697, antiderivative size = 12, 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} \[ \frac{a x^2}{2}+b x \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)*x,x]

[Out]

b*x + (a*x^2)/2

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

Mathematica [A]  time = 0.0004465, size = 12, normalized size = 1. \[ \frac{a x^2}{2}+b x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)*x,x]

[Out]

b*x + (a*x^2)/2

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Maple [A]  time = 0., size = 11, normalized size = 0.9 \begin{align*} bx+{\frac{a{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.952817, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{2} \, a x^{2} + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2 + b*x

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Fricas [A]  time = 1.38619, size = 23, normalized size = 1.92 \begin{align*} \frac{1}{2} \, a x^{2} + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2 + b*x

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Sympy [A]  time = 0.051414, size = 8, normalized size = 0.67 \begin{align*} \frac{a x^{2}}{2} + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**2/2 + b*x

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Giac [A]  time = 1.19271, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{2} \, a x^{2} + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2 + b*x