\(\int (a+\frac {b}{x}) x \, dx\) [1551]

Optimal result

Integrand size = 9, antiderivative size = 12 \[ \int \left (a+\frac {b}{x}\right ) x \, dx=b x+\frac {a x^2}{2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {14} \[ \int \left (a+\frac {b}{x}\right ) x \, dx=\frac {a x^2}{2}+b x \]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x}\right ) x \, dx=b x+\frac {a x^2}{2} \]

[In]

Integrate[(a + b/x)*x,x]

[Out]

b*x + (a*x^2)/2

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92

method	result	size
gosper	\(\frac {x \left (a x +2 b \right )}{2}\)	\(11\)
default	\(b x +\frac {1}{2} a \,x^{2}\)	\(11\)
norman	\(b x +\frac {1}{2} a \,x^{2}\)	\(11\)
risch	\(b x +\frac {1}{2} a \,x^{2}\)	\(11\)
parallelrisch	\(b x +\frac {1}{2} a \,x^{2}\)	\(11\)
parts	\(b x +\frac {1}{2} a \,x^{2}\)	\(11\)

[In]

int((a+b/x)*x,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \left (a+\frac {b}{x}\right ) x \, dx=\frac {1}{2} \, a x^{2} + b x \]

[In]

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

[Out]

1/2*a*x^2 + b*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \left (a+\frac {b}{x}\right ) x \, dx=\frac {a x^{2}}{2} + b x \]

[In]

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

[Out]

a*x**2/2 + b*x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \left (a+\frac {b}{x}\right ) x \, dx=\frac {1}{2} \, a x^{2} + b x \]

[In]

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

[Out]

1/2*a*x^2 + b*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \left (a+\frac {b}{x}\right ) x \, dx=\frac {1}{2} \, a x^{2} + b x \]

[In]

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

[Out]

1/2*a*x^2 + b*x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \left (a+\frac {b}{x}\right ) x \, dx=\frac {a\,x^2}{2}+b\,x \]

[In]

int(x*(a + b/x),x)

[Out]

b*x + (a*x^2)/2