\(\int \frac {a+\frac {b}{x^2}}{x^2} \, dx\) [1812]

Optimal result

Integrand size = 11, antiderivative size = 15 \[ \int \frac {a+\frac {b}{x^2}}{x^2} \, dx=-\frac {b}{3 x^3}-\frac {a}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \[ \int \frac {a+\frac {b}{x^2}}{x^2} \, dx=-\frac {a}{x}-\frac {b}{3 x^3} \]

[In]

Int[(a + b/x^2)/x^2,x]

[Out]

-1/3*b/x^3 - a/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*} \text {integral}& = \int \left (\frac {b}{x^4}+\frac {a}{x^2}\right ) \, dx \\ & = -\frac {b}{3 x^3}-\frac {a}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {a+\frac {b}{x^2}}{x^2} \, dx=-\frac {b}{3 x^3}-\frac {a}{x} \]

[In]

Integrate[(a + b/x^2)/x^2,x]

[Out]

-1/3*b/x^3 - a/x

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method	result	size
gosper	\(-\frac {3 a \,x^{2}+b}{3 x^{3}}\)	\(14\)
default	\(-\frac {b}{3 x^{3}}-\frac {a}{x}\)	\(14\)
norman	\(\frac {-a \,x^{2}-\frac {b}{3}}{x^{3}}\)	\(15\)
risch	\(\frac {-a \,x^{2}-\frac {b}{3}}{x^{3}}\)	\(15\)
parallelrisch	\(\frac {-3 a \,x^{2}-b}{3 x^{3}}\)	\(16\)

[In]

int((a+b/x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

-1/3*(3*a*x^2+b)/x^3

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+\frac {b}{x^2}}{x^2} \, dx=-\frac {3 \, a x^{2} + b}{3 \, x^{3}} \]

[In]

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

[Out]

-1/3*(3*a*x^2 + b)/x^3

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {a+\frac {b}{x^2}}{x^2} \, dx=\frac {- 3 a x^{2} - b}{3 x^{3}} \]

[In]

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

[Out]

(-3*a*x**2 - b)/(3*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+\frac {b}{x^2}}{x^2} \, dx=-\frac {3 \, a x^{2} + b}{3 \, x^{3}} \]

[In]

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

[Out]

-1/3*(3*a*x^2 + b)/x^3

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+\frac {b}{x^2}}{x^2} \, dx=-\frac {3 \, a x^{2} + b}{3 \, x^{3}} \]

[In]

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

[Out]

-1/3*(3*a*x^2 + b)/x^3

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+\frac {b}{x^2}}{x^2} \, dx=-\frac {3\,a\,x^2+b}{3\,x^3} \]

[In]

int((a + b/x^2)/x^2,x)

[Out]

-(b + 3*a*x^2)/(3*x^3)