\(\int \frac {-3+e^5+x^2}{x^2} \, dx\) [613]

Optimal result

Integrand size = 12, antiderivative size = 13 \[ \int \frac {-3+e^5+x^2}{x^2} \, dx=\frac {3-e^5}{x}+x \]

[Out]

(3-exp(5))/x+x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {14} \[ \int \frac {-3+e^5+x^2}{x^2} \, dx=x+\frac {3-e^5}{x} \]

[In]

Int[(-3 + E^5 + x^2)/x^2,x]

[Out]

(3 - E^5)/x + 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 (1+\frac {-3+e^5}{x^2}\right ) \, dx \\ & = \frac {3-e^5}{x}+x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {-3+e^5+x^2}{x^2} \, dx=\frac {3-e^5}{x}+x \]

[In]

Integrate[(-3 + E^5 + x^2)/x^2,x]

[Out]

(3 - E^5)/x + x

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

method	result	size
default	\(x -\frac {{\mathrm e}^{5}-3}{x}\)	\(12\)
norman	\(\frac {x^{2}+3-{\mathrm e}^{5}}{x}\)	\(14\)
gosper	\(-\frac {-x^{2}+{\mathrm e}^{5}-3}{x}\)	\(15\)
risch	\(\frac {3}{x}-\frac {{\mathrm e}^{5}}{x}+x\)	\(15\)
parallelrisch	\(-\frac {-x^{2}+{\mathrm e}^{5}-3}{x}\)	\(15\)

[In]

int((exp(5)+x^2-3)/x^2,x,method=_RETURNVERBOSE)

[Out]

x-(exp(5)-3)/x

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {-3+e^5+x^2}{x^2} \, dx=\frac {x^{2} - e^{5} + 3}{x} \]

[In]

integrate((exp(5)+x^2-3)/x^2,x, algorithm="fricas")

[Out]

(x^2 - e^5 + 3)/x

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.54 \[ \int \frac {-3+e^5+x^2}{x^2} \, dx=x + \frac {3 - e^{5}}{x} \]

[In]

integrate((exp(5)+x**2-3)/x**2,x)

[Out]

x + (3 - exp(5))/x

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {-3+e^5+x^2}{x^2} \, dx=x - \frac {e^{5} - 3}{x} \]

[In]

integrate((exp(5)+x^2-3)/x^2,x, algorithm="maxima")

[Out]

x - (e^5 - 3)/x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {-3+e^5+x^2}{x^2} \, dx=x - \frac {e^{5} - 3}{x} \]

[In]

integrate((exp(5)+x^2-3)/x^2,x, algorithm="giac")

[Out]

x - (e^5 - 3)/x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {-3+e^5+x^2}{x^2} \, dx=x-\frac {{\mathrm {e}}^5-3}{x} \]

[In]

int((exp(5) + x^2 - 3)/x^2,x)

[Out]

x - (exp(5) - 3)/x