\(\int \frac {-4+x^2}{2+x} \, dx\) [466]

Optimal result

Integrand size = 11, antiderivative size = 11 \[ \int \frac {-4+x^2}{2+x} \, dx=-2 x+\frac {x^2}{2} \]

[Out]

-2*x+1/2*x^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, 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 = {641} \[ \int \frac {-4+x^2}{2+x} \, dx=\frac {x^2}{2}-2 x \]

[In]

Int[(-4 + x^2)/(2 + x),x]

[Out]

-2*x + x^2/2

Rule 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rubi steps \begin{align*} \text {integral}& = \int (-2+x) \, dx \\ & = -2 x+\frac {x^2}{2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-4+x^2}{2+x} \, dx=-2 x+\frac {x^2}{2} \]

[In]

Integrate[(-4 + x^2)/(2 + x),x]

[Out]

-2*x + x^2/2

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64

method	result	size
gosper	\(\frac {x \left (x -4\right )}{2}\)	\(7\)
meijerg	\(-\frac {x \left (-\frac {3 x}{2}+6\right )}{3}\)	\(9\)
default	\(-2 x +\frac {1}{2} x^{2}\)	\(10\)
norman	\(-2 x +\frac {1}{2} x^{2}\)	\(10\)
risch	\(-2 x +\frac {1}{2} x^{2}\)	\(10\)
parallelrisch	\(-2 x +\frac {1}{2} x^{2}\)	\(10\)
parts	\(-2 x +\frac {1}{2} x^{2}\)	\(10\)

[In]

int((x^2-4)/(x+2),x,method=_RETURNVERBOSE)

[Out]

1/2*x*(x-4)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {-4+x^2}{2+x} \, dx=\frac {1}{2} \, x^{2} - 2 \, x \]

[In]

integrate((x^2-4)/(2+x),x, algorithm="fricas")

[Out]

1/2*x^2 - 2*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64 \[ \int \frac {-4+x^2}{2+x} \, dx=\frac {x^{2}}{2} - 2 x \]

[In]

integrate((x**2-4)/(2+x),x)

[Out]

x**2/2 - 2*x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {-4+x^2}{2+x} \, dx=\frac {1}{2} \, x^{2} - 2 \, x \]

[In]

integrate((x^2-4)/(2+x),x, algorithm="maxima")

[Out]

1/2*x^2 - 2*x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {-4+x^2}{2+x} \, dx=\frac {1}{2} \, x^{2} - 2 \, x \]

[In]

integrate((x^2-4)/(2+x),x, algorithm="giac")

[Out]

1/2*x^2 - 2*x

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int \frac {-4+x^2}{2+x} \, dx=\frac {x\,\left (x-4\right )}{2} \]

[In]

int((x^2 - 4)/(x + 2),x)

[Out]

(x*(x - 4))/2