3.46.40 \(\int \frac {8-4 x^2}{5 x^2} \, dx\)

Optimal. Leaf size=23 \[ e^2+\frac {4}{5} \left (2-e^5-\frac {2}{x}-x\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 0.57, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 14} \begin {gather*} -\frac {4 x}{5}-\frac {8}{5 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8 - 4*x^2)/(5*x^2),x]

[Out]

-8/(5*x) - (4*x)/5

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {8-4 x^2}{x^2} \, dx\\ &=\frac {1}{5} \int \left (-4+\frac {8}{x^2}\right ) \, dx\\ &=-\frac {8}{5 x}-\frac {4 x}{5}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 13, normalized size = 0.57 \begin {gather*} -\frac {8}{5 x}-\frac {4 x}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 - 4*x^2)/(5*x^2),x]

[Out]

-8/(5*x) - (4*x)/5

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 10, normalized size = 0.43 \begin {gather*} -\frac {4 \, {\left (x^{2} + 2\right )}}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-4*x^2+8)/x^2,x, algorithm="fricas")

[Out]

-4/5*(x^2 + 2)/x

________________________________________________________________________________________

giac [A]  time = 0.12, size = 9, normalized size = 0.39 \begin {gather*} -\frac {4}{5} \, x - \frac {8}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-4*x^2+8)/x^2,x, algorithm="giac")

[Out]

-4/5*x - 8/5/x

________________________________________________________________________________________

maple [A]  time = 0.02, size = 10, normalized size = 0.43

method	result	size
default	\(-\frac {4 x}{5}-\frac {8}{5 x}\)	\(10\)
risch	\(-\frac {4 x}{5}-\frac {8}{5 x}\)	\(10\)
gosper	\(-\frac {4 \left (x^{2}+2\right )}{5 x}\)	\(11\)
norman	\(\frac {-\frac {8}{5}-\frac {4 x^{2}}{5}}{x}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(-4*x^2+8)/x^2,x,method=_RETURNVERBOSE)

[Out]

-4/5*x-8/5/x

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 9, normalized size = 0.39 \begin {gather*} -\frac {4}{5} \, x - \frac {8}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-4*x^2+8)/x^2,x, algorithm="maxima")

[Out]

-4/5*x - 8/5/x

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 10, normalized size = 0.43 \begin {gather*} -\frac {4\,\left (x^2+2\right )}{5\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((4*x^2)/5 - 8/5)/x^2,x)

[Out]

-(4*(x^2 + 2))/(5*x)

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 10, normalized size = 0.43 \begin {gather*} - \frac {4 x}{5} - \frac {8}{5 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-4*x**2+8)/x**2,x)

[Out]

-4*x/5 - 8/(5*x)

________________________________________________________________________________________