\(\int \frac {-1+5 x^2}{x^2} \, dx\) [9200]

Optimal result

Integrand size = 11, antiderivative size = 13 \[ \int \frac {-1+5 x^2}{x^2} \, dx=-5-\frac {2}{e^4}+\frac {1}{x}+5 x \]

[Out]

1/x+5*x-5-2/exp(4)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.54, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \[ \int \frac {-1+5 x^2}{x^2} \, dx=5 x+\frac {1}{x} \]

[In]

Int[(-1 + 5*x^2)/x^2,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.54 \[ \int \frac {-1+5 x^2}{x^2} \, dx=\frac {1}{x}+5 x \]

[In]

Integrate[(-1 + 5*x^2)/x^2,x]

[Out]

x^(-1) + 5*x

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62

method	result	size
default	\(5 x +\frac {1}{x}\)	\(8\)
risch	\(5 x +\frac {1}{x}\)	\(8\)
gosper	\(\frac {5 x^{2}+1}{x}\)	\(12\)
norman	\(\frac {5 x^{2}+1}{x}\)	\(12\)
parallelrisch	\(\frac {5 x^{2}+1}{x}\)	\(12\)

[In]

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

[Out]

5*x+1/x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {-1+5 x^2}{x^2} \, dx=\frac {5 \, x^{2} + 1}{x} \]

[In]

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

[Out]

(5*x^2 + 1)/x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.38 \[ \int \frac {-1+5 x^2}{x^2} \, dx=5 x + \frac {1}{x} \]

[In]

integrate((5*x**2-1)/x**2,x)

[Out]

5*x + 1/x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.54 \[ \int \frac {-1+5 x^2}{x^2} \, dx=5 \, x + \frac {1}{x} \]

[In]

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

[Out]

5*x + 1/x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.54 \[ \int \frac {-1+5 x^2}{x^2} \, dx=5 \, x + \frac {1}{x} \]

[In]

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

[Out]

5*x + 1/x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.54 \[ \int \frac {-1+5 x^2}{x^2} \, dx=5\,x+\frac {1}{x} \]

[In]

int((5*x^2 - 1)/x^2,x)

[Out]

5*x + 1/x