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

Optimal. Leaf size=13 \[ -5-\frac {2}{e^4}+\frac {1}{x}+5 x \]

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Rubi [A]  time = 0.00, antiderivative size = 7, normalized size of antiderivative = 0.54, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \begin {gather*} 5 x+\frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[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 {gather*} \begin {aligned} \text {integral} &=\int \left (5-\frac {1}{x^2}\right ) \, dx\\ &=\frac {1}{x}+5 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 7, normalized size = 0.54 \begin {gather*} \frac {1}{x}+5 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 5*x^2)/x^2,x]

[Out]

x^(-1) + 5*x

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fricas [A]  time = 0.65, size = 11, normalized size = 0.85 \begin {gather*} \frac {5 \, x^{2} + 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*x^2 + 1)/x

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giac [A]  time = 0.15, size = 7, normalized size = 0.54 \begin {gather*} 5 \, x + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x + 1/x

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maple [A]  time = 0.02, size = 8, normalized size = 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	\(\frac {5 x^{2}+1}{x}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x+1/x

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maxima [A]  time = 0.36, size = 7, normalized size = 0.54 \begin {gather*} 5 \, x + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x + 1/x

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mupad [B]  time = 0.03, size = 7, normalized size = 0.54 \begin {gather*} 5\,x+\frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^2 - 1)/x^2,x)

[Out]

5*x + 1/x

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sympy [A]  time = 0.11, size = 5, normalized size = 0.38 \begin {gather*} 5 x + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x + 1/x

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