3.10.94 \(\int \frac {e+4 x^2}{x^2} \, dx\)

Optimal. Leaf size=20 \[ 8 x+\frac {-e+3 x-4 x^2}{x} \]

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 0.50, 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*} 4 x-\frac {e}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E + 4*x^2)/x^2,x]

[Out]

-(E/x) + 4*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 (4+\frac {e}{x^2}\right ) \, dx\\ &=-\frac {e}{x}+4 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.50 \begin {gather*} -\frac {e}{x}+4 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E + 4*x^2)/x^2,x]

[Out]

-(E/x) + 4*x

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fricas [A]  time = 0.52, size = 14, normalized size = 0.70 \begin {gather*} \frac {4 \, x^{2} - e}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*x^2 - e)/x

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giac [A]  time = 0.20, size = 11, normalized size = 0.55 \begin {gather*} 4 \, x - \frac {e}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x - e/x

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maple [A]  time = 0.02, size = 12, normalized size = 0.60

method	result	size
default	\(-\frac {{\mathrm e}}{x}+4 x\)	\(12\)
risch	\(-\frac {{\mathrm e}}{x}+4 x\)	\(12\)
gosper	\(-\frac {-4 x^{2}+{\mathrm e}}{x}\)	\(14\)
norman	\(\frac {4 x^{2}-{\mathrm e}}{x}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1)+4*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

-exp(1)/x+4*x

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maxima [A]  time = 0.51, size = 11, normalized size = 0.55 \begin {gather*} 4 \, x - \frac {e}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x - e/x

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mupad [B]  time = 0.03, size = 11, normalized size = 0.55 \begin {gather*} 4\,x-\frac {\mathrm {e}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1) + 4*x^2)/x^2,x)

[Out]

4*x - exp(1)/x

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sympy [A]  time = 0.07, size = 7, normalized size = 0.35 \begin {gather*} 4 x - \frac {e}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(1)+4*x**2)/x**2,x)

[Out]

4*x - E/x

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