3.6.52 \(\int \frac {-101+e^2}{3 x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac {-x+\frac {1}{3} \left (5-e^2+24 (4-x)+x\right )}{x} \]

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 0.52, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 30} \begin {gather*} \frac {101-e^2}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-101 + E^2)/(3*x^2),x]

[Out]

(101 - E^2)/(3*x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \left (-101+e^2\right ) \int \frac {1}{x^2} \, dx\\ &=\frac {101-e^2}{3 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.44 \begin {gather*} -\frac {-101+e^2}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-101 + E^2)/(3*x^2),x]

[Out]

-1/3*(-101 + E^2)/x

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fricas [A]  time = 0.72, size = 9, normalized size = 0.33 \begin {gather*} -\frac {e^{2} - 101}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(exp(2)-101)/x^2,x, algorithm="fricas")

[Out]

-1/3*(e^2 - 101)/x

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giac [A]  time = 0.51, size = 9, normalized size = 0.33 \begin {gather*} -\frac {e^{2} - 101}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(exp(2)-101)/x^2,x, algorithm="giac")

[Out]

-1/3*(e^2 - 101)/x

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maple [A]  time = 0.02, size = 10, normalized size = 0.37

method	result	size
gosper	\(-\frac {{\mathrm e}^{2}-101}{3 x}\)	\(10\)
norman	\(\frac {-\frac {{\mathrm e}^{2}}{3}+\frac {101}{3}}{x}\)	\(11\)
default	\(-\frac {\frac {{\mathrm e}^{2}}{3}-\frac {101}{3}}{x}\)	\(12\)
risch	\(-\frac {{\mathrm e}^{2}}{3 x}+\frac {101}{3 x}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(exp(2)-101)/x^2,x,method=_RETURNVERBOSE)

[Out]

-1/3/x*(exp(2)-101)

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maxima [A]  time = 0.40, size = 9, normalized size = 0.33 \begin {gather*} -\frac {e^{2} - 101}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(exp(2)-101)/x^2,x, algorithm="maxima")

[Out]

-1/3*(e^2 - 101)/x

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mupad [B]  time = 0.03, size = 11, normalized size = 0.41 \begin {gather*} -\frac {\frac {{\mathrm {e}}^2}{3}-\frac {101}{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2)/3 - 101/3)/x^2,x)

[Out]

-(exp(2)/3 - 101/3)/x

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sympy [A]  time = 0.05, size = 10, normalized size = 0.37 \begin {gather*} - \frac {- \frac {101}{3} + \frac {e^{2}}{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(exp(2)-101)/x**2,x)

[Out]

-(-101/3 + exp(2)/3)/x

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