3.93.24 \(\int \frac {-10+2 e^4+2 x^4}{x^3} \, dx\)

Optimal. Leaf size=14 \[ \frac {5-e^4+x^4}{x^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14} \begin {gather*} x^2+\frac {5-e^4}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-10 + 2*E^4 + 2*x^4)/x^3,x]

[Out]

(5 - E^4)/x^2 + x^2

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 (\frac {2 \left (-5+e^4\right )}{x^3}+2 x\right ) \, dx\\ &=\frac {5-e^4}{x^2}+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {5-e^4+x^4}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10 + 2*E^4 + 2*x^4)/x^3,x]

[Out]

(5 - E^4 + x^4)/x^2

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fricas [A]  time = 0.91, size = 13, normalized size = 0.93 \begin {gather*} \frac {x^{4} - e^{4} + 5}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(4)+2*x^4-10)/x^3,x, algorithm="fricas")

[Out]

(x^4 - e^4 + 5)/x^2

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giac [A]  time = 0.22, size = 13, normalized size = 0.93 \begin {gather*} x^{2} - \frac {e^{4} - 5}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(4)+2*x^4-10)/x^3,x, algorithm="giac")

[Out]

x^2 - (e^4 - 5)/x^2

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maple [A]  time = 0.03, size = 14, normalized size = 1.00

method	result	size
default	\(x^{2}-\frac {{\mathrm e}^{4}-5}{x^{2}}\)	\(14\)
norman	\(\frac {5+x^{4}-{\mathrm e}^{4}}{x^{2}}\)	\(14\)
gosper	\(-\frac {-x^{4}+{\mathrm e}^{4}-5}{x^{2}}\)	\(15\)
risch	\(x^{2}-\frac {{\mathrm e}^{4}}{x^{2}}+\frac {5}{x^{2}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(4)+2*x^4-10)/x^3,x,method=_RETURNVERBOSE)

[Out]

x^2-(exp(4)-5)/x^2

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maxima [A]  time = 0.35, size = 13, normalized size = 0.93 \begin {gather*} x^{2} - \frac {e^{4} - 5}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(4)+2*x^4-10)/x^3,x, algorithm="maxima")

[Out]

x^2 - (e^4 - 5)/x^2

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mupad [B]  time = 7.45, size = 13, normalized size = 0.93 \begin {gather*} x^2-\frac {{\mathrm {e}}^4-5}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(4) + 2*x^4 - 10)/x^3,x)

[Out]

x^2 - (exp(4) - 5)/x^2

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sympy [A]  time = 0.10, size = 10, normalized size = 0.71 \begin {gather*} x^{2} + \frac {5 - e^{4}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(4)+2*x**4-10)/x**3,x)

[Out]

x**2 + (5 - exp(4))/x**2

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