3.90.95 \(\int \frac {-9-2 e^{\frac {1}{5} (2+e^3)} x+x^2+x^4+2 x^5}{x^4} \, dx\)

Optimal. Leaf size=30 \[ x+x^2-\frac {-e^{\frac {1}{5} \left (2+e^3\right )}-\frac {3}{x}+x}{x^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {14} \begin {gather*} \frac {3}{x^3}+x^2+\frac {e^{\frac {1}{5} \left (2+e^3\right )}}{x^2}+x-\frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 - 2*E^((2 + E^3)/5)*x + x^2 + x^4 + 2*x^5)/x^4,x]

[Out]

3/x^3 + E^((2 + E^3)/5)/x^2 - x^(-1) + x + 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 (1-\frac {9}{x^4}-\frac {2 e^{\frac {2}{5}+\frac {e^3}{5}}}{x^3}+\frac {1}{x^2}+2 x\right ) \, dx\\ &=\frac {3}{x^3}+\frac {e^{\frac {1}{5} \left (2+e^3\right )}}{x^2}-\frac {1}{x}+x+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 30, normalized size = 1.00 \begin {gather*} \frac {3}{x^3}+\frac {e^{\frac {1}{5} \left (2+e^3\right )}}{x^2}-\frac {1}{x}+x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 - 2*E^((2 + E^3)/5)*x + x^2 + x^4 + 2*x^5)/x^4,x]

[Out]

3/x^3 + E^((2 + E^3)/5)/x^2 - x^(-1) + x + x^2

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fricas [A]  time = 0.44, size = 26, normalized size = 0.87 \begin {gather*} \frac {x^{5} + x^{4} - x^{2} + x e^{\left (\frac {1}{5} \, e^{3} + \frac {2}{5}\right )} + 3}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(2/5+1/5*exp(3))+2*x^5+x^4+x^2-9)/x^4,x, algorithm="fricas")

[Out]

(x^5 + x^4 - x^2 + x*e^(1/5*e^3 + 2/5) + 3)/x^3

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giac [A]  time = 0.17, size = 25, normalized size = 0.83 \begin {gather*} x^{2} + x - \frac {x^{2} - x e^{\left (\frac {1}{5} \, e^{3} + \frac {2}{5}\right )} - 3}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(2/5+1/5*exp(3))+2*x^5+x^4+x^2-9)/x^4,x, algorithm="giac")

[Out]

x^2 + x - (x^2 - x*e^(1/5*e^3 + 2/5) - 3)/x^3

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maple [A]  time = 0.10, size = 26, normalized size = 0.87

method	result	size
risch	\(x^{2}+x +\frac {-x^{2}+x \,{\mathrm e}^{\frac {2}{5}+\frac {{\mathrm e}^{3}}{5}}+3}{x^{3}}\)	\(26\)
gosper	\(\frac {x^{5}+x^{4}+x \,{\mathrm e}^{\frac {2}{5}+\frac {{\mathrm e}^{3}}{5}}-x^{2}+3}{x^{3}}\)	\(27\)
default	\(x^{2}+x +\frac {{\mathrm e}^{\frac {2}{5}+\frac {{\mathrm e}^{3}}{5}}}{x^{2}}-\frac {1}{x}+\frac {3}{x^{3}}\)	\(27\)
norman	\(\frac {3+x^{4}+x^{5}+{\mathrm e}^{\frac {2}{5}} {\mathrm e}^{\frac {{\mathrm e}^{3}}{5}} x -x^{2}}{x^{3}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x*exp(2/5+1/5*exp(3))+2*x^5+x^4+x^2-9)/x^4,x,method=_RETURNVERBOSE)

[Out]

x^2+x+(-x^2+x*exp(2/5+1/5*exp(3))+3)/x^3

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maxima [A]  time = 0.35, size = 25, normalized size = 0.83 \begin {gather*} x^{2} + x - \frac {x^{2} - x e^{\left (\frac {1}{5} \, e^{3} + \frac {2}{5}\right )} - 3}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(2/5+1/5*exp(3))+2*x^5+x^4+x^2-9)/x^4,x, algorithm="maxima")

[Out]

x^2 + x - (x^2 - x*e^(1/5*e^3 + 2/5) - 3)/x^3

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mupad [B]  time = 8.27, size = 26, normalized size = 0.87 \begin {gather*} \frac {x^5+x^4-x^2+{\mathrm {e}}^{\frac {{\mathrm {e}}^3}{5}+\frac {2}{5}}\,x+3}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - 2*x*exp(exp(3)/5 + 2/5) + x^4 + 2*x^5 - 9)/x^4,x)

[Out]

(x*exp(exp(3)/5 + 2/5) - x^2 + x^4 + x^5 + 3)/x^3

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sympy [A]  time = 0.27, size = 26, normalized size = 0.87 \begin {gather*} x^{2} + x + \frac {- x^{2} + x e^{\frac {2}{5}} e^{\frac {e^{3}}{5}} + 3}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(2/5+1/5*exp(3))+2*x**5+x**4+x**2-9)/x**4,x)

[Out]

x**2 + x + (-x**2 + x*exp(2/5)*exp(exp(3)/5) + 3)/x**3

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