3.25.50 \(\int \frac {x^5+e^{\frac {9-6 x+x^2}{x^4}} (-180+90 x-10 x^2)}{x^5} \, dx\)

Optimal. Leaf size=17 \[ 5 e^{\frac {(3-x)^2}{x^4}}+x \]

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Rubi [A]  time = 0.18, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {14, 6706} \begin {gather*} 5 e^{\frac {(3-x)^2}{x^4}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^5 + E^((9 - 6*x + x^2)/x^4)*(-180 + 90*x - 10*x^2))/x^5,x]

[Out]

5*E^((3 - x)^2/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]]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-\frac {10 e^{\frac {(-3+x)^2}{x^4}} (-6+x) (-3+x)}{x^5}\right ) \, dx\\ &=x-10 \int \frac {e^{\frac {(-3+x)^2}{x^4}} (-6+x) (-3+x)}{x^5} \, dx\\ &=5 e^{\frac {(3-x)^2}{x^4}}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 20, normalized size = 1.18 \begin {gather*} 5 e^{\frac {9}{x^4}-\frac {6}{x^3}+\frac {1}{x^2}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^5 + E^((9 - 6*x + x^2)/x^4)*(-180 + 90*x - 10*x^2))/x^5,x]

[Out]

5*E^(9/x^4 - 6/x^3 + x^(-2)) + x

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fricas [A]  time = 0.77, size = 17, normalized size = 1.00 \begin {gather*} x + 5 \, e^{\left (\frac {x^{2} - 6 \, x + 9}{x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^2+90*x-180)*exp((x^2-6*x+9)/x^4)+x^5)/x^5,x, algorithm="fricas")

[Out]

x + 5*e^((x^2 - 6*x + 9)/x^4)

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giac [A]  time = 0.46, size = 19, normalized size = 1.12 \begin {gather*} x + 5 \, e^{\left (\frac {1}{x^{2}} - \frac {6}{x^{3}} + \frac {9}{x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^2+90*x-180)*exp((x^2-6*x+9)/x^4)+x^5)/x^5,x, algorithm="giac")

[Out]

x + 5*e^(1/x^2 - 6/x^3 + 9/x^4)

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maple [A]  time = 0.09, size = 15, normalized size = 0.88

method	result	size
risch	\(x +5 \,{\mathrm e}^{\frac {\left (x -3\right )^{2}}{x^{4}}}\)	\(15\)
norman	\(\frac {x^{5}+5 x^{4} {\mathrm e}^{\frac {x^{2}-6 x +9}{x^{4}}}}{x^{4}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-10*x^2+90*x-180)*exp((x^2-6*x+9)/x^4)+x^5)/x^5,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.56, size = 19, normalized size = 1.12 \begin {gather*} x + 5 \, e^{\left (\frac {1}{x^{2}} - \frac {6}{x^{3}} + \frac {9}{x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^2+90*x-180)*exp((x^2-6*x+9)/x^4)+x^5)/x^5,x, algorithm="maxima")

[Out]

x + 5*e^(1/x^2 - 6/x^3 + 9/x^4)

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mupad [B]  time = 1.38, size = 20, normalized size = 1.18 \begin {gather*} x+5\,{\mathrm {e}}^{\frac {1}{x^2}}\,{\mathrm {e}}^{-\frac {6}{x^3}}\,{\mathrm {e}}^{\frac {9}{x^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((x^2 - 6*x + 9)/x^4)*(10*x^2 - 90*x + 180) - x^5)/x^5,x)

[Out]

x + 5*exp(1/x^2)*exp(-6/x^3)*exp(9/x^4)

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sympy [A]  time = 0.16, size = 15, normalized size = 0.88 \begin {gather*} x + 5 e^{\frac {x^{2} - 6 x + 9}{x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x**2+90*x-180)*exp((x**2-6*x+9)/x**4)+x**5)/x**5,x)

[Out]

x + 5*exp((x**2 - 6*x + 9)/x**4)

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