3.11.10 \(\int \frac {2 x^3-6 x^4+e^{\frac {16-8 x-9 x^2}{x^2}} (-96+24 x)}{x^3} \, dx\)

Optimal. Leaf size=26 \[ -x+3 \left (e^{-10+\left (1-\frac {4}{x}\right )^2}+x-x^2\right ) \]

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Rubi [A]  time = 0.20, antiderivative size = 28, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {14, 6706} \begin {gather*} 3 e^{\frac {16}{x^2}-\frac {8}{x}-9}-\frac {1}{3} (1-3 x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x^3 - 6*x^4 + E^((16 - 8*x - 9*x^2)/x^2)*(-96 + 24*x))/x^3,x]

[Out]

3*E^(-9 + 16/x^2 - 8/x) - (1 - 3*x)^2/3

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 (\frac {24 e^{-9+\frac {16}{x^2}-\frac {8}{x}} (-4+x)}{x^3}-2 (-1+3 x)\right ) \, dx\\ &=-\frac {1}{3} (1-3 x)^2+24 \int \frac {e^{-9+\frac {16}{x^2}-\frac {8}{x}} (-4+x)}{x^3} \, dx\\ &=3 e^{-9+\frac {16}{x^2}-\frac {8}{x}}-\frac {1}{3} (1-3 x)^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 25, normalized size = 0.96 \begin {gather*} 3 e^{-9+\frac {16}{x^2}-\frac {8}{x}}+2 x-3 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^3 - 6*x^4 + E^((16 - 8*x - 9*x^2)/x^2)*(-96 + 24*x))/x^3,x]

[Out]

3*E^(-9 + 16/x^2 - 8/x) + 2*x - 3*x^2

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fricas [A]  time = 0.52, size = 27, normalized size = 1.04 \begin {gather*} -3 \, x^{2} + 2 \, x + 3 \, e^{\left (-\frac {9 \, x^{2} + 8 \, x - 16}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x-96)*exp((-9*x^2-8*x+16)/x^2)-6*x^4+2*x^3)/x^3,x, algorithm="fricas")

[Out]

-3*x^2 + 2*x + 3*e^(-(9*x^2 + 8*x - 16)/x^2)

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giac [A]  time = 0.33, size = 24, normalized size = 0.92 \begin {gather*} -3 \, x^{2} + 2 \, x + 3 \, e^{\left (-\frac {8}{x} + \frac {16}{x^{2}} - 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x-96)*exp((-9*x^2-8*x+16)/x^2)-6*x^4+2*x^3)/x^3,x, algorithm="giac")

[Out]

-3*x^2 + 2*x + 3*e^(-8/x + 16/x^2 - 9)

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maple [A]  time = 0.04, size = 25, normalized size = 0.96

method	result	size
derivativedivides	\(-3 x^{2}+2 x +3 \,{\mathrm e}^{-9-\frac {8}{x}+\frac {16}{x^{2}}}\)	\(25\)
default	\(-3 x^{2}+2 x +3 \,{\mathrm e}^{-9-\frac {8}{x}+\frac {16}{x^{2}}}\)	\(25\)
risch	\(-3 x^{2}+2 x +3 \,{\mathrm e}^{-\frac {9 x^{2}+8 x -16}{x^{2}}}\)	\(28\)
norman	\(\frac {2 x^{3}-3 x^{4}+3 x^{2} {\mathrm e}^{\frac {-9 x^{2}-8 x +16}{x^{2}}}}{x^{2}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x-96)*exp((-9*x^2-8*x+16)/x^2)-6*x^4+2*x^3)/x^3,x,method=_RETURNVERBOSE)

[Out]

-3*x^2+2*x+3*exp(-9-8/x+16/x^2)

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maxima [A]  time = 0.53, size = 24, normalized size = 0.92 \begin {gather*} -3 \, x^{2} + 2 \, x + 3 \, e^{\left (-\frac {8}{x} + \frac {16}{x^{2}} - 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x-96)*exp((-9*x^2-8*x+16)/x^2)-6*x^4+2*x^3)/x^3,x, algorithm="maxima")

[Out]

-3*x^2 + 2*x + 3*e^(-8/x + 16/x^2 - 9)

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mupad [B]  time = 0.70, size = 25, normalized size = 0.96 \begin {gather*} 2\,x-3\,x^2+3\,{\mathrm {e}}^{-9}\,{\mathrm {e}}^{-\frac {8}{x}}\,{\mathrm {e}}^{\frac {16}{x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(8*x + 9*x^2 - 16)/x^2)*(24*x - 96) + 2*x^3 - 6*x^4)/x^3,x)

[Out]

2*x - 3*x^2 + 3*exp(-9)*exp(-8/x)*exp(16/x^2)

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sympy [A]  time = 0.14, size = 24, normalized size = 0.92 \begin {gather*} - 3 x^{2} + 2 x + 3 e^{\frac {- 9 x^{2} - 8 x + 16}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x-96)*exp((-9*x**2-8*x+16)/x**2)-6*x**4+2*x**3)/x**3,x)

[Out]

-3*x**2 + 2*x + 3*exp((-9*x**2 - 8*x + 16)/x**2)

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