3.25.59 \(\int \frac {-1-x^2-130050 e^{2-2 e^x+x} x^2+2 x^3}{x^2} \, dx\)

Optimal. Leaf size=21 \[ 65025 e^{2-2 e^x}+\frac {1}{x}-x+x^2 \]

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Rubi [A]  time = 0.04, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {14, 2282, 2194} \begin {gather*} x^2-x+65025 e^{2-2 e^x}+\frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - x^2 - 130050*E^(2 - 2*E^x + x)*x^2 + 2*x^3)/x^2,x]

[Out]

65025*E^(2 - 2*E^x) + 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]]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-130050 e^{2-2 e^x+x}+\frac {-1-x^2+2 x^3}{x^2}\right ) \, dx\\ &=-\left (130050 \int e^{2-2 e^x+x} \, dx\right )+\int \frac {-1-x^2+2 x^3}{x^2} \, dx\\ &=-\left (130050 \operatorname {Subst}\left (\int e^{2-2 x} \, dx,x,e^x\right )\right )+\int \left (-1-\frac {1}{x^2}+2 x\right ) \, dx\\ &=65025 e^{2-2 e^x}+\frac {1}{x}-x+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 21, normalized size = 1.00 \begin {gather*} 65025 e^{2-2 e^x}+\frac {1}{x}-x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - x^2 - 130050*E^(2 - 2*E^x + x)*x^2 + 2*x^3)/x^2,x]

[Out]

65025*E^(2 - 2*E^x) + x^(-1) - x + x^2

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fricas [A]  time = 0.66, size = 33, normalized size = 1.57 \begin {gather*} \frac {{\left (65025 \, x e^{\left (x - 2 \, e^{x} + 2\right )} + {\left (x^{3} - x^{2} + 1\right )} e^{x}\right )} e^{\left (-x\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-130050*x^2*exp(x)*exp(1-exp(x))^2+2*x^3-x^2-1)/x^2,x, algorithm="fricas")

[Out]

(65025*x*e^(x - 2*e^x + 2) + (x^3 - x^2 + 1)*e^x)*e^(-x)/x

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giac [A]  time = 0.38, size = 35, normalized size = 1.67 \begin {gather*} \frac {{\left (x^{3} e^{x} - x^{2} e^{x} + 65025 \, x e^{\left (x - 2 \, e^{x} + 2\right )} + e^{x}\right )} e^{\left (-x\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-130050*x^2*exp(x)*exp(1-exp(x))^2+2*x^3-x^2-1)/x^2,x, algorithm="giac")

[Out]

(x^3*e^x - x^2*e^x + 65025*x*e^(x - 2*e^x + 2) + e^x)*e^(-x)/x

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maple [A]  time = 0.03, size = 20, normalized size = 0.95

method	result	size
risch	\(\frac {1}{x}+65025 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+2}+x^{2}-x\)	\(20\)
default	\(x^{2}-x +\frac {1}{x}+65025 \,{\mathrm e}^{2} {\mathrm e}^{-2 \,{\mathrm e}^{x}}\)	\(22\)
norman	\(\frac {1+x^{3}-x^{2}+65025 x \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+2}}{x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-130050*x^2*exp(x)*exp(1-exp(x))^2+2*x^3-x^2-1)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/x+65025*exp(-2*exp(x)+2)+x^2-x

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maxima [A]  time = 0.49, size = 19, normalized size = 0.90 \begin {gather*} x^{2} - x + \frac {1}{x} + 65025 \, e^{\left (-2 \, e^{x} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-130050*x^2*exp(x)*exp(1-exp(x))^2+2*x^3-x^2-1)/x^2,x, algorithm="maxima")

[Out]

x^2 - x + 1/x + 65025*e^(-2*e^x + 2)

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mupad [B]  time = 1.38, size = 19, normalized size = 0.90 \begin {gather*} 65025\,{\mathrm {e}}^{2-2\,{\mathrm {e}}^x}-x+\frac {1}{x}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 - 2*x^3 + 130050*x^2*exp(2 - 2*exp(x))*exp(x) + 1)/x^2,x)

[Out]

65025*exp(2 - 2*exp(x)) - x + 1/x + x^2

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sympy [A]  time = 0.14, size = 17, normalized size = 0.81 \begin {gather*} x^{2} - x + 65025 e^{2 - 2 e^{x}} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-130050*x**2*exp(x)*exp(1-exp(x))**2+2*x**3-x**2-1)/x**2,x)

[Out]

x**2 - x + 65025*exp(2 - 2*exp(x)) + 1/x

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