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

Optimal. Leaf size=20 \[ 9-\frac {-2-e^x}{x}+x+x (2+x) \]

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Rubi [A]  time = 0.04, antiderivative size = 19, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {14, 2197} \begin {gather*} x^2+3 x+\frac {e^x}{x}+\frac {2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2/x + E^x/x + 3*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 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 19, normalized size = 0.95 \begin {gather*} \frac {2}{x}+\frac {e^x}{x}+3 x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2/x + E^x/x + 3*x + x^2

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fricas [A]  time = 0.58, size = 16, normalized size = 0.80 \begin {gather*} \frac {x^{3} + 3 \, x^{2} + e^{x} + 2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^3 + 3*x^2 + e^x + 2)/x

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giac [A]  time = 0.21, size = 16, normalized size = 0.80 \begin {gather*} \frac {x^{3} + 3 \, x^{2} + e^{x} + 2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^3 + 3*x^2 + e^x + 2)/x

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maple [A]  time = 0.02, size = 17, normalized size = 0.85

method	result	size
norman	\(\frac {2+x^{3}+3 x^{2}+{\mathrm e}^{x}}{x}\)	\(17\)
default	\(3 x +x^{2}+\frac {2}{x}+\frac {{\mathrm e}^{x}}{x}\)	\(19\)
risch	\(3 x +x^{2}+\frac {2}{x}+\frac {{\mathrm e}^{x}}{x}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2+x^3+3*x^2+exp(x))/x

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maxima [C]  time = 0.44, size = 21, normalized size = 1.05 \begin {gather*} x^{2} + 3 \, x + \frac {2}{x} + {\rm Ei}\relax (x) - \Gamma \left (-1, -x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + 3*x + 2/x + Ei(x) - gamma(-1, -x)

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mupad [B]  time = 0.05, size = 14, normalized size = 0.70 \begin {gather*} \frac {{\mathrm {e}}^x+2}{x}+x\,\left (x+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(x) + 2)/x + x*(x + 3)

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sympy [A]  time = 0.10, size = 14, normalized size = 0.70 \begin {gather*} x^{2} + 3 x + \frac {e^{x}}{x} + \frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2 + 3*x + exp(x)/x + 2/x

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