3.690 \(\int e^x (-5 x+x^2) \, dx\)

Optimal. Leaf size=19 \[ e^x x^2-7 e^x x+7 e^x \]

[Out]

7*E^x - 7*E^x*x + E^x*x^2

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Rubi [A]  time = 0.0429315, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {1593, 2196, 2176, 2194} \[ e^x x^2-7 e^x x+7 e^x \]

Antiderivative was successfully verified.

[In]

Int[E^x*(-5*x + x^2),x]

[Out]

7*E^x - 7*E^x*x + E^x*x^2

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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]

Rubi steps

\begin{align*} \int e^x \left (-5 x+x^2\right ) \, dx &=\int e^x (-5+x) x \, dx\\ &=\int \left (-5 e^x x+e^x x^2\right ) \, dx\\ &=-\left (5 \int e^x x \, dx\right )+\int e^x x^2 \, dx\\ &=-5 e^x x+e^x x^2-2 \int e^x x \, dx+5 \int e^x \, dx\\ &=5 e^x-7 e^x x+e^x x^2+2 \int e^x \, dx\\ &=7 e^x-7 e^x x+e^x x^2\\ \end{align*}

Mathematica [A]  time = 0.0269181, size = 12, normalized size = 0.63 \[ e^x \left (x^2-7 x+7\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*(-5*x + x^2),x]

[Out]

E^x*(7 - 7*x + x^2)

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Maple [A]  time = 0.022, size = 12, normalized size = 0.6 \begin{align*}{{\rm e}^{x}} \left ({x}^{2}-7\,x+7 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*(x^2-5*x),x)

[Out]

exp(x)*(x^2-7*x+7)

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Maxima [A]  time = 0.980786, size = 26, normalized size = 1.37 \begin{align*}{\left (x^{2} - 2 \, x + 2\right )} e^{x} - 5 \,{\left (x - 1\right )} e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(x^2-5*x),x, algorithm="maxima")

[Out]

(x^2 - 2*x + 2)*e^x - 5*(x - 1)*e^x

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Fricas [A]  time = 0.721356, size = 28, normalized size = 1.47 \begin{align*}{\left (x^{2} - 7 \, x + 7\right )} e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(x^2-5*x),x, algorithm="fricas")

[Out]

(x^2 - 7*x + 7)*e^x

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Sympy [A]  time = 0.0871, size = 10, normalized size = 0.53 \begin{align*} \left (x^{2} - 7 x + 7\right ) e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(x**2-5*x),x)

[Out]

(x**2 - 7*x + 7)*exp(x)

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Giac [A]  time = 1.26269, size = 15, normalized size = 0.79 \begin{align*}{\left (x^{2} - 7 \, x + 7\right )} e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*(x^2-5*x),x, algorithm="giac")

[Out]

(x^2 - 7*x + 7)*e^x