### 3.17 $$\int e^x x^2 \, dx$$

Optimal. Leaf size=19 $e^x x^2-2 e^x x+2 e^x$

[Out]

2*E^x - 2*E^x*x + E^x*x^2

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Rubi [A]  time = 0.0201952, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {2176, 2194} $e^x x^2-2 e^x x+2 e^x$

Antiderivative was successfully veriﬁed.

[In]

Int[E^x*x^2,x]

[Out]

2*E^x - 2*E^x*x + E^x*x^2

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 x^2 \, dx &=e^x x^2-2 \int e^x x \, dx\\ &=-2 e^x x+e^x x^2+2 \int e^x \, dx\\ &=2 e^x-2 e^x x+e^x x^2\\ \end{align*}

Mathematica [A]  time = 0.0047733, size = 12, normalized size = 0.63 $e^x \left (x^2-2 x+2\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^x*x^2,x]

[Out]

E^x*(2 - 2*x + x^2)

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Maple [A]  time = 0.002, size = 12, normalized size = 0.6 \begin{align*} \left ({x}^{2}-2\,x+2 \right ){{\rm e}^{x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.934845, size = 15, normalized size = 0.79 \begin{align*}{\left (x^{2} - 2 \, x + 2\right )} e^{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2 - 2*x + 2)*e^x

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Fricas [A]  time = 1.88894, size = 28, normalized size = 1.47 \begin{align*}{\left (x^{2} - 2 \, x + 2\right )} e^{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2 - 2*x + 2)*e^x

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Sympy [A]  time = 0.078241, size = 10, normalized size = 0.53 \begin{align*} \left (x^{2} - 2 x + 2\right ) e^{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x**2 - 2*x + 2)*exp(x)

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Giac [A]  time = 1.04944, size = 15, normalized size = 0.79 \begin{align*}{\left (x^{2} - 2 \, x + 2\right )} e^{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2 - 2*x + 2)*e^x