3.1.76 \(\int e^x x^2 \, dx\) [76]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {2207, 2225} \begin {gather*} e^x x^2-2 e^x x+2 e^x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*E^x - 2*E^x*x + E^x*x^2

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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*}

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Mathematica [A]
time = 0.01, size = 12, normalized size = 0.63 \begin {gather*} e^x \left (2-2 x+x^2\right ) \end {gather*}

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 = 0.01, size = 17, normalized size = 0.89

method	result	size
gosper	\(\left (x^{2}-2 x +2\right ) {\mathrm e}^{x}\)	\(12\)
risch	\(\left (x^{2}-2 x +2\right ) {\mathrm e}^{x}\)	\(12\)
default	\(2 \,{\mathrm e}^{x}-2 \,{\mathrm e}^{x} x +{\mathrm e}^{x} x^{2}\)	\(17\)
norman	\(2 \,{\mathrm e}^{x}-2 \,{\mathrm e}^{x} x +{\mathrm e}^{x} x^{2}\)	\(17\)
meijerg	\(-2+\frac {\left (3 x^{2}-6 x +6\right ) {\mathrm e}^{x}}{3}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*x^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.42, size = 11, normalized size = 0.58 \begin {gather*} {\left (x^{2} - 2 \, x + 2\right )} e^{x} \end {gather*}

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 = 0.77, size = 11, normalized size = 0.58 \begin {gather*} {\left (x^{2} - 2 \, x + 2\right )} e^{x} \end {gather*}

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 = 0.02, size = 10, normalized size = 0.53 \begin {gather*} \left (x^{2} - 2 x + 2\right ) e^{x} \end {gather*}

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 = 0.45, size = 11, normalized size = 0.58 \begin {gather*} {\left (x^{2} - 2 \, x + 2\right )} e^{x} \end {gather*}

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

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Mupad [B]
time = 0.02, size = 11, normalized size = 0.58 \begin {gather*} {\mathrm {e}}^x\,\left (x^2-2\,x+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)*(x^2 - 2*x + 2)

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