3.1.13 \(\int (e^{x^2}+2 e^{x^2} x^2) \, dx\) [13]

Optimal. Leaf size=7 \[ e^{x^2} x \]

[Out]

exp(x^2)*x

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Rubi [A]
time = 0.01, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2235, 2243} \begin {gather*} e^{x^2} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x^2 + 2*E^x^2*x^2,x]

[Out]

E^x^2*x

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2243

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
- n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rubi steps

\begin {align*} \int \left (e^{x^2}+2 e^{x^2} x^2\right ) \, dx &=2 \int e^{x^2} x^2 \, dx+\int e^{x^2} \, dx\\ &=e^{x^2} x+\frac {1}{2} \sqrt {\pi } \text {erfi}(x)-\int e^{x^2} \, dx\\ &=e^{x^2} x\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 7, normalized size = 1.00 \begin {gather*} e^{x^2} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x^2 + 2*E^x^2*x^2,x]

[Out]

E^x^2*x

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Maple [A]
time = 0.01, size = 7, normalized size = 1.00

method	result	size
gosper	\({\mathrm e}^{x^{2}} x\)	\(7\)
default	\({\mathrm e}^{x^{2}} x\)	\(7\)
norman	\({\mathrm e}^{x^{2}} x\)	\(7\)
risch	\({\mathrm e}^{x^{2}} x\)	\(7\)
meijerg	\(\frac {\erfi \left (x \right ) \sqrt {\pi }}{2}+i \left (-i x \,{\mathrm e}^{x^{2}}+\frac {i \erfi \left (x \right ) \sqrt {\pi }}{2}\right )\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x^2)*x

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Maxima [A]
time = 4.37, size = 6, normalized size = 0.86 \begin {gather*} x e^{\left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(x^2)

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Fricas [A]
time = 1.39, size = 6, normalized size = 0.86 \begin {gather*} x e^{\left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(x^2)

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Sympy [A]
time = 0.02, size = 5, normalized size = 0.71 \begin {gather*} x e^{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*exp(x**2)

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Giac [A]
time = 0.45, size = 6, normalized size = 0.86 \begin {gather*} x e^{\left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(x^2)

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Mupad [B]
time = 0.15, size = 6, normalized size = 0.86 \begin {gather*} x\,{\mathrm {e}}^{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^2) + 2*x^2*exp(x^2),x)

[Out]

x*exp(x^2)

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