3.76 \(\int e^{x^2} (1+2 x^2) \, dx\)

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

[Out]

E^x^2*x

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Rubi [A]  time = 0.0298354, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2226, 2204, 2212} \[ e^{x^2} x \]

Antiderivative was successfully verified.

[In]

Int[E^x^2*(1 + 2*x^2),x]

[Out]

E^x^2*x

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rule 2204

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 2212

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 e^{x^2} \left (1+2 x^2\right ) \, dx &=\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*}

Mathematica [A]  time = 0.0043513, size = 7, normalized size = 1. \[ e^{x^2} x \]

Antiderivative was successfully verified.

[In]

Integrate[E^x^2*(1 + 2*x^2),x]

[Out]

E^x^2*x

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Maple [A]  time = 0., size = 7, normalized size = 1. \begin{align*}{{\rm e}^{{x}^{2}}}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^2)*(2*x^2+1),x)

[Out]

exp(x^2)*x

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Maxima [A]  time = 0.989386, size = 8, normalized size = 1.14 \begin{align*} x e^{\left (x^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(x^2)

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Fricas [A]  time = 2.09897, size = 15, normalized size = 2.14 \begin{align*} x e^{\left (x^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(x^2)

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Sympy [A]  time = 0.080341, size = 5, normalized size = 0.71 \begin{align*} x e^{x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x**2)*(2*x**2+1),x)

[Out]

x*exp(x**2)

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Giac [A]  time = 1.05528, size = 8, normalized size = 1.14 \begin{align*} x e^{\left (x^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(x^2)