### 3.45 $$\int e^{x^2} x^3 \, dx$$

Optimal. Leaf size=22 $\frac{1}{2} e^{x^2} x^2-\frac{e^{x^2}}{2}$

[Out]

-E^x^2/2 + (E^x^2*x^2)/2

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Rubi [A]  time = 0.019292, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {2212, 2209} $\frac{1}{2} e^{x^2} x^2-\frac{e^{x^2}}{2}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^x^2*x^3,x]

[Out]

-E^x^2/2 + (E^x^2*x^2)/2

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])

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int e^{x^2} x^3 \, dx &=\frac{1}{2} e^{x^2} x^2-\int e^{x^2} x \, dx\\ &=-\frac{e^{x^2}}{2}+\frac{1}{2} e^{x^2} x^2\\ \end{align*}

Mathematica [A]  time = 0.0016353, size = 14, normalized size = 0.64 $\frac{1}{2} e^{x^2} \left (x^2-1\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^x^2*x^3,x]

[Out]

(E^x^2*(-1 + x^2))/2

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Maple [A]  time = 0.002, size = 12, normalized size = 0.6 \begin{align*}{\frac{ \left ({x}^{2}-1 \right ){{\rm e}^{{x}^{2}}}}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^2)*x^3,x)

[Out]

1/2*(x^2-1)*exp(x^2)

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Maxima [A]  time = 0.941293, size = 15, normalized size = 0.68 \begin{align*} \frac{1}{2} \,{\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^2)*x^3,x, algorithm="maxima")

[Out]

1/2*(x^2 - 1)*e^(x^2)

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Fricas [A]  time = 2.32601, size = 31, normalized size = 1.41 \begin{align*} \frac{1}{2} \,{\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^2)*x^3,x, algorithm="fricas")

[Out]

1/2*(x^2 - 1)*e^(x^2)

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Sympy [A]  time = 0.077963, size = 10, normalized size = 0.45 \begin{align*} \frac{\left (x^{2} - 1\right ) e^{x^{2}}}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x**2)*x**3,x)

[Out]

(x**2 - 1)*exp(x**2)/2

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Giac [A]  time = 1.06031, size = 15, normalized size = 0.68 \begin{align*} \frac{1}{2} \,{\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^2)*x^3,x, algorithm="giac")

[Out]

1/2*(x^2 - 1)*e^(x^2)