3.1.54 \(\int e^{x^2} x^5 \, dx\) [54]

Optimal. Leaf size=28 \[ e^{x^2}-e^{x^2} x^2+\frac {1}{2} e^{x^2} x^4 \]

[Out]

exp(x^2)-exp(x^2)*x^2+1/2*exp(x^2)*x^4

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Rubi [A]
time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2243, 2240} \begin {gather*} -e^{x^2} x^2+e^{x^2}+\frac {1}{2} e^{x^2} x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x^2*x^5,x]

[Out]

E^x^2 - E^x^2*x^2 + (E^x^2*x^4)/2

Rule 2240

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]

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 e^{x^2} x^5 \, dx &=\frac {1}{2} e^{x^2} x^4-2 \int e^{x^2} x^3 \, dx\\ &=-e^{x^2} x^2+\frac {1}{2} e^{x^2} x^4+2 \int e^{x^2} x \, dx\\ &=e^{x^2}-e^{x^2} x^2+\frac {1}{2} e^{x^2} x^4\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 19, normalized size = 0.68 \begin {gather*} \frac {1}{2} e^{x^2} \left (2-2 x^2+x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x^2*x^5,x]

[Out]

(E^x^2*(2 - 2*x^2 + x^4))/2

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Mathics [A]
time = 1.74, size = 18, normalized size = 0.64 \begin {gather*} \left (1-x^2+\frac {x^4}{2}\right ) E^{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^5*E^(x^2),x]')

[Out]

(1 - x ^ 2 + x ^ 4 / 2) E ^ x ^ 2

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Maple [A]
time = 0.01, size = 24, normalized size = 0.86

method	result	size
gosper	\(\frac {\left (x^{4}-2 x^{2}+2\right ) {\mathrm e}^{x^{2}}}{2}\)	\(17\)
risch	\(\left (\frac {1}{2} x^{4}-x^{2}+1\right ) {\mathrm e}^{x^{2}}\)	\(18\)
meijerg	\(-1+\frac {\left (3 x^{4}-6 x^{2}+6\right ) {\mathrm e}^{x^{2}}}{6}\)	\(21\)
default	\({\mathrm e}^{x^{2}}-{\mathrm e}^{x^{2}} x^{2}+\frac {{\mathrm e}^{x^{2}} x^{4}}{2}\)	\(24\)
norman	\({\mathrm e}^{x^{2}}-{\mathrm e}^{x^{2}} x^{2}+\frac {{\mathrm e}^{x^{2}} x^{4}}{2}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^2)*x^5,x,method=_RETURNVERBOSE)

[Out]

exp(x^2)-exp(x^2)*x^2+1/2*exp(x^2)*x^4

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Maxima [A]
time = 0.27, size = 16, normalized size = 0.57 \begin {gather*} \frac {1}{2} \, {\left (x^{4} - 2 \, x^{2} + 2\right )} e^{\left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^2)*x^5,x, algorithm="maxima")

[Out]

1/2*(x^4 - 2*x^2 + 2)*e^(x^2)

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Fricas [A]
time = 0.34, size = 16, normalized size = 0.57 \begin {gather*} \frac {1}{2} \, {\left (x^{4} - 2 \, x^{2} + 2\right )} e^{\left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^2)*x^5,x, algorithm="fricas")

[Out]

1/2*(x^4 - 2*x^2 + 2)*e^(x^2)

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Sympy [A]
time = 0.06, size = 15, normalized size = 0.54 \begin {gather*} \frac {\left (x^{4} - 2 x^{2} + 2\right ) e^{x^{2}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x**2)*x**5,x)

[Out]

(x**4 - 2*x**2 + 2)*exp(x**2)/2

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Giac [A]
time = 0.00, size = 18, normalized size = 0.64 \begin {gather*} \frac {1}{2} \left (x^{4}-2 x^{2}+2\right ) \mathrm {e}^{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^2)*x^5,x)

[Out]

1/2*(x^4 - 2*x^2 + 2)*e^(x^2)

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Mupad [B]
time = 0.03, size = 16, normalized size = 0.57 \begin {gather*} \frac {{\mathrm {e}}^{x^2}\,\left (x^4-2\,x^2+2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*exp(x^2),x)

[Out]

(exp(x^2)*(x^4 - 2*x^2 + 2))/2

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