3.65.34 \(\int (2+50 x+e^{4+2 x^2} (1+4 x^2)) \, dx\)

Optimal. Leaf size=21 \[ 5+2 x+e^{4+2 x^2} x+25 x^2 \]

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Rubi [A]  time = 0.06, antiderivative size = 20, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2226, 2204, 2212} \begin {gather*} 25 x^2+e^{2 x^2+4} x+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2 + 50*x + E^(4 + 2*x^2)*(1 + 4*x^2),x]

[Out]

2*x + E^(4 + 2*x^2)*x + 25*x^2

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])

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 x+25 x^2+\int e^{4+2 x^2} \left (1+4 x^2\right ) \, dx\\ &=2 x+25 x^2+\int \left (e^{4+2 x^2}+4 e^{4+2 x^2} x^2\right ) \, dx\\ &=2 x+25 x^2+4 \int e^{4+2 x^2} x^2 \, dx+\int e^{4+2 x^2} \, dx\\ &=2 x+e^{4+2 x^2} x+25 x^2+\frac {1}{2} e^4 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} x\right )-\int e^{4+2 x^2} \, dx\\ &=2 x+e^{4+2 x^2} x+25 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 0.95 \begin {gather*} 2 x+e^{4+2 x^2} x+25 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2 + 50*x + E^(4 + 2*x^2)*(1 + 4*x^2),x]

[Out]

2*x + E^(4 + 2*x^2)*x + 25*x^2

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fricas [A]  time = 0.62, size = 19, normalized size = 0.90 \begin {gather*} 25 \, x^{2} + x e^{\left (2 \, x^{2} + 4\right )} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+1)*exp(x^2+2)^2+50*x+2,x, algorithm="fricas")

[Out]

25*x^2 + x*e^(2*x^2 + 4) + 2*x

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giac [A]  time = 0.18, size = 19, normalized size = 0.90 \begin {gather*} 25 \, x^{2} + x e^{\left (2 \, x^{2} + 4\right )} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+1)*exp(x^2+2)^2+50*x+2,x, algorithm="giac")

[Out]

25*x^2 + x*e^(2*x^2 + 4) + 2*x

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maple [A]  time = 0.04, size = 20, normalized size = 0.95

method	result	size
norman	\(x \,{\mathrm e}^{2 x^{2}+4}+2 x +25 x^{2}\)	\(20\)
risch	\(x \,{\mathrm e}^{2 x^{2}+4}+2 x +25 x^{2}\)	\(20\)
default	\(2 x +\frac {{\mathrm e}^{4} \sqrt {2}\, \sqrt {\pi }\, \erfi \left (\sqrt {2}\, x \right )}{4}+4 \,{\mathrm e}^{4} \left (\frac {x \,{\mathrm e}^{2 x^{2}}}{4}-\frac {\sqrt {2}\, \sqrt {\pi }\, \erfi \left (\sqrt {2}\, x \right )}{16}\right )+25 x^{2}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2+1)*exp(x^2+2)^2+50*x+2,x,method=_RETURNVERBOSE)

[Out]

x*exp(x^2+2)^2+2*x+25*x^2

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maxima [A]  time = 0.38, size = 19, normalized size = 0.90 \begin {gather*} 25 \, x^{2} + x e^{\left (2 \, x^{2} + 4\right )} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+1)*exp(x^2+2)^2+50*x+2,x, algorithm="maxima")

[Out]

25*x^2 + x*e^(2*x^2 + 4) + 2*x

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mupad [B]  time = 0.07, size = 15, normalized size = 0.71 \begin {gather*} x\,\left (25\,x+{\mathrm {e}}^{2\,x^2+4}+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(50*x + exp(2*x^2 + 4)*(4*x^2 + 1) + 2,x)

[Out]

x*(25*x + exp(2*x^2 + 4) + 2)

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sympy [A]  time = 0.08, size = 17, normalized size = 0.81 \begin {gather*} 25 x^{2} + x e^{2 x^{2} + 4} + 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**2+1)*exp(x**2+2)**2+50*x+2,x)

[Out]

25*x**2 + x*exp(2*x**2 + 4) + 2*x

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