\(\int (2+50 x+e^{4+2 x^2} (1+4 x^2)) \, dx\) [6434]

Optimal result

Integrand size = 22, antiderivative size = 21 \[ \int \left (2+50 x+e^{4+2 x^2} \left (1+4 x^2\right )\right ) \, dx=5+2 x+e^{4+2 x^2} x+25 x^2 \]

[Out]

5+2*x+x*exp(x^2+2)^2+25*x^2

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2258, 2235, 2243} \[ \int \left (2+50 x+e^{4+2 x^2} \left (1+4 x^2\right )\right ) \, dx=25 x^2+e^{2 x^2+4} x+2 x \]

[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 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])

Rule 2258

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \left (2+50 x+e^{4+2 x^2} \left (1+4 x^2\right )\right ) \, dx=2 x+e^{4+2 x^2} x+25 x^2 \]

[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

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \left (2+50 x+e^{4+2 x^2} \left (1+4 x^2\right )\right ) \, dx=25 \, x^{2} + x e^{\left (2 \, x^{2} + 4\right )} + 2 \, x \]

[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

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \left (2+50 x+e^{4+2 x^2} \left (1+4 x^2\right )\right ) \, dx=25 x^{2} + x e^{2 x^{2} + 4} + 2 x \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \left (2+50 x+e^{4+2 x^2} \left (1+4 x^2\right )\right ) \, dx=25 \, x^{2} + x e^{\left (2 \, x^{2} + 4\right )} + 2 \, x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \left (2+50 x+e^{4+2 x^2} \left (1+4 x^2\right )\right ) \, dx=25 \, x^{2} + x e^{\left (2 \, x^{2} + 4\right )} + 2 \, x \]

[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

Mupad [B] (verification not implemented)

Time = 12.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \left (2+50 x+e^{4+2 x^2} \left (1+4 x^2\right )\right ) \, dx=x\,\left (25\,x+{\mathrm {e}}^{2\,x^2+4}+2\right ) \]

[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)