3.1.27 \(\int (12+e^{2 (x-4 x^2)} (2-16 x)+50 x+12 x^2+16 x^3+e^{x-4 x^2} (14-86 x-12 x^2-32 x^3)) \, dx\)

Optimal. Leaf size=20 \[ \left (e^{x-4 x^2}+x+2 \left (3+x^2\right )\right )^2 \]

________________________________________________________________________________________

Rubi [B]  time = 0.31, antiderivative size = 67, normalized size of antiderivative = 3.35, number of steps used = 26, number of rules used = 7, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.123, Rules used = {2244, 2236, 6742, 2234, 2205, 2240, 2241} \begin {gather*} 4 x^4+4 x^3+4 e^{x-4 x^2} x^2+25 x^2+2 e^{x-4 x^2} x+e^{2 x-8 x^2}+12 e^{x-4 x^2}+12 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[12 + E^(2*(x - 4*x^2))*(2 - 16*x) + 50*x + 12*x^2 + 16*x^3 + E^(x - 4*x^2)*(14 - 86*x - 12*x^2 - 32*x^3),x
]

[Out]

E^(2*x - 8*x^2) + 12*E^(x - 4*x^2) + 12*x + 2*E^(x - 4*x^2)*x + 25*x^2 + 4*E^(x - 4*x^2)*x^2 + 4*x^3 + 4*x^4

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 2240

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] - Dist[(b*e - 2*c*d)/(2*c), Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] &&
NeQ[b*e - 2*c*d, 0]

Rule 2241

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*F^(a + b*x + c*x^2))/(2*c*Log[F]), x] + (-Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2)
, x], x] - Dist[((m - 1)*e^2)/(2*c*Log[F]), Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x], x]) /; FreeQ[{F, a,
b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]

Rule 2244

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=12 x+25 x^2+4 x^3+4 x^4+\int e^{2 \left (x-4 x^2\right )} (2-16 x) \, dx+\int e^{x-4 x^2} \left (14-86 x-12 x^2-32 x^3\right ) \, dx\\ &=12 x+25 x^2+4 x^3+4 x^4+\int e^{2 x-8 x^2} (2-16 x) \, dx+\int \left (14 e^{x-4 x^2}-86 e^{x-4 x^2} x-12 e^{x-4 x^2} x^2-32 e^{x-4 x^2} x^3\right ) \, dx\\ &=e^{2 x-8 x^2}+12 x+25 x^2+4 x^3+4 x^4-12 \int e^{x-4 x^2} x^2 \, dx+14 \int e^{x-4 x^2} \, dx-32 \int e^{x-4 x^2} x^3 \, dx-86 \int e^{x-4 x^2} x \, dx\\ &=e^{2 x-8 x^2}+\frac {43}{4} e^{x-4 x^2}+12 x+\frac {3}{2} e^{x-4 x^2} x+25 x^2+4 e^{x-4 x^2} x^2+4 x^3+4 x^4-\frac {3}{2} \int e^{x-4 x^2} \, dx-\frac {3}{2} \int e^{x-4 x^2} x \, dx-4 \int e^{x-4 x^2} x^2 \, dx-8 \int e^{x-4 x^2} x \, dx-\frac {43}{4} \int e^{x-4 x^2} \, dx+\left (14 \sqrt [16]{e}\right ) \int e^{-\frac {1}{16} (1-8 x)^2} \, dx\\ &=e^{2 x-8 x^2}+\frac {191}{16} e^{x-4 x^2}+12 x+2 e^{x-4 x^2} x+25 x^2+4 e^{x-4 x^2} x^2+4 x^3+4 x^4-\frac {7}{2} \sqrt [16]{e} \sqrt {\pi } \text {erf}\left (\frac {1}{4} (1-8 x)\right )-\frac {3}{16} \int e^{x-4 x^2} \, dx-\frac {1}{2} \int e^{x-4 x^2} \, dx-\frac {1}{2} \int e^{x-4 x^2} x \, dx-\frac {1}{2} \left (3 \sqrt [16]{e}\right ) \int e^{-\frac {1}{16} (1-8 x)^2} \, dx-\frac {1}{4} \left (43 \sqrt [16]{e}\right ) \int e^{-\frac {1}{16} (1-8 x)^2} \, dx-\int e^{x-4 x^2} \, dx\\ &=e^{2 x-8 x^2}+12 e^{x-4 x^2}+12 x+2 e^{x-4 x^2} x+25 x^2+4 e^{x-4 x^2} x^2+4 x^3+4 x^4-\frac {7}{16} \sqrt [16]{e} \sqrt {\pi } \text {erf}\left (\frac {1}{4} (1-8 x)\right )-\frac {1}{16} \int e^{x-4 x^2} \, dx-\frac {1}{16} \left (3 \sqrt [16]{e}\right ) \int e^{-\frac {1}{16} (1-8 x)^2} \, dx-\frac {1}{2} \sqrt [16]{e} \int e^{-\frac {1}{16} (1-8 x)^2} \, dx-\sqrt [16]{e} \int e^{-\frac {1}{16} (1-8 x)^2} \, dx\\ &=e^{2 x-8 x^2}+12 e^{x-4 x^2}+12 x+2 e^{x-4 x^2} x+25 x^2+4 e^{x-4 x^2} x^2+4 x^3+4 x^4-\frac {1}{64} \sqrt [16]{e} \sqrt {\pi } \text {erf}\left (\frac {1}{4} (1-8 x)\right )-\frac {1}{16} \sqrt [16]{e} \int e^{-\frac {1}{16} (1-8 x)^2} \, dx\\ &=e^{2 x-8 x^2}+12 e^{x-4 x^2}+12 x+2 e^{x-4 x^2} x+25 x^2+4 e^{x-4 x^2} x^2+4 x^3+4 x^4\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.16, size = 51, normalized size = 2.55 \begin {gather*} e^{2 x-8 x^2}+12 x+25 x^2+4 x^3+4 x^4-2 e^{x-4 x^2} \left (-6-x-2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[12 + E^(2*(x - 4*x^2))*(2 - 16*x) + 50*x + 12*x^2 + 16*x^3 + E^(x - 4*x^2)*(14 - 86*x - 12*x^2 - 32*
x^3),x]

[Out]

E^(2*x - 8*x^2) + 12*x + 25*x^2 + 4*x^3 + 4*x^4 - 2*E^(x - 4*x^2)*(-6 - x - 2*x^2)

________________________________________________________________________________________

fricas [B]  time = 0.83, size = 47, normalized size = 2.35 \begin {gather*} 4 \, x^{4} + 4 \, x^{3} + 25 \, x^{2} + 2 \, {\left (2 \, x^{2} + x + 6\right )} e^{\left (-4 \, x^{2} + x\right )} + 12 \, x + e^{\left (-8 \, x^{2} + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x+2)*exp(-2*x^2+1/2*x)^4+(-32*x^3-12*x^2-86*x+14)*exp(-2*x^2+1/2*x)^2+16*x^3+12*x^2+50*x+12,x,
algorithm="fricas")

[Out]

4*x^4 + 4*x^3 + 25*x^2 + 2*(2*x^2 + x + 6)*e^(-4*x^2 + x) + 12*x + e^(-8*x^2 + 2*x)

________________________________________________________________________________________

giac [B]  time = 0.41, size = 51, normalized size = 2.55 \begin {gather*} 4 \, x^{4} + 4 \, x^{3} + 25 \, x^{2} + \frac {1}{16} \, {\left ({\left (8 \, x - 1\right )}^{2} + 48 \, x + 191\right )} e^{\left (-4 \, x^{2} + x\right )} + 12 \, x + e^{\left (-8 \, x^{2} + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x+2)*exp(-2*x^2+1/2*x)^4+(-32*x^3-12*x^2-86*x+14)*exp(-2*x^2+1/2*x)^2+16*x^3+12*x^2+50*x+12,x,
algorithm="giac")

[Out]

4*x^4 + 4*x^3 + 25*x^2 + 1/16*((8*x - 1)^2 + 48*x + 191)*e^(-4*x^2 + x) + 12*x + e^(-8*x^2 + 2*x)

________________________________________________________________________________________

maple [B]  time = 0.06, size = 49, normalized size = 2.45

method	result	size
risch	\({\mathrm e}^{-2 x \left (4 x -1\right )}+\left (4 x^{2}+2 x +12\right ) {\mathrm e}^{-x \left (4 x -1\right )}+4 x^{4}+4 x^{3}+25 x^{2}+12 x\)	\(49\)
default	\(12 x +{\mathrm e}^{-8 x^{2}+2 x}+12 \,{\mathrm e}^{-4 x^{2}+x}+2 x \,{\mathrm e}^{-4 x^{2}+x}+4 x^{2} {\mathrm e}^{-4 x^{2}+x}+25 x^{2}+4 x^{3}+4 x^{4}\)	\(64\)
norman	\(12 x +{\mathrm e}^{-8 x^{2}+2 x}+12 \,{\mathrm e}^{-4 x^{2}+x}+2 x \,{\mathrm e}^{-4 x^{2}+x}+4 x^{2} {\mathrm e}^{-4 x^{2}+x}+25 x^{2}+4 x^{3}+4 x^{4}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-16*x+2)*exp(-2*x^2+1/2*x)^4+(-32*x^3-12*x^2-86*x+14)*exp(-2*x^2+1/2*x)^2+16*x^3+12*x^2+50*x+12,x,method=
_RETURNVERBOSE)

[Out]

exp(-2*x*(4*x-1))+(4*x^2+2*x+12)*exp(-x*(4*x-1))+4*x^4+4*x^3+25*x^2+12*x

________________________________________________________________________________________

maxima [B]  time = 0.38, size = 47, normalized size = 2.35 \begin {gather*} 4 \, x^{4} + 4 \, x^{3} + 25 \, x^{2} + 2 \, {\left (2 \, x^{2} + x + 6\right )} e^{\left (-4 \, x^{2} + x\right )} + 12 \, x + e^{\left (-8 \, x^{2} + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x+2)*exp(-2*x^2+1/2*x)^4+(-32*x^3-12*x^2-86*x+14)*exp(-2*x^2+1/2*x)^2+16*x^3+12*x^2+50*x+12,x,
algorithm="maxima")

[Out]

4*x^4 + 4*x^3 + 25*x^2 + 2*(2*x^2 + x + 6)*e^(-4*x^2 + x) + 12*x + e^(-8*x^2 + 2*x)

________________________________________________________________________________________

mupad [B]  time = 0.11, size = 63, normalized size = 3.15 \begin {gather*} 12\,x+12\,{\mathrm {e}}^{x-4\,x^2}+{\mathrm {e}}^{2\,x-8\,x^2}+2\,x\,{\mathrm {e}}^{x-4\,x^2}+4\,x^2\,{\mathrm {e}}^{x-4\,x^2}+25\,x^2+4\,x^3+4\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(50*x - exp(2*x - 8*x^2)*(16*x - 2) - exp(x - 4*x^2)*(86*x + 12*x^2 + 32*x^3 - 14) + 12*x^2 + 16*x^3 + 12,x
)

[Out]

12*x + 12*exp(x - 4*x^2) + exp(2*x - 8*x^2) + 2*x*exp(x - 4*x^2) + 4*x^2*exp(x - 4*x^2) + 25*x^2 + 4*x^3 + 4*x
^4

________________________________________________________________________________________

sympy [B]  time = 0.12, size = 46, normalized size = 2.30 \begin {gather*} 4 x^{4} + 4 x^{3} + 25 x^{2} + 12 x + \left (4 x^{2} + 2 x + 12\right ) e^{- 4 x^{2} + x} + e^{- 8 x^{2} + 2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x+2)*exp(-2*x**2+1/2*x)**4+(-32*x**3-12*x**2-86*x+14)*exp(-2*x**2+1/2*x)**2+16*x**3+12*x**2+50*
x+12,x)

[Out]

4*x**4 + 4*x**3 + 25*x**2 + 12*x + (4*x**2 + 2*x + 12)*exp(-4*x**2 + x) + exp(-8*x**2 + 2*x)

________________________________________________________________________________________