3.5.23 \(\int (-99+40 x-3 x^2+e^{8 x-16 x^2} (-25-200 x+800 x^2)+e^{4 x-8 x^2} (100+380 x-1640 x^2+160 x^3)) \, dx\)

Optimal. Leaf size=26 \[ x-x \left (-5 \left (2-e^{4 x-8 x^2}\right )+x\right )^2 \]

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Rubi [B]  time = 0.39, antiderivative size = 71, normalized size of antiderivative = 2.73, number of steps used = 25, number of rules used = 6, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.102, Rules used = {2288, 6742, 2234, 2205, 2240, 2241} \begin {gather*} -x^3-10 e^{4 x-8 x^2} x^2+20 x^2+100 e^{4 x-8 x^2} x-\frac {25 e^{8 x-16 x^2} \left (x-4 x^2\right )}{1-4 x}-99 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-99 + 40*x - 3*x^2 + E^(8*x - 16*x^2)*(-25 - 200*x + 800*x^2) + E^(4*x - 8*x^2)*(100 + 380*x - 1640*x^2 +
160*x^3),x]

[Out]

-99*x + 100*E^(4*x - 8*x^2)*x + 20*x^2 - 10*E^(4*x - 8*x^2)*x^2 - x^3 - (25*E^(8*x - 16*x^2)*(x - 4*x^2))/(1 -
4*x)

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 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 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, 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} &=-99 x+20 x^2-x^3+\int e^{8 x-16 x^2} \left (-25-200 x+800 x^2\right ) \, dx+\int e^{4 x-8 x^2} \left (100+380 x-1640 x^2+160 x^3\right ) \, dx\\ &=-99 x+20 x^2-x^3-\frac {25 e^{8 x-16 x^2} \left (x-4 x^2\right )}{1-4 x}+\int \left (100 e^{4 x-8 x^2}+380 e^{4 x-8 x^2} x-1640 e^{4 x-8 x^2} x^2+160 e^{4 x-8 x^2} x^3\right ) \, dx\\ &=-99 x+20 x^2-x^3-\frac {25 e^{8 x-16 x^2} \left (x-4 x^2\right )}{1-4 x}+100 \int e^{4 x-8 x^2} \, dx+160 \int e^{4 x-8 x^2} x^3 \, dx+380 \int e^{4 x-8 x^2} x \, dx-1640 \int e^{4 x-8 x^2} x^2 \, dx\\ &=-\frac {95}{4} e^{4 x-8 x^2}-99 x+\frac {205}{2} e^{4 x-8 x^2} x+20 x^2-10 e^{4 x-8 x^2} x^2-x^3-\frac {25 e^{8 x-16 x^2} \left (x-4 x^2\right )}{1-4 x}+20 \int e^{4 x-8 x^2} x \, dx+40 \int e^{4 x-8 x^2} x^2 \, dx+95 \int e^{4 x-8 x^2} \, dx-\frac {205}{2} \int e^{4 x-8 x^2} \, dx-410 \int e^{4 x-8 x^2} x \, dx+\left (100 \sqrt {e}\right ) \int e^{-\frac {1}{32} (4-16 x)^2} \, dx\\ &=\frac {5}{8} e^{4 x-8 x^2}-99 x+100 e^{4 x-8 x^2} x+20 x^2-10 e^{4 x-8 x^2} x^2-x^3-\frac {25 e^{8 x-16 x^2} \left (x-4 x^2\right )}{1-4 x}-25 \sqrt {\frac {e \pi }{2}} \text {erf}\left (\frac {1-4 x}{\sqrt {2}}\right )+\frac {5}{2} \int e^{4 x-8 x^2} \, dx+5 \int e^{4 x-8 x^2} \, dx+10 \int e^{4 x-8 x^2} x \, dx-\frac {205}{2} \int e^{4 x-8 x^2} \, dx+\left (95 \sqrt {e}\right ) \int e^{-\frac {1}{32} (4-16 x)^2} \, dx-\frac {1}{2} \left (205 \sqrt {e}\right ) \int e^{-\frac {1}{32} (4-16 x)^2} \, dx\\ &=-99 x+100 e^{4 x-8 x^2} x+20 x^2-10 e^{4 x-8 x^2} x^2-x^3-\frac {25 e^{8 x-16 x^2} \left (x-4 x^2\right )}{1-4 x}-\frac {185}{8} \sqrt {\frac {e \pi }{2}} \text {erf}\left (\frac {1-4 x}{\sqrt {2}}\right )+\frac {5}{2} \int e^{4 x-8 x^2} \, dx+\frac {1}{2} \left (5 \sqrt {e}\right ) \int e^{-\frac {1}{32} (4-16 x)^2} \, dx+\left (5 \sqrt {e}\right ) \int e^{-\frac {1}{32} (4-16 x)^2} \, dx-\frac {1}{2} \left (205 \sqrt {e}\right ) \int e^{-\frac {1}{32} (4-16 x)^2} \, dx\\ &=-99 x+100 e^{4 x-8 x^2} x+20 x^2-10 e^{4 x-8 x^2} x^2-x^3-\frac {25 e^{8 x-16 x^2} \left (x-4 x^2\right )}{1-4 x}+\frac {5}{8} \sqrt {\frac {e \pi }{2}} \text {erf}\left (\frac {1-4 x}{\sqrt {2}}\right )+\frac {1}{2} \left (5 \sqrt {e}\right ) \int e^{-\frac {1}{32} (4-16 x)^2} \, dx\\ &=-99 x+100 e^{4 x-8 x^2} x+20 x^2-10 e^{4 x-8 x^2} x^2-x^3-\frac {25 e^{8 x-16 x^2} \left (x-4 x^2\right )}{1-4 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 39, normalized size = 1.50 \begin {gather*} x \left (-99-25 e^{-8 x (-1+2 x)}-10 e^{4 (1-2 x) x} (-10+x)+20 x-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-99 + 40*x - 3*x^2 + E^(8*x - 16*x^2)*(-25 - 200*x + 800*x^2) + E^(4*x - 8*x^2)*(100 + 380*x - 1640*
x^2 + 160*x^3),x]

[Out]

x*(-99 - 25/E^(8*x*(-1 + 2*x)) - 10*E^(4*(1 - 2*x)*x)*(-10 + x) + 20*x - x^2)

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fricas [B]  time = 0.78, size = 46, normalized size = 1.77 \begin {gather*} -x^{3} + 20 \, x^{2} - 10 \, {\left (x^{2} - 10 \, x\right )} e^{\left (-8 \, x^{2} + 4 \, x\right )} - 25 \, x e^{\left (-16 \, x^{2} + 8 \, x\right )} - 99 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((800*x^2-200*x-25)*exp(-8*x^2+4*x)^2+(160*x^3-1640*x^2+380*x+100)*exp(-8*x^2+4*x)-3*x^2+40*x-99,x, a
lgorithm="fricas")

[Out]

-x^3 + 20*x^2 - 10*(x^2 - 10*x)*e^(-8*x^2 + 4*x) - 25*x*e^(-16*x^2 + 8*x) - 99*x

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giac [B]  time = 0.50, size = 51, normalized size = 1.96 \begin {gather*} -x^{3} + 20 \, x^{2} - \frac {5}{8} \, {\left ({\left (4 \, x - 1\right )}^{2} - 152 \, x - 1\right )} e^{\left (-8 \, x^{2} + 4 \, x\right )} - 25 \, x e^{\left (-16 \, x^{2} + 8 \, x\right )} - 99 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((800*x^2-200*x-25)*exp(-8*x^2+4*x)^2+(160*x^3-1640*x^2+380*x+100)*exp(-8*x^2+4*x)-3*x^2+40*x-99,x, a
lgorithm="giac")

[Out]

-x^3 + 20*x^2 - 5/8*((4*x - 1)^2 - 152*x - 1)*e^(-8*x^2 + 4*x) - 25*x*e^(-16*x^2 + 8*x) - 99*x

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maple [B]  time = 0.06, size = 46, normalized size = 1.77

method	result	size
risch	\(-25 \,{\mathrm e}^{-8 x \left (2 x -1\right )} x +\left (-10 x^{2}+100 x \right ) {\mathrm e}^{-4 x \left (2 x -1\right )}-x^{3}+20 x^{2}-99 x\)	\(46\)
default	\(-99 x -25 x \,{\mathrm e}^{-16 x^{2}+8 x}+100 \,{\mathrm e}^{-8 x^{2}+4 x} x -10 \,{\mathrm e}^{-8 x^{2}+4 x} x^{2}+20 x^{2}-x^{3}\)	\(56\)
norman	\(-99 x -25 x \,{\mathrm e}^{-16 x^{2}+8 x}+100 \,{\mathrm e}^{-8 x^{2}+4 x} x -10 \,{\mathrm e}^{-8 x^{2}+4 x} x^{2}+20 x^{2}-x^{3}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((800*x^2-200*x-25)*exp(-8*x^2+4*x)^2+(160*x^3-1640*x^2+380*x+100)*exp(-8*x^2+4*x)-3*x^2+40*x-99,x,method=_
RETURNVERBOSE)

[Out]

-25*exp(-8*x*(2*x-1))*x+(-10*x^2+100*x)*exp(-4*x*(2*x-1))-x^3+20*x^2-99*x

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maxima [B]  time = 0.42, size = 46, normalized size = 1.77 \begin {gather*} -x^{3} + 20 \, x^{2} - 10 \, {\left (x^{2} - 10 \, x\right )} e^{\left (-8 \, x^{2} + 4 \, x\right )} - 25 \, x e^{\left (-16 \, x^{2} + 8 \, x\right )} - 99 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((800*x^2-200*x-25)*exp(-8*x^2+4*x)^2+(160*x^3-1640*x^2+380*x+100)*exp(-8*x^2+4*x)-3*x^2+40*x-99,x, a
lgorithm="maxima")

[Out]

-x^3 + 20*x^2 - 10*(x^2 - 10*x)*e^(-8*x^2 + 4*x) - 25*x*e^(-16*x^2 + 8*x) - 99*x

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mupad [B]  time = 0.12, size = 48, normalized size = 1.85 \begin {gather*} -x\,\left (25\,{\mathrm {e}}^{8\,x-16\,x^2}-100\,{\mathrm {e}}^{4\,x-8\,x^2}-20\,x+10\,x\,{\mathrm {e}}^{4\,x-8\,x^2}+x^2+99\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(40*x - exp(8*x - 16*x^2)*(200*x - 800*x^2 + 25) + exp(4*x - 8*x^2)*(380*x - 1640*x^2 + 160*x^3 + 100) - 3*
x^2 - 99,x)

[Out]

-x*(25*exp(8*x - 16*x^2) - 100*exp(4*x - 8*x^2) - 20*x + 10*x*exp(4*x - 8*x^2) + x^2 + 99)

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sympy [B]  time = 0.14, size = 42, normalized size = 1.62 \begin {gather*} - x^{3} + 20 x^{2} - 25 x e^{- 16 x^{2} + 8 x} - 99 x + \left (- 10 x^{2} + 100 x\right ) e^{- 8 x^{2} + 4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((800*x**2-200*x-25)*exp(-8*x**2+4*x)**2+(160*x**3-1640*x**2+380*x+100)*exp(-8*x**2+4*x)-3*x**2+40*x-
99,x)

[Out]

-x**3 + 20*x**2 - 25*x*exp(-16*x**2 + 8*x) - 99*x + (-10*x**2 + 100*x)*exp(-8*x**2 + 4*x)

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