3.81.48 \(\int (1+e^{8 x} (66+10 e^{2 x}+12 x-16 x^2)) \, dx\)

Optimal. Leaf size=29 \[ e^5+x-2 e^{8 x} \left (-4-\frac {e^{2 x}}{2}-x+x^2\right ) \]

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Rubi [A]  time = 0.16, antiderivative size = 32, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6741, 12, 6742, 2194, 2176} \begin {gather*} -2 e^{8 x} x^2+2 e^{8 x} x+x+8 e^{8 x}+e^{10 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + E^(8*x)*(66 + 10*E^(2*x) + 12*x - 16*x^2),x]

[Out]

8*E^(8*x) + E^(10*x) + x + 2*E^(8*x)*x - 2*E^(8*x)*x^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x+\int e^{8 x} \left (66+10 e^{2 x}+12 x-16 x^2\right ) \, dx\\ &=x+\int 2 e^{8 x} \left (33+5 e^{2 x}+6 x-8 x^2\right ) \, dx\\ &=x+2 \int e^{8 x} \left (33+5 e^{2 x}+6 x-8 x^2\right ) \, dx\\ &=x+2 \int \left (33 e^{8 x}+5 e^{10 x}+6 e^{8 x} x-8 e^{8 x} x^2\right ) \, dx\\ &=x+10 \int e^{10 x} \, dx+12 \int e^{8 x} x \, dx-16 \int e^{8 x} x^2 \, dx+66 \int e^{8 x} \, dx\\ &=\frac {33 e^{8 x}}{4}+e^{10 x}+x+\frac {3}{2} e^{8 x} x-2 e^{8 x} x^2-\frac {3}{2} \int e^{8 x} \, dx+4 \int e^{8 x} x \, dx\\ &=\frac {129 e^{8 x}}{16}+e^{10 x}+x+2 e^{8 x} x-2 e^{8 x} x^2-\frac {1}{2} \int e^{8 x} \, dx\\ &=8 e^{8 x}+e^{10 x}+x+2 e^{8 x} x-2 e^{8 x} x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 22, normalized size = 0.76 \begin {gather*} e^{10 x}+x+2 e^{8 x} \left (4+x-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + E^(8*x)*(66 + 10*E^(2*x) + 12*x - 16*x^2),x]

[Out]

E^(10*x) + x + 2*E^(8*x)*(4 + x - x^2)

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fricas [A]  time = 0.65, size = 20, normalized size = 0.69 \begin {gather*} -2 \, {\left (x^{2} - x - 4\right )} e^{\left (8 \, x\right )} + x + e^{\left (10 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*exp(x)^2-16*x^2+12*x+66)*exp(8*x)+1,x, algorithm="fricas")

[Out]

-2*(x^2 - x - 4)*e^(8*x) + x + e^(10*x)

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giac [A]  time = 0.22, size = 20, normalized size = 0.69 \begin {gather*} -2 \, {\left (x^{2} - x - 4\right )} e^{\left (8 \, x\right )} + x + e^{\left (10 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*exp(x)^2-16*x^2+12*x+66)*exp(8*x)+1,x, algorithm="giac")

[Out]

-2*(x^2 - x - 4)*e^(8*x) + x + e^(10*x)

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maple [A]  time = 0.02, size = 22, normalized size = 0.76

method	result	size
risch	\({\mathrm e}^{10 x}+\left (-2 x^{2}+2 x +8\right ) {\mathrm e}^{8 x}+x\)	\(22\)
default	\(x +8 \,{\mathrm e}^{8 x}+{\mathrm e}^{10 x}+2 x \,{\mathrm e}^{8 x}-2 \,{\mathrm e}^{8 x} x^{2}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*exp(x)^2-16*x^2+12*x+66)*exp(8*x)+1,x,method=_RETURNVERBOSE)

[Out]

exp(10*x)+(-2*x^2+2*x+8)*exp(8*x)+x

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maxima [A]  time = 0.37, size = 39, normalized size = 1.34 \begin {gather*} -\frac {1}{16} \, {\left (32 \, x^{2} - 8 \, x + 1\right )} e^{\left (8 \, x\right )} + \frac {3}{16} \, {\left (8 \, x - 1\right )} e^{\left (8 \, x\right )} + x + e^{\left (10 \, x\right )} + \frac {33}{4} \, e^{\left (8 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*exp(x)^2-16*x^2+12*x+66)*exp(8*x)+1,x, algorithm="maxima")

[Out]

-1/16*(32*x^2 - 8*x + 1)*e^(8*x) + 3/16*(8*x - 1)*e^(8*x) + x + e^(10*x) + 33/4*e^(8*x)

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mupad [B]  time = 5.39, size = 28, normalized size = 0.97 \begin {gather*} x+8\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+2\,x\,{\mathrm {e}}^{8\,x}-2\,x^2\,{\mathrm {e}}^{8\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(8*x)*(12*x + 10*exp(2*x) - 16*x^2 + 66) + 1,x)

[Out]

x + 8*exp(8*x) + exp(10*x) + 2*x*exp(8*x) - 2*x^2*exp(8*x)

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sympy [A]  time = 0.14, size = 20, normalized size = 0.69 \begin {gather*} x + \left (- 2 x^{2} + 2 x + 8\right ) e^{8 x} + e^{10 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*exp(x)**2-16*x**2+12*x+66)*exp(8*x)+1,x)

[Out]

x + (-2*x**2 + 2*x + 8)*exp(8*x) + exp(10*x)

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