3.69.79 \(\int (1+512 e^{2 x}+e^x (-192-192 x)+72 x) \, dx\)

Optimal. Leaf size=13 \[ x+\left (-16 e^x+6 x\right )^2 \]

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Rubi [B]  time = 0.04, antiderivative size = 27, normalized size of antiderivative = 2.08, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2194, 2176} \begin {gather*} 36 x^2+x+192 e^x+256 e^{2 x}-192 e^x (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + 512*E^(2*x) + E^x*(-192 - 192*x) + 72*x,x]

[Out]

192*E^x + 256*E^(2*x) + x + 36*x^2 - 192*E^x*(1 + 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x+36 x^2+512 \int e^{2 x} \, dx+\int e^x (-192-192 x) \, dx\\ &=256 e^{2 x}+x+36 x^2-192 e^x (1+x)+192 \int e^x \, dx\\ &=192 e^x+256 e^{2 x}+x+36 x^2-192 e^x (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.54 \begin {gather*} 256 e^{2 x}+x-192 e^x x+36 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + 512*E^(2*x) + E^x*(-192 - 192*x) + 72*x,x]

[Out]

256*E^(2*x) + x - 192*E^x*x + 36*x^2

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fricas [A]  time = 2.21, size = 18, normalized size = 1.38 \begin {gather*} 36 \, x^{2} - 192 \, x e^{x} + x + 256 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(512*exp(x)^2+(-192*x-192)*exp(x)+72*x+1,x, algorithm="fricas")

[Out]

36*x^2 - 192*x*e^x + x + 256*e^(2*x)

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giac [A]  time = 0.12, size = 18, normalized size = 1.38 \begin {gather*} 36 \, x^{2} - 192 \, x e^{x} + x + 256 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(512*exp(x)^2+(-192*x-192)*exp(x)+72*x+1,x, algorithm="giac")

[Out]

36*x^2 - 192*x*e^x + x + 256*e^(2*x)

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maple [A]  time = 0.02, size = 19, normalized size = 1.46

method	result	size
default	\(x -192 \,{\mathrm e}^{x} x +36 x^{2}+256 \,{\mathrm e}^{2 x}\)	\(19\)
norman	\(x -192 \,{\mathrm e}^{x} x +36 x^{2}+256 \,{\mathrm e}^{2 x}\)	\(19\)
risch	\(x -192 \,{\mathrm e}^{x} x +36 x^{2}+256 \,{\mathrm e}^{2 x}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(512*exp(x)^2+(-192*x-192)*exp(x)+72*x+1,x,method=_RETURNVERBOSE)

[Out]

x-192*exp(x)*x+36*x^2+256*exp(x)^2

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maxima [A]  time = 0.43, size = 18, normalized size = 1.38 \begin {gather*} 36 \, x^{2} - 192 \, x e^{x} + x + 256 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(512*exp(x)^2+(-192*x-192)*exp(x)+72*x+1,x, algorithm="maxima")

[Out]

36*x^2 - 192*x*e^x + x + 256*e^(2*x)

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mupad [B]  time = 4.10, size = 18, normalized size = 1.38 \begin {gather*} x+256\,{\mathrm {e}}^{2\,x}-192\,x\,{\mathrm {e}}^x+36\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(72*x + 512*exp(2*x) - exp(x)*(192*x + 192) + 1,x)

[Out]

x + 256*exp(2*x) - 192*x*exp(x) + 36*x^2

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sympy [A]  time = 0.10, size = 19, normalized size = 1.46 \begin {gather*} 36 x^{2} - 192 x e^{x} + x + 256 e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(512*exp(x)**2+(-192*x-192)*exp(x)+72*x+1,x)

[Out]

36*x**2 - 192*x*exp(x) + x + 256*exp(2*x)

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