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

Optimal. Leaf size=13 $$x+{(-16e^x+6x)}^2$$

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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} $$\begin{array}{c}36x^2+x+192e^x+256e^{2x}-192e^x(x+1)\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =x+36x^2+512\int e^{2x}dx+\int e^x(-192-192x)dx\\ & =256e^{2x}+x+36x^2-192e^x(1+x)+192\int e^{x}dx\\ & =192e^x+256e^{2x}+x+36x^2-192e^x(1+x)\end{array}\end{array}$$

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

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 $$\begin{array}{c}36x^2-192x{e}^x+x+256e^{\left(2x\right)}\end{array}$$

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 $$\begin{array}{c}36x^2-192x{e}^x+x+256e^{\left(2x\right)}\end{array}$$

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	$x-192{\mathrm{e}}^{x}x+36x^2+256{\mathrm{e}}^{2x}$	$19$
norman	$x-192{\mathrm{e}}^{x}x+36x^2+256{\mathrm{e}}^{2x}$	$19$
risch	$x-192{\mathrm{e}}^{x}x+36x^2+256{\mathrm{e}}^{2x}$	$19$

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 $$\begin{array}{c}36x^2-192x{e}^x+x+256e^{\left(2x\right)}\end{array}$$

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 $$\begin{array}{c}x+256{\mathrm{e}}^{2x}-192x{\mathrm{e}}^x+36x^2\end{array}$$

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 $$\begin{array}{c}36x^2-192x{e}^x+x+256e^{2x}\end{array}$$

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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