\(\int (1+512 e^{2 x}+e^x (-192-192 x)+72 x) \, dx\) [6879]

Optimal result

Integrand size = 21, antiderivative size = 13 \[ \int \left (1+512 e^{2 x}+e^x (-192-192 x)+72 x\right ) \, dx=x+\left (-16 e^x+6 x\right )^2 \]

[Out]

x+(6*x-16*exp(x))^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(27\) vs. \(2(13)=26\).

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.08, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2225, 2207} \[ \int \left (1+512 e^{2 x}+e^x (-192-192 x)+72 x\right ) \, dx=36 x^2+x+192 e^x+256 e^{2 x}-192 e^x (x+1) \]

[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 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.54 \[ \int \left (1+512 e^{2 x}+e^x (-192-192 x)+72 x\right ) \, dx=256 e^{2 x}+x-192 e^x x+36 x^2 \]

[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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 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	\(x -192 \,{\mathrm e}^{x} x +36 x^{2}+256 \,{\mathrm e}^{2 x}\)	\(19\)
parallelrisch	\(x -192 \,{\mathrm e}^{x} x +36 x^{2}+256 \,{\mathrm e}^{2 x}\)	\(19\)
parts	\(x -192 \,{\mathrm e}^{x} x +36 x^{2}+256 \,{\mathrm e}^{2 x}\)	\(19\)

[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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38 \[ \int \left (1+512 e^{2 x}+e^x (-192-192 x)+72 x\right ) \, dx=36 \, x^{2} - 192 \, x e^{x} + x + 256 \, e^{\left (2 \, x\right )} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.46 \[ \int \left (1+512 e^{2 x}+e^x (-192-192 x)+72 x\right ) \, dx=36 x^{2} - 192 x e^{x} + x + 256 e^{2 x} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38 \[ \int \left (1+512 e^{2 x}+e^x (-192-192 x)+72 x\right ) \, dx=36 \, x^{2} - 192 \, x e^{x} + x + 256 \, e^{\left (2 \, x\right )} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38 \[ \int \left (1+512 e^{2 x}+e^x (-192-192 x)+72 x\right ) \, dx=36 \, x^{2} - 192 \, x e^{x} + x + 256 \, e^{\left (2 \, x\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 12.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38 \[ \int \left (1+512 e^{2 x}+e^x (-192-192 x)+72 x\right ) \, dx=x+256\,{\mathrm {e}}^{2\,x}-192\,x\,{\mathrm {e}}^x+36\,x^2 \]

[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