\(\int \frac {1}{9} (9+e^{\frac {1}{9} (160 x-32 x^2)} (160-64 x)) \, dx\) [4545]

Optimal result

Integrand size = 27, antiderivative size = 14 \[ \int \frac {1}{9} \left (9+e^{\frac {1}{9} \left (160 x-32 x^2\right )} (160-64 x)\right ) \, dx=e^{\frac {32}{9} (5-x) x}+x \]

[Out]

exp(x*(160/9-32/9*x))+x

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.21, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2276, 2268} \[ \int \frac {1}{9} \left (9+e^{\frac {1}{9} \left (160 x-32 x^2\right )} (160-64 x)\right ) \, dx=e^{\frac {160 x}{9}-\frac {32 x^2}{9}}+x \]

[In]

Int[(9 + E^((160*x - 32*x^2)/9)*(160 - 64*x))/9,x]

[Out]

E^((160*x)/9 - (32*x^2)/9) + x

Rule 12

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

Rule 2268

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] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 2276

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} \int \left (9+e^{\frac {1}{9} \left (160 x-32 x^2\right )} (160-64 x)\right ) \, dx \\ & = x+\frac {1}{9} \int e^{\frac {1}{9} \left (160 x-32 x^2\right )} (160-64 x) \, dx \\ & = x+\frac {1}{9} \int e^{\frac {160 x}{9}-\frac {32 x^2}{9}} (160-64 x) \, dx \\ & = e^{\frac {160 x}{9}-\frac {32 x^2}{9}}+x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{9} \left (9+e^{\frac {1}{9} \left (160 x-32 x^2\right )} (160-64 x)\right ) \, dx=e^{-\frac {32}{9} (-5+x) x}+x \]

[In]

Integrate[(9 + E^((160*x - 32*x^2)/9)*(160 - 64*x))/9,x]

[Out]

E^((-32*(-5 + x)*x)/9) + x

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71

method	result	size
risch	\(x +{\mathrm e}^{-\frac {32 \left (-5+x \right ) x}{9}}\)	\(10\)
default	\(x +{\mathrm e}^{-\frac {32}{9} x^{2}+\frac {160}{9} x}\)	\(13\)
norman	\(x +{\mathrm e}^{-\frac {32}{9} x^{2}+\frac {160}{9} x}\)	\(13\)
parallelrisch	\(x +{\mathrm e}^{-\frac {32}{9} x^{2}+\frac {160}{9} x}\)	\(13\)
parts	\(x +{\mathrm e}^{-\frac {32}{9} x^{2}+\frac {160}{9} x}\)	\(13\)

[In]

int(1/9*(-64*x+160)*exp(-32/9*x^2+160/9*x)+1,x,method=_RETURNVERBOSE)

[Out]

x+exp(-32/9*(-5+x)*x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{9} \left (9+e^{\frac {1}{9} \left (160 x-32 x^2\right )} (160-64 x)\right ) \, dx=x + e^{\left (-\frac {32}{9} \, x^{2} + \frac {160}{9} \, x\right )} \]

[In]

integrate(1/9*(-64*x+160)*exp(-32/9*x^2+160/9*x)+1,x, algorithm="fricas")

[Out]

x + e^(-32/9*x^2 + 160/9*x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{9} \left (9+e^{\frac {1}{9} \left (160 x-32 x^2\right )} (160-64 x)\right ) \, dx=x + e^{- \frac {32 x^{2}}{9} + \frac {160 x}{9}} \]

[In]

integrate(1/9*(-64*x+160)*exp(-32/9*x**2+160/9*x)+1,x)

[Out]

x + exp(-32*x**2/9 + 160*x/9)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{9} \left (9+e^{\frac {1}{9} \left (160 x-32 x^2\right )} (160-64 x)\right ) \, dx=x + e^{\left (-\frac {32}{9} \, x^{2} + \frac {160}{9} \, x\right )} \]

[In]

integrate(1/9*(-64*x+160)*exp(-32/9*x^2+160/9*x)+1,x, algorithm="maxima")

[Out]

x + e^(-32/9*x^2 + 160/9*x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{9} \left (9+e^{\frac {1}{9} \left (160 x-32 x^2\right )} (160-64 x)\right ) \, dx=x + e^{\left (-\frac {32}{9} \, x^{2} + \frac {160}{9} \, x\right )} \]

[In]

integrate(1/9*(-64*x+160)*exp(-32/9*x^2+160/9*x)+1,x, algorithm="giac")

[Out]

x + e^(-32/9*x^2 + 160/9*x)

Mupad [B] (verification not implemented)

Time = 12.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{9} \left (9+e^{\frac {1}{9} \left (160 x-32 x^2\right )} (160-64 x)\right ) \, dx=x+{\mathrm {e}}^{\frac {160\,x}{9}-\frac {32\,x^2}{9}} \]

[In]

int(1 - (exp((160*x)/9 - (32*x^2)/9)*(64*x - 160))/9,x)

[Out]

x + exp((160*x)/9 - (32*x^2)/9)