\(\int (81+e^x+40 x) \, dx\) [7309]

Optimal result

Integrand size = 8, antiderivative size = 18 \[ \int \left (81+e^x+40 x\right ) \, dx=-8+e^4+e^x+x+5 (4+2 x)^2 \]

[Out]

5*(4+2*x)^2+exp(4)+exp(x)+x-8

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2225} \[ \int \left (81+e^x+40 x\right ) \, dx=20 x^2+81 x+e^x \]

[In]

Int[81 + E^x + 40*x,x]

[Out]

E^x + 81*x + 20*x^2

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}& = 81 x+20 x^2+\int e^x \, dx \\ & = e^x+81 x+20 x^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \left (81+e^x+40 x\right ) \, dx=e^x+81 x+20 x^2 \]

[In]

Integrate[81 + E^x + 40*x,x]

[Out]

E^x + 81*x + 20*x^2

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67

method	result	size
default	\(20 x^{2}+81 x +{\mathrm e}^{x}\)	\(12\)
norman	\(20 x^{2}+81 x +{\mathrm e}^{x}\)	\(12\)
risch	\(20 x^{2}+81 x +{\mathrm e}^{x}\)	\(12\)
parallelrisch	\(20 x^{2}+81 x +{\mathrm e}^{x}\)	\(12\)
parts	\(20 x^{2}+81 x +{\mathrm e}^{x}\)	\(12\)

[In]

int(exp(x)+40*x+81,x,method=_RETURNVERBOSE)

[Out]

20*x^2+81*x+exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \left (81+e^x+40 x\right ) \, dx=20 \, x^{2} + 81 \, x + e^{x} \]

[In]

integrate(exp(x)+40*x+81,x, algorithm="fricas")

[Out]

20*x^2 + 81*x + e^x

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.56 \[ \int \left (81+e^x+40 x\right ) \, dx=20 x^{2} + 81 x + e^{x} \]

[In]

integrate(exp(x)+40*x+81,x)

[Out]

20*x**2 + 81*x + exp(x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \left (81+e^x+40 x\right ) \, dx=20 \, x^{2} + 81 \, x + e^{x} \]

[In]

integrate(exp(x)+40*x+81,x, algorithm="maxima")

[Out]

20*x^2 + 81*x + e^x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \left (81+e^x+40 x\right ) \, dx=20 \, x^{2} + 81 \, x + e^{x} \]

[In]

integrate(exp(x)+40*x+81,x, algorithm="giac")

[Out]

20*x^2 + 81*x + e^x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \left (81+e^x+40 x\right ) \, dx=81\,x+{\mathrm {e}}^x+20\,x^2 \]

[In]

int(40*x + exp(x) + 81,x)

[Out]

81*x + exp(x) + 20*x^2