\(\int (1+5 e^x+6 x-3 x^2) \, dx\) [22]

Optimal result

Integrand size = 15, antiderivative size = 16 \[ \int \left (1+5 e^x+6 x-3 x^2\right ) \, dx=-2+5 e^x+x-(-3+x) x^2 \]

[Out]

5*exp(x)-2-x^2*(-3+x)+x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2225} \[ \int \left (1+5 e^x+6 x-3 x^2\right ) \, dx=-x^3+3 x^2+x+5 e^x \]

[In]

Int[1 + 5*E^x + 6*x - 3*x^2,x]

[Out]

5*E^x + x + 3*x^2 - x^3

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+3 x^2-x^3+5 \int e^x \, dx \\ & = 5 e^x+x+3 x^2-x^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \left (1+5 e^x+6 x-3 x^2\right ) \, dx=5 e^x+x+3 x^2-x^3 \]

[In]

Integrate[1 + 5*E^x + 6*x - 3*x^2,x]

[Out]

5*E^x + x + 3*x^2 - x^3

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

method	result	size
default	\(-x^{3}+3 x^{2}+x +5 \,{\mathrm e}^{x}\)	\(17\)
norman	\(-x^{3}+3 x^{2}+x +5 \,{\mathrm e}^{x}\)	\(17\)
risch	\(-x^{3}+3 x^{2}+x +5 \,{\mathrm e}^{x}\)	\(17\)
parallelrisch	\(-x^{3}+3 x^{2}+x +5 \,{\mathrm e}^{x}\)	\(17\)
parts	\(-x^{3}+3 x^{2}+x +5 \,{\mathrm e}^{x}\)	\(17\)

[In]

int(5*exp(x)-3*x^2+6*x+1,x,method=_RETURNVERBOSE)

[Out]

-x^3+3*x^2+x+5*exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (1+5 e^x+6 x-3 x^2\right ) \, dx=-x^{3} + 3 \, x^{2} + x + 5 \, e^{x} \]

[In]

integrate(5*exp(x)-3*x^2+6*x+1,x, algorithm="fricas")

[Out]

-x^3 + 3*x^2 + x + 5*e^x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \left (1+5 e^x+6 x-3 x^2\right ) \, dx=- x^{3} + 3 x^{2} + x + 5 e^{x} \]

[In]

integrate(5*exp(x)-3*x**2+6*x+1,x)

[Out]

-x**3 + 3*x**2 + x + 5*exp(x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (1+5 e^x+6 x-3 x^2\right ) \, dx=-x^{3} + 3 \, x^{2} + x + 5 \, e^{x} \]

[In]

integrate(5*exp(x)-3*x^2+6*x+1,x, algorithm="maxima")

[Out]

-x^3 + 3*x^2 + x + 5*e^x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (1+5 e^x+6 x-3 x^2\right ) \, dx=-x^{3} + 3 \, x^{2} + x + 5 \, e^{x} \]

[In]

integrate(5*exp(x)-3*x^2+6*x+1,x, algorithm="giac")

[Out]

-x^3 + 3*x^2 + x + 5*e^x

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (1+5 e^x+6 x-3 x^2\right ) \, dx=x+5\,{\mathrm {e}}^x+3\,x^2-x^3 \]

[In]

int(6*x + 5*exp(x) - 3*x^2 + 1,x)

[Out]

x + 5*exp(x) + 3*x^2 - x^3