\(\int (5-4 e^{3+x}-2 e^{5/4} x+9 x^2) \, dx\) [4892]

Optimal result

Integrand size = 22, antiderivative size = 33 \[ \int \left (5-4 e^{3+x}-2 e^{5/4} x+9 x^2\right ) \, dx=-x \left (\frac {4 \left (5+e^{3+x}\right )}{x}+e^{5/4} x\right )+x \left (5+3 x^2\right ) \]

[Out]

x*(3*x^2+5)-x*(4*(5+exp(3+x))/x+x*exp(5/4))

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2225} \[ \int \left (5-4 e^{3+x}-2 e^{5/4} x+9 x^2\right ) \, dx=3 x^3-e^{5/4} x^2+5 x-4 e^{x+3} \]

[In]

Int[5 - 4*E^(3 + x) - 2*E^(5/4)*x + 9*x^2,x]

[Out]

-4*E^(3 + x) + 5*x - E^(5/4)*x^2 + 3*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}& = 5 x-e^{5/4} x^2+3 x^3-4 \int e^{3+x} \, dx \\ & = -4 e^{3+x}+5 x-e^{5/4} x^2+3 x^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \left (5-4 e^{3+x}-2 e^{5/4} x+9 x^2\right ) \, dx=-4 e^{3+x}+5 x-e^{5/4} x^2+3 x^3 \]

[In]

Integrate[5 - 4*E^(3 + x) - 2*E^(5/4)*x + 9*x^2,x]

[Out]

-4*E^(3 + x) + 5*x - E^(5/4)*x^2 + 3*x^3

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70

method	result	size
default	\(3 x^{3}+5 x -{\mathrm e}^{\frac {5}{4}} x^{2}-4 \,{\mathrm e}^{3+x}\)	\(23\)
norman	\(3 x^{3}+5 x -{\mathrm e}^{\frac {5}{4}} x^{2}-4 \,{\mathrm e}^{3+x}\)	\(23\)
risch	\(3 x^{3}+5 x -{\mathrm e}^{\frac {5}{4}} x^{2}-4 \,{\mathrm e}^{3+x}\)	\(23\)
parallelrisch	\(3 x^{3}+5 x -{\mathrm e}^{\frac {5}{4}} x^{2}-4 \,{\mathrm e}^{3+x}\)	\(23\)
parts	\(3 x^{3}+5 x -{\mathrm e}^{\frac {5}{4}} x^{2}-4 \,{\mathrm e}^{3+x}\)	\(23\)
derivativedivides	\(15+5 x +3 x^{3}-2 \,{\mathrm e}^{\frac {5}{4}} \left (-9-3 x +\frac {\left (3+x \right )^{2}}{2}\right )-4 \,{\mathrm e}^{3+x}\)	\(33\)

[In]

int(-4*exp(3+x)-2*x*exp(5/4)+9*x^2+5,x,method=_RETURNVERBOSE)

[Out]

3*x^3+5*x-exp(5/4)*x^2-4*exp(3+x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67 \[ \int \left (5-4 e^{3+x}-2 e^{5/4} x+9 x^2\right ) \, dx=3 \, x^{3} - x^{2} e^{\frac {5}{4}} + 5 \, x - 4 \, e^{\left (x + 3\right )} \]

[In]

integrate(-4*exp(3+x)-2*x*exp(5/4)+9*x^2+5,x, algorithm="fricas")

[Out]

3*x^3 - x^2*e^(5/4) + 5*x - 4*e^(x + 3)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67 \[ \int \left (5-4 e^{3+x}-2 e^{5/4} x+9 x^2\right ) \, dx=3 x^{3} - x^{2} e^{\frac {5}{4}} + 5 x - 4 e^{x + 3} \]

[In]

integrate(-4*exp(3+x)-2*x*exp(5/4)+9*x**2+5,x)

[Out]

3*x**3 - x**2*exp(5/4) + 5*x - 4*exp(x + 3)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67 \[ \int \left (5-4 e^{3+x}-2 e^{5/4} x+9 x^2\right ) \, dx=3 \, x^{3} - x^{2} e^{\frac {5}{4}} + 5 \, x - 4 \, e^{\left (x + 3\right )} \]

[In]

integrate(-4*exp(3+x)-2*x*exp(5/4)+9*x^2+5,x, algorithm="maxima")

[Out]

3*x^3 - x^2*e^(5/4) + 5*x - 4*e^(x + 3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67 \[ \int \left (5-4 e^{3+x}-2 e^{5/4} x+9 x^2\right ) \, dx=3 \, x^{3} - x^{2} e^{\frac {5}{4}} + 5 \, x - 4 \, e^{\left (x + 3\right )} \]

[In]

integrate(-4*exp(3+x)-2*x*exp(5/4)+9*x^2+5,x, algorithm="giac")

[Out]

3*x^3 - x^2*e^(5/4) + 5*x - 4*e^(x + 3)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67 \[ \int \left (5-4 e^{3+x}-2 e^{5/4} x+9 x^2\right ) \, dx=5\,x-4\,{\mathrm {e}}^{x+3}-x^2\,{\mathrm {e}}^{5/4}+3\,x^3 \]

[In]

int(9*x^2 - 2*x*exp(5/4) - 4*exp(x + 3) + 5,x)

[Out]

5*x - 4*exp(x + 3) - x^2*exp(5/4) + 3*x^3