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

Optimal. Leaf size=33 \[ -x \left (\frac {4 \left (5+e^{3+x}\right )}{x}+e^{5/4} x\right )+x \left (5+3 x^2\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 0.79, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2194} \begin {gather*} 3 x^3-e^{5/4} x^2+5 x-4 e^{x+3} \end {gather*}

Antiderivative was successfully verified.

[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 2194

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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 26, normalized size = 0.79 \begin {gather*} -4 e^{3+x}+5 x-e^{5/4} x^2+3 x^3 \end {gather*}

Antiderivative was successfully verified.

[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

________________________________________________________________________________________

fricas [A]  time = 1.20, size = 22, normalized size = 0.67 \begin {gather*} 3 \, x^{3} - x^{2} e^{\frac {5}{4}} + 5 \, x - 4 \, e^{\left (x + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [A]  time = 0.22, size = 22, normalized size = 0.67 \begin {gather*} 3 \, x^{3} - x^{2} e^{\frac {5}{4}} + 5 \, x - 4 \, e^{\left (x + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 23, normalized size = 0.70

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 22, normalized size = 0.67 \begin {gather*} 3 \, x^{3} - x^{2} e^{\frac {5}{4}} + 5 \, x - 4 \, e^{\left (x + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 3.32, size = 22, normalized size = 0.67 \begin {gather*} 5\,x-4\,{\mathrm {e}}^{x+3}-x^2\,{\mathrm {e}}^{5/4}+3\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 22, normalized size = 0.67 \begin {gather*} 3 x^{3} - x^{2} e^{\frac {5}{4}} + 5 x - 4 e^{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________