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

Optimal. Leaf size=16 \[ -2+5 e^x+x-(-3+x) x^2 \]

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.06, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2194} \begin {gather*} -x^3+3 x^2+x+5 e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + 5*E^x + 6*x - 3*x^2,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.06 \begin {gather*} 5 e^x+x+3 x^2-x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + 5*E^x + 6*x - 3*x^2,x]

[Out]

5*E^x + x + 3*x^2 - x^3

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fricas [A]  time = 0.64, size = 16, normalized size = 1.00 \begin {gather*} -x^{3} + 3 \, x^{2} + x + 5 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^3 + 3*x^2 + x + 5*e^x

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giac [A]  time = 0.35, size = 16, normalized size = 1.00 \begin {gather*} -x^{3} + 3 \, x^{2} + x + 5 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^3 + 3*x^2 + x + 5*e^x

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maple [A]  time = 0.02, size = 17, normalized size = 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	\(-x^{3}+3 x^{2}+x +5 \,{\mathrm e}^{x}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 16, normalized size = 1.00 \begin {gather*} -x^{3} + 3 \, x^{2} + x + 5 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^3 + 3*x^2 + x + 5*e^x

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mupad [B]  time = 0.22, size = 16, normalized size = 1.00 \begin {gather*} x+5\,{\mathrm {e}}^x+3\,x^2-x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6*x + 5*exp(x) - 3*x^2 + 1,x)

[Out]

x + 5*exp(x) + 3*x^2 - x^3

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sympy [A]  time = 0.07, size = 14, normalized size = 0.88 \begin {gather*} - x^{3} + 3 x^{2} + x + 5 e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**3 + 3*x**2 + x + 5*exp(x)

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