3.50.61 \(\int (1-e^x+2 e^4 x+3 x^2) \, dx\)

Optimal. Leaf size=24 \[ -e^x+x^2 \left (e^4+x+\frac {\frac {4}{3}+x}{x^2}\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 0.71, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2194} \begin {gather*} x^3+e^4 x^2+x-e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 - E^x + 2*E^4*x + 3*x^2,x]

[Out]

-E^x + x + E^4*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+e^4 x^2+x^3-\int e^x \, dx\\ &=-e^x+x+e^4 x^2+x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 0.71 \begin {gather*} -e^x+x+e^4 x^2+x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 - E^x + 2*E^4*x + 3*x^2,x]

[Out]

-E^x + x + E^4*x^2 + x^3

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fricas [A]  time = 1.11, size = 15, normalized size = 0.62 \begin {gather*} x^{3} + x^{2} e^{4} + x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3 + x^2*e^4 + x - e^x

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giac [A]  time = 0.20, size = 15, normalized size = 0.62 \begin {gather*} x^{3} + x^{2} e^{4} + x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3 + x^2*e^4 + x - e^x

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maple [A]  time = 0.02, size = 16, normalized size = 0.67

method	result	size
risch	\(x^{3}+x +x^{2} {\mathrm e}^{4}-{\mathrm e}^{x}\)	\(16\)
default	\(x^{3}+x +x^{2} {\mathrm e}^{4}-{\mathrm e}^{x}\)	\(18\)
norman	\(x^{3}+x +x^{2} {\mathrm e}^{4}-{\mathrm e}^{x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.34, size = 15, normalized size = 0.62 \begin {gather*} x^{3} + x^{2} e^{4} + x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3 + x^2*e^4 + x - e^x

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mupad [B]  time = 0.05, size = 15, normalized size = 0.62 \begin {gather*} x-{\mathrm {e}}^x+x^2\,{\mathrm {e}}^4+x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x*exp(4) - exp(x) + 3*x^2 + 1,x)

[Out]

x - exp(x) + x^2*exp(4) + x^3

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sympy [A]  time = 0.08, size = 14, normalized size = 0.58 \begin {gather*} x^{3} + x^{2} e^{4} + x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3 + x**2*exp(4) + x - exp(x)

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