3.74.9 \(\int (81+e^x+40 x) \, dx\)

Optimal. Leaf size=18 \[ -8+e^4+e^x+x+5 (4+2 x)^2 \]

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2194} \begin {gather*} 20 x^2+81 x+e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[81 + E^x + 40*x,x]

[Out]

E^x + 81*x + 20*x^2

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} &=81 x+20 x^2+\int e^x \, dx\\ &=e^x+81 x+20 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.67 \begin {gather*} e^x+81 x+20 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[81 + E^x + 40*x,x]

[Out]

E^x + 81*x + 20*x^2

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fricas [A]  time = 0.56, size = 11, normalized size = 0.61 \begin {gather*} 20 \, x^{2} + 81 \, x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)+40*x+81,x, algorithm="fricas")

[Out]

20*x^2 + 81*x + e^x

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giac [A]  time = 0.14, size = 11, normalized size = 0.61 \begin {gather*} 20 \, x^{2} + 81 \, x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)+40*x+81,x, algorithm="giac")

[Out]

20*x^2 + 81*x + e^x

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maple [A]  time = 0.01, size = 12, normalized size = 0.67

method	result	size
default	\(20 x^{2}+81 x +{\mathrm e}^{x}\)	\(12\)
norman	\(20 x^{2}+81 x +{\mathrm e}^{x}\)	\(12\)
risch	\(20 x^{2}+81 x +{\mathrm e}^{x}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)+40*x+81,x,method=_RETURNVERBOSE)

[Out]

20*x^2+81*x+exp(x)

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maxima [A]  time = 0.35, size = 11, normalized size = 0.61 \begin {gather*} 20 \, x^{2} + 81 \, x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)+40*x+81,x, algorithm="maxima")

[Out]

20*x^2 + 81*x + e^x

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mupad [B]  time = 0.04, size = 11, normalized size = 0.61 \begin {gather*} 81\,x+{\mathrm {e}}^x+20\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(40*x + exp(x) + 81,x)

[Out]

81*x + exp(x) + 20*x^2

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sympy [A]  time = 0.07, size = 10, normalized size = 0.56 \begin {gather*} 20 x^{2} + 81 x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)+40*x+81,x)

[Out]

20*x**2 + 81*x + exp(x)

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