3.42.4 \(\int \frac {1}{3} (253-3 e^x+96 x) \, dx\)

Optimal. Leaf size=23 \[ -e^x+\frac {x}{3}+4 (x+4 (2+x) (3+x)) \]

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 0.70, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2194} \begin {gather*} 16 x^2+\frac {253 x}{3}-e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(253 - 3*E^x + 96*x)/3,x]

[Out]

-E^x + (253*x)/3 + 16*x^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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} &=\frac {1}{3} \int \left (253-3 e^x+96 x\right ) \, dx\\ &=\frac {253 x}{3}+16 x^2-\int e^x \, dx\\ &=-e^x+\frac {253 x}{3}+16 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.70 \begin {gather*} -e^x+\frac {253 x}{3}+16 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(253 - 3*E^x + 96*x)/3,x]

[Out]

-E^x + (253*x)/3 + 16*x^2

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fricas [A]  time = 0.65, size = 13, normalized size = 0.57 \begin {gather*} 16 \, x^{2} + \frac {253}{3} \, x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(x)+32*x+253/3,x, algorithm="fricas")

[Out]

16*x^2 + 253/3*x - e^x

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giac [A]  time = 0.15, size = 13, normalized size = 0.57 \begin {gather*} 16 \, x^{2} + \frac {253}{3} \, x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(x)+32*x+253/3,x, algorithm="giac")

[Out]

16*x^2 + 253/3*x - e^x

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maple [A]  time = 0.02, size = 14, normalized size = 0.61

method	result	size
default	\(\frac {253 x}{3}+16 x^{2}-{\mathrm e}^{x}\)	\(14\)
norman	\(\frac {253 x}{3}+16 x^{2}-{\mathrm e}^{x}\)	\(14\)
risch	\(\frac {253 x}{3}+16 x^{2}-{\mathrm e}^{x}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(x)+32*x+253/3,x,method=_RETURNVERBOSE)

[Out]

253/3*x+16*x^2-exp(x)

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maxima [A]  time = 0.38, size = 13, normalized size = 0.57 \begin {gather*} 16 \, x^{2} + \frac {253}{3} \, x - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(x)+32*x+253/3,x, algorithm="maxima")

[Out]

16*x^2 + 253/3*x - e^x

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mupad [B]  time = 3.11, size = 13, normalized size = 0.57 \begin {gather*} \frac {253\,x}{3}-{\mathrm {e}}^x+16\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(32*x - exp(x) + 253/3,x)

[Out]

(253*x)/3 - exp(x) + 16*x^2

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sympy [A]  time = 0.07, size = 12, normalized size = 0.52 \begin {gather*} 16 x^{2} + \frac {253 x}{3} - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(x)+32*x+253/3,x)

[Out]

16*x**2 + 253*x/3 - exp(x)

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