3.72.34 \(\int \frac {1}{2} (26+5 e^x-6 x) \, dx\)

Optimal. Leaf size=30 \[ 4+2 \left (e^x-(3-x)^2\right )+x+\frac {1}{2} \left (5+e^x+x^2\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 0.60, 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*} -\frac {3 x^2}{2}+13 x+\frac {5 e^x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(26 + 5*E^x - 6*x)/2,x]

[Out]

(5*E^x)/2 + 13*x - (3*x^2)/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}{2} \int \left (26+5 e^x-6 x\right ) \, dx\\ &=13 x-\frac {3 x^2}{2}+\frac {5 \int e^x \, dx}{2}\\ &=\frac {5 e^x}{2}+13 x-\frac {3 x^2}{2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 0.60 \begin {gather*} \frac {5 e^x}{2}+13 x-\frac {3 x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(26 + 5*E^x - 6*x)/2,x]

[Out]

(5*E^x)/2 + 13*x - (3*x^2)/2

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fricas [A]  time = 0.57, size = 13, normalized size = 0.43 \begin {gather*} -\frac {3}{2} \, x^{2} + 13 \, x + \frac {5}{2} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/2*exp(x)-3*x+13,x, algorithm="fricas")

[Out]

-3/2*x^2 + 13*x + 5/2*e^x

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giac [A]  time = 0.15, size = 13, normalized size = 0.43 \begin {gather*} -\frac {3}{2} \, x^{2} + 13 \, x + \frac {5}{2} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/2*exp(x)-3*x+13,x, algorithm="giac")

[Out]

-3/2*x^2 + 13*x + 5/2*e^x

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maple [A]  time = 0.01, size = 14, normalized size = 0.47

method	result	size
default	\(13 x -\frac {3 x^{2}}{2}+\frac {5 \,{\mathrm e}^{x}}{2}\)	\(14\)
norman	\(13 x -\frac {3 x^{2}}{2}+\frac {5 \,{\mathrm e}^{x}}{2}\)	\(14\)
risch	\(13 x -\frac {3 x^{2}}{2}+\frac {5 \,{\mathrm e}^{x}}{2}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5/2*exp(x)-3*x+13,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.35, size = 13, normalized size = 0.43 \begin {gather*} -\frac {3}{2} \, x^{2} + 13 \, x + \frac {5}{2} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/2*exp(x)-3*x+13,x, algorithm="maxima")

[Out]

-3/2*x^2 + 13*x + 5/2*e^x

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mupad [B]  time = 0.04, size = 13, normalized size = 0.43 \begin {gather*} 13\,x+\frac {5\,{\mathrm {e}}^x}{2}-\frac {3\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*exp(x))/2 - 3*x + 13,x)

[Out]

13*x + (5*exp(x))/2 - (3*x^2)/2

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sympy [A]  time = 0.08, size = 15, normalized size = 0.50 \begin {gather*} - \frac {3 x^{2}}{2} + 13 x + \frac {5 e^{x}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/2*exp(x)-3*x+13,x)

[Out]

-3*x**2/2 + 13*x + 5*exp(x)/2

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