3.24.54 \(\int \frac {1}{2} (-5+3 e^{-3 x/2}+4 x) \, dx\)

Optimal. Leaf size=19 \[ 1-e^{-3 x/2}-\frac {5 x}{2}+x^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2194} \begin {gather*} x^2-\frac {5 x}{2}-e^{-3 x/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 + 3/E^((3*x)/2) + 4*x)/2,x]

[Out]

-E^((-3*x)/2) - (5*x)/2 + 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}{2} \int \left (-5+3 e^{-3 x/2}+4 x\right ) \, dx\\ &=-\frac {5 x}{2}+x^2+\frac {3}{2} \int e^{-3 x/2} \, dx\\ &=-e^{-3 x/2}-\frac {5 x}{2}+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.16 \begin {gather*} \frac {1}{2} \left (-2 e^{-3 x/2}-5 x+2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + 3/E^((3*x)/2) + 4*x)/2,x]

[Out]

(-2/E^((3*x)/2) - 5*x + 2*x^2)/2

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fricas [A]  time = 0.58, size = 13, normalized size = 0.68 \begin {gather*} x^{2} - \frac {5}{2} \, x - e^{\left (-\frac {3}{2} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 - 5/2*x - e^(-3/2*x)

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giac [A]  time = 0.20, size = 13, normalized size = 0.68 \begin {gather*} x^{2} - \frac {5}{2} \, x - e^{\left (-\frac {3}{2} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 - 5/2*x - e^(-3/2*x)

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maple [A]  time = 0.07, size = 14, normalized size = 0.74

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.48, size = 13, normalized size = 0.68 \begin {gather*} x^{2} - \frac {5}{2} \, x - e^{\left (-\frac {3}{2} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 - 5/2*x - e^(-3/2*x)

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mupad [B]  time = 0.05, size = 13, normalized size = 0.68 \begin {gather*} x^2-{\mathrm {e}}^{-\frac {3\,x}{2}}-\frac {5\,x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 - exp(-(3*x)/2) - (5*x)/2

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sympy [A]  time = 0.08, size = 15, normalized size = 0.79 \begin {gather*} x^{2} - \frac {5 x}{2} - e^{- \frac {3 x}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2 - 5*x/2 - exp(-3*x/2)

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