3.9.26 \(\int (e^{\frac {1}{2} (3+2 x)} (-3-3 x)+6 x) \, dx\)

Optimal. Leaf size=24 \[ 3 x \left (-e^{\frac {-2 x+x \left (\frac {7}{2}+x\right )}{x}}+x\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.46, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2176, 2194} \begin {gather*} 3 x^2+3 e^{\frac {1}{2} (2 x+3)}-3 e^{\frac {1}{2} (2 x+3)} (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((3 + 2*x)/2)*(-3 - 3*x) + 6*x,x]

[Out]

3*E^((3 + 2*x)/2) + 3*x^2 - 3*E^((3 + 2*x)/2)*(1 + x)

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.67 \begin {gather*} -3 e^{\frac {3}{2}+x} x+3 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^((3 + 2*x)/2)*(-3 - 3*x) + 6*x,x]

[Out]

-3*E^(3/2 + x)*x + 3*x^2

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fricas [A]  time = 0.58, size = 13, normalized size = 0.54 \begin {gather*} 3 \, x^{2} - 3 \, x e^{\left (x + \frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x-3)*exp(x+3/2)+6*x,x, algorithm="fricas")

[Out]

3*x^2 - 3*x*e^(x + 3/2)

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giac [A]  time = 0.37, size = 13, normalized size = 0.54 \begin {gather*} 3 \, x^{2} - 3 \, x e^{\left (x + \frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x-3)*exp(x+3/2)+6*x,x, algorithm="giac")

[Out]

3*x^2 - 3*x*e^(x + 3/2)

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maple [A]  time = 0.03, size = 14, normalized size = 0.58

method	result	size
norman	\(3 x^{2}-3 \,{\mathrm e}^{x +\frac {3}{2}} x\)	\(14\)
risch	\(3 x^{2}-3 \,{\mathrm e}^{x +\frac {3}{2}} x\)	\(14\)
default	\(-3 \,{\mathrm e}^{x +\frac {3}{2}} \left (x +\frac {3}{2}\right )+\frac {9 \,{\mathrm e}^{x +\frac {3}{2}}}{2}+3 x^{2}\)	\(22\)
derivativedivides	\(-9 x -\frac {27}{2}-3 \,{\mathrm e}^{x +\frac {3}{2}} \left (x +\frac {3}{2}\right )+\frac {9 \,{\mathrm e}^{x +\frac {3}{2}}}{2}+3 \left (x +\frac {3}{2}\right )^{2}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x-3)*exp(x+3/2)+6*x,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.40, size = 25, normalized size = 1.04 \begin {gather*} 3 \, x^{2} - 3 \, {\left (x e^{\frac {3}{2}} - e^{\frac {3}{2}}\right )} e^{x} - 3 \, e^{\left (x + \frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x-3)*exp(x+3/2)+6*x,x, algorithm="maxima")

[Out]

3*x^2 - 3*(x*e^(3/2) - e^(3/2))*e^x - 3*e^(x + 3/2)

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mupad [B]  time = 0.05, size = 13, normalized size = 0.54 \begin {gather*} 3\,x^2-3\,x\,{\mathrm {e}}^{3/2}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6*x - exp(x + 3/2)*(3*x + 3),x)

[Out]

3*x^2 - 3*x*exp(3/2)*exp(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x-3)*exp(x+3/2)+6*x,x)

[Out]

3*x**2 - 3*x*exp(x + 3/2)

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