3.32.20 \(\int \frac {8+e^{\frac {3}{4+2 x}} (9-3 x)+8 x+2 x^2}{-24-16 x+2 x^2+2 x^3} \, dx\)

Optimal. Leaf size=17 \[ -3+e^{\frac {3}{4+2 x}}+\log (-3+x) \]

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Rubi [A]  time = 0.10, antiderivative size = 18, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 2, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {6688, 2209} \begin {gather*} e^{\frac {3}{2 (x+2)}}+\log (3-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8 + E^(3/(4 + 2*x))*(9 - 3*x) + 8*x + 2*x^2)/(-24 - 16*x + 2*x^2 + 2*x^3),x]

[Out]

E^(3/(2*(2 + x))) + Log[3 - x]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{-3+x}-\frac {3 e^{\frac {3}{4+2 x}}}{2 (2+x)^2}\right ) \, dx\\ &=\log (3-x)-\frac {3}{2} \int \frac {e^{\frac {3}{4+2 x}}}{(2+x)^2} \, dx\\ &=e^{\frac {3}{2 (2+x)}}+\log (3-x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 0.94 \begin {gather*} e^{\frac {3}{4+2 x}}+\log (-3+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 + E^(3/(4 + 2*x))*(9 - 3*x) + 8*x + 2*x^2)/(-24 - 16*x + 2*x^2 + 2*x^3),x]

[Out]

E^(3/(4 + 2*x)) + Log[-3 + x]

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fricas [A]  time = 0.62, size = 13, normalized size = 0.76 \begin {gather*} e^{\left (\frac {3}{2 \, {\left (x + 2\right )}}\right )} + \log \left (x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+9)*exp(3/(2*x+4))+2*x^2+8*x+8)/(2*x^3+2*x^2-16*x-24),x, algorithm="fricas")

[Out]

e^(3/2/(x + 2)) + log(x - 3)

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giac [B]  time = 0.29, size = 29, normalized size = 1.71 \begin {gather*} e^{\left (\frac {3}{2 \, {\left (x + 2\right )}}\right )} - \log \left (\frac {3}{2 \, {\left (x + 2\right )}}\right ) + \log \left (\frac {15}{x + 2} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+9)*exp(3/(2*x+4))+2*x^2+8*x+8)/(2*x^3+2*x^2-16*x-24),x, algorithm="giac")

[Out]

e^(3/2/(x + 2)) - log(3/2/(x + 2)) + log(15/(x + 2) - 3)

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maple [A]  time = 0.14, size = 14, normalized size = 0.82

method	result	size
risch	\(\ln \left (x -3\right )+{\mathrm e}^{\frac {3}{2 \left (2+x \right )}}\)	\(14\)
derivativedivides	\(\ln \left (-3+\frac {15}{2+x}\right )-\ln \left (\frac {3}{2 \left (2+x \right )}\right )+{\mathrm e}^{\frac {3}{2 \left (2+x \right )}}\)	\(30\)
default	\(\ln \left (-3+\frac {15}{2+x}\right )-\ln \left (\frac {3}{2 \left (2+x \right )}\right )+{\mathrm e}^{\frac {3}{2 \left (2+x \right )}}\)	\(30\)
norman	\(\frac {x \,{\mathrm e}^{\frac {3}{2 x +4}}+2 \,{\mathrm e}^{\frac {3}{2 x +4}}}{2+x}+\ln \left (x -3\right )\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x+9)*exp(3/(2*x+4))+2*x^2+8*x+8)/(2*x^3+2*x^2-16*x-24),x,method=_RETURNVERBOSE)

[Out]

ln(x-3)+exp(3/2/(2+x))

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maxima [A]  time = 0.51, size = 13, normalized size = 0.76 \begin {gather*} e^{\left (\frac {3}{2 \, {\left (x + 2\right )}}\right )} + \log \left (x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+9)*exp(3/(2*x+4))+2*x^2+8*x+8)/(2*x^3+2*x^2-16*x-24),x, algorithm="maxima")

[Out]

e^(3/2/(x + 2)) + log(x - 3)

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mupad [B]  time = 1.90, size = 15, normalized size = 0.88 \begin {gather*} \ln \left (x-3\right )+{\mathrm {e}}^{\frac {3}{2\,x+4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x + 2*x^2 - exp(3/(2*x + 4))*(3*x - 9) + 8)/(16*x - 2*x^2 - 2*x^3 + 24),x)

[Out]

log(x - 3) + exp(3/(2*x + 4))

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sympy [A]  time = 0.15, size = 12, normalized size = 0.71 \begin {gather*} e^{\frac {3}{2 x + 4}} + \log {\left (x - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+9)*exp(3/(2*x+4))+2*x**2+8*x+8)/(2*x**3+2*x**2-16*x-24),x)

[Out]

exp(3/(2*x + 4)) + log(x - 3)

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