3.95.26 \(\int \frac {-37-16 x-2 x^2+e^5 (-16-8 x-x^2)}{16+8 x+x^2} \, dx\)

Optimal. Leaf size=17 \[ 2-\left (2+e^5\right ) x+\frac {5}{4+x} \]

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {27, 1850} \begin {gather*} \frac {5}{x+4}-\left (2+e^5\right ) x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-37 - 16*x - 2*x^2 + E^5*(-16 - 8*x - x^2))/(16 + 8*x + x^2),x]

[Out]

-((2 + E^5)*x) + 5/(4 + x)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-37-16 x-2 x^2+e^5 \left (-16-8 x-x^2\right )}{(4+x)^2} \, dx\\ &=\int \left (-2-e^5-\frac {5}{(4+x)^2}\right ) \, dx\\ &=-\left (\left (2+e^5\right ) x\right )+\frac {5}{4+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 1.06 \begin {gather*} \frac {5}{4+x}-\left (2+e^5\right ) (4+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-37 - 16*x - 2*x^2 + E^5*(-16 - 8*x - x^2))/(16 + 8*x + x^2),x]

[Out]

5/(4 + x) - (2 + E^5)*(4 + x)

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fricas [A]  time = 0.54, size = 27, normalized size = 1.59 \begin {gather*} -\frac {2 \, x^{2} + {\left (x^{2} + 4 \, x\right )} e^{5} + 8 \, x - 5}{x + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-8*x-16)*exp(5)-2*x^2-16*x-37)/(x^2+8*x+16),x, algorithm="fricas")

[Out]

-(2*x^2 + (x^2 + 4*x)*e^5 + 8*x - 5)/(x + 4)

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giac [A]  time = 0.18, size = 16, normalized size = 0.94 \begin {gather*} -x e^{5} - 2 \, x + \frac {5}{x + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-8*x-16)*exp(5)-2*x^2-16*x-37)/(x^2+8*x+16),x, algorithm="giac")

[Out]

-x*e^5 - 2*x + 5/(x + 4)

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maple [A]  time = 0.62, size = 17, normalized size = 1.00

method	result	size
default	\(-x \,{\mathrm e}^{5}-2 x +\frac {5}{4+x}\)	\(17\)
risch	\(-x \,{\mathrm e}^{5}-2 x +\frac {5}{4+x}\)	\(17\)
norman	\(\frac {\left (-{\mathrm e}^{5}-2\right ) x^{2}+37+16 \,{\mathrm e}^{5}}{4+x}\)	\(23\)
gosper	\(-\frac {x^{2} {\mathrm e}^{5}+2 x^{2}-16 \,{\mathrm e}^{5}-37}{4+x}\)	\(25\)
meijerg	\(-\frac {37 x}{16 \left (1+\frac {x}{4}\right )}+4 \left (-{\mathrm e}^{5}-2\right ) \left (\frac {x \left (\frac {3 x}{4}+6\right )}{3 x +12}-2 \ln \left (1+\frac {x}{4}\right )\right )+4 \left (-2 \,{\mathrm e}^{5}-4\right ) \left (-\frac {x}{4 \left (1+\frac {x}{4}\right )}+\ln \left (1+\frac {x}{4}\right )\right )-\frac {{\mathrm e}^{5} x}{1+\frac {x}{4}}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2-8*x-16)*exp(5)-2*x^2-16*x-37)/(x^2+8*x+16),x,method=_RETURNVERBOSE)

[Out]

-x*exp(5)-2*x+5/(4+x)

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maxima [A]  time = 0.35, size = 15, normalized size = 0.88 \begin {gather*} -x {\left (e^{5} + 2\right )} + \frac {5}{x + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-8*x-16)*exp(5)-2*x^2-16*x-37)/(x^2+8*x+16),x, algorithm="maxima")

[Out]

-x*(e^5 + 2) + 5/(x + 4)

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mupad [B]  time = 7.38, size = 15, normalized size = 0.88 \begin {gather*} \frac {5}{x+4}-x\,\left ({\mathrm {e}}^5+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*x + exp(5)*(8*x + x^2 + 16) + 2*x^2 + 37)/(8*x + x^2 + 16),x)

[Out]

5/(x + 4) - x*(exp(5) + 2)

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sympy [A]  time = 0.11, size = 10, normalized size = 0.59 \begin {gather*} - x \left (2 + e^{5}\right ) + \frac {5}{x + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2-8*x-16)*exp(5)-2*x**2-16*x-37)/(x**2+8*x+16),x)

[Out]

-x*(2 + exp(5)) + 5/(x + 4)

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