3.68.51 \(\int \frac {-34-8 x-x^2+e^5 (16+8 x+x^2)}{-16-8 x-x^2+e^5 (16+8 x+x^2)} \, dx\)

Optimal. Leaf size=23 \[ x-\frac {2 (5-x)}{\left (1-e^5\right ) (4+x)} \]

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Rubi [A]  time = 0.05, antiderivative size = 18, normalized size of antiderivative = 0.78, number of steps used = 5, number of rules used = 4, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {1984, 27, 12, 683} \begin {gather*} x-\frac {18}{\left (1-e^5\right ) (x+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-34 - 8*x - x^2 + E^5*(16 + 8*x + x^2))/(-16 - 8*x - x^2 + E^5*(16 + 8*x + x^2)),x]

[Out]

x - 18/((1 - E^5)*(4 + x))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 1984

Int[(u_)^(p_.)*(v_)^(q_.), x_Symbol] :> Int[ExpandToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{p, q}, x] &&
QuadraticQ[{u, v}, x] &&  !QuadraticMatchQ[{u, v}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 \left (17-8 e^5\right )-8 \left (1-e^5\right ) x-\left (1-e^5\right ) x^2}{-16 \left (1-e^5\right )-8 \left (1-e^5\right ) x+\left (-1+e^5\right ) x^2} \, dx\\ &=\int \frac {-2 \left (17-8 e^5\right )-8 \left (1-e^5\right ) x-\left (1-e^5\right ) x^2}{\left (-1+e^5\right ) (4+x)^2} \, dx\\ &=\frac {\int \frac {-2 \left (17-8 e^5\right )-8 \left (1-e^5\right ) x-\left (1-e^5\right ) x^2}{(4+x)^2} \, dx}{-1+e^5}\\ &=\frac {\int \left (-1+e^5-\frac {18}{(4+x)^2}\right ) \, dx}{-1+e^5}\\ &=x-\frac {18}{\left (1-e^5\right ) (4+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 1.22 \begin {gather*} -\frac {-\frac {18}{4+x}+\left (1-e^5\right ) (4+x)}{-1+e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-34 - 8*x - x^2 + E^5*(16 + 8*x + x^2))/(-16 - 8*x - x^2 + E^5*(16 + 8*x + x^2)),x]

[Out]

-((-18/(4 + x) + (1 - E^5)*(4 + x))/(-1 + E^5))

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fricas [A]  time = 0.59, size = 34, normalized size = 1.48 \begin {gather*} -\frac {x^{2} - {\left (x^{2} + 4 \, x\right )} e^{5} + 4 \, x - 18}{{\left (x + 4\right )} e^{5} - x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+8*x+16)*exp(5)-x^2-8*x-34)/((x^2+8*x+16)*exp(5)-x^2-8*x-16),x, algorithm="fricas")

[Out]

-(x^2 - (x^2 + 4*x)*e^5 + 4*x - 18)/((x + 4)*e^5 - x - 4)

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giac [A]  time = 0.17, size = 29, normalized size = 1.26 \begin {gather*} \frac {x e^{5} - x}{e^{5} - 1} + \frac {18}{{\left (x + 4\right )} {\left (e^{5} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+8*x+16)*exp(5)-x^2-8*x-34)/((x^2+8*x+16)*exp(5)-x^2-8*x-16),x, algorithm="giac")

[Out]

(x*e^5 - x)/(e^5 - 1) + 18/((x + 4)*(e^5 - 1))

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maple [A]  time = 0.10, size = 20, normalized size = 0.87

method	result	size
risch	\(x +\frac {18}{x \,{\mathrm e}^{5}+4 \,{\mathrm e}^{5}-x -4}\)	\(20\)
default	\(\frac {x \,{\mathrm e}^{5}-x +\frac {18}{4+x}}{{\mathrm e}^{5}-1}\)	\(23\)
norman	\(\frac {x^{2}-\frac {2 \left (8 \,{\mathrm e}^{5}-17\right )}{{\mathrm e}^{5}-1}}{4+x}\)	\(25\)
gosper	\(\frac {x^{2} {\mathrm e}^{5}-x^{2}-16 \,{\mathrm e}^{5}+34}{x \,{\mathrm e}^{5}+4 \,{\mathrm e}^{5}-x -4}\)	\(34\)
meijerg	\(-\frac {17 x}{8 \left ({\mathrm e}^{5}-1\right ) \left (1+\frac {x}{4}\right )}+\frac {x \left (\frac {3 x}{4}+6\right )}{3+\frac {3 x}{4}}-8 \ln \left (1+\frac {x}{4}\right )+\frac {\left (8 \,{\mathrm e}^{5}-8\right ) \left (-\frac {x}{4 \left (1+\frac {x}{4}\right )}+\ln \left (1+\frac {x}{4}\right )\right )}{{\mathrm e}^{5}-1}+\frac {{\mathrm e}^{5} x}{\left ({\mathrm e}^{5}-1\right ) \left (1+\frac {x}{4}\right )}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2+8*x+16)*exp(5)-x^2-8*x-34)/((x^2+8*x+16)*exp(5)-x^2-8*x-16),x,method=_RETURNVERBOSE)

[Out]

x+18/(x*exp(5)+4*exp(5)-x-4)

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maxima [A]  time = 0.40, size = 18, normalized size = 0.78 \begin {gather*} x + \frac {18}{x {\left (e^{5} - 1\right )} + 4 \, e^{5} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+8*x+16)*exp(5)-x^2-8*x-34)/((x^2+8*x+16)*exp(5)-x^2-8*x-16),x, algorithm="maxima")

[Out]

x + 18/(x*(e^5 - 1) + 4*e^5 - 4)

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mupad [B]  time = 0.09, size = 15, normalized size = 0.65 \begin {gather*} x+\frac {18}{\left ({\mathrm {e}}^5-1\right )\,\left (x+4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x - exp(5)*(8*x + x^2 + 16) + x^2 + 34)/(8*x - exp(5)*(8*x + x^2 + 16) + x^2 + 16),x)

[Out]

x + 18/((exp(5) - 1)*(x + 4))

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sympy [A]  time = 0.19, size = 15, normalized size = 0.65 \begin {gather*} x + \frac {18}{x \left (-1 + e^{5}\right ) - 4 + 4 e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2+8*x+16)*exp(5)-x**2-8*x-34)/((x**2+8*x+16)*exp(5)-x**2-8*x-16),x)

[Out]

x + 18/(x*(-1 + exp(5)) - 4 + 4*exp(5))

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