\(\int \frac {-67+4 e^5}{-17+34 x-17 x^2+e^5 (4-8 x+4 x^2)} \, dx\) [3882]

Optimal result

Integrand size = 34, antiderivative size = 23 \[ \int \frac {-67+4 e^5}{-17+34 x-17 x^2+e^5 \left (4-8 x+4 x^2\right )} \, dx=-2+\frac {1+\frac {50}{17-4 e^5}}{1-x} \]

[Out]

(25/(17/2-2*exp(5))+1)/(1-x)-2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {12, 2006, 27, 32} \[ \int \frac {-67+4 e^5}{-17+34 x-17 x^2+e^5 \left (4-8 x+4 x^2\right )} \, dx=\frac {67-4 e^5}{\left (17-4 e^5\right ) (1-x)} \]

[In]

Int[(-67 + 4*E^5)/(-17 + 34*x - 17*x^2 + E^5*(4 - 8*x + 4*x^2)),x]

[Out]

(67 - 4*E^5)/((17 - 4*E^5)*(1 - 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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2006

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && QuadraticQ[u, x] &&  !QuadraticMatch
Q[u, x]

Rubi steps \begin{align*} \text {integral}& = \left (-67+4 e^5\right ) \int \frac {1}{-17+34 x-17 x^2+e^5 \left (4-8 x+4 x^2\right )} \, dx \\ & = \left (-67+4 e^5\right ) \int \frac {1}{-17+4 e^5+2 \left (17-4 e^5\right ) x-\left (17-4 e^5\right ) x^2} \, dx \\ & = \left (-67+4 e^5\right ) \int \frac {1}{\left (-17+4 e^5\right ) (-1+x)^2} \, dx \\ & = \frac {\left (67-4 e^5\right ) \int \frac {1}{(-1+x)^2} \, dx}{17-4 e^5} \\ & = \frac {67-4 e^5}{\left (17-4 e^5\right ) (1-x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-67+4 e^5}{-17+34 x-17 x^2+e^5 \left (4-8 x+4 x^2\right )} \, dx=-\frac {-67+4 e^5}{\left (-17+4 e^5\right ) (-1+x)} \]

[In]

Integrate[(-67 + 4*E^5)/(-17 + 34*x - 17*x^2 + E^5*(4 - 8*x + 4*x^2)),x]

[Out]

-((-67 + 4*E^5)/((-17 + 4*E^5)*(-1 + x)))

Maple [A] (verified)

Time = 2.58 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {4 \,{\mathrm e}^{5}-67}{\left (4 \,{\mathrm e}^{5}-17\right ) \left (-1+x \right )}\)	\(22\)
norman	\(-\frac {4 \,{\mathrm e}^{5}-67}{\left (4 \,{\mathrm e}^{5}-17\right ) \left (-1+x \right )}\)	\(22\)
parallelrisch	\(-\frac {4 \,{\mathrm e}^{5}-67}{\left (4 \,{\mathrm e}^{5}-17\right ) \left (-1+x \right )}\)	\(22\)
gosper	\(-\frac {4 \,{\mathrm e}^{5}-67}{4 x \,{\mathrm e}^{5}-4 \,{\mathrm e}^{5}-17 x +17}\)	\(25\)
risch	\(-\frac {{\mathrm e}^{5}}{x \,{\mathrm e}^{5}-{\mathrm e}^{5}-\frac {17 x}{4}+\frac {17}{4}}+\frac {67}{4 \left (x \,{\mathrm e}^{5}-{\mathrm e}^{5}-\frac {17 x}{4}+\frac {17}{4}\right )}\)	\(38\)
meijerg	\(\frac {4 \,{\mathrm e}^{5} x}{\left (4 \,{\mathrm e}^{5}-17\right ) \left (1-x \right )}-\frac {67 x}{\left (4 \,{\mathrm e}^{5}-17\right ) \left (1-x \right )}\)	\(40\)

[In]

int((4*exp(5)-67)/((4*x^2-8*x+4)*exp(5)-17*x^2+34*x-17),x,method=_RETURNVERBOSE)

[Out]

-(4*exp(5)-67)/(4*exp(5)-17)/(-1+x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-67+4 e^5}{-17+34 x-17 x^2+e^5 \left (4-8 x+4 x^2\right )} \, dx=-\frac {4 \, e^{5} - 67}{4 \, {\left (x - 1\right )} e^{5} - 17 \, x + 17} \]

[In]

integrate((4*exp(5)-67)/((4*x^2-8*x+4)*exp(5)-17*x^2+34*x-17),x, algorithm="fricas")

[Out]

-(4*e^5 - 67)/(4*(x - 1)*e^5 - 17*x + 17)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-67+4 e^5}{-17+34 x-17 x^2+e^5 \left (4-8 x+4 x^2\right )} \, dx=- \frac {-67 + 4 e^{5}}{x \left (-17 + 4 e^{5}\right ) - 4 e^{5} + 17} \]

[In]

integrate((4*exp(5)-67)/((4*x**2-8*x+4)*exp(5)-17*x**2+34*x-17),x)

[Out]

-(-67 + 4*exp(5))/(x*(-17 + 4*exp(5)) - 4*exp(5) + 17)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {-67+4 e^5}{-17+34 x-17 x^2+e^5 \left (4-8 x+4 x^2\right )} \, dx=-\frac {4 \, e^{5} - 67}{x {\left (4 \, e^{5} - 17\right )} - 4 \, e^{5} + 17} \]

[In]

integrate((4*exp(5)-67)/((4*x^2-8*x+4)*exp(5)-17*x^2+34*x-17),x, algorithm="maxima")

[Out]

-(4*e^5 - 67)/(x*(4*e^5 - 17) - 4*e^5 + 17)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-67+4 e^5}{-17+34 x-17 x^2+e^5 \left (4-8 x+4 x^2\right )} \, dx=-\frac {4 \, e^{5} - 67}{{\left (x - 1\right )} {\left (4 \, e^{5} - 17\right )}} \]

[In]

integrate((4*exp(5)-67)/((4*x^2-8*x+4)*exp(5)-17*x^2+34*x-17),x, algorithm="giac")

[Out]

-(4*e^5 - 67)/((x - 1)*(4*e^5 - 17))

Mupad [B] (verification not implemented)

Time = 9.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-67+4 e^5}{-17+34 x-17 x^2+e^5 \left (4-8 x+4 x^2\right )} \, dx=-\frac {4\,{\mathrm {e}}^5-67}{\left (4\,{\mathrm {e}}^5-17\right )\,\left (x-1\right )} \]

[In]

int((4*exp(5) - 67)/(34*x + exp(5)*(4*x^2 - 8*x + 4) - 17*x^2 - 17),x)

[Out]

-(4*exp(5) - 67)/((4*exp(5) - 17)*(x - 1))