\(\int \frac {-2-x+e^5 (-1024 x^5+2048 x^6-320 x^7-608 x^8-96 x^9)}{e^5 (256 x^2+128 x^3+16 x^4)} \, dx\) [7085]

Optimal result

Integrand size = 57, antiderivative size = 29 \[ \int \frac {-2-x+e^5 \left (-1024 x^5+2048 x^6-320 x^7-608 x^8-96 x^9\right )}{e^5 \left (256 x^2+128 x^3+16 x^4\right )} \, dx=\frac {1}{32 e^5 x (4+x)}-\left (x^2-x^3\right )^2 \]

[Out]

1/exp(5)/x/(128+32*x)-(-x^3+x^2)^2

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {12, 1608, 27, 1634} \[ \int \frac {-2-x+e^5 \left (-1024 x^5+2048 x^6-320 x^7-608 x^8-96 x^9\right )}{e^5 \left (256 x^2+128 x^3+16 x^4\right )} \, dx=-x^6+2 x^5-x^4-\frac {1}{128 e^5 (x+4)}+\frac {1}{128 e^5 x} \]

[In]

Int[(-2 - x + E^5*(-1024*x^5 + 2048*x^6 - 320*x^7 - 608*x^8 - 96*x^9))/(E^5*(256*x^2 + 128*x^3 + 16*x^4)),x]

[Out]

1/(128*E^5*x) - x^4 + 2*x^5 - x^6 - 1/(128*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 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-2-x+e^5 \left (-1024 x^5+2048 x^6-320 x^7-608 x^8-96 x^9\right )}{256 x^2+128 x^3+16 x^4} \, dx}{e^5} \\ & = \frac {\int \frac {-2-x+e^5 \left (-1024 x^5+2048 x^6-320 x^7-608 x^8-96 x^9\right )}{x^2 \left (256+128 x+16 x^2\right )} \, dx}{e^5} \\ & = \frac {\int \frac {-2-x+e^5 \left (-1024 x^5+2048 x^6-320 x^7-608 x^8-96 x^9\right )}{16 x^2 (4+x)^2} \, dx}{e^5} \\ & = \frac {\int \frac {-2-x+e^5 \left (-1024 x^5+2048 x^6-320 x^7-608 x^8-96 x^9\right )}{x^2 (4+x)^2} \, dx}{16 e^5} \\ & = \frac {\int \left (-\frac {1}{8 x^2}-64 e^5 x^3+160 e^5 x^4-96 e^5 x^5+\frac {1}{8 (4+x)^2}\right ) \, dx}{16 e^5} \\ & = \frac {1}{128 e^5 x}-x^4+2 x^5-x^6-\frac {1}{128 e^5 (4+x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {-2-x+e^5 \left (-1024 x^5+2048 x^6-320 x^7-608 x^8-96 x^9\right )}{e^5 \left (256 x^2+128 x^3+16 x^4\right )} \, dx=-\frac {16 e^5 (-1+x)^2 x^4-\frac {1}{2 x (4+x)}}{16 e^5} \]

[In]

Integrate[(-2 - x + E^5*(-1024*x^5 + 2048*x^6 - 320*x^7 - 608*x^8 - 96*x^9))/(E^5*(256*x^2 + 128*x^3 + 16*x^4)
),x]

[Out]

-1/16*(16*E^5*(-1 + x)^2*x^4 - 1/(2*x*(4 + x)))/E^5

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00

method	result	size
risch	\(-x^{6}+2 x^{5}-x^{4}+\frac {{\mathrm e}^{-5}}{32 x \left (4+x \right )}\)	\(29\)
norman	\(\frac {-4 x^{5}+7 x^{6}-2 x^{7}-x^{8}+\frac {{\mathrm e}^{-5}}{32}}{\left (4+x \right ) x}\)	\(37\)
default	\(\frac {{\mathrm e}^{-5} \left (-16 x^{6} {\mathrm e}^{5}+32 x^{5} {\mathrm e}^{5}-16 x^{4} {\mathrm e}^{5}+\frac {1}{8 x}-\frac {1}{8 \left (4+x \right )}\right )}{16}\)	\(41\)
gosper	\(-\frac {\left (32 x^{8} {\mathrm e}^{5}+64 x^{7} {\mathrm e}^{5}-224 x^{6} {\mathrm e}^{5}+128 x^{5} {\mathrm e}^{5}-1\right ) {\mathrm e}^{-5}}{32 x \left (4+x \right )}\)	\(45\)
parallelrisch	\(-\frac {\left (32 x^{8} {\mathrm e}^{5}+64 x^{7} {\mathrm e}^{5}-224 x^{6} {\mathrm e}^{5}+128 x^{5} {\mathrm e}^{5}-1\right ) {\mathrm e}^{-5}}{32 x \left (4+x \right )}\)	\(45\)

[In]

int(((-96*x^9-608*x^8-320*x^7+2048*x^6-1024*x^5)*exp(5)-x-2)/(16*x^4+128*x^3+256*x^2)/exp(5),x,method=_RETURNV
ERBOSE)

[Out]

-x^6+2*x^5-x^4+1/32*exp(-5)/x/(4+x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {-2-x+e^5 \left (-1024 x^5+2048 x^6-320 x^7-608 x^8-96 x^9\right )}{e^5 \left (256 x^2+128 x^3+16 x^4\right )} \, dx=-\frac {{\left (32 \, {\left (x^{8} + 2 \, x^{7} - 7 \, x^{6} + 4 \, x^{5}\right )} e^{5} - 1\right )} e^{\left (-5\right )}}{32 \, {\left (x^{2} + 4 \, x\right )}} \]

[In]

integrate(((-96*x^9-608*x^8-320*x^7+2048*x^6-1024*x^5)*exp(5)-x-2)/(16*x^4+128*x^3+256*x^2)/exp(5),x, algorith
m="fricas")

[Out]

-1/32*(32*(x^8 + 2*x^7 - 7*x^6 + 4*x^5)*e^5 - 1)*e^(-5)/(x^2 + 4*x)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-2-x+e^5 \left (-1024 x^5+2048 x^6-320 x^7-608 x^8-96 x^9\right )}{e^5 \left (256 x^2+128 x^3+16 x^4\right )} \, dx=- x^{6} + 2 x^{5} - x^{4} + \frac {1}{32 x^{2} e^{5} + 128 x e^{5}} \]

[In]

integrate(((-96*x**9-608*x**8-320*x**7+2048*x**6-1024*x**5)*exp(5)-x-2)/(16*x**4+128*x**3+256*x**2)/exp(5),x)

[Out]

-x**6 + 2*x**5 - x**4 + 1/(32*x**2*exp(5) + 128*x*exp(5))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {-2-x+e^5 \left (-1024 x^5+2048 x^6-320 x^7-608 x^8-96 x^9\right )}{e^5 \left (256 x^2+128 x^3+16 x^4\right )} \, dx=-\frac {1}{32} \, {\left (32 \, x^{6} e^{5} - 64 \, x^{5} e^{5} + 32 \, x^{4} e^{5} - \frac {1}{x^{2} + 4 \, x}\right )} e^{\left (-5\right )} \]

[In]

integrate(((-96*x^9-608*x^8-320*x^7+2048*x^6-1024*x^5)*exp(5)-x-2)/(16*x^4+128*x^3+256*x^2)/exp(5),x, algorith
m="maxima")

[Out]

-1/32*(32*x^6*e^5 - 64*x^5*e^5 + 32*x^4*e^5 - 1/(x^2 + 4*x))*e^(-5)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {-2-x+e^5 \left (-1024 x^5+2048 x^6-320 x^7-608 x^8-96 x^9\right )}{e^5 \left (256 x^2+128 x^3+16 x^4\right )} \, dx=-\frac {1}{32} \, {\left (32 \, x^{6} e^{5} - 64 \, x^{5} e^{5} + 32 \, x^{4} e^{5} - \frac {1}{x^{2} + 4 \, x}\right )} e^{\left (-5\right )} \]

[In]

integrate(((-96*x^9-608*x^8-320*x^7+2048*x^6-1024*x^5)*exp(5)-x-2)/(16*x^4+128*x^3+256*x^2)/exp(5),x, algorith
m="giac")

[Out]

-1/32*(32*x^6*e^5 - 64*x^5*e^5 + 32*x^4*e^5 - 1/(x^2 + 4*x))*e^(-5)

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {-2-x+e^5 \left (-1024 x^5+2048 x^6-320 x^7-608 x^8-96 x^9\right )}{e^5 \left (256 x^2+128 x^3+16 x^4\right )} \, dx=\frac {1}{2\,\left (16\,{\mathrm {e}}^5\,x^2+64\,{\mathrm {e}}^5\,x\right )}-x^4+2\,x^5-x^6 \]

[In]

int(-(exp(-5)*(x + exp(5)*(1024*x^5 - 2048*x^6 + 320*x^7 + 608*x^8 + 96*x^9) + 2))/(256*x^2 + 128*x^3 + 16*x^4
),x)

[Out]

1/(2*(64*x*exp(5) + 16*x^2*exp(5))) - x^4 + 2*x^5 - x^6