\(\int \frac {32 e^8+128 x^4+e^4 (24+128 x^2)}{9 x^3+16 e^8 x^3+48 x^5+64 x^7+e^4 (24 x^3+64 x^5)} \, dx\) [2232]

Optimal result

Integrand size = 64, antiderivative size = 26 \[ \int \frac {32 e^8+128 x^4+e^4 \left (24+128 x^2\right )}{9 x^3+16 e^8 x^3+48 x^5+64 x^7+e^4 \left (24 x^3+64 x^5\right )} \, dx=-\frac {4}{4 x^2+\frac {3 x}{\frac {e^4}{x}+2 x}} \]

[Out]

-4/(3*x/(2*x+exp(4)/x)+4*x^2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {6, 2099, 267} \[ \int \frac {32 e^8+128 x^4+e^4 \left (24+128 x^2\right )}{9 x^3+16 e^8 x^3+48 x^5+64 x^7+e^4 \left (24 x^3+64 x^5\right )} \, dx=-\frac {4 e^4}{\left (3+4 e^4\right ) x^2}-\frac {24}{\left (3+4 e^4\right ) \left (8 x^2+4 e^4+3\right )} \]

[In]

Int[(32*E^8 + 128*x^4 + E^4*(24 + 128*x^2))/(9*x^3 + 16*E^8*x^3 + 48*x^5 + 64*x^7 + E^4*(24*x^3 + 64*x^5)),x]

[Out]

(-4*E^4)/((3 + 4*E^4)*x^2) - 24/((3 + 4*E^4)*(3 + 4*E^4 + 8*x^2))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {32 e^8+128 x^4+e^4 \left (24+128 x^2\right )}{\left (9+16 e^8\right ) x^3+48 x^5+64 x^7+e^4 \left (24 x^3+64 x^5\right )} \, dx \\ & = \int \left (\frac {8 e^4}{\left (3+4 e^4\right ) x^3}+\frac {384 x}{\left (3+4 e^4\right ) \left (3+4 e^4+8 x^2\right )^2}\right ) \, dx \\ & = -\frac {4 e^4}{\left (3+4 e^4\right ) x^2}+\frac {384 \int \frac {x}{\left (3+4 e^4+8 x^2\right )^2} \, dx}{3+4 e^4} \\ & = -\frac {4 e^4}{\left (3+4 e^4\right ) x^2}-\frac {24}{\left (3+4 e^4\right ) \left (3+4 e^4+8 x^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {32 e^8+128 x^4+e^4 \left (24+128 x^2\right )}{9 x^3+16 e^8 x^3+48 x^5+64 x^7+e^4 \left (24 x^3+64 x^5\right )} \, dx=-\frac {4 \left (e^4+2 x^2\right )}{x^2 \left (3+4 e^4+8 x^2\right )} \]

[In]

Integrate[(32*E^8 + 128*x^4 + E^4*(24 + 128*x^2))/(9*x^3 + 16*E^8*x^3 + 48*x^5 + 64*x^7 + E^4*(24*x^3 + 64*x^5
)),x]

[Out]

(-4*(E^4 + 2*x^2))/(x^2*(3 + 4*E^4 + 8*x^2))

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04

method	result	size
gosper	\(-\frac {4 \left (2 x^{2}+{\mathrm e}^{4}\right )}{x^{2} \left (8 x^{2}+4 \,{\mathrm e}^{4}+3\right )}\)	\(27\)
norman	\(\frac {-8 x^{2}-4 \,{\mathrm e}^{4}}{x^{2} \left (8 x^{2}+4 \,{\mathrm e}^{4}+3\right )}\)	\(28\)
risch	\(\frac {-8 x^{2}-4 \,{\mathrm e}^{4}}{x^{2} \left (8 x^{2}+4 \,{\mathrm e}^{4}+3\right )}\)	\(29\)
parallelrisch	\(\frac {-64 x^{2}-32 \,{\mathrm e}^{4}}{8 x^{2} \left (8 x^{2}+4 \,{\mathrm e}^{4}+3\right )}\)	\(29\)

[In]

int((32*exp(4)^2+(128*x^2+24)*exp(4)+128*x^4)/(16*x^3*exp(4)^2+(64*x^5+24*x^3)*exp(4)+64*x^7+48*x^5+9*x^3),x,m
ethod=_RETURNVERBOSE)

[Out]

-4/x^2*(2*x^2+exp(4))/(8*x^2+4*exp(4)+3)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {32 e^8+128 x^4+e^4 \left (24+128 x^2\right )}{9 x^3+16 e^8 x^3+48 x^5+64 x^7+e^4 \left (24 x^3+64 x^5\right )} \, dx=-\frac {4 \, {\left (2 \, x^{2} + e^{4}\right )}}{8 \, x^{4} + 4 \, x^{2} e^{4} + 3 \, x^{2}} \]

[In]

integrate((32*exp(4)^2+(128*x^2+24)*exp(4)+128*x^4)/(16*x^3*exp(4)^2+(64*x^5+24*x^3)*exp(4)+64*x^7+48*x^5+9*x^
3),x, algorithm="fricas")

[Out]

-4*(2*x^2 + e^4)/(8*x^4 + 4*x^2*e^4 + 3*x^2)

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {32 e^8+128 x^4+e^4 \left (24+128 x^2\right )}{9 x^3+16 e^8 x^3+48 x^5+64 x^7+e^4 \left (24 x^3+64 x^5\right )} \, dx=\frac {- 8 x^{2} - 4 e^{4}}{8 x^{4} + x^{2} \cdot \left (3 + 4 e^{4}\right )} \]

[In]

integrate((32*exp(4)**2+(128*x**2+24)*exp(4)+128*x**4)/(16*x**3*exp(4)**2+(64*x**5+24*x**3)*exp(4)+64*x**7+48*
x**5+9*x**3),x)

[Out]

(-8*x**2 - 4*exp(4))/(8*x**4 + x**2*(3 + 4*exp(4)))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {32 e^8+128 x^4+e^4 \left (24+128 x^2\right )}{9 x^3+16 e^8 x^3+48 x^5+64 x^7+e^4 \left (24 x^3+64 x^5\right )} \, dx=-\frac {4 \, {\left (2 \, x^{2} + e^{4}\right )}}{8 \, x^{4} + x^{2} {\left (4 \, e^{4} + 3\right )}} \]

[In]

integrate((32*exp(4)^2+(128*x^2+24)*exp(4)+128*x^4)/(16*x^3*exp(4)^2+(64*x^5+24*x^3)*exp(4)+64*x^7+48*x^5+9*x^
3),x, algorithm="maxima")

[Out]

-4*(2*x^2 + e^4)/(8*x^4 + x^2*(4*e^4 + 3))

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {32 e^8+128 x^4+e^4 \left (24+128 x^2\right )}{9 x^3+16 e^8 x^3+48 x^5+64 x^7+e^4 \left (24 x^3+64 x^5\right )} \, dx=-\frac {4 \, {\left (2 \, x^{2} + e^{4}\right )}}{8 \, x^{4} + 4 \, x^{2} e^{4} + 3 \, x^{2}} \]

[In]

integrate((32*exp(4)^2+(128*x^2+24)*exp(4)+128*x^4)/(16*x^3*exp(4)^2+(64*x^5+24*x^3)*exp(4)+64*x^7+48*x^5+9*x^
3),x, algorithm="giac")

[Out]

-4*(2*x^2 + e^4)/(8*x^4 + 4*x^2*e^4 + 3*x^2)

Mupad [B] (verification not implemented)

Time = 8.96 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {32 e^8+128 x^4+e^4 \left (24+128 x^2\right )}{9 x^3+16 e^8 x^3+48 x^5+64 x^7+e^4 \left (24 x^3+64 x^5\right )} \, dx=-\frac {4\,\left (2\,x^2+{\mathrm {e}}^4\right )}{x^2\,\left (8\,x^2+4\,{\mathrm {e}}^4+3\right )} \]

[In]

int((32*exp(8) + exp(4)*(128*x^2 + 24) + 128*x^4)/(exp(4)*(24*x^3 + 64*x^5) + 16*x^3*exp(8) + 9*x^3 + 48*x^5 +
64*x^7),x)

[Out]

-(4*(exp(4) + 2*x^2))/(x^2*(4*exp(4) + 8*x^2 + 3))