\(\int -\frac {288 e^{15} x^2}{e^{30}+32 e^{15} x^3+256 x^6} \, dx\) [6850]

Optimal result

Integrand size = 27, antiderivative size = 15 \[ \int -\frac {288 e^{15} x^2}{e^{30}+32 e^{15} x^3+256 x^6} \, dx=\frac {6 x}{x+\frac {16 x^4}{e^{15}}} \]

[Out]

6*x/(x+16*exp(-3)/exp(3)^4*x^4)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 28, 267} \[ \int -\frac {288 e^{15} x^2}{e^{30}+32 e^{15} x^3+256 x^6} \, dx=\frac {6 e^{15}}{16 x^3+e^{15}} \]

[In]

Int[(-288*E^15*x^2)/(E^30 + 32*E^15*x^3 + 256*x^6),x]

[Out]

(6*E^15)/(E^15 + 16*x^3)

Rule 12

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

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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]

Rubi steps \begin{align*} \text {integral}& = -\left (\left (288 e^{15}\right ) \int \frac {x^2}{e^{30}+32 e^{15} x^3+256 x^6} \, dx\right ) \\ & = -\left (\left (73728 e^{15}\right ) \int \frac {x^2}{\left (16 e^{15}+256 x^3\right )^2} \, dx\right ) \\ & = \frac {6 e^{15}}{e^{15}+16 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int -\frac {288 e^{15} x^2}{e^{30}+32 e^{15} x^3+256 x^6} \, dx=\frac {6 e^{15}}{e^{15}+16 x^3} \]

[In]

Integrate[(-288*E^15*x^2)/(E^30 + 32*E^15*x^3 + 256*x^6),x]

[Out]

(6*E^15)/(E^15 + 16*x^3)

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {6 \,{\mathrm e}^{15}}{{\mathrm e}^{15}+16 x^{3}}\)	\(15\)
gosper	\(\frac {6 \,{\mathrm e}^{15}}{{\mathrm e}^{15}+16 x^{3}}\)	\(19\)
norman	\(\frac {6 \,{\mathrm e}^{15}}{{\mathrm e}^{15}+16 x^{3}}\)	\(19\)
parallelrisch	\(\frac {6 \,{\mathrm e}^{15}}{{\mathrm e}^{15}+16 x^{3}}\)	\(19\)

[In]

int(-288*x^2*exp(3)^5/(exp(3)^10+32*x^3*exp(3)^5+256*x^6),x,method=_RETURNVERBOSE)

[Out]

6*exp(15)/(exp(15)+16*x^3)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int -\frac {288 e^{15} x^2}{e^{30}+32 e^{15} x^3+256 x^6} \, dx=\frac {6 \, e^{15}}{16 \, x^{3} + e^{15}} \]

[In]

integrate(-288*x^2*exp(3)^5/(exp(3)^10+32*x^3*exp(3)^5+256*x^6),x, algorithm="fricas")

[Out]

6*e^15/(16*x^3 + e^15)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int -\frac {288 e^{15} x^2}{e^{30}+32 e^{15} x^3+256 x^6} \, dx=\frac {288 e^{15}}{768 x^{3} + 48 e^{15}} \]

[In]

integrate(-288*x**2*exp(3)**5/(exp(3)**10+32*x**3*exp(3)**5+256*x**6),x)

[Out]

288*exp(15)/(768*x**3 + 48*exp(15))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int -\frac {288 e^{15} x^2}{e^{30}+32 e^{15} x^3+256 x^6} \, dx=\frac {6 \, e^{15}}{16 \, x^{3} + e^{15}} \]

[In]

integrate(-288*x^2*exp(3)^5/(exp(3)^10+32*x^3*exp(3)^5+256*x^6),x, algorithm="maxima")

[Out]

6*e^15/(16*x^3 + e^15)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int -\frac {288 e^{15} x^2}{e^{30}+32 e^{15} x^3+256 x^6} \, dx=\frac {6 \, e^{15}}{16 \, x^{3} + e^{15}} \]

[In]

integrate(-288*x^2*exp(3)^5/(exp(3)^10+32*x^3*exp(3)^5+256*x^6),x, algorithm="giac")

[Out]

6*e^15/(16*x^3 + e^15)

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int -\frac {288 e^{15} x^2}{e^{30}+32 e^{15} x^3+256 x^6} \, dx=\frac {6\,{\mathrm {e}}^{15}}{16\,x^3+{\mathrm {e}}^{15}} \]

[In]

int(-(288*x^2*exp(15))/(exp(30) + 32*x^3*exp(15) + 256*x^6),x)

[Out]

(6*exp(15))/(exp(15) + 16*x^3)