3.69.50 $\int -\frac{288e^{15}{x}^2}{e^{30}+32e^{15}{x}^3+256x^6}dx$

Optimal. Leaf size=15 $$\frac{6x}{x+\frac{16x^4}{e^{15}}}$$

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, integrand size = 27, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.111, Rules used = {12, 28, 261} $$\begin{array}{c}\frac{6e^{15}}{16x^3+e^{15}}\end{array}$$

Antiderivative was successfully verified.

[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 261

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =-(\left(288e^{15}\right)\int \frac{x^2}{e^{30}+32e^{15}{x}^3+256x^6}dx)\\ & =-(\left(73728e^{15}\right)\int \frac{x^2}{{(16e^{15}+256x^3)}^2}dx)\\ & =\frac{6e^{15}}{e^{15}+16x^3}\end{array}\end{array}$$

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.07 $$\begin{array}{c}\frac{6e^{15}}{e^{15}+16x^3}\end{array}$$

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.50, size = 14, normalized size = 0.93 $$\begin{array}{c}\frac{6e^{15}}{16x^3+e^{15}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\text{Exception raised: NotImplementedError}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -288*exp(15)/48/sqrt(-exp(15)^2+exp(30))
*atan((16*sageVARx^3+exp(15))/sqrt(-exp(15)^2+exp(30)))

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maple [A]  time = 0.05, size = 15, normalized size = 1.00

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.36, size = 14, normalized size = 0.93 $$\begin{array}{c}\frac{6e^{15}}{16x^3+e^{15}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.12, size = 14, normalized size = 0.93 $$\begin{array}{c}\frac{6{\mathrm{e}}^{15}}{16x^3+{\mathrm{e}}^{15}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.15, size = 14, normalized size = 0.93 $$\begin{array}{c}\frac{288e^{15}}{768x^3+48e^{15}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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))

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