3.34.84 \(\int \frac {9 e^{2 e}+15 e^e x^2+2 x^4}{9 e^{2 e}+6 e^e x^2+x^4} \, dx\)

Optimal. Leaf size=24 \[ x+\frac {x}{-5+\frac {3 \left (\frac {e^e}{x}+2 x\right )}{x}} \]

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Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 4, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {28, 1157, 21, 8} \begin {gather*} 2 x-\frac {3 e^e x}{x^2+3 e^e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(9*E^(2*E) + 15*E^E*x^2 + 2*x^4)/(9*E^(2*E) + 6*E^E*x^2 + x^4),x]

[Out]

2*x - (3*E^E*x)/(3*E^E + x^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + 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 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 e^{2 e}+15 e^e x^2+2 x^4}{\left (3 e^e+x^2\right )^2} \, dx\\ &=-\frac {3 e^e x}{3 e^e+x^2}-\frac {1}{6} e^{-e} \int \frac {-36 e^{2 e}-12 e^e x^2}{3 e^e+x^2} \, dx\\ &=-\frac {3 e^e x}{3 e^e+x^2}+2 \int 1 \, dx\\ &=2 x-\frac {3 e^e x}{3 e^e+x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 21, normalized size = 0.88 \begin {gather*} 2 x-\frac {3 e^e x}{3 e^e+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9*E^(2*E) + 15*E^E*x^2 + 2*x^4)/(9*E^(2*E) + 6*E^E*x^2 + x^4),x]

[Out]

2*x - (3*E^E*x)/(3*E^E + x^2)

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fricas [A]  time = 0.47, size = 24, normalized size = 1.00 \begin {gather*} \frac {2 \, x^{3} + 3 \, x e^{e}}{x^{2} + 3 \, e^{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*exp(exp(1))^2+15*x^2*exp(exp(1))+2*x^4)/(9*exp(exp(1))^2+6*x^2*exp(exp(1))+x^4),x, algorithm="fri
cas")

[Out]

(2*x^3 + 3*x*e^e)/(x^2 + 3*e^e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*exp(exp(1))^2+15*x^2*exp(exp(1))+2*x^4)/(9*exp(exp(1))^2+6*x^2*exp(exp(1))+x^4),x, algorithm="gia
c")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (-sqrt(-3*sqrt(exp(exp(1))^2-exp(2*exp(1
)))+3*exp(exp(1)))*sqrt(-exp(2*exp(1))+exp(exp(1))^2)*exp(exp(1))+sqrt(-3*sqrt(exp(exp(1))^2-exp(2*exp(1)))+3*
exp(exp(1)))*exp(2*ex

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maple [A]  time = 0.06, size = 22, normalized size = 0.92

method	result	size
risch	\(2 x -\frac {x \,{\mathrm e}^{{\mathrm e}}}{\frac {x^{2}}{3}+{\mathrm e}^{{\mathrm e}}}\)	\(22\)
gosper	\(\frac {x \left (2 x^{2}+3 \,{\mathrm e}^{{\mathrm e}}\right )}{x^{2}+3 \,{\mathrm e}^{{\mathrm e}}}\)	\(25\)
norman	\(\frac {2 x^{3}+3 x \,{\mathrm e}^{{\mathrm e}}}{x^{2}+3 \,{\mathrm e}^{{\mathrm e}}}\)	\(25\)
default	\(2 x -\frac {{\mathrm e}^{2 \,{\mathrm e}} {\mathrm e}^{-{\mathrm e}} x}{\frac {x^{2}}{3}+{\mathrm e}^{{\mathrm e}}}\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*exp(exp(1))^2+15*x^2*exp(exp(1))+2*x^4)/(9*exp(exp(1))^2+6*x^2*exp(exp(1))+x^4),x,method=_RETURNVERBOSE
)

[Out]

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

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maxima [A]  time = 0.35, size = 21, normalized size = 0.88 \begin {gather*} 2 \, x - \frac {3 \, x e^{e}}{x^{2} + 3 \, e^{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*exp(exp(1))^2+15*x^2*exp(exp(1))+2*x^4)/(9*exp(exp(1))^2+6*x^2*exp(exp(1))+x^4),x, algorithm="max
ima")

[Out]

2*x - 3*x*e^e/(x^2 + 3*e^e)

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mupad [B]  time = 0.09, size = 24, normalized size = 1.00 \begin {gather*} \frac {x\,\left (2\,x^2+3\,{\mathrm {e}}^{\mathrm {e}}\right )}{x^2+3\,{\mathrm {e}}^{\mathrm {e}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*exp(2*exp(1)) + 15*x^2*exp(exp(1)) + 2*x^4)/(9*exp(2*exp(1)) + 6*x^2*exp(exp(1)) + x^4),x)

[Out]

(x*(3*exp(exp(1)) + 2*x^2))/(3*exp(exp(1)) + x^2)

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sympy [A]  time = 0.18, size = 20, normalized size = 0.83 \begin {gather*} 2 x - \frac {3 x e^{e}}{x^{2} + 3 e^{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*exp(exp(1))**2+15*x**2*exp(exp(1))+2*x**4)/(9*exp(exp(1))**2+6*x**2*exp(exp(1))+x**4),x)

[Out]

2*x - 3*x*exp(E)/(x**2 + 3*exp(E))

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