3.84.77 \(\int \frac {3 x^2+3 x^4-e x^4}{1+6 x^2+4 x^3+9 x^4+e^2 x^4+12 x^5+4 x^6+e (-2 x^2-6 x^4-4 x^5)} \, dx\)

Optimal. Leaf size=15 \[ \frac {x}{3-e+\frac {1}{x^2}+2 x} \]

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Rubi [B]  time = 0.37, antiderivative size = 53, normalized size of antiderivative = 3.53, number of steps used = 6, number of rules used = 5, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {6, 1593, 6688, 2102, 1588} \begin {gather*} -\frac {(3-e) x^2}{2 \left (2 x^3+(3-e) x^2+1\right )}-\frac {1}{2 \left (2 x^3+(3-e) x^2+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3*x^2 + 3*x^4 - E*x^4)/(1 + 6*x^2 + 4*x^3 + 9*x^4 + E^2*x^4 + 12*x^5 + 4*x^6 + E*(-2*x^2 - 6*x^4 - 4*x^5)
),x]

[Out]

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

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 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rule 1593

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

Rule 2102

Int[(Pm_)*(Qn_)^(p_.), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*x^(m - n
+ 1)*Qn^(p + 1))/((m + n*p + 1)*Coeff[Qn, x, n]), x] + Dist[1/((m + n*p + 1)*Coeff[Qn, x, n]), Int[ExpandToSum
[(m + n*p + 1)*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*x^(m - n)*((m - n + 1)*Qn + (p + 1)*x*D[Qn, x]), x]*Qn^p,
x], x] /; LtQ[1, n, m + 1] && m + n*p + 1 < 0] /; FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && LtQ[p, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 x^2+(3-e) x^4}{1+6 x^2+4 x^3+9 x^4+e^2 x^4+12 x^5+4 x^6+e \left (-2 x^2-6 x^4-4 x^5\right )} \, dx\\ &=\int \frac {3 x^2+(3-e) x^4}{1+6 x^2+4 x^3+\left (9+e^2\right ) x^4+12 x^5+4 x^6+e \left (-2 x^2-6 x^4-4 x^5\right )} \, dx\\ &=\int \frac {x^2 \left (3+(3-e) x^2\right )}{1+6 x^2+4 x^3+\left (9+e^2\right ) x^4+12 x^5+4 x^6+e \left (-2 x^2-6 x^4-4 x^5\right )} \, dx\\ &=\int \frac {x^2 \left (3-(-3+e) x^2\right )}{\left (1+(3-e) x^2+2 x^3\right )^2} \, dx\\ &=-\frac {(3-e) x^2}{2 \left (1+(3-e) x^2+2 x^3\right )}-\frac {1}{2} \int \frac {-2 (3-e) x-6 x^2}{\left (1+(3-e) x^2+2 x^3\right )^2} \, dx\\ &=-\frac {1}{2 \left (1+(3-e) x^2+2 x^3\right )}-\frac {(3-e) x^2}{2 \left (1+(3-e) x^2+2 x^3\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 27, normalized size = 1.80 \begin {gather*} \frac {-1+(-3+e) x^2}{2-2 (-3+e) x^2+4 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x^2 + 3*x^4 - E*x^4)/(1 + 6*x^2 + 4*x^3 + 9*x^4 + E^2*x^4 + 12*x^5 + 4*x^6 + E*(-2*x^2 - 6*x^4 -
4*x^5)),x]

[Out]

(-1 + (-3 + E)*x^2)/(2 - 2*(-3 + E)*x^2 + 4*x^3)

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fricas [B]  time = 0.52, size = 36, normalized size = 2.40 \begin {gather*} \frac {x^{2} e - 3 \, x^{2} - 1}{2 \, {\left (2 \, x^{3} - x^{2} e + 3 \, x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4*exp(1)+3*x^4+3*x^2)/(x^4*exp(1)^2+(-4*x^5-6*x^4-2*x^2)*exp(1)+4*x^6+12*x^5+9*x^4+4*x^3+6*x^2+1
),x, algorithm="fricas")

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4*exp(1)+3*x^4+3*x^2)/(x^4*exp(1)^2+(-4*x^5-6*x^4-2*x^2)*exp(1)+4*x^6+12*x^5+9*x^4+4*x^3+6*x^2+1
),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.23, size = 34, normalized size = 2.27

method	result	size
norman	\(\frac {\left (\frac {3}{2}-\frac {{\mathrm e}}{2}\right ) x^{2}+\frac {1}{2}}{x^{2} {\mathrm e}-2 x^{3}-3 x^{2}-1}\)	\(34\)
risch	\(\frac {\left (\frac {3}{2}-\frac {{\mathrm e}}{2}\right ) x^{2}+\frac {1}{2}}{x^{2} {\mathrm e}-2 x^{3}-3 x^{2}-1}\)	\(34\)
gosper	\(-\frac {x^{2} {\mathrm e}-3 x^{2}-1}{2 \left (x^{2} {\mathrm e}-2 x^{3}-3 x^{2}-1\right )}\)	\(36\)
default	\(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (1+4 \textit {\_Z}^{6}+\left (-4 \,{\mathrm e}+12\right ) \textit {\_Z}^{5}+\left ({\mathrm e}^{2}-6 \,{\mathrm e}+9\right ) \textit {\_Z}^{4}+4 \textit {\_Z}^{3}+\left (-2 \,{\mathrm e}+6\right ) \textit {\_Z}^{2}\right )}{\sum }\frac {\left (\left ({\mathrm e}-3\right ) \textit {\_R}^{4}-3 \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3} {\mathrm e}^{2}-5 \textit {\_R}^{4} {\mathrm e}+6 \textit {\_R}^{5}-6 \textit {\_R}^{3} {\mathrm e}+15 \textit {\_R}^{4}+9 \textit {\_R}^{3}-\textit {\_R} \,{\mathrm e}+3 \textit {\_R}^{2}+3 \textit {\_R}}\right )}{4}\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^4*exp(1)+3*x^4+3*x^2)/(x^4*exp(1)^2+(-4*x^5-6*x^4-2*x^2)*exp(1)+4*x^6+12*x^5+9*x^4+4*x^3+6*x^2+1),x,me
thod=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.36, size = 30, normalized size = 2.00 \begin {gather*} \frac {x^{2} {\left (e - 3\right )} - 1}{2 \, {\left (2 \, x^{3} - x^{2} {\left (e - 3\right )} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4*exp(1)+3*x^4+3*x^2)/(x^4*exp(1)^2+(-4*x^5-6*x^4-2*x^2)*exp(1)+4*x^6+12*x^5+9*x^4+4*x^3+6*x^2+1
),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.13, size = 31, normalized size = 2.07 \begin {gather*} \frac {x^2\,\left (\frac {\mathrm {e}}{2}-\frac {3}{2}\right )-\frac {1}{2}}{2\,x^3+\left (3-\mathrm {e}\right )\,x^2+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2 - x^4*exp(1) + 3*x^4)/(x^4*exp(2) - exp(1)*(2*x^2 + 6*x^4 + 4*x^5) + 6*x^2 + 4*x^3 + 9*x^4 + 12*x^5
+ 4*x^6 + 1),x)

[Out]

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

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sympy [A]  time = 1.11, size = 26, normalized size = 1.73 \begin {gather*} \frac {x^{2} \left (-3 + e\right ) - 1}{4 x^{3} + x^{2} \left (6 - 2 e\right ) + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**4*exp(1)+3*x**4+3*x**2)/(x**4*exp(1)**2+(-4*x**5-6*x**4-2*x**2)*exp(1)+4*x**6+12*x**5+9*x**4+4*
x**3+6*x**2+1),x)

[Out]

(x**2*(-3 + E) - 1)/(4*x**3 + x**2*(6 - 2*E) + 2)

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