3.56.68 \(\int \frac {663552 x-2097152 x^5}{6561+41472 x^2+107008 x^4+65536 e^{10} x^4+131072 x^6+65536 x^8+e^5 (41472 x^2+131072 x^4+131072 x^6)} \, dx\)

Optimal. Leaf size=19 \[ \frac {16}{1+e^5+\frac {81}{256 x^2}+x^2} \]

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Rubi [A]  time = 0.17, antiderivative size = 24, normalized size of antiderivative = 1.26, number of steps used = 5, number of rules used = 5, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6, 1593, 6688, 12, 1588} \begin {gather*} \frac {4096 x^2}{256 x^4+256 \left (1+e^5\right ) x^2+81} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(663552*x - 2097152*x^5)/(6561 + 41472*x^2 + 107008*x^4 + 65536*E^10*x^4 + 131072*x^6 + 65536*x^8 + E^5*(4
1472*x^2 + 131072*x^4 + 131072*x^6)),x]

[Out]

(4096*x^2)/(81 + 256*(1 + E^5)*x^2 + 256*x^4)

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 {663552 x-2097152 x^5}{6561+41472 x^2+\left (107008+65536 e^{10}\right ) x^4+131072 x^6+65536 x^8+e^5 \left (41472 x^2+131072 x^4+131072 x^6\right )} \, dx\\ &=\int \frac {x \left (663552-2097152 x^4\right )}{6561+41472 x^2+\left (107008+65536 e^{10}\right ) x^4+131072 x^6+65536 x^8+e^5 \left (41472 x^2+131072 x^4+131072 x^6\right )} \, dx\\ &=\int \frac {8192 x \left (81-256 x^4\right )}{\left (81+256 \left (1+e^5\right ) x^2+256 x^4\right )^2} \, dx\\ &=8192 \int \frac {x \left (81-256 x^4\right )}{\left (81+256 \left (1+e^5\right ) x^2+256 x^4\right )^2} \, dx\\ &=\frac {4096 x^2}{81+256 \left (1+e^5\right ) x^2+256 x^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 24, normalized size = 1.26 \begin {gather*} \frac {8192 x^2}{162+512 \left (1+e^5\right ) x^2+512 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(663552*x - 2097152*x^5)/(6561 + 41472*x^2 + 107008*x^4 + 65536*E^10*x^4 + 131072*x^6 + 65536*x^8 +
E^5*(41472*x^2 + 131072*x^4 + 131072*x^6)),x]

[Out]

(8192*x^2)/(162 + 512*(1 + E^5)*x^2 + 512*x^4)

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fricas [A]  time = 0.64, size = 26, normalized size = 1.37 \begin {gather*} \frac {4096 \, x^{2}}{256 \, x^{4} + 256 \, x^{2} e^{5} + 256 \, x^{2} + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2097152*x^5+663552*x)/(65536*x^4*exp(5)^2+(131072*x^6+131072*x^4+41472*x^2)*exp(5)+65536*x^8+13107
2*x^6+107008*x^4+41472*x^2+6561),x, algorithm="fricas")

[Out]

4096*x^2/(256*x^4 + 256*x^2*e^5 + 256*x^2 + 81)

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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((-2097152*x^5+663552*x)/(65536*x^4*exp(5)^2+(131072*x^6+131072*x^4+41472*x^2)*exp(5)+65536*x^8+13107
2*x^6+107008*x^4+41472*x^2+6561),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.18, size = 22, normalized size = 1.16

method	result	size
risch	\(\frac {16 x^{2}}{x^{4}+x^{2} {\mathrm e}^{5}+x^{2}+\frac {81}{256}}\)	\(22\)
gosper	\(\frac {4096 x^{2}}{256 x^{4}+256 x^{2} {\mathrm e}^{5}+256 x^{2}+81}\)	\(27\)
norman	\(\frac {4096 x^{2}}{256 x^{4}+256 x^{2} {\mathrm e}^{5}+256 x^{2}+81}\)	\(27\)
default	\(8 \left (\munderset {\textit {\_R} =\RootOf \left (6561+65536 \textit {\_Z}^{4}+\left (131072 \,{\mathrm e}^{5}+131072\right ) \textit {\_Z}^{3}+\left (65536 \,{\mathrm e}^{10}+131072 \,{\mathrm e}^{5}+107008\right ) \textit {\_Z}^{2}+\left (41472 \,{\mathrm e}^{5}+41472\right ) \textit {\_Z} \right )}{\sum }\frac {\left (-256 \textit {\_R}^{2}+81\right ) \ln \left (x^{2}-\textit {\_R} \right )}{81+512 \textit {\_R}^{3}+768 \textit {\_R}^{2} {\mathrm e}^{5}+256 \textit {\_R} \,{\mathrm e}^{10}+768 \textit {\_R}^{2}+512 \textit {\_R} \,{\mathrm e}^{5}+418 \textit {\_R} +81 \,{\mathrm e}^{5}}\right )\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2097152*x^5+663552*x)/(65536*x^4*exp(5)^2+(131072*x^6+131072*x^4+41472*x^2)*exp(5)+65536*x^8+131072*x^6+
107008*x^4+41472*x^2+6561),x,method=_RETURNVERBOSE)

[Out]

16*x^2/(x^4+x^2*exp(5)+x^2+81/256)

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maxima [A]  time = 0.36, size = 23, normalized size = 1.21 \begin {gather*} \frac {4096 \, x^{2}}{256 \, x^{4} + 256 \, x^{2} {\left (e^{5} + 1\right )} + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2097152*x^5+663552*x)/(65536*x^4*exp(5)^2+(131072*x^6+131072*x^4+41472*x^2)*exp(5)+65536*x^8+13107
2*x^6+107008*x^4+41472*x^2+6561),x, algorithm="maxima")

[Out]

4096*x^2/(256*x^4 + 256*x^2*(e^5 + 1) + 81)

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mupad [B]  time = 3.49, size = 24, normalized size = 1.26 \begin {gather*} \frac {4096\,x^2}{256\,x^4+\left (256\,{\mathrm {e}}^5+256\right )\,x^2+81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((663552*x - 2097152*x^5)/(65536*x^4*exp(10) + exp(5)*(41472*x^2 + 131072*x^4 + 131072*x^6) + 41472*x^2 + 1
07008*x^4 + 131072*x^6 + 65536*x^8 + 6561),x)

[Out]

(4096*x^2)/(x^2*(256*exp(5) + 256) + 256*x^4 + 81)

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sympy [A]  time = 0.92, size = 20, normalized size = 1.05 \begin {gather*} \frac {4096 x^{2}}{256 x^{4} + x^{2} \left (256 + 256 e^{5}\right ) + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2097152*x**5+663552*x)/(65536*x**4*exp(5)**2+(131072*x**6+131072*x**4+41472*x**2)*exp(5)+65536*x**
8+131072*x**6+107008*x**4+41472*x**2+6561),x)

[Out]

4096*x**2/(256*x**4 + x**2*(256 + 256*exp(5)) + 81)

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