3.101.74 \(\int \frac {16 (4-12 e^{16} x)^4}{81 e^{64} x^4 (-25 x+75 e^{16} x^2+\frac {(4-12 e^{16} x)^4 (10 x-30 e^{16} x^2)}{81 e^{64} x^4}+\frac {(4-12 e^{16} x)^8 (-x+3 e^{16} x^2)}{6561 e^{128} x^8})} \, dx\)

Optimal. Leaf size=25 \[ \frac {4}{5-\left (-4+\frac {4}{3 e^{16} x}\right )^4}+\log (15) \]

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Rubi [B]  time = 1.01, antiderivative size = 169, normalized size of antiderivative = 6.76, number of steps used = 7, number of rules used = 4, integrand size = 98, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {12, 6688, 2102, 1588} \begin {gather*} -\frac {110592 e^{48} x^3}{251 \left (20331 e^{64} x^4-27648 e^{48} x^3+13824 e^{32} x^2-3072 e^{16} x+256\right )}+\frac {55296 e^{32} x^2}{251 \left (20331 e^{64} x^4-27648 e^{48} x^3+13824 e^{32} x^2-3072 e^{16} x+256\right )}-\frac {12288 e^{16} x}{251 \left (20331 e^{64} x^4-27648 e^{48} x^3+13824 e^{32} x^2-3072 e^{16} x+256\right )}+\frac {1024}{251 \left (20331 e^{64} x^4-27648 e^{48} x^3+13824 e^{32} x^2-3072 e^{16} x+256\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(16*(4 - 12*E^16*x)^4)/(81*E^64*x^4*(-25*x + 75*E^16*x^2 + ((4 - 12*E^16*x)^4*(10*x - 30*E^16*x^2))/(81*E^
64*x^4) + ((4 - 12*E^16*x)^8*(-x + 3*E^16*x^2))/(6561*E^128*x^8))),x]

[Out]

1024/(251*(256 - 3072*E^16*x + 13824*E^32*x^2 - 27648*E^48*x^3 + 20331*E^64*x^4)) - (12288*E^16*x)/(251*(256 -
3072*E^16*x + 13824*E^32*x^2 - 27648*E^48*x^3 + 20331*E^64*x^4)) + (55296*E^32*x^2)/(251*(256 - 3072*E^16*x +
13824*E^32*x^2 - 27648*E^48*x^3 + 20331*E^64*x^4)) - (110592*E^48*x^3)/(251*(256 - 3072*E^16*x + 13824*E^32*x
^2 - 27648*E^48*x^3 + 20331*E^64*x^4))

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 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} &=\frac {16 \int \frac {\left (4-12 e^{16} x\right )^4}{x^4 \left (-25 x+75 e^{16} x^2+\frac {\left (4-12 e^{16} x\right )^4 \left (10 x-30 e^{16} x^2\right )}{81 e^{64} x^4}+\frac {\left (4-12 e^{16} x\right )^8 \left (-x+3 e^{16} x^2\right )}{6561 e^{128} x^8}\right )} \, dx}{81 e^{64}}\\ &=\frac {16 \int \frac {1679616 e^{128} x^3 \left (-1+3 e^{16} x\right )^3}{\left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )^2} \, dx}{81 e^{64}}\\ &=\left (331776 e^{64}\right ) \int \frac {x^3 \left (-1+3 e^{16} x\right )^3}{\left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )^2} \, dx\\ &=-\frac {110592 e^{48} x^3}{251 \left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )}-\frac {4096}{251} \int \frac {-20736 e^{48} x^2+186219 e^{64} x^3-556227 e^{80} x^4+548937 e^{96} x^5}{\left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )^2} \, dx\\ &=\frac {55296 e^{32} x^2}{251 \left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )}-\frac {110592 e^{48} x^3}{251 \left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )}+\frac {2048 \int \frac {-281055744 e^{96} x+2529501696 e^{112} x^2-7572036978 e^{128} x^3+7440292098 e^{144} x^4}{\left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )^2} \, dx}{5103081 e^{64}}\\ &=-\frac {12288 e^{16} x}{251 \left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )}+\frac {55296 e^{32} x^2}{251 \left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )}-\frac {110592 e^{48} x^3}{251 \left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )}-\frac {2048 \int \frac {-1904714777088 e^{144}[phone] e^{160} x-51427298981376 e^{176} x^2+50422859548146 e^{192} x^3}{\left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )^2} \, dx}{311252219433 e^{128}}\\ &=\frac {1024}{251 \left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )}-\frac {12288 e^{16} x}{251 \left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )}+\frac {55296 e^{32} x^2}{251 \left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )}-\frac {110592 e^{48} x^3}{251 \left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.04, size = 62, normalized size = 2.48 \begin {gather*} \frac {1024 \left (1-12 e^{16} x+54 e^{32} x^2-108 e^{48} x^3\right )}{251 \left (256-3072 e^{16} x+13824 e^{32} x^2-27648 e^{48} x^3+20331 e^{64} x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16*(4 - 12*E^16*x)^4)/(81*E^64*x^4*(-25*x + 75*E^16*x^2 + ((4 - 12*E^16*x)^4*(10*x - 30*E^16*x^2))/
(81*E^64*x^4) + ((4 - 12*E^16*x)^8*(-x + 3*E^16*x^2))/(6561*E^128*x^8))),x]

[Out]

(1024*(1 - 12*E^16*x + 54*E^32*x^2 - 108*E^48*x^3))/(251*(256 - 3072*E^16*x + 13824*E^32*x^2 - 27648*E^48*x^3
+ 20331*E^64*x^4))

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fricas [B]  time = 1.64, size = 53, normalized size = 2.12 \begin {gather*} -\frac {1024 \, {\left (108 \, x^{3} e^{48} - 54 \, x^{2} e^{32} + 12 \, x e^{16} - 1\right )}}{251 \, {\left (20331 \, x^{4} e^{64} - 27648 \, x^{3} e^{48} + 13824 \, x^{2} e^{32} - 3072 \, x e^{16} + 256\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16/81*(-12*x*exp(16)+4)^4/x^4/exp(16)^4/(1/6561*(3*x^2*exp(16)-x)*(-12*x*exp(16)+4)^8/x^8/exp(16)^8+
1/81*(-30*x^2*exp(16)+10*x)*(-12*x*exp(16)+4)^4/x^4/exp(16)^4+75*x^2*exp(16)-25*x),x, algorithm="fricas")

[Out]

-1024/251*(108*x^3*e^48 - 54*x^2*e^32 + 12*x*e^16 - 1)/(20331*x^4*e^64 - 27648*x^3*e^48 + 13824*x^2*e^32 - 307
2*x*e^16 + 256)

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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(16/81*(-12*x*exp(16)+4)^4/x^4/exp(16)^4/(1/6561*(3*x^2*exp(16)-x)*(-12*x*exp(16)+4)^8/x^8/exp(16)^8+
1/81*(-30*x^2*exp(16)+10*x)*(-12*x*exp(16)+4)^4/x^4/exp(16)^4+75*x^2*exp(16)-25*x),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.45, size = 46, normalized size = 1.84

method	result	size
norman	\(-\frac {324 \,{\mathrm e}^{64} x^{4}}{20331 x^{4} {\mathrm e}^{64}-27648 \,{\mathrm e}^{48} x^{3}+13824 \,{\mathrm e}^{32} x^{2}-3072 x \,{\mathrm e}^{16}+256}\)	\(46\)
risch	\(\frac {16 \,{\mathrm e}^{-64} \left (-\frac {6912 \,{\mathrm e}^{112} x^{3}}{63001}+\frac {3456 \,{\mathrm e}^{96} x^{2}}{63001}-\frac {768 \,{\mathrm e}^{80} x}{63001}+\frac {64 \,{\mathrm e}^{64}}{63001}\right )}{81 \left (x^{4} {\mathrm e}^{64}-\frac {1024 \,{\mathrm e}^{48} x^{3}}{753}+\frac {512 \,{\mathrm e}^{32} x^{2}}{753}-\frac {1024 x \,{\mathrm e}^{16}}{6777}+\frac {256}{20331}\right )}\)	\(58\)
gosper	\(-\frac {1024 \left (108 \,{\mathrm e}^{48} x^{3}-54 \,{\mathrm e}^{32} x^{2}+12 x \,{\mathrm e}^{16}-1\right )}{251 \left (20331 x^{4} {\mathrm e}^{64}-27648 \,{\mathrm e}^{48} x^{3}+13824 \,{\mathrm e}^{32} x^{2}-3072 x \,{\mathrm e}^{16}+256\right )}\)	\(64\)
default	\(13824 \,{\mathrm e}^{64} \left (\munderset {\textit {\_R} =\RootOf \left (413349561 \textit {\_Z}^{8} {\mathrm e}^{128}-1124222976 \,{\mathrm e}^{112} \textit {\_Z}^{7}[phone] \,{\mathrm e}^{96} \textit {\_Z}^{6}-889325568 \,{\mathrm e}^{80} \textit {\_Z}^{5}[phone] \textit {\_Z}^{4} {\mathrm e}^{64}-99090432 \,{\mathrm e}^{48} \textit {\_Z}^{3}[phone] \,{\mathrm e}^{32} \textit {\_Z}^{2}-1572864 \textit {\_Z} \,{\mathrm e}^{16}+65536\right )}{\sum }\frac {\left (-27 \,{\mathrm e}^{48} \textit {\_R}^{6}+27 \,{\mathrm e}^{32} \textit {\_R}^{5}-9 \,{\mathrm e}^{16} \textit {\_R}^{4}+\textit {\_R}^{3}\right ) \ln \left (x -\textit {\_R} \right )}{-137783187 \textit {\_R}^{7} {\mathrm e}^{128}[phone] \,{\mathrm e}^{112} \textit {\_R}^{6}-331630848 \,{\mathrm e}^{96} \textit {\_R}^{5}[phone] \,{\mathrm e}^{80} \textit {\_R}^{4}-61896960 \textit {\_R}^{3} {\mathrm e}^{64}[phone] \,{\mathrm e}^{48} \textit {\_R}^{2}-1376256 \,{\mathrm e}^{32} \textit {\_R} +65536 \,{\mathrm e}^{16}}\right )\)	\(157\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(16/81*(-12*x*exp(16)+4)^4/x^4/exp(16)^4/(1/6561*(3*x^2*exp(16)-x)*(-12*x*exp(16)+4)^8/x^8/exp(16)^8+1/81*(
-30*x^2*exp(16)+10*x)*(-12*x*exp(16)+4)^4/x^4/exp(16)^4+75*x^2*exp(16)-25*x),x,method=_RETURNVERBOSE)

[Out]

-324*exp(16)^4*x^4/(20331*x^4*exp(16)^4-27648*exp(16)^3*x^3+13824*exp(16)^2*x^2-3072*x*exp(16)+256)

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maxima [B]  time = 0.45, size = 58, normalized size = 2.32 \begin {gather*} -\frac {1024 \, {\left (108 \, x^{3} e^{112} - 54 \, x^{2} e^{96} + 12 \, x e^{80} - e^{64}\right )} e^{\left (-64\right )}}{251 \, {\left (20331 \, x^{4} e^{64} - 27648 \, x^{3} e^{48} + 13824 \, x^{2} e^{32} - 3072 \, x e^{16} + 256\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16/81*(-12*x*exp(16)+4)^4/x^4/exp(16)^4/(1/6561*(3*x^2*exp(16)-x)*(-12*x*exp(16)+4)^8/x^8/exp(16)^8+
1/81*(-30*x^2*exp(16)+10*x)*(-12*x*exp(16)+4)^4/x^4/exp(16)^4+75*x^2*exp(16)-25*x),x, algorithm="maxima")

[Out]

-1024/251*(108*x^3*e^112 - 54*x^2*e^96 + 12*x*e^80 - e^64)*e^(-64)/(20331*x^4*e^64 - 27648*x^3*e^48 + 13824*x^
2*e^32 - 3072*x*e^16 + 256)

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mupad [B]  time = 6.63, size = 53, normalized size = 2.12 \begin {gather*} -\frac {\frac {110592\,{\mathrm {e}}^{48}\,x^3}{251}-\frac {55296\,{\mathrm {e}}^{32}\,x^2}{251}+\frac {12288\,{\mathrm {e}}^{16}\,x}{251}-\frac {1024}{251}}{20331\,{\mathrm {e}}^{64}\,x^4-27648\,{\mathrm {e}}^{48}\,x^3+13824\,{\mathrm {e}}^{32}\,x^2-3072\,{\mathrm {e}}^{16}\,x+256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*exp(-64)*(12*x*exp(16) - 4)^4)/(81*x^4*(25*x - 75*x^2*exp(16) + (exp(-128)*(x - 3*x^2*exp(16))*(12*x*
exp(16) - 4)^8)/(6561*x^8) - (exp(-64)*(12*x*exp(16) - 4)^4*(10*x - 30*x^2*exp(16)))/(81*x^4))),x)

[Out]

-((12288*x*exp(16))/251 - (55296*x^2*exp(32))/251 + (110592*x^3*exp(48))/251 - 1024/251)/(13824*x^2*exp(32) -
3072*x*exp(16) - 27648*x^3*exp(48) + 20331*x^4*exp(64) + 256)

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sympy [B]  time = 0.93, size = 58, normalized size = 2.32 \begin {gather*} \frac {- 110592 x^{3} e^{48} + 55296 x^{2} e^{32} - 12288 x e^{16} + 1024}{5103081 x^{4} e^{64} - 6939648 x^{3} e^{48} + 3469824 x^{2} e^{32} - 771072 x e^{16} + 64256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16/81*(-12*x*exp(16)+4)**4/x**4/exp(16)**4/(1/6561*(3*x**2*exp(16)-x)*(-12*x*exp(16)+4)**8/x**8/exp(
16)**8+1/81*(-30*x**2*exp(16)+10*x)*(-12*x*exp(16)+4)**4/x**4/exp(16)**4+75*x**2*exp(16)-25*x),x)

[Out]

(-110592*x**3*exp(48) + 55296*x**2*exp(32) - 12288*x*exp(16) + 1024)/(5103081*x**4*exp(64) - 6939648*x**3*exp(
48) + 3469824*x**2*exp(32) - 771072*x*exp(16) + 64256)

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