3.79.62 \(\int \frac {384 x^3+(-576 x^2+352 x^3) \log (9)+(192 x-352 x^2) \log ^2(9)}{e^{32} (7776+71280 x+261360 x^2+479160 x^3+439230 x^4+161051 x^5)} \, dx\)

Optimal. Leaf size=24 \[ \frac {x^2 (-x+\log (9))^2}{e^{32} \left (-3-\frac {11 x}{2}\right )^4} \]

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Rubi [B]  time = 0.13, antiderivative size = 95, normalized size of antiderivative = 3.96, number of steps used = 3, number of rules used = 2, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {12, 2074} \begin {gather*} \frac {16 \left (216+121 \log ^2(9)+396 \log (9)\right )}{14641 e^{32} (11 x+6)^2}-\frac {192 \left (72+121 \log ^2(9)+198 \log (9)\right )}{14641 e^{32} (11 x+6)^3}-\frac {32 (12+11 \log (9))}{14641 e^{32} (11 x+6)}+\frac {576 (6+11 \log (9))^2}{14641 e^{32} (11 x+6)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(384*x^3 + (-576*x^2 + 352*x^3)*Log[9] + (192*x - 352*x^2)*Log[9]^2)/(E^32*(7776 + 71280*x + 261360*x^2 +
479160*x^3 + 439230*x^4 + 161051*x^5)),x]

[Out]

(576*(6 + 11*Log[9])^2)/(14641*E^32*(6 + 11*x)^4) - (32*(12 + 11*Log[9]))/(14641*E^32*(6 + 11*x)) - (192*(72 +
198*Log[9] + 121*Log[9]^2))/(14641*E^32*(6 + 11*x)^3) + (16*(216 + 396*Log[9] + 121*Log[9]^2))/(14641*E^32*(6
+ 11*x)^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {384 x^3+\left (-576 x^2+352 x^3\right ) \log (9)+\left (192 x-352 x^2\right ) \log ^2(9)}{7776+71280 x+261360 x^2+479160 x^3+439230 x^4+161051 x^5} \, dx}{e^{32}}\\ &=\frac {\int \left (-\frac {2304 (6+11 \log (9))^2}{1331 (6+11 x)^5}+\frac {32 (12+11 \log (9))}{1331 (6+11 x)^2}+\frac {576 \left (72+198 \log (9)+121 \log ^2(9)\right )}{1331 (6+11 x)^4}-\frac {32 \left (216+396 \log (9)+121 \log ^2(9)\right )}{1331 (6+11 x)^3}\right ) \, dx}{e^{32}}\\ &=\frac {576 (6+11 \log (9))^2}{14641 e^{32} (6+11 x)^4}-\frac {32 (12+11 \log (9))}{14641 e^{32} (6+11 x)}-\frac {192 \left (72+198 \log (9)+121 \log ^2(9)\right )}{14641 e^{32} (6+11 x)^3}+\frac {16 \left (216+396 \log (9)+121 \log ^2(9)\right )}{14641 e^{32} (6+11 x)^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 43, normalized size = 1.79 \begin {gather*} -\frac {16 \left (1296+9504 x+2662 x^3 (12+11 \log (9))-121 x^2 \left (-216+121 \log ^2(9)\right )\right )}{14641 e^{32} (6+11 x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(384*x^3 + (-576*x^2 + 352*x^3)*Log[9] + (192*x - 352*x^2)*Log[9]^2)/(E^32*(7776 + 71280*x + 261360*
x^2 + 479160*x^3 + 439230*x^4 + 161051*x^5)),x]

[Out]

(-16*(1296 + 9504*x + 2662*x^3*(12 + 11*Log[9]) - 121*x^2*(-216 + 121*Log[9]^2)))/(14641*E^32*(6 + 11*x)^4)

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fricas [B]  time = 0.78, size = 57, normalized size = 2.38 \begin {gather*} -\frac {64 \, {\left (14641 \, x^{3} \log \relax (3) - 14641 \, x^{2} \log \relax (3)^{2} + 7986 \, x^{3} + 6534 \, x^{2} + 2376 \, x + 324\right )} e^{\left (-32\right )}}{14641 \, {\left (14641 \, x^{4} + 31944 \, x^{3} + 26136 \, x^{2} + 9504 \, x + 1296\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-352*x^2+192*x)*log(3)^2+2*(352*x^3-576*x^2)*log(3)+384*x^3)/(161051*x^5+439230*x^4+479160*x^3+2
61360*x^2+71280*x+7776)/exp(16)^2,x, algorithm="fricas")

[Out]

-64/14641*(14641*x^3*log(3) - 14641*x^2*log(3)^2 + 7986*x^3 + 6534*x^2 + 2376*x + 324)*e^(-32)/(14641*x^4 + 31
944*x^3 + 26136*x^2 + 9504*x + 1296)

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giac [A]  time = 0.19, size = 42, normalized size = 1.75 \begin {gather*} -\frac {64 \, {\left (14641 \, x^{3} \log \relax (3) - 14641 \, x^{2} \log \relax (3)^{2} + 7986 \, x^{3} + 6534 \, x^{2} + 2376 \, x + 324\right )} e^{\left (-32\right )}}{14641 \, {\left (11 \, x + 6\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-352*x^2+192*x)*log(3)^2+2*(352*x^3-576*x^2)*log(3)+384*x^3)/(161051*x^5+439230*x^4+479160*x^3+2
61360*x^2+71280*x+7776)/exp(16)^2,x, algorithm="giac")

[Out]

-64/14641*(14641*x^3*log(3) - 14641*x^2*log(3)^2 + 7986*x^3 + 6534*x^2 + 2376*x + 324)*e^(-32)/(11*x + 6)^4

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maple [A]  time = 0.10, size = 51, normalized size = 2.12

method	result	size
risch	\(\frac {{\mathrm e}^{-32} \left (\left (-\frac {384}{161051}-\frac {64 \ln \relax (3)}{14641}\right ) x^{3}+\left (\frac {64 \ln \relax (3)^{2}}{14641}-\frac {3456}{1771561}\right ) x^{2}-\frac {13824 x}{19487171}-\frac {20736}{214358881}\right )}{x^{4}+\frac {24}{11} x^{3}+\frac {216}{121} x^{2}+\frac {864}{1331} x +\frac {1296}{14641}}\)	\(51\)
norman	\(\frac {\left (-\frac {64 \left (11 \ln \relax (3)+6\right ) {\mathrm e}^{-16} x^{3}}{11}+\frac {64 \left (121 \ln \relax (3)^{2}-54\right ) {\mathrm e}^{-16} x^{2}}{121}-\frac {13824 x \,{\mathrm e}^{-16}}{1331}-\frac {20736 \,{\mathrm e}^{-16}}{14641}\right ) {\mathrm e}^{-16}}{\left (11 x +6\right )^{4}}\)	\(59\)
gosper	\(\frac {64 \left (14641 x^{2} \ln \relax (3)^{2}-14641 x^{3} \ln \relax (3)-7986 x^{3}-6534 x^{2}-2376 x -324\right ) {\mathrm e}^{-32}}{14641 \left (14641 x^{4}+31944 x^{3}+26136 x^{2}+9504 x +1296\right )}\)	\(60\)
default	\(2 \,{\mathrm e}^{-32} \left (-\frac {8 \left (-\frac {144 \ln \relax (3)^{2}}{121}-\frac {864 \ln \relax (3)}{1331}-\frac {1296}{14641}\right )}{\left (11 x +6\right )^{4}}-\frac {32 \left (\frac {36 \ln \relax (3)^{2}}{121}+\frac {324 \ln \relax (3)}{1331}+\frac {648}{14641}\right )}{3 \left (11 x +6\right )^{3}}-\frac {16 \left (-\frac {2 \ln \relax (3)^{2}}{121}-\frac {36 \ln \relax (3)}{1331}-\frac {108}{14641}\right )}{\left (11 x +6\right )^{2}}-\frac {32 \left (\frac {\ln \relax (3)}{1331}+\frac {6}{14641}\right )}{11 x +6}\right )\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*(-352*x^2+192*x)*ln(3)^2+2*(352*x^3-576*x^2)*ln(3)+384*x^3)/(161051*x^5+439230*x^4+479160*x^3+261360*x^
2+71280*x+7776)/exp(16)^2,x,method=_RETURNVERBOSE)

[Out]

exp(-32)*((-384/161051-64/14641*ln(3))*x^3+(64/14641*ln(3)^2-3456/1771561)*x^2-13824/19487171*x-20736/21435888
1)/(x^4+24/11*x^3+216/121*x^2+864/1331*x+1296/14641)

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maxima [B]  time = 0.35, size = 55, normalized size = 2.29 \begin {gather*} -\frac {64 \, {\left (1331 \, x^{3} {\left (11 \, \log \relax (3) + 6\right )} - 121 \, {\left (121 \, \log \relax (3)^{2} - 54\right )} x^{2} + 2376 \, x + 324\right )} e^{\left (-32\right )}}{14641 \, {\left (14641 \, x^{4} + 31944 \, x^{3} + 26136 \, x^{2} + 9504 \, x + 1296\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-352*x^2+192*x)*log(3)^2+2*(352*x^3-576*x^2)*log(3)+384*x^3)/(161051*x^5+439230*x^4+479160*x^3+2
61360*x^2+71280*x+7776)/exp(16)^2,x, algorithm="maxima")

[Out]

-64/14641*(1331*x^3*(11*log(3) + 6) - 121*(121*log(3)^2 - 54)*x^2 + 2376*x + 324)*e^(-32)/(14641*x^4 + 31944*x
^3 + 26136*x^2 + 9504*x + 1296)

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mupad [B]  time = 5.35, size = 83, normalized size = 3.46 \begin {gather*} \frac {64\,{\mathrm {e}}^{-32}\,\left (198\,\ln \relax (3)+121\,{\ln \relax (3)}^2+54\right )}{14641\,{\left (11\,x+6\right )}^2}-\frac {768\,{\mathrm {e}}^{-32}\,\left (99\,\ln \relax (3)+121\,{\ln \relax (3)}^2+18\right )}{14641\,{\left (11\,x+6\right )}^3}-\frac {64\,{\mathrm {e}}^{-32}\,\left (11\,\ln \relax (3)+6\right )}{14641\,\left (11\,x+6\right )}+\frac {2304\,{\mathrm {e}}^{-32}\,{\left (11\,\ln \relax (3)+3\right )}^2}{14641\,{\left (11\,x+6\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-32)*(4*log(3)^2*(192*x - 352*x^2) - 2*log(3)*(576*x^2 - 352*x^3) + 384*x^3))/(71280*x + 261360*x^2 +
479160*x^3 + 439230*x^4 + 161051*x^5 + 7776),x)

[Out]

(64*exp(-32)*(198*log(3) + 121*log(3)^2 + 54))/(14641*(11*x + 6)^2) - (768*exp(-32)*(99*log(3) + 121*log(3)^2
+ 18))/(14641*(11*x + 6)^3) - (64*exp(-32)*(11*log(3) + 6))/(14641*(11*x + 6)) + (2304*exp(-32)*(11*log(3) + 3
)^2)/(14641*(11*x + 6)^4)

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sympy [B]  time = 1.48, size = 65, normalized size = 2.71 \begin {gather*} \frac {x^{3} \left (- 937024 \log {\relax (3 )} - 511104\right ) + x^{2} \left (-418176 + 937024 \log {\relax (3 )}^{2}\right ) - 152064 x - 20736}{214358881 x^{4} e^{32} + 467692104 x^{3} e^{32} + 382657176 x^{2} e^{32} + 139148064 x e^{32} + 18974736 e^{32}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-352*x**2+192*x)*ln(3)**2+2*(352*x**3-576*x**2)*ln(3)+384*x**3)/(161051*x**5+439230*x**4+479160*
x**3+261360*x**2+71280*x+7776)/exp(16)**2,x)

[Out]

(x**3*(-937024*log(3) - 511104) + x**2*(-418176 + 937024*log(3)**2) - 152064*x - 20736)/(214358881*x**4*exp(32
) + 467692104*x**3*exp(32) + 382657176*x**2*exp(32) + 139148064*x*exp(32) + 18974736*exp(32))

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