3.20.59 \(\int \frac {-179159040-414720 x+165888 x^2-165888 x^2 \log (3)}{-2015539200+820212480 x-86252688 x^2+572857 x^3-1306 x^4+x^5+(-806215680 x+166841856 x^2-1132704 x^3+2602 x^4-2 x^5) \log (3)+(-80621568 x^2+559872 x^3-1296 x^4+x^5) \log ^2(3)} \, dx\)

Optimal. Leaf size=25 \[ 4+\frac {4 x}{\left (-3+\frac {x}{144}\right )^2 (5-x+x \log (3))} \]

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Rubi [B]  time = 0.21, antiderivative size = 64, normalized size of antiderivative = 2.56, number of steps used = 3, number of rules used = 2, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6, 2074} \begin {gather*} \frac {414720 (1-\log (3))}{(427-432 \log (3))^2 (5-x (1-\log (3)))}-\frac {414720}{(432-x) (427-432 \log (3))^2}-\frac {35831808}{(432-x)^2 (427-432 \log (3))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-179159040 - 414720*x + 165888*x^2 - 165888*x^2*Log[3])/(-2015539200 + 820212480*x - 86252688*x^2 + 57285
7*x^3 - 1306*x^4 + x^5 + (-806215680*x + 166841856*x^2 - 1132704*x^3 + 2602*x^4 - 2*x^5)*Log[3] + (-80621568*x
^2 + 559872*x^3 - 1296*x^4 + x^5)*Log[3]^2),x]

[Out]

-414720/((432 - x)*(427 - 432*Log[3])^2) - 35831808/((432 - x)^2*(427 - 432*Log[3])) + (414720*(1 - Log[3]))/(
(5 - x*(1 - Log[3]))*(427 - 432*Log[3])^2)

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 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} &=\int \frac {-179159040-414720 x+x^2 (165888-165888 \log (3))}{-2015539200+820212480 x-86252688 x^2+572857 x^3-1306 x^4+x^5+\left (-806215680 x+166841856 x^2-1132704 x^3+2602 x^4-2 x^5\right ) \log (3)+\left (-80621568 x^2+559872 x^3-1296 x^4+x^5\right ) \log ^2(3)} \, dx\\ &=\int \left (\frac {414720 (1-\log (3))^2}{(5-x (1-\log (3)))^2 (427-432 \log (3))^2}-\frac {414720}{(-432+x)^2 (-427+432 \log (3))^2}-\frac {71663616}{(-432+x)^3 (-427+432 \log (3))}\right ) \, dx\\ &=-\frac {414720}{(432-x) (427-432 \log (3))^2}-\frac {35831808}{(432-x)^2 (427-432 \log (3))}+\frac {414720 (1-\log (3))}{(5-x (1-\log (3))) (427-432 \log (3))^2}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 54, normalized size = 2.16 \begin {gather*} -\frac {82944 x \left (77854483-41753664 \log (9)-1119744 \log ^2(3) (-67+36 \log (9))+432 \log (3) (-353683+189864 \log (9))\right )}{(-432+x)^2 (5+x (-1+\log (3))) (-427+432 \log (3))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-179159040 - 414720*x + 165888*x^2 - 165888*x^2*Log[3])/(-2015539200 + 820212480*x - 86252688*x^2 +
572857*x^3 - 1306*x^4 + x^5 + (-806215680*x + 166841856*x^2 - 1132704*x^3 + 2602*x^4 - 2*x^5)*Log[3] + (-8062
1568*x^2 + 559872*x^3 - 1296*x^4 + x^5)*Log[3]^2),x]

[Out]

(-82944*x*(77854483 - 41753664*Log[9] - 1119744*Log[3]^2*(-67 + 36*Log[9]) + 432*Log[3]*(-353683 + 189864*Log[
9])))/((-432 + x)^2*(5 + x*(-1 + Log[3]))*(-427 + 432*Log[3])^3)

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fricas [A]  time = 0.56, size = 34, normalized size = 1.36 \begin {gather*} -\frac {82944 \, x}{x^{3} - 869 \, x^{2} - {\left (x^{3} - 864 \, x^{2} + 186624 \, x\right )} \log \relax (3) + 190944 \, x - 933120} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-165888*x^2*log(3)+165888*x^2-414720*x-179159040)/((x^5-1296*x^4+559872*x^3-80621568*x^2)*log(3)^2+
(-2*x^5+2602*x^4-1132704*x^3+166841856*x^2-806215680*x)*log(3)+x^5-1306*x^4+572857*x^3-86252688*x^2+820212480*
x-2015539200),x, algorithm="fricas")

[Out]

-82944*x/(x^3 - 869*x^2 - (x^3 - 864*x^2 + 186624*x)*log(3) + 190944*x - 933120)

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giac [B]  time = 0.32, size = 62, normalized size = 2.48 \begin {gather*} -\frac {414720 \, {\left (\log \relax (3) - 1\right )}}{{\left (x \log \relax (3) - x + 5\right )} {\left (186624 \, \log \relax (3)^{2} - 368928 \, \log \relax (3) + 182329\right )}} + \frac {82944 \, {\left (5 \, x + 186624 \, \log \relax (3) - 186624\right )}}{{\left (186624 \, \log \relax (3)^{2} - 368928 \, \log \relax (3) + 182329\right )} {\left (x - 432\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-165888*x^2*log(3)+165888*x^2-414720*x-179159040)/((x^5-1296*x^4+559872*x^3-80621568*x^2)*log(3)^2+
(-2*x^5+2602*x^4-1132704*x^3+166841856*x^2-806215680*x)*log(3)+x^5-1306*x^4+572857*x^3-86252688*x^2+820212480*
x-2015539200),x, algorithm="giac")

[Out]

-414720*(log(3) - 1)/((x*log(3) - x + 5)*(186624*log(3)^2 - 368928*log(3) + 182329)) + 82944*(5*x + 186624*log
(3) - 186624)/((186624*log(3)^2 - 368928*log(3) + 182329)*(x - 432)^2)

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maple [A]  time = 0.17, size = 20, normalized size = 0.80

method	result	size
norman	\(\frac {82944 x}{\left (x -432\right )^{2} \left (5+x \ln \relax (3)-x \right )}\)	\(20\)
gosper	\(\frac {82944 x}{x^{3} \ln \relax (3)-864 x^{2} \ln \relax (3)-x^{3}+186624 x \ln \relax (3)+869 x^{2}-190944 x +933120}\)	\(39\)
risch	\(\frac {82944 x}{x^{3} \ln \relax (3)-864 x^{2} \ln \relax (3)-x^{3}+186624 x \ln \relax (3)+869 x^{2}-190944 x +933120}\)	\(39\)
default	\(\frac {414720}{\left (-427+432 \ln \relax (3)\right )^{2} \left (x -432\right )}+\frac {35831808}{\left (-427+432 \ln \relax (3)\right ) \left (x -432\right )^{2}}-\frac {414720 \left (\ln \relax (3)-1\right )}{\left (-427+432 \ln \relax (3)\right )^{2} \left (5+x \ln \relax (3)-x \right )}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-165888*x^2*ln(3)+165888*x^2-414720*x-179159040)/((x^5-1296*x^4+559872*x^3-80621568*x^2)*ln(3)^2+(-2*x^5+
2602*x^4-1132704*x^3+166841856*x^2-806215680*x)*ln(3)+x^5-1306*x^4+572857*x^3-86252688*x^2+820212480*x-2015539
200),x,method=_RETURNVERBOSE)

[Out]

82944*x/(x-432)^2/(5+x*ln(3)-x)

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maxima [A]  time = 0.61, size = 35, normalized size = 1.40 \begin {gather*} \frac {82944 \, x}{x^{3} {\left (\log \relax (3) - 1\right )} - x^{2} {\left (864 \, \log \relax (3) - 869\right )} + 864 \, x {\left (216 \, \log \relax (3) - 221\right )} + 933120} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-165888*x^2*log(3)+165888*x^2-414720*x-179159040)/((x^5-1296*x^4+559872*x^3-80621568*x^2)*log(3)^2+
(-2*x^5+2602*x^4-1132704*x^3+166841856*x^2-806215680*x)*log(3)+x^5-1306*x^4+572857*x^3-86252688*x^2+820212480*
x-2015539200),x, algorithm="maxima")

[Out]

82944*x/(x^3*(log(3) - 1) - x^2*(864*log(3) - 869) + 864*x*(216*log(3) - 221) + 933120)

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mupad [B]  time = 3.32, size = 342, normalized size = 13.68 \begin {gather*} \sum _{k=1}^5\ln \left (-{\left (\ln \relax (3)-1\right )}^6\,\left (194567695554960\,\mathrm {root}\left (-55037657088\,{\left (432\,\ln \relax (3)-427\right )}^5\,{\left (\ln \relax (3)-1\right )}^4,z,k\right )-625131806240\,x-142004620247040\,\ln \relax (3)-\mathrm {root}\left (-55037657088\,{\left (432\,\ln \relax (3)-427\right )}^5\,{\left (\ln \relax (3)-1\right )}^4,z,k\right )\,\ln \relax (3)\,784264884560640-\mathrm {root}\left (-55037657088\,{\left (432\,\ln \relax (3)-427\right )}^5\,{\left (\ln \relax (3)-1\right )}^4,z,k\right )\,x\,38747319789787+1869451038720\,x\,\ln \relax (3)+\mathrm {root}\left (-55037657088\,{\left (432\,\ln \relax (3)-427\right )}^5\,{\left (\ln \relax (3)-1\right )}^4,z,k\right )\,{\ln \relax (3)}^2\,1185438977372160-\mathrm {root}\left (-55037657088\,{\left (432\,\ln \relax (3)-427\right )}^5\,{\left (\ln \relax (3)-1\right )}^4,z,k\right )\,{\ln \relax (3)}^3\,796354049802240+\mathrm {root}\left (-55037657088\,{\left (432\,\ln \relax (3)-427\right )}^5\,{\left (\ln \relax (3)-1\right )}^4,z,k\right )\,{\ln \relax (3)}^4\,200612260085760-1863492894720\,x\,{\ln \relax (3)}^2+619173642240\,x\,{\ln \relax (3)}^3+137869331005440\,{\ln \relax (3)}^2-44580502241280\,{\ln \relax (3)}^3+\mathrm {root}\left (-55037657088\,{\left (432\,\ln \relax (3)-427\right )}^5\,{\left (\ln \relax (3)-1\right )}^4,z,k\right )\,x\,\ln \relax (3)\,195093853290000-\mathrm {root}\left (-55037657088\,{\left (432\,\ln \relax (3)-427\right )}^5\,{\left (\ln \relax (3)-1\right )}^4,z,k\right )\,x\,{\ln \relax (3)}^2\,392919963367680+\mathrm {root}\left (-55037657088\,{\left (432\,\ln \relax (3)-427\right )}^5\,{\left (\ln \relax (3)-1\right )}^4,z,k\right )\,x\,{\ln \relax (3)}^3\,395670097244160-\mathrm {root}\left (-55037657088\,{\left (432\,\ln \relax (3)-427\right )}^5\,{\left (\ln \relax (3)-1\right )}^4,z,k\right )\,x\,{\ln \relax (3)}^4\,199219119390720+\mathrm {root}\left (-55037657088\,{\left (432\,\ln \relax (3)-427\right )}^5\,{\left (\ln \relax (3)-1\right )}^4,z,k\right )\,x\,{\ln \relax (3)}^5\,40122452017152+48715782842880\right )\,1479074071160291328\right )\,\mathrm {root}\left (-55037657088\,{\left (432\,\ln \relax (3)-427\right )}^5\,{\left (\ln \relax (3)-1\right )}^4,z,k\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((414720*x + 165888*x^2*log(3) - 165888*x^2 + 179159040)/(log(3)*(806215680*x - 166841856*x^2 + 1132704*x^3
- 2602*x^4 + 2*x^5) - 820212480*x + log(3)^2*(80621568*x^2 - 559872*x^3 + 1296*x^4 - x^5) + 86252688*x^2 - 57
2857*x^3 + 1306*x^4 - x^5 + 2015539200),x)

[Out]

symsum(log(-1479074071160291328*(log(3) - 1)^6*(194567695554960*root(-55037657088*(432*log(3) - 427)^5*(log(3)
- 1)^4, z, k) - 625131806240*x - 142004620247040*log(3) - 784264884560640*root(-55037657088*(432*log(3) - 427
)^5*(log(3) - 1)^4, z, k)*log(3) - 38747319789787*root(-55037657088*(432*log(3) - 427)^5*(log(3) - 1)^4, z, k)
*x + 1869451038720*x*log(3) + 1185438977372160*root(-55037657088*(432*log(3) - 427)^5*(log(3) - 1)^4, z, k)*lo
g(3)^2 - 796354049802240*root(-55037657088*(432*log(3) - 427)^5*(log(3) - 1)^4, z, k)*log(3)^3 + 2006122600857
60*root(-55037657088*(432*log(3) - 427)^5*(log(3) - 1)^4, z, k)*log(3)^4 - 1863492894720*x*log(3)^2 + 61917364
2240*x*log(3)^3 + 137869331005440*log(3)^2 - 44580502241280*log(3)^3 + 195093853290000*root(-55037657088*(432*
log(3) - 427)^5*(log(3) - 1)^4, z, k)*x*log(3) - 392919963367680*root(-55037657088*(432*log(3) - 427)^5*(log(3
) - 1)^4, z, k)*x*log(3)^2 + 395670097244160*root(-55037657088*(432*log(3) - 427)^5*(log(3) - 1)^4, z, k)*x*lo
g(3)^3 - 199219119390720*root(-55037657088*(432*log(3) - 427)^5*(log(3) - 1)^4, z, k)*x*log(3)^4 + 40122452017
152*root(-55037657088*(432*log(3) - 427)^5*(log(3) - 1)^4, z, k)*x*log(3)^5 + 48715782842880))*root(-550376570
88*(432*log(3) - 427)^5*(log(3) - 1)^4, z, k), k, 1, 5)

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sympy [A]  time = 1.47, size = 31, normalized size = 1.24 \begin {gather*} \frac {82944 x}{x^{3} \left (-1 + \log {\relax (3 )}\right ) + x^{2} \left (869 - 864 \log {\relax (3 )}\right ) + x \left (-190944 + 186624 \log {\relax (3 )}\right ) + 933120} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-165888*x**2*ln(3)+165888*x**2-414720*x-179159040)/((x**5-1296*x**4+559872*x**3-80621568*x**2)*ln(3
)**2+(-2*x**5+2602*x**4-1132704*x**3+166841856*x**2-806215680*x)*ln(3)+x**5-1306*x**4+572857*x**3-86252688*x**
2+820212480*x-2015539200),x)

[Out]

82944*x/(x**3*(-1 + log(3)) + x**2*(869 - 864*log(3)) + x*(-190944 + 186624*log(3)) + 933120)

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