3.103.86 \(\int \frac {405+810 x+270 \log (4)+(405+810 x+270 \log (4)) \log (5)}{2916+972 x+1053 x^2+162 x^3+81 x^4+(648 x+108 x^2+108 x^3) \log (4)+(108+18 x+54 x^2) \log ^2(4)+12 x \log ^3(4)+\log ^4(4)+(5832+972 x+972 x^2+648 x \log (4)+108 \log ^2(4)) \log (5)+2916 \log ^2(5)} \, dx\)
Optimal. Leaf size=27 \[ 3-\frac {5}{6+\frac {x+\left (x+\frac {\log (4)}{3}\right )^2}{1+\log (5)}} \]
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Rubi [A] time = 0.09, antiderivative size = 34, normalized size of antiderivative = 1.26,
number of steps used = 4, number of rules used = 4, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used
= {6, 1680, 12, 261} \begin {gather*} -\frac {60 (1+\log (5))}{12 \left (x+\frac {1}{324} (162+108 \log (4))\right )^2+69+72 \log (5)-4 \log (4)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(405 + 810*x + 270*Log[4] + (405 + 810*x + 270*Log[4])*Log[5])/(2916 + 972*x + 1053*x^2 + 162*x^3 + 81*x^4
+ (648*x + 108*x^2 + 108*x^3)*Log[4] + (108 + 18*x + 54*x^2)*Log[4]^2 + 12*x*Log[4]^3 + Log[4]^4 + (5832 + 97
2*x + 972*x^2 + 648*x*Log[4] + 108*Log[4]^2)*Log[5] + 2916*Log[5]^2),x]
[Out]
(-60*(1 + Log[5]))/(69 - 4*Log[4] + 12*(x + (162 + 108*Log[4])/324)^2 + 72*Log[5])
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 261
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
Rule 1680
Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] && !IGtQ[p, 0]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {405+810 x+270 \log (4)+(405+810 x+270 \log (4)) \log (5)}{2916+1053 x^2+162 x^3+81 x^4+\left (648 x+108 x^2+108 x^3\right ) \log (4)+\left (108+18 x+54 x^2\right ) \log ^2(4)+\log ^4(4)+x \left (972+12 \log ^3(4)\right )+\left (5832+972 x+972 x^2+648 x \log (4)+108 \log ^2(4)\right ) \log (5)+2916 \log ^2(5)} \, dx\\ &=\operatorname {Subst}\left (\int \frac {1440 x (1+\log (5))}{\left (69+12 x^2-4 \log (4)+72 \log (5)\right )^2} \, dx,x,x+\frac {1}{324} (162+108 \log (4))\right )\\ &=(1440 (1+\log (5))) \operatorname {Subst}\left (\int \frac {x}{\left (69+12 x^2-4 \log (4)+72 \log (5)\right )^2} \, dx,x,x+\frac {1}{324} (162+108 \log (4))\right )\\ &=-\frac {180 (1+\log (5))}{207-12 \log (4)+216 \log (5)+(3+6 x+\log (16))^2}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.04, size = 69, normalized size = 2.56 \begin {gather*} \frac {15 (1+\log (5)) \left (207-6 \log (4)+4 \log ^2(4)+216 \log (5)-3 \log (16)-2 \log (4) \log (16)\right )}{(-69+4 \log (4)-72 \log (5)) \left (54+9 x+9 x^2+6 x \log (4)+\log ^2(4)+54 \log (5)\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(405 + 810*x + 270*Log[4] + (405 + 810*x + 270*Log[4])*Log[5])/(2916 + 972*x + 1053*x^2 + 162*x^3 +
81*x^4 + (648*x + 108*x^2 + 108*x^3)*Log[4] + (108 + 18*x + 54*x^2)*Log[4]^2 + 12*x*Log[4]^3 + Log[4]^4 + (583
2 + 972*x + 972*x^2 + 648*x*Log[4] + 108*Log[4]^2)*Log[5] + 2916*Log[5]^2),x]
[Out]
(15*(1 + Log[5])*(207 - 6*Log[4] + 4*Log[4]^2 + 216*Log[5] - 3*Log[16] - 2*Log[4]*Log[16]))/((-69 + 4*Log[4] -
72*Log[5])*(54 + 9*x + 9*x^2 + 6*x*Log[4] + Log[4]^2 + 54*Log[5]))
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fricas [A] time = 1.45, size = 33, normalized size = 1.22 \begin {gather*} -\frac {45 \, {\left (\log \relax (5) + 1\right )}}{9 \, x^{2} + 12 \, x \log \relax (2) + 4 \, \log \relax (2)^{2} + 9 \, x + 54 \, \log \relax (5) + 54} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((540*log(2)+810*x+405)*log(5)+540*log(2)+810*x+405)/(2916*log(5)^2+(432*log(2)^2+1296*x*log(2)+972*
x^2+972*x+5832)*log(5)+16*log(2)^4+96*x*log(2)^3+4*(54*x^2+18*x+108)*log(2)^2+2*(108*x^3+108*x^2+648*x)*log(2)
+81*x^4+162*x^3+1053*x^2+972*x+2916),x, algorithm="fricas")
[Out]
-45*(log(5) + 1)/(9*x^2 + 12*x*log(2) + 4*log(2)^2 + 9*x + 54*log(5) + 54)
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giac [B] time = 0.18, size = 71, normalized size = 2.63 \begin {gather*} -\frac {45 \, {\left (\log \relax (5)^{2} + 2 \, \log \relax (5) + 1\right )}}{4 \, \log \relax (5) \log \relax (2)^{2} + 9 \, x^{2} + 3 \, {\left (3 \, x^{2} + 4 \, x \log \relax (2) + 3 \, x\right )} \log \relax (5) + 54 \, \log \relax (5)^{2} + 12 \, x \log \relax (2) + 4 \, \log \relax (2)^{2} + 9 \, x + 108 \, \log \relax (5) + 54} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((540*log(2)+810*x+405)*log(5)+540*log(2)+810*x+405)/(2916*log(5)^2+(432*log(2)^2+1296*x*log(2)+972*
x^2+972*x+5832)*log(5)+16*log(2)^4+96*x*log(2)^3+4*(54*x^2+18*x+108)*log(2)^2+2*(108*x^3+108*x^2+648*x)*log(2)
+81*x^4+162*x^3+1053*x^2+972*x+2916),x, algorithm="giac")
[Out]
-45*(log(5)^2 + 2*log(5) + 1)/(4*log(5)*log(2)^2 + 9*x^2 + 3*(3*x^2 + 4*x*log(2) + 3*x)*log(5) + 54*log(5)^2 +
12*x*log(2) + 4*log(2)^2 + 9*x + 108*log(5) + 54)
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maple [A] time = 0.10, size = 34, normalized size = 1.26
|
|
|
| method |
result |
size |
|
|
|
| gosper |
\(-\frac {45 \left (\ln \relax (5)+1\right )}{4 \ln \relax (2)^{2}+12 x \ln \relax (2)+9 x^{2}+54 \ln \relax (5)+9 x +54}\) |
\(34\) |
| norman |
\(\frac {-45-45 \ln \relax (5)}{4 \ln \relax (2)^{2}+12 x \ln \relax (2)+9 x^{2}+54 \ln \relax (5)+9 x +54}\) |
\(35\) |
| risch |
\(-\frac {5 \ln \relax (5)}{6 \left (\frac {2 \ln \relax (2)^{2}}{27}+\frac {2 x \ln \relax (2)}{9}+\frac {x^{2}}{6}+\ln \relax (5)+\frac {x}{6}+1\right )}-\frac {5}{6 \left (\frac {2 \ln \relax (2)^{2}}{27}+\frac {2 x \ln \relax (2)}{9}+\frac {x^{2}}{6}+\ln \relax (5)+\frac {x}{6}+1\right )}\) |
\(58\) |
| default |
\(\frac {\left (135 \ln \relax (5)+135\right ) \left (-648+\left (4 \ln \relax (2)+3\right ) \left (12 \ln \relax (2)+9\right )-48 \ln \relax (2)^{2}-648 \ln \relax (5)\right )}{\left (1944 \ln \relax (5)-216 \ln \relax (2)+1863\right ) \left (4 \ln \relax (2)^{2}+12 x \ln \relax (2)+9 x^{2}+54 \ln \relax (5)+9 x +54\right )}\) |
\(72\) |
|
|
|
| |
| |
| |
| |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((540*ln(2)+810*x+405)*ln(5)+540*ln(2)+810*x+405)/(2916*ln(5)^2+(432*ln(2)^2+1296*x*ln(2)+972*x^2+972*x+58
32)*ln(5)+16*ln(2)^4+96*x*ln(2)^3+4*(54*x^2+18*x+108)*ln(2)^2+2*(108*x^3+108*x^2+648*x)*ln(2)+81*x^4+162*x^3+1
053*x^2+972*x+2916),x,method=_RETURNVERBOSE)
[Out]
-45*(ln(5)+1)/(4*ln(2)^2+12*x*ln(2)+9*x^2+54*ln(5)+9*x+54)
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maxima [A] time = 0.36, size = 34, normalized size = 1.26 \begin {gather*} -\frac {45 \, {\left (\log \relax (5) + 1\right )}}{9 \, x^{2} + 3 \, x {\left (4 \, \log \relax (2) + 3\right )} + 4 \, \log \relax (2)^{2} + 54 \, \log \relax (5) + 54} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((540*log(2)+810*x+405)*log(5)+540*log(2)+810*x+405)/(2916*log(5)^2+(432*log(2)^2+1296*x*log(2)+972*
x^2+972*x+5832)*log(5)+16*log(2)^4+96*x*log(2)^3+4*(54*x^2+18*x+108)*log(2)^2+2*(108*x^3+108*x^2+648*x)*log(2)
+81*x^4+162*x^3+1053*x^2+972*x+2916),x, algorithm="maxima")
[Out]
-45*(log(5) + 1)/(9*x^2 + 3*x*(4*log(2) + 3) + 4*log(2)^2 + 54*log(5) + 54)
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mupad [B] time = 9.87, size = 2558, normalized size = 94.74 result too large to
display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((810*x + 540*log(2) + log(5)*(810*x + 540*log(2) + 405) + 405)/(972*x + 2*log(2)*(648*x + 108*x^2 + 108*x^
3) + 4*log(2)^2*(18*x + 54*x^2 + 108) + 96*x*log(2)^3 + log(5)*(972*x + 1296*x*log(2) + 432*log(2)^2 + 972*x^2
+ 5832) + 16*log(2)^4 + 2916*log(5)^2 + 1053*x^2 + 162*x^3 + 81*x^4 + 2916),x)
[Out]
symsum(log(4009802061150*root(418066920000*log(2)*log(5)^3 - 9447840000*log(2)^2*log(5) + 210804930000*log(2)*
log(5)^4 - 2361960000*log(2)^2*log(5)^4 + 42515280000*log(2)*log(5)^5 - 14171760000*log(2)^2*log(5)^2 - 944784
0000*log(2)^2*log(5)^3 + 414523980000*log(2)*log(5)^2 + 205490520000*log(2)*log(5) - 2790397400625*log(5)^4 -
1069525012500*log(5) - 1131969330000*log(5)^5 - 2361960000*log(2)^2 - 191318760000*log(5)^6 - 2712342003750*lo
g(5)^2 + 40743810000*log(2) - 3668271502500*log(5)^3 - 175707680625, z, k) + 3486784401000*x + 2324522934000*l
og(2) + 5230176601500*log(5) + 4881498161400*root(418066920000*log(2)*log(5)^3 - 9447840000*log(2)^2*log(5) +
210804930000*log(2)*log(5)^4 - 2361960000*log(2)^2*log(5)^4 + 42515280000*log(2)*log(5)^5 - 14171760000*log(2)
^2*log(5)^2 - 9447840000*log(2)^2*log(5)^3 + 414523980000*log(2)*log(5)^2 + 205490520000*log(2)*log(5) - 27903
97400625*log(5)^4 - 1069525012500*log(5) - 1131969330000*log(5)^5 - 2361960000*log(2)^2 - 191318760000*log(5)^
6 - 2712342003750*log(5)^2 + 40743810000*log(2) - 3668271502500*log(5)^3 - 175707680625, z, k)*log(2) + 122037
45403500*root(418066920000*log(2)*log(5)^3 - 9447840000*log(2)^2*log(5) + 210804930000*log(2)*log(5)^4 - 23619
60000*log(2)^2*log(5)^4 + 42515280000*log(2)*log(5)^5 - 14171760000*log(2)^2*log(5)^2 - 9447840000*log(2)^2*lo
g(5)^3 + 414523980000*log(2)*log(5)^2 + 205490520000*log(2)*log(5) - 2790397400625*log(5)^4 - 1069525012500*lo
g(5) - 1131969330000*log(5)^5 - 2361960000*log(2)^2 - 191318760000*log(5)^6 - 2712342003750*log(5)^2 + 4074381
0000*log(2) - 3668271502500*log(5)^3 - 175707680625, z, k)*log(5) + 8019604122300*root(418066920000*log(2)*log
(5)^3 - 9447840000*log(2)^2*log(5) + 210804930000*log(2)*log(5)^4 - 2361960000*log(2)^2*log(5)^4 + 42515280000
*log(2)*log(5)^5 - 14171760000*log(2)^2*log(5)^2 - 9447840000*log(2)^2*log(5)^3 + 414523980000*log(2)*log(5)^2
+ 205490520000*log(2)*log(5) - 2790397400625*log(5)^4 - 1069525012500*log(5) - 1131969330000*log(5)^5 - 23619
60000*log(2)^2 - 191318760000*log(5)^6 - 2712342003750*log(5)^2 + 40743810000*log(2) - 3668271502500*log(5)^3
- 175707680625, z, k)*x + 6973568802000*log(2)*log(5) + 10460353203000*x*log(5) - 619872782400*root(4180669200
00*log(2)*log(5)^3 - 9447840000*log(2)^2*log(5) + 210804930000*log(2)*log(5)^4 - 2361960000*log(2)^2*log(5)^4
+ 42515280000*log(2)*log(5)^5 - 14171760000*log(2)^2*log(5)^2 - 9447840000*log(2)^2*log(5)^3 + 414523980000*lo
g(2)*log(5)^2 + 205490520000*log(2)*log(5) - 2790397400625*log(5)^4 - 1069525012500*log(5) - 1131969330000*log
(5)^5 - 2361960000*log(2)^2 - 191318760000*log(5)^6 - 2712342003750*log(5)^2 + 40743810000*log(2) - 3668271502
500*log(5)^3 - 175707680625, z, k)*log(2)^2 + 12378084623550*root(418066920000*log(2)*log(5)^3 - 9447840000*lo
g(2)^2*log(5) + 210804930000*log(2)*log(5)^4 - 2361960000*log(2)^2*log(5)^4 + 42515280000*log(2)*log(5)^5 - 14
171760000*log(2)^2*log(5)^2 - 9447840000*log(2)^2*log(5)^3 + 414523980000*log(2)*log(5)^2 + 205490520000*log(2
)*log(5) - 2790397400625*log(5)^4 - 1069525012500*log(5) - 1131969330000*log(5)^5 - 2361960000*log(2)^2 - 1913
18760000*log(5)^6 - 2712342003750*log(5)^2 + 40743810000*log(2) - 3668271502500*log(5)^3 - 175707680625, z, k)
*log(5)^2 + 4184141281200*root(418066920000*log(2)*log(5)^3 - 9447840000*log(2)^2*log(5) + 210804930000*log(2)
*log(5)^4 - 2361960000*log(2)^2*log(5)^4 + 42515280000*log(2)*log(5)^5 - 14171760000*log(2)^2*log(5)^2 - 94478
40000*log(2)^2*log(5)^3 + 414523980000*log(2)*log(5)^2 + 205490520000*log(2)*log(5) - 2790397400625*log(5)^4 -
1069525012500*log(5) - 1131969330000*log(5)^5 - 2361960000*log(2)^2 - 191318760000*log(5)^6 - 2712342003750*l
og(5)^2 + 40743810000*log(2) - 3668271502500*log(5)^3 - 175707680625, z, k)*log(5)^3 + 6973568802000*log(2)*lo
g(5)^2 + 2324522934000*log(2)*log(5)^3 + 10460353203000*x*log(5)^2 + 3486784401000*x*log(5)^3 + 5230176601500*
log(5)^2 + 1743392200500*log(5)^3 - 619872782400*root(418066920000*log(2)*log(5)^3 - 9447840000*log(2)^2*log(5
) + 210804930000*log(2)*log(5)^4 - 2361960000*log(2)^2*log(5)^4 + 42515280000*log(2)*log(5)^5 - 14171760000*lo
g(2)^2*log(5)^2 - 9447840000*log(2)^2*log(5)^3 + 414523980000*log(2)*log(5)^2 + 205490520000*log(2)*log(5) - 2
790397400625*log(5)^4 - 1069525012500*log(5) - 1131969330000*log(5)^5 - 2361960000*log(2)^2 - 191318760000*log
(5)^6 - 2712342003750*log(5)^2 + 40743810000*log(2) - 3668271502500*log(5)^3 - 175707680625, z, k)*log(2)^2*lo
g(5)^2 + 15341851364400*root(418066920000*log(2)*log(5)^3 - 9447840000*log(2)^2*log(5) + 210804930000*log(2)*l
og(5)^4 - 2361960000*log(2)^2*log(5)^4 + 42515280000*log(2)*log(5)^5 - 14171760000*log(2)^2*log(5)^2 - 9447840
000*log(2)^2*log(5)^3 + 414523980000*log(2)*log(5)^2 + 205490520000*log(2)*log(5) - 2790397400625*log(5)^4 - 1
069525012500*log(5) - 1131969330000*log(5)^5 - 2361960000*log(2)^2 - 191318760000*log(5)^6 - 2712342003750*log
(5)^2 + 40743810000*log(2) - 3668271502500*log(5)^3 - 175707680625, z, k)*log(2)*log(5) - 929809173600*root(41
8066920000*log(2)*log(5)^3 - 9447840000*log(2)^2*log(5) + 210804930000*log(2)*log(5)^4 - 2361960000*log(2)^2*l
og(5)^4 + 42515280000*log(2)*log(5)^5 - 14171760000*log(2)^2*log(5)^2 - 9447840000*log(2)^2*log(5)^3 + 4145239
80000*log(2)*log(5)^2 + 205490520000*log(2)*log(5) - 2790397400625*log(5)^4 - 1069525012500*log(5) - 113196933
0000*log(5)^5 - 2361960000*log(2)^2 - 191318760000*log(5)^6 - 2712342003750*log(5)^2 + 40743810000*log(2) - 36
68271502500*log(5)^3 - 175707680625, z, k)*x*log(2) + 24407490807000*root(418066920000*log(2)*log(5)^3 - 94478
40000*log(2)^2*log(5) + 210804930000*log(2)*log(5)^4 - 2361960000*log(2)^2*log(5)^4 + 42515280000*log(2)*log(5
)^5 - 14171760000*log(2)^2*log(5)^2 - 9447840000*log(2)^2*log(5)^3 + 414523980000*log(2)*log(5)^2 + 2054905200
00*log(2)*log(5) - 2790397400625*log(5)^4 - 1069525012500*log(5) - 1131969330000*log(5)^5 - 2361960000*log(2)^
2 - 191318760000*log(5)^6 - 2712342003750*log(5)^2 + 40743810000*log(2) - 3668271502500*log(5)^3 - 17570768062
5, z, k)*x*log(5) + 16039208244600*root(418066920000*log(2)*log(5)^3 - 9447840000*log(2)^2*log(5) + 2108049300
00*log(2)*log(5)^4 - 2361960000*log(2)^2*log(5)^4 + 42515280000*log(2)*log(5)^5 - 14171760000*log(2)^2*log(5)^
2 - 9447840000*log(2)^2*log(5)^3 + 414523980000*log(2)*log(5)^2 + 205490520000*log(2)*log(5) - 2790397400625*l
og(5)^4 - 1069525012500*log(5) - 1131969330000*log(5)^5 - 2361960000*log(2)^2 - 191318760000*log(5)^6 - 271234
2003750*log(5)^2 + 40743810000*log(2) - 3668271502500*log(5)^3 - 175707680625, z, k)*log(2)*log(5)^2 - 1239745
564800*root(418066920000*log(2)*log(5)^3 - 9447840000*log(2)^2*log(5) + 210804930000*log(2)*log(5)^4 - 2361960
000*log(2)^2*log(5)^4 + 42515280000*log(2)*log(5)^5 - 14171760000*log(2)^2*log(5)^2 - 9447840000*log(2)^2*log(
5)^3 + 414523980000*log(2)*log(5)^2 + 205490520000*log(2)*log(5) - 2790397400625*log(5)^4 - 1069525012500*log(
5) - 1131969330000*log(5)^5 - 2361960000*log(2)^2 - 191318760000*log(5)^6 - 2712342003750*log(5)^2 + 407438100
00*log(2) - 3668271502500*log(5)^3 - 175707680625, z, k)*log(2)^2*log(5) + 5578855041600*root(418066920000*log
(2)*log(5)^3 - 9447840000*log(2)^2*log(5) + 210804930000*log(2)*log(5)^4 - 2361960000*log(2)^2*log(5)^4 + 4251
5280000*log(2)*log(5)^5 - 14171760000*log(2)^2*log(5)^2 - 9447840000*log(2)^2*log(5)^3 + 414523980000*log(2)*l
og(5)^2 + 205490520000*log(2)*log(5) - 2790397400625*log(5)^4 - 1069525012500*log(5) - 1131969330000*log(5)^5
- 2361960000*log(2)^2 - 191318760000*log(5)^6 - 2712342003750*log(5)^2 + 40743810000*log(2) - 3668271502500*lo
g(5)^3 - 175707680625, z, k)*log(2)*log(5)^3 + 24756169247100*root(418066920000*log(2)*log(5)^3 - 9447840000*l
og(2)^2*log(5) + 210804930000*log(2)*log(5)^4 - 2361960000*log(2)^2*log(5)^4 + 42515280000*log(2)*log(5)^5 - 1
4171760000*log(2)^2*log(5)^2 - 9447840000*log(2)^2*log(5)^3 + 414523980000*log(2)*log(5)^2 + 205490520000*log(
2)*log(5) - 2790397400625*log(5)^4 - 1069525012500*log(5) - 1131969330000*log(5)^5 - 2361960000*log(2)^2 - 191
318760000*log(5)^6 - 2712342003750*log(5)^2 + 40743810000*log(2) - 3668271502500*log(5)^3 - 175707680625, z, k
)*x*log(5)^2 + 8368282562400*root(418066920000*log(2)*log(5)^3 - 9447840000*log(2)^2*log(5) + 210804930000*log
(2)*log(5)^4 - 2361960000*log(2)^2*log(5)^4 + 42515280000*log(2)*log(5)^5 - 14171760000*log(2)^2*log(5)^2 - 94
47840000*log(2)^2*log(5)^3 + 414523980000*log(2)*log(5)^2 + 205490520000*log(2)*log(5) - 2790397400625*log(5)^
4 - 1069525012500*log(5) - 1131969330000*log(5)^5 - 2361960000*log(2)^2 - 191318760000*log(5)^6 - 271234200375
0*log(5)^2 + 40743810000*log(2) - 3668271502500*log(5)^3 - 175707680625, z, k)*x*log(5)^3 - 1859618347200*root
(418066920000*log(2)*log(5)^3 - 9447840000*log(2)^2*log(5) + 210804930000*log(2)*log(5)^4 - 2361960000*log(2)^
2*log(5)^4 + 42515280000*log(2)*log(5)^5 - 14171760000*log(2)^2*log(5)^2 - 9447840000*log(2)^2*log(5)^3 + 4145
23980000*log(2)*log(5)^2 + 205490520000*log(2)*log(5) - 2790397400625*log(5)^4 - 1069525012500*log(5) - 113196
9330000*log(5)^5 - 2361960000*log(2)^2 - 191318760000*log(5)^6 - 2712342003750*log(5)^2 + 40743810000*log(2) -
3668271502500*log(5)^3 - 175707680625, z, k)*x*log(2)*log(5) - 929809173600*root(418066920000*log(2)*log(5)^3
- 9447840000*log(2)^2*log(5) + 210804930000*log(2)*log(5)^4 - 2361960000*log(2)^2*log(5)^4 + 42515280000*log(
2)*log(5)^5 - 14171760000*log(2)^2*log(5)^2 - 9447840000*log(2)^2*log(5)^3 + 414523980000*log(2)*log(5)^2 + 20
5490520000*log(2)*log(5) - 2790397400625*log(5)^4 - 1069525012500*log(5) - 1131969330000*log(5)^5 - 2361960000
*log(2)^2 - 191318760000*log(5)^6 - 2712342003750*log(5)^2 + 40743810000*log(2) - 3668271502500*log(5)^3 - 175
707680625, z, k)*x*log(2)*log(5)^2 + 1743392200500)*root(418066920000*log(2)*log(5)^3 - 9447840000*log(2)^2*lo
g(5) + 210804930000*log(2)*log(5)^4 - 2361960000*log(2)^2*log(5)^4 + 42515280000*log(2)*log(5)^5 - 14171760000
*log(2)^2*log(5)^2 - 9447840000*log(2)^2*log(5)^3 + 414523980000*log(2)*log(5)^2 + 205490520000*log(2)*log(5)
- 2790397400625*log(5)^4 - 1069525012500*log(5) - 1131969330000*log(5)^5 - 2361960000*log(2)^2 - 191318760000*
log(5)^6 - 2712342003750*log(5)^2 + 40743810000*log(2) - 3668271502500*log(5)^3 - 175707680625, z, k), k, 1, 4
)
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sympy [A] time = 1.40, size = 34, normalized size = 1.26 \begin {gather*} \frac {- 45 \log {\relax (5 )} - 45}{9 x^{2} + x \left (12 \log {\relax (2 )} + 9\right ) + 4 \log {\relax (2 )}^{2} + 54 + 54 \log {\relax (5 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((540*ln(2)+810*x+405)*ln(5)+540*ln(2)+810*x+405)/(2916*ln(5)**2+(432*ln(2)**2+1296*x*ln(2)+972*x**2
+972*x+5832)*ln(5)+16*ln(2)**4+96*x*ln(2)**3+4*(54*x**2+18*x+108)*ln(2)**2+2*(108*x**3+108*x**2+648*x)*ln(2)+8
1*x**4+162*x**3+1053*x**2+972*x+2916),x)
[Out]
(-45*log(5) - 45)/(9*x**2 + x*(12*log(2) + 9) + 4*log(2)**2 + 54 + 54*log(5))
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