∫ −24x2−30x6+12 log(5)________________ 4x2−16x3+16x4−4x7+8x8+x12+(−8x+16x2+4x6) log(5)+4 log2(5) dx [5689]

Optimal. Leaf size=23 \[ \frac {3}{-1+2 x+\frac {\frac {x^6}{2}+\log (5)}{x}} \]

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Rubi [F]
time = 0.18, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-24 x^2-30 x^6+12 \log (5)}{4 x^2-16 x^3+16 x^4-4 x^7+8 x^8+x^{12}+\left (-8 x+16 x^2+4 x^6\right ) \log (5)+4 \log ^2(5)} \, dx \end {gather*}

Int[(-24*x^2 - 30*x^6 + 12*Log[5])/(4*x^2 - 16*x^3 + 16*x^4 - 4*x^7 + 8*x^8 + x^12 + (-8*x + 16*x^2 + 4*x^6)*L

72*Log[5]*Defer[Int][(-2*x + 4*x^2 + x^6 + Log[25])^(-2), x] - 60*Defer[Int][x/(-2*x + 4*x^2 + x^6 + Log[25])^

2, x] + 96*Defer[Int][x^2/(-2*x + 4*x^2 + x^6 + Log[25])^2, x] - 30*Defer[Int][(-2*x + 4*x^2 + x^6 + Log[25])^

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {12 \left (-5 x+8 x^2+6 \log (5)\right )}{\left (-2 x+4 x^2+x^6+\log (25)\right )^2}-\frac {30}{-2 x+4 x^2+x^6+\log (25)}\right ) \, dx\\ &=12 \int \frac {-5 x+8 x^2+6 \log (5)}{\left (-2 x+4 x^2+x^6+\log (25)\right )^2} \, dx-30 \int \frac {1}{-2 x+4 x^2+x^6+\log (25)} \, dx\\ &=12 \int \left (-\frac {5 x}{\left (-2 x+4 x^2+x^6+\log (25)\right )^2}+\frac {8 x^2}{\left (-2 x+4 x^2+x^6+\log (25)\right )^2}+\frac {6 \log (5)}{\left (-2 x+4 x^2+x^6+\log (25)\right )^2}\right ) \, dx-30 \int \frac {1}{-2 x+4 x^2+x^6+\log (25)} \, dx\\ &=-\left (30 \int \frac {1}{-2 x+4 x^2+x^6+\log (25)} \, dx\right )-60 \int \frac {x}{\left (-2 x+4 x^2+x^6+\log (25)\right )^2} \, dx+96 \int \frac {x^2}{\left (-2 x+4 x^2+x^6+\log (25)\right )^2} \, dx+(72 \log (5)) \int \frac {1}{\left (-2 x+4 x^2+x^6+\log (25)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.03, size = 19, normalized size = 0.83 \begin {gather*} \frac {6 x}{-2 x+4 x^2+x^6+\log (25)} \end {gather*}

Integrate[(-24*x^2 - 30*x^6 + 12*Log[5])/(4*x^2 - 16*x^3 + 16*x^4 - 4*x^7 + 8*x^8 + x^12 + (-8*x + 16*x^2 + 4*

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method	result	size

gosper	\(\frac {6 x}{x^{6}+4 x^{2}+2 \ln \left (5\right )-2 x}\)	\(22\)
default	\(\frac {3 x}{\frac {x^{6}}{2}+2 x^{2}+\ln \left (5\right )-x}\)	\(22\)
norman	\(\frac {6 x}{x^{6}+4 x^{2}+2 \ln \left (5\right )-2 x}\)	\(22\)
risch	\(\frac {3 x}{\frac {x^{6}}{2}+2 x^{2}+\ln \left (5\right )-x}\)	\(22\)

int((12*ln(5)-30*x^6-24*x^2)/(4*ln(5)^2+(4*x^6+16*x^2-8*x)*ln(5)+x^12+8*x^8-4*x^7+16*x^4-16*x^3+4*x^2),x,metho

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Maxima [A]
time = 0.27, size = 21, normalized size = 0.91 \begin {gather*} \frac {6 \, x}{x^{6} + 4 \, x^{2} - 2 \, x + 2 \, \log \left (5\right )} \end {gather*}

integrate((12*log(5)-30*x^6-24*x^2)/(4*log(5)^2+(4*x^6+16*x^2-8*x)*log(5)+x^12+8*x^8-4*x^7+16*x^4-16*x^3+4*x^2

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Fricas [A]
time = 0.41, size = 21, normalized size = 0.91 \begin {gather*} \frac {6 \, x}{x^{6} + 4 \, x^{2} - 2 \, x + 2 \, \log \left (5\right )} \end {gather*}

integrate((12*log(5)-30*x^6-24*x^2)/(4*log(5)^2+(4*x^6+16*x^2-8*x)*log(5)+x^12+8*x^8-4*x^7+16*x^4-16*x^3+4*x^2

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Sympy [A]
time = 1.08, size = 19, normalized size = 0.83 \begin {gather*} \frac {6 x}{x^{6} + 4 x^{2} - 2 x + 2 \log {\left (5 \right )}} \end {gather*}

integrate((12*ln(5)-30*x**6-24*x**2)/(4*ln(5)**2+(4*x**6+16*x**2-8*x)*ln(5)+x**12+8*x**8-4*x**7+16*x**4-16*x**

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Giac [A]
time = 0.42, size = 21, normalized size = 0.91 \begin {gather*} \frac {6 \, x}{x^{6} + 4 \, x^{2} - 2 \, x + 2 \, \log \left (5\right )} \end {gather*}

integrate((12*log(5)-30*x^6-24*x^2)/(4*log(5)^2+(4*x^6+16*x^2-8*x)*log(5)+x^12+8*x^8-4*x^7+16*x^4-16*x^3+4*x^2

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Mupad [B]
time = 13.06, size = 2500, normalized size = 108.70 \begin {gather*} \text {Too large to display} \end {gather*}

int(-(24*x^2 - 12*log(5) + 30*x^6)/(log(5)*(16*x^2 - 8*x + 4*x^6) + 4*log(5)^2 + 4*x^2 - 16*x^3 + 16*x^4 - 4*x

symsum((log(247669456896) + log(1141803243*log(5) + 761202162*root(70198242049209139200*log(5)^9 + 56740497585

4624702464*log(5)^5 - 3313942840733220864*log(5)^2 - 178020791885626343424*log(5)^8 + 59817780906615373824*log

(5)^3 - 328638925913793232896*log(5)^4 + 340716998789642059776*log(5)^7 - 44926874911493849088*log(5)^10 - 599

086626906216333312*log(5)^6 - 4738381338321616896*log(5)^12, z, k)*log(5) - 761202162*root(7019824204920913920

0*log(5)^9 + 567404975854624702464*log(5)^5 - 3313942840733220864*log(5)^2 - 178020791885626343424*log(5)^8 +

59817780906615373824*log(5)^3 - 328638925913793232896*log(5)^4 + 340716998789642059776*log(5)^7 - 449268749114

93849088*log(5)^10 - 599086626906216333312*log(5)^6 - 4738381338321616896*log(5)^12, z, k)*x - 9134425944*x*lo

g(5) - 7806409296*root(70198242049209139200*log(5)^9 + 567404975854624702464*log(5)^5 - 3313942840733220864*lo

g(5)^2 - 178020791885626343424*log(5)^8 + 59817780906615373824*log(5)^3 - 328638925913793232896*log(5)^4 + 340

716998789642059776*log(5)^7 - 44926874911493849088*log(5)^10 - 599086626906216333312*log(5)^6 - 47383813383216

16896*log(5)^12, z, k)*log(5)^2 + 16542228480*root(70198242049209139200*log(5)^9 + 567404975854624702464*log(5

)^5 - 3313942840733220864*log(5)^2 - 178020791885626343424*log(5)^8 + 59817780906615373824*log(5)^3 - 32863892

5913793232896*log(5)^4 + 340716998789642059776*log(5)^7 - 44926874911493849088*log(5)^10 - 5990866269062163333

12*log(5)^6 - 4738381338321616896*log(5)^12, z, k)*log(5)^3 - 24779575296*root(70198242049209139200*log(5)^9 +

567404975854624702464*log(5)^5 - 3313942840733220864*log(5)^2 - 178020791885626343424*log(5)^8 + 598177809066

15373824*log(5)^3 - 328638925913793232896*log(5)^4 + 340716998789642059776*log(5)^7 - 44926874911493849088*log

(5)^10 - 599086626906216333312*log(5)^6 - 4738381338321616896*log(5)^12, z, k)*log(5)^4 + 17252352000*root(701

98242049209139200*log(5)^9 + 567404975854624702464*log(5)^5 - 3313942840733220864*log(5)^2 - 17802079188562634

3424*log(5)^8 + 59817780906615373824*log(5)^3 - 328638925913793232896*log(5)^4 + 340716998789642059776*log(5)^

7 - 44926874911493849088*log(5)^10 - 599086626906216333312*log(5)^6 - 4738381338321616896*log(5)^12, z, k)*log

(5)^5 - 8554330944*root(70198242049209139200*log(5)^9 + 567404975854624702464*log(5)^5 - 3313942840733220864*l

og(5)^2 - 178020791885626343424*log(5)^8 + 59817780906615373824*log(5)^3 - 328638925913793232896*log(5)^4 + 34

0716998789642059776*log(5)^7 - 44926874911493849088*log(5)^10 - 599086626906216333312*log(5)^6 - 4738381338321

616896*log(5)^12, z, k)*log(5)^6 + 1119744000*root(70198242049209139200*log(5)^9 + 567404975854624702464*log(5

)^5 - 3313942840733220864*log(5)^2 - 178020791885626343424*log(5)^8 + 59817780906615373824*log(5)^3 - 32863892

5913793232896*log(5)^4 + 340716998789642059776*log(5)^7 - 44926874911493849088*log(5)^10 - 5990866269062163333

12*log(5)^6 - 4738381338321616896*log(5)^12, z, k)*log(5)^7 - 1612431360*root(70198242049209139200*log(5)^9 +

567404975854624702464*log(5)^5 - 3313942840733220864*log(5)^2 - 178020791885626343424*log(5)^8 + 5981778090661

5373824*log(5)^3 - 328638925913793232896*log(5)^4 + 340716998789642059776*log(5)^7 - 44926874911493849088*log(

5)^10 - 599086626906216333312*log(5)^6 - 4738381338321616896*log(5)^12, z, k)*log(5)^8 + 18968640000*x*log(5)^

2 - 21762957312*x*log(5)^3 + 23224320000*x*log(5)^4 - 22114827360*x*log(5)^5 + 6998400000*x*log(5)^6 - 8062156

800*x*log(5)^7 + 6877453944*log(5)^2 - 18119393280*log(5)^3 + 22654162944*log(5)^4 - 5971968000*log(5)^5 + 137

06296416*log(5)^6 - 5038848000*log(5)^7 + 2418647040*log(5)^8 + 10695145944*root(70198242049209139200*log(5)^9

+ 567404975854624702464*log(5)^5 - 3313942840733220864*log(5)^2 - 178020791885626343424*log(5)^8 + 5981778090

6615373824*log(5)^3 - 328638925913793232896*log(5)^4 + 340716998789642059776*log(5)^7 - 44926874911493849088*l

og(5)^10 - 599086626906216333312*log(5)^6 - 4738381338321616896*log(5)^12, z, k)*x*log(5) - 44636254784*root(7

0198242049209139200*log(5)^9 + 567404975854624702464*log(5)^5 - 3313942840733220864*log(5)^2 - 178020791885626

343424*log(5)^8 + 59817780906615373824*log(5)^3 - 328638925913793232896*log(5)^4 + 340716998789642059776*log(5

)^7 - 44926874911493849088*log(5)^10 - 599086626906216333312*log(5)^6 - 4738381338321616896*log(5)^12, z, k)*x

*log(5)^2 + 70059921408*root(70198242049209139200*log(5)^9 + 567404975854624702464*log(5)^5 - 3313942840733220

864*log(5)^2 - 178020791885626343424*log(5)^8 + 59817780906615373824*log(5)^3 - 328638925913793232896*log(5)^4

+ 340716998789642059776*log(5)^7 - 44926874911493849088*log(5)^10 - 599086626906216333312*log(5)^6 - 47383813

38321616896*log(5)^12, z, k)*x*log(5)^3 - 64525271040*root(70198242049209139200*log(5)^9 + 5674049758546247024

64*log(5)^5 - 3313942840733220864*log(5)^2 - 178020791885626343424*log(5)^8 + 59817780906615373824*log(5)^3 -

328638925913793232896*log(5)^4 + 340716998789642059776*log(5)^7 - 44926874911493849088*log(5)^10 - 59908662690

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3.57.89 \(\int \frac {-24 x^2-30 x^6+12 \log (5)}{4 x^2-16 x^3+16 x^4-4 x^7+8 x^8+x^{12}+(-8 x+16 x^2+4 x^6) \log (5)+4 \log ^2(5)} \, dx\) [5689]