∫ (512+128x−512x2+128x4) log7(x)+(−640−224x+1152x2−416x4) log8(x)____________________________________________________________ 640000x11+800000x12−2800000x13−3100000x14+6012500x15+5400625x16−8112500x17−5450000x18+7403125x19+3450000x20−4665000x21−1393750x22+2025000x23+350000x24−593750x25−50000x26+112500x27+3125x28−12500x29+625x31 dx

Optimal. Leaf size=22 \[ \frac {16 \log ^8(x)}{625 x^{10} \left (x+\left (-2+x^2\right )^2\right )^4} \]

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Rubi [F] time = 9.59, 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 {\left (512+128 x-512 x^2+128 x^4\right ) \log ^7(x)+\left (-640-224 x+1152 x^2-416 x^4\right ) \log ^8(x)}{640000 x^{11}+800000 x^{12}-2800000 x^{13}-3100000 x^{14}+6012500 x^{15}+5400625 x^{16}-8112500 x^{17}-5450000 x^{18}+7403125 x^{19}+3450000 x^{20}-4665000 x^{21}-1393750 x^{22}+2025000 x^{23}+350000 x^{24}-593750 x^{25}-50000 x^{26}+112500 x^{27}+3125 x^{28}-12500 x^{29}+625 x^{31}} \, dx \end {gather*}

Int[((512 + 128*x - 512*x^2 + 128*x^4)*Log[x]^7 + (-640 - 224*x + 1152*x^2 - 416*x^4)*Log[x]^8)/(640000*x^11 +

800000*x^12 - 2800000*x^13 - 3100000*x^14 + 6012500*x^15 + 5400625*x^16 - 8112500*x^17 - 5450000*x^18 + 74031

25*x^19 + 3450000*x^20 - 4665000*x^21 - 1393750*x^22 + 2025000*x^23 + 350000*x^24 - 593750*x^25 - 50000*x^26 +

112500*x^27 + 3125*x^28 - 12500*x^29 + 625*x^31),x]

(116870431227*Log[x]^8)/16384000000000 + Log[x]^8/(10000*x^10) - Log[x]^8/(10000*x^9) + (37*Log[x]^8)/(80000*x

^8) - (17*Log[x]^8)/(32000*x^7) + (3299*Log[x]^8)/(2560000*x^6) - (2047*Log[x]^8)/(1280000*x^5) + (28981*Log[x

]^8)/(10240000*x^4) - (74143*Log[x]^8)/(20480000*x^3) + (706721*Log[x]^8)/(131072000*x^2) - (4520791*Log[x]^8)

/(655360000*x) + (16*Log[x]^8)/(390625*(1 + x)^4) + (672*Log[x]^8)/(1953125*(1 + x)^3) + (3504*Log[x]^8)/(1953

125*(1 + x)^2) - (68608*x*Log[x]^8)/(9765625*(1 + x)) - (3216893*Defer[Int][Log[x]^7/(4 - 3*x - x^2 + x^3)^4,

x])/12800000000 + (1908449*Defer[Int][(x*Log[x]^7)/(4 - 3*x - x^2 + x^3)^4, x])/12800000000 - (108397*Defer[In

t][(x^2*Log[x]^7)/(4 - 3*x - x^2 + x^3)^4, x])/12800000000 - (50053763*Defer[Int][Log[x]^7/(4 - 3*x - x^2 + x^

3)^3, x])/256000000000 + (19088719*Defer[Int][(x*Log[x]^7)/(4 - 3*x - x^2 + x^3)^3, x])/256000000000 - (185137

67*Defer[Int][(x^2*Log[x]^7)/(4 - 3*x - x^2 + x^3)^3, x])/256000000000 + (403919387*Defer[Int][Log[x]^7/(4 - 3

*x - x^2 + x^3)^2, x])/512000000000 - (10150667*Defer[Int][(x*Log[x]^7)/(4 - 3*x - x^2 + x^3)^2, x])/102400000

000 - (204251131*Defer[Int][(x^2*Log[x]^7)/(4 - 3*x - x^2 + x^3)^2, x])/512000000000 + (23762938581*Defer[Int]

[Log[x]^7/(4 - 3*x - x^2 + x^3), x])/10240000000000 - (1600204193*Defer[Int][(x*Log[x]^7)/(4 - 3*x - x^2 + x^3

), x])/10240000000000 - (8826538471*Defer[Int][(x^2*Log[x]^7)/(4 - 3*x - x^2 + x^3), x])/10240000000000 - (850

1009*Defer[Int][Log[x]^8/(4 - 3*x - x^2 + x^3)^5, x])/6400000000 + (9580037*Defer[Int][(x*Log[x]^8)/(4 - 3*x -

x^2 + x^3)^5, x])/6400000000 - (2589761*Defer[Int][(x^2*Log[x]^8)/(4 - 3*x - x^2 + x^3)^5, x])/6400000000 - (

314035551*Defer[Int][Log[x]^8/(4 - 3*x - x^2 + x^3)^4, x])/256000000000 + (248443403*Defer[Int][(x*Log[x]^8)/(

4 - 3*x - x^2 + x^3)^4, x])/256000000000 - (42784119*Defer[Int][(x^2*Log[x]^8)/(4 - 3*x - x^2 + x^3)^4, x])/25

6000000000 - (34305931*Defer[Int][Log[x]^8/(4 - 3*x - x^2 + x^3)^3, x])/204800000000 + (256946579*Defer[Int][(

x*Log[x]^8)/(4 - 3*x - x^2 + x^3)^3, x])/1024000000000 - (173032851*Defer[Int][(x^2*Log[x]^8)/(4 - 3*x - x^2 +

x^3)^3, x])/1024000000000 + (6304985049*Defer[Int][Log[x]^8/(4 - 3*x - x^2 + x^3)^2, x])/10240000000000 + (66

2749603*Defer[Int][(x*Log[x]^8)/(4 - 3*x - x^2 + x^3)^2, x])/10240000000000 - (2244939789*Defer[Int][(x^2*Log[

x]^8)/(4 - 3*x - x^2 + x^3)^2, x])/10240000000000 + (1982512041*Defer[Int][Log[x]^8/(4 - 3*x - x^2 + x^3), x])

/10240000000000 + (1303342833*Defer[Int][(x*Log[x]^8)/(4 - 3*x - x^2 + x^3), x])/10240000000000

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {32 \log ^7(x) \left (4 \left (4+x-4 x^2+x^4\right )-\left (20+7 x-36 x^2+13 x^4\right ) \log (x)\right )}{625 x^{11} \left (4+x-4 x^2+x^4\right )^5} \, dx\\ &=\frac {32}{625} \int \frac {\log ^7(x) \left (4 \left (4+x-4 x^2+x^4\right )-\left (20+7 x-36 x^2+13 x^4\right ) \log (x)\right )}{x^{11} \left (4+x-4 x^2+x^4\right )^5} \, dx\\ &=\frac {32}{625} \int \left (\frac {4 \log ^7(x)}{x^{11} \left (4+x-4 x^2+x^4\right )^4}-\frac {\left (20+7 x-36 x^2+13 x^4\right ) \log ^8(x)}{x^{11} \left (4+x-4 x^2+x^4\right )^5}\right ) \, dx\\ &=-\left (\frac {32}{625} \int \frac {\left (20+7 x-36 x^2+13 x^4\right ) \log ^8(x)}{x^{11} \left (4+x-4 x^2+x^4\right )^5} \, dx\right )+\frac {128}{625} \int \frac {\log ^7(x)}{x^{11} \left (4+x-4 x^2+x^4\right )^4} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [C] time = 91.72, size = 466003, normalized size = 21181.95 \begin {gather*} \text {Result too large to show} \end {gather*}

Integrate[((512 + 128*x - 512*x^2 + 128*x^4)*Log[x]^7 + (-640 - 224*x + 1152*x^2 - 416*x^4)*Log[x]^8)/(640000*

x^11 + 800000*x^12 - 2800000*x^13 - 3100000*x^14 + 6012500*x^15 + 5400625*x^16 - 8112500*x^17 - 5450000*x^18 +

7403125*x^19 + 3450000*x^20 - 4665000*x^21 - 1393750*x^22 + 2025000*x^23 + 350000*x^24 - 593750*x^25 - 50000*

x^26 + 112500*x^27 + 3125*x^28 - 12500*x^29 + 625*x^31),x]

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fricas [B] time = 0.53, size = 87, normalized size = 3.95 \begin {gather*} \frac {16 \, \log \relax (x)^{8}}{625 \, {\left (x^{26} - 16 \, x^{24} + 4 \, x^{23} + 112 \, x^{22} - 48 \, x^{21} - 442 \, x^{20} + 240 \, x^{19} + 1072 \, x^{18} - 636 \, x^{17} - 1648 \, x^{16} + 944 \, x^{15} + 1601 \, x^{14} - 752 \, x^{13} - 928 \, x^{12} + 256 \, x^{11} + 256 \, x^{10}\right )}} \end {gather*}

integrate(((-416*x^4+1152*x^2-224*x-640)*log(x)^8+(128*x^4-512*x^2+128*x+512)*log(x)^7)/(625*x^31-12500*x^29+3

125*x^28+112500*x^27-50000*x^26-593750*x^25+350000*x^24+2025000*x^23-1393750*x^22-4665000*x^21+3450000*x^20+74

03125*x^19-5450000*x^18-8112500*x^17+5400625*x^16+6012500*x^15-3100000*x^14-2800000*x^13+800000*x^12+640000*x^

16/625*log(x)^8/(x^26 - 16*x^24 + 4*x^23 + 112*x^22 - 48*x^21 - 442*x^20 + 240*x^19 + 1072*x^18 - 636*x^17 - 1

648*x^16 + 944*x^15 + 1601*x^14 - 752*x^13 - 928*x^12 + 256*x^11 + 256*x^10)

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giac [B] time = 0.27, size = 208, normalized size = 9.45 \begin {gather*} \frac {1}{655360000} \, {\left (\frac {4520791 \, x^{15} - 3533605 \, x^{14} - 69960080 \, x^{13} + 72766060 \, x^{12} + 455281020 \, x^{11} - 574439424 \, x^{10} - 1586688070 \, x^{9} + 2342625650 \, x^{8} + 3147073760 \, x^{7} - 5412734100 \, x^{6} - 3490610724 \, x^{5} + 7169196160 \, x^{4} + 1960274855 \, x^{3} - 5125415925 \, x^{2} - 405863120 \, x + 1562340832}{x^{16} - 16 \, x^{14} + 4 \, x^{13} + 112 \, x^{12} - 48 \, x^{11} - 442 \, x^{10} + 240 \, x^{9} + 1072 \, x^{8} - 636 \, x^{7} - 1648 \, x^{6} + 944 \, x^{5} + 1601 \, x^{4} - 752 \, x^{3} - 928 \, x^{2} + 256 \, x + 256} - \frac {4520791 \, x^{9} - 3533605 \, x^{8} + 2372576 \, x^{7} - 1854784 \, x^{6} + 1048064 \, x^{5} - 844544 \, x^{4} + 348160 \, x^{3} - 303104 \, x^{2} + 65536 \, x - 65536}{x^{10}}\right )} \log \relax (x)^{8} \end {gather*}

integrate(((-416*x^4+1152*x^2-224*x-640)*log(x)^8+(128*x^4-512*x^2+128*x+512)*log(x)^7)/(625*x^31-12500*x^29+3

125*x^28+112500*x^27-50000*x^26-593750*x^25+350000*x^24+2025000*x^23-1393750*x^22-4665000*x^21+3450000*x^20+74

03125*x^19-5450000*x^18-8112500*x^17+5400625*x^16+6012500*x^15-3100000*x^14-2800000*x^13+800000*x^12+640000*x^

1/655360000*((4520791*x^15 - 3533605*x^14 - 69960080*x^13 + 72766060*x^12 + 455281020*x^11 - 574439424*x^10 -

1586688070*x^9 + 2342625650*x^8 + 3147073760*x^7 - 5412734100*x^6 - 3490610724*x^5 + 7169196160*x^4 + 19602748

55*x^3 - 5125415925*x^2 - 405863120*x + 1562340832)/(x^16 - 16*x^14 + 4*x^13 + 112*x^12 - 48*x^11 - 442*x^10 +

240*x^9 + 1072*x^8 - 636*x^7 - 1648*x^6 + 944*x^5 + 1601*x^4 - 752*x^3 - 928*x^2 + 256*x + 256) - (4520791*x^

9 - 3533605*x^8 + 2372576*x^7 - 1854784*x^6 + 1048064*x^5 - 844544*x^4 + 348160*x^3 - 303104*x^2 + 65536*x - 6

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method	result	size

risch	\(\frac {16 \ln \relax (x )^{8}}{625 x^{10} \left (x^{16}-16 x^{14}+4 x^{13}+112 x^{12}-48 x^{11}-442 x^{10}+240 x^{9}+1072 x^{8}-636 x^{7}-1648 x^{6}+944 x^{5}+1601 x^{4}-752 x^{3}-928 x^{2}+256 x +256\right )}\)	\(85\)

int(((-416*x^4+1152*x^2-224*x-640)*ln(x)^8+(128*x^4-512*x^2+128*x+512)*ln(x)^7)/(625*x^31-12500*x^29+3125*x^28

+112500*x^27-50000*x^26-593750*x^25+350000*x^24+2025000*x^23-1393750*x^22-4665000*x^21+3450000*x^20+7403125*x^

19-5450000*x^18-8112500*x^17+5400625*x^16+6012500*x^15-3100000*x^14-2800000*x^13+800000*x^12+640000*x^11),x,me

16/625/x^10/(x^16-16*x^14+4*x^13+112*x^12-48*x^11-442*x^10+240*x^9+1072*x^8-636*x^7-1648*x^6+944*x^5+1601*x^4-

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maxima [B] time = 0.48, size = 87, normalized size = 3.95 \begin {gather*} \frac {16 \, \log \relax (x)^{8}}{625 \, {\left (x^{26} - 16 \, x^{24} + 4 \, x^{23} + 112 \, x^{22} - 48 \, x^{21} - 442 \, x^{20} + 240 \, x^{19} + 1072 \, x^{18} - 636 \, x^{17} - 1648 \, x^{16} + 944 \, x^{15} + 1601 \, x^{14} - 752 \, x^{13} - 928 \, x^{12} + 256 \, x^{11} + 256 \, x^{10}\right )}} \end {gather*}

integrate(((-416*x^4+1152*x^2-224*x-640)*log(x)^8+(128*x^4-512*x^2+128*x+512)*log(x)^7)/(625*x^31-12500*x^29+3

125*x^28+112500*x^27-50000*x^26-593750*x^25+350000*x^24+2025000*x^23-1393750*x^22-4665000*x^21+3450000*x^20+74

03125*x^19-5450000*x^18-8112500*x^17+5400625*x^16+6012500*x^15-3100000*x^14-2800000*x^13+800000*x^12+640000*x^

16/625*log(x)^8/(x^26 - 16*x^24 + 4*x^23 + 112*x^22 - 48*x^21 - 442*x^20 + 240*x^19 + 1072*x^18 - 636*x^17 - 1

648*x^16 + 944*x^15 + 1601*x^14 - 752*x^13 - 928*x^12 + 256*x^11 + 256*x^10)

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mupad [B] time = 5.91, size = 89, normalized size = 4.05 \begin {gather*} \frac {16\,{\ln \relax (x)}^8}{625\,\left (x^{26}-16\,x^{24}+4\,x^{23}+112\,x^{22}-48\,x^{21}-442\,x^{20}+240\,x^{19}+1072\,x^{18}-636\,x^{17}-1648\,x^{16}+944\,x^{15}+1601\,x^{14}-752\,x^{13}-928\,x^{12}+256\,x^{11}+256\,x^{10}\right )} \end {gather*}

int((log(x)^7*(128*x - 512*x^2 + 128*x^4 + 512) - log(x)^8*(224*x - 1152*x^2 + 416*x^4 + 640))/(640000*x^11 +

800000*x^12 - 2800000*x^13 - 3100000*x^14 + 6012500*x^15 + 5400625*x^16 - 8112500*x^17 - 5450000*x^18 + 740312

5*x^19 + 3450000*x^20 - 4665000*x^21 - 1393750*x^22 + 2025000*x^23 + 350000*x^24 - 593750*x^25 - 50000*x^26 +

112500*x^27 + 3125*x^28 - 12500*x^29 + 625*x^31),x)

(16*log(x)^8)/(625*(256*x^10 + 256*x^11 - 928*x^12 - 752*x^13 + 1601*x^14 + 944*x^15 - 1648*x^16 - 636*x^17 +

1072*x^18 + 240*x^19 - 442*x^20 - 48*x^21 + 112*x^22 + 4*x^23 - 16*x^24 + x^26))

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sympy [B] time = 0.48, size = 87, normalized size = 3.95 \begin {gather*} \frac {16 \log {\relax (x )}^{8}}{625 x^{26} - 10000 x^{24} + 2500 x^{23} + 70000 x^{22} - 30000 x^{21} - 276250 x^{20} + 150000 x^{19} + 670000 x^{18} - 397500 x^{17} - 1030000 x^{16} + 590000 x^{15} + 1000625 x^{14} - 470000 x^{13} - 580000 x^{12} + 160000 x^{11} + 160000 x^{10}} \end {gather*}

integrate(((-416*x**4+1152*x**2-224*x-640)*ln(x)**8+(128*x**4-512*x**2+128*x+512)*ln(x)**7)/(625*x**31-12500*x

**29+3125*x**28+112500*x**27-50000*x**26-593750*x**25+350000*x**24+2025000*x**23-1393750*x**22-4665000*x**21+3

450000*x**20+7403125*x**19-5450000*x**18-8112500*x**17+5400625*x**16+6012500*x**15-3100000*x**14-2800000*x**13

16*log(x)**8/(625*x**26 - 10000*x**24 + 2500*x**23 + 70000*x**22 - 30000*x**21 - 276250*x**20 + 150000*x**19 +

670000*x**18 - 397500*x**17 - 1030000*x**16 + 590000*x**15 + 1000625*x**14 - 470000*x**13 - 580000*x**12 + 16

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