3.78.38 $\int \frac{(512+128x-512x^2+128x^4){\log }^7(x)+(-640-224x+1152x^2-416x^4){\log }^8(x)}{640000x^{11}+800000x^{12}-2800000x^{13}-3100000x^{14}+6012500x^{15}+5400625x^{16}-8112500x^{17}-5450000x^{18}+7403125x^{19}+3450000x^{20}-4665000x^{21}-1393750x^{22}+2025000x^{23}+350000x^{24}-593750x^{25}-50000x^{26}+112500x^{27}+3125x^{28}-12500x^{29}+625x^{31}}dx$

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

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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 = {} $$\begin{array}{c}\int \frac{(512+128x-512x^2+128x^4){\log }^7(x)+(-640-224x+1152x^2-416x^4){\log }^8(x)}{640000x^{11}+800000x^{12}-2800000x^{13}-3100000x^{14}+6012500x^{15}+5400625x^{16}-8112500x^{17}-5450000x^{18}+7403125x^{19}+3450000x^{20}-4665000x^{21}-1393750x^{22}+2025000x^{23}+350000x^{24}-593750x^{25}-50000x^{26}+112500x^{27}+3125x^{28}-12500x^{29}+625x^{31}}dx\end{array}$$

Verification is not applicable to the result.

[In]

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]

[Out]

(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

Rubi steps

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

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

Warning: Unable to verify antiderivative.

[In]

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]

[Out]

Result too large to show

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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^
11),x, algorithm="fricas")

[Out]

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 $$\begin{array}{c}\frac{1}{655360000}(\frac{4520791x^{15}-3533605x^{14}-69960080x^{13}+72766060x^{12}+455281020x^{11}-574439424x^{10}-1586688070x^9+2342625650x^8+3147073760x^7-5412734100x^6-3490610724x^5+7169196160x^4+1960274855x^3-5125415925x^2-405863120x+1562340832}{x^{16}-16x^{14}+4x^{13}+112x^{12}-48x^{11}-442x^{10}+240x^9+1072x^8-636x^7-1648x^6+944x^5+1601x^4-752x^3-928x^2+256x+256}-\frac{4520791x^9-3533605x^8+2372576x^7-1854784x^6+1048064x^5-844544x^4+348160x^3-303104x^2+65536x-65536}{x^{10}})\log (x{)}^8\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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^
11),x, algorithm="giac")

[Out]

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
5536)/x^10)*log(x)^8

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maple [B]  time = 0.20, size = 85, normalized size = 3.86

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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
thod=_RETURNVERBOSE)

[Out]

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-
752*x^3-928*x^2+256*x+256)*ln(x)^8

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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^
11),x, algorithm="maxima")

[Out]

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 $$\begin{array}{c}\frac{16{\ln (x)}^8}{625(x^{26}-16x^{24}+4x^{23}+112x^{22}-48x^{21}-442x^{20}+240x^{19}+1072x^{18}-636x^{17}-1648x^{16}+944x^{15}+1601x^{14}-752x^{13}-928x^{12}+256x^{11}+256x^{10})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

(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 $$\begin{array}{c}\frac{16\log {(x)}^8}{625x^{26}-10000x^{24}+2500x^{23}+70000x^{22}-30000x^{21}-276250x^{20}+150000x^{19}+670000x^{18}-397500x^{17}-1030000x^{16}+590000x^{15}+1000625x^{14}-470000x^{13}-580000x^{12}+160000x^{11}+160000x^{10}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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
+800000*x**12+640000*x**11),x)

[Out]

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
0000*x**11 + 160000*x**10)

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