3.71.40 \(\int \frac {720+720 x+364 x^2-588 x^3-48 x^4-52 x^5+24 x^6+16 x^7-4 x^8+(1200 x+1512 x^2-732 x^3+368 x^4+60 x^5-40 x^6-40 x^7+8 x^8) \log (x)}{3600 x-2400 x^2+40 x^3-480 x^4+89 x^5+190 x^6-9 x^7+4 x^8-9 x^9-2 x^{10}+x^{11}} \, dx\)

Optimal. Leaf size=34 \[ \frac {\left (1+x \left (\frac {5}{3-x}+x\right )\right ) \log (x)}{5-\frac {1}{4} x^2 (1+x)^2} \]

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Rubi [F]  time = 9.12, 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 {720+720 x+364 x^2-588 x^3-48 x^4-52 x^5+24 x^6+16 x^7-4 x^8+\left (1200 x+1512 x^2-732 x^3+368 x^4+60 x^5-40 x^6-40 x^7+8 x^8\right ) \log (x)}{3600 x-2400 x^2+40 x^3-480 x^4+89 x^5+190 x^6-9 x^7+4 x^8-9 x^9-2 x^{10}+x^{11}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(720 + 720*x + 364*x^2 - 588*x^3 - 48*x^4 - 52*x^5 + 24*x^6 + 16*x^7 - 4*x^8 + (1200*x + 1512*x^2 - 732*x^
3 + 368*x^4 + 60*x^5 - 40*x^6 - 40*x^7 + 8*x^8)*Log[x])/(3600*x - 2400*x^2 + 40*x^3 - 480*x^4 + 89*x^5 + 190*x
^6 - 9*x^7 + 4*x^8 - 9*x^9 - 2*x^10 + x^11),x]

[Out]

-1/98098880*((1 + 2*x)*(1076651 - 638651*(1 + 2*x)^2))/(20 - x^2 - 2*x^3 - x^4) - (3*(1 + 2*x)*(971889 - 24844
9*(1 + 2*x)^2))/(49049440*(20 - x^2 - 2*x^3 - x^4)) + (66249 - 2569*(1 + 2*x)^2)/(76880*(20 - x^2 - 2*x^3 - x^
4)) + (63*(439 + 41*(1 + 2*x)^2))/(15376*(20 - x^2 - 2*x^3 - x^4)) - (441*(5003 + 117*(1 + 2*x)^2))/(307520*(2
0 - x^2 - 2*x^3 - x^4)) + (9*(2967 + 137*(1 + 2*x)^2))/(15376*(20 - x^2 - 2*x^3 - x^4)) + (91*(7111 + 489*(1 +
2*x)^2))/(307520*(20 - x^2 - 2*x^3 - x^4)) - (3*(22211 + 1549*(1 + 2*x)^2))/(76880*(20 - x^2 - 2*x^3 - x^4))
- (13*(2029 + 2291*(1 + 2*x)^2))/(307520*(20 - x^2 - 2*x^3 - x^4)) + (45*(1 + 2*x)*(73789 + 3459*(1 + 2*x)^2))
/(4904944*(20 - x^2 - 2*x^3 - x^4)) + (3*(115951 + 3649*(1 + 2*x)^2))/(153760*(20 - x^2 - 2*x^3 - x^4)) + (9*(
1 + 2*x)*(208583 + 9113*(1 + 2*x)^2))/(4904944*(20 - x^2 - 2*x^3 - x^4)) - (3*(65257 + 10823*(1 + 2*x)^2))/(30
7520*(20 - x^2 - 2*x^3 - x^4)) - (3*(1 + 2*x)*(1903251 + 26269*(1 + 2*x)^2))/(24524720*(20 - x^2 - 2*x^3 - x^4
)) + (91*(1 + 2*x)*(305671 + 37129*(1 + 2*x)^2))/(98098880*(20 - x^2 - 2*x^3 - x^4)) - (147*(1 + 2*x)*(981369
+ 59431*(1 + 2*x)^2))/(98098880*(20 - x^2 - 2*x^3 - x^4)) - (13*(1 + 2*x)*(2591029 + 233931*(1 + 2*x)^2))/(980
98880*(20 - x^2 - 2*x^3 - x^4)) + ((1 + 2*x)*(18980129 + 457791*(1 + 2*x)^2))/(24524720*(20 - x^2 - 2*x^3 - x^
4)) + (27*(39777 - 4612*Sqrt[5])*Sqrt[5/(-1 + 8*Sqrt[5])]*ArcTan[(1 + 2*x)/Sqrt[-1 + 8*Sqrt[5]]])/2452472 + (9
*(3352 - 9435*Sqrt[5])*ArcTan[(1 + 2*x)/Sqrt[-1 + 8*Sqrt[5]]])/(595820*Sqrt[-1 + 8*Sqrt[5]]) + (91*(2722 - 235
5*Sqrt[5])*ArcTan[(1 + 2*x)/Sqrt[-1 + 8*Sqrt[5]]])/(476656*Sqrt[-1 + 8*Sqrt[5]]) + (3*(9871557 - 610388*Sqrt[5
])*ArcTan[(1 + 2*x)/Sqrt[-1 + 8*Sqrt[5]]])/(12262360*Sqrt[5*(-1 + 8*Sqrt[5])]) - (39*(1490457 - 311908*Sqrt[5]
)*ArcTan[(1 + 2*x)/Sqrt[-1 + 8*Sqrt[5]]])/(49049440*Sqrt[5*(-1 + 8*Sqrt[5])]) - (147*(1620631 - 237724*Sqrt[5]
)*ArcTan[(1 + 2*x)/Sqrt[-1 + 8*Sqrt[5]]])/(49049440*Sqrt[5*(-1 + 8*Sqrt[5])]) + (91*(551329 - 148516*Sqrt[5])*
ArcTan[(1 + 2*x)/Sqrt[-1 + 8*Sqrt[5]]])/(49049440*Sqrt[5*(-1 + 8*Sqrt[5])]) - ((650777 - 134136*Sqrt[5])*ArcTa
n[(1 + 2*x)/Sqrt[-1 + 8*Sqrt[5]]])/(119164*Sqrt[5*(-1 + 8*Sqrt[5])]) - (3*(2920549 - 105076*Sqrt[5])*ArcTan[(1
+ 2*x)/Sqrt[-1 + 8*Sqrt[5]]])/(12262360*Sqrt[5*(-1 + 8*Sqrt[5])]) + ((530671 - 86022*Sqrt[5])*ArcTan[(1 + 2*x
)/Sqrt[-1 + 8*Sqrt[5]]])/(476656*Sqrt[5*(-1 + 8*Sqrt[5])]) + (9*(335657 - 36452*Sqrt[5])*ArcTan[(1 + 2*x)/Sqrt
[-1 + 8*Sqrt[5]]])/(2452472*Sqrt[5*(-1 + 8*Sqrt[5])]) + (441*(9171 - 2288*Sqrt[5])*ArcTan[(1 + 2*x)/Sqrt[-1 +
8*Sqrt[5]]])/(476656*Sqrt[5*(-1 + 8*Sqrt[5])]) + (351*(3963 - 1420*Sqrt[5])*ArcTan[(1 + 2*x)/Sqrt[-1 + 8*Sqrt[
5]]])/(476656*Sqrt[5*(-1 + 8*Sqrt[5])]) - (243*(1359 - 986*Sqrt[5])*ArcTan[(1 + 2*x)/Sqrt[-1 + 8*Sqrt[5]]])/(2
38328*Sqrt[5*(-1 + 8*Sqrt[5])]) + (81*(2189 - 618*Sqrt[5])*ArcTan[(1 + 2*x)/Sqrt[-1 + 8*Sqrt[5]]])/(119164*Sqr
t[5*(-1 + 8*Sqrt[5])]) - (3*(836711 + 993796*Sqrt[5])*ArcTan[(1 + 2*x)/Sqrt[-1 + 8*Sqrt[5]]])/(24524720*Sqrt[5
*(-1 + 8*Sqrt[5])]) - ((18349 + 2554604*Sqrt[5])*ArcTan[(1 + 2*x)/Sqrt[-1 + 8*Sqrt[5]]])/(49049440*Sqrt[5*(-1
+ 8*Sqrt[5])]) - (9*Sqrt[(5*(-374872511 + 217966480*Sqrt[5]))/319]*ArcTan[(1 + 2*x)/Sqrt[-1 + 8*Sqrt[5]]])/119
164 - ((18349 - 2554604*Sqrt[5])*ArcTanh[(1 + 2*x)/Sqrt[1 + 8*Sqrt[5]]])/(49049440*Sqrt[5*(1 + 8*Sqrt[5])]) -
(3*(836711 - 993796*Sqrt[5])*ArcTanh[(1 + 2*x)/Sqrt[1 + 8*Sqrt[5]]])/(24524720*Sqrt[5*(1 + 8*Sqrt[5])]) + (81*
(2189 + 618*Sqrt[5])*ArcTanh[(1 + 2*x)/Sqrt[1 + 8*Sqrt[5]]])/(119164*Sqrt[5*(1 + 8*Sqrt[5])]) - (243*(1359 + 9
86*Sqrt[5])*ArcTanh[(1 + 2*x)/Sqrt[1 + 8*Sqrt[5]]])/(238328*Sqrt[5*(1 + 8*Sqrt[5])]) - (9*Sqrt[5/(1 + 8*Sqrt[5
])]*(4793 + 1052*Sqrt[5])*ArcTanh[(1 + 2*x)/Sqrt[1 + 8*Sqrt[5]]])/119164 + (351*(3963 + 1420*Sqrt[5])*ArcTanh[
(1 + 2*x)/Sqrt[1 + 8*Sqrt[5]]])/(476656*Sqrt[5*(1 + 8*Sqrt[5])]) + (441*(9171 + 2288*Sqrt[5])*ArcTanh[(1 + 2*x
)/Sqrt[1 + 8*Sqrt[5]]])/(476656*Sqrt[5*(1 + 8*Sqrt[5])]) - (91*(2722 + 2355*Sqrt[5])*ArcTanh[(1 + 2*x)/Sqrt[1
+ 8*Sqrt[5]]])/(476656*Sqrt[1 + 8*Sqrt[5]]) + (27*Sqrt[5/(1 + 8*Sqrt[5])]*(39777 + 4612*Sqrt[5])*ArcTanh[(1 +
2*x)/Sqrt[1 + 8*Sqrt[5]]])/2452472 - (9*(3352 + 9435*Sqrt[5])*ArcTanh[(1 + 2*x)/Sqrt[1 + 8*Sqrt[5]]])/(595820*
Sqrt[1 + 8*Sqrt[5]]) + (9*(335657 + 36452*Sqrt[5])*ArcTanh[(1 + 2*x)/Sqrt[1 + 8*Sqrt[5]]])/(2452472*Sqrt[5*(1
+ 8*Sqrt[5])]) + ((530671 + 86022*Sqrt[5])*ArcTanh[(1 + 2*x)/Sqrt[1 + 8*Sqrt[5]]])/(476656*Sqrt[5*(1 + 8*Sqrt[
5])]) - (3*(2920549 + 105076*Sqrt[5])*ArcTanh[(1 + 2*x)/Sqrt[1 + 8*Sqrt[5]]])/(12262360*Sqrt[5*(1 + 8*Sqrt[5])
]) - ((650777 + 134136*Sqrt[5])*ArcTanh[(1 + 2*x)/Sqrt[1 + 8*Sqrt[5]]])/(119164*Sqrt[5*(1 + 8*Sqrt[5])]) + (91
*(551329 + 148516*Sqrt[5])*ArcTanh[(1 + 2*x)/Sqrt[1 + 8*Sqrt[5]]])/(49049440*Sqrt[5*(1 + 8*Sqrt[5])]) - (147*(
1620631 + 237724*Sqrt[5])*ArcTanh[(1 + 2*x)/Sqrt[1 + 8*Sqrt[5]]])/(49049440*Sqrt[5*(1 + 8*Sqrt[5])]) - (39*(14
90457 + 311908*Sqrt[5])*ArcTanh[(1 + 2*x)/Sqrt[1 + 8*Sqrt[5]]])/(49049440*Sqrt[5*(1 + 8*Sqrt[5])]) + (3*(98715
57 + 610388*Sqrt[5])*ArcTanh[(1 + 2*x)/Sqrt[1 + 8*Sqrt[5]]])/(12262360*Sqrt[5*(1 + 8*Sqrt[5])]) + Log[x]/5 - (
5*x*Log[x])/(31*(3 - x)) - (9*(5048 - 1525*Sqrt[5])*Log[2*Sqrt[5] - x - x^2])/1191640 + (729*(220 + 47*Sqrt[5]
)*Log[2*Sqrt[5] - x - x^2])/1191640 + (63*(120 + 113*Sqrt[5])*Log[2*Sqrt[5] - x - x^2])/238328 - (351*(256 + 1
77*Sqrt[5])*Log[2*Sqrt[5] - x - x^2])/953312 - (81*(530 + 419*Sqrt[5])*Log[2*Sqrt[5] - x - x^2])/1191640 + (24
3*(970 + 513*Sqrt[5])*Log[2*Sqrt[5] - x - x^2])/2383280 - (441*(1900 + 1629*Sqrt[5])*Log[2*Sqrt[5] - x - x^2])
/4766560 + (91*(2210 + 2001*Sqrt[5])*Log[2*Sqrt[5] - x - x^2])/4766560 - ((255150 + 308257*Sqrt[5])*Log[2*Sqrt
[5] - x - x^2])/4766560 - ((255150 - 308257*Sqrt[5])*Log[2*Sqrt[5] + x + x^2])/4766560 + (91*(2210 - 2001*Sqrt
[5])*Log[2*Sqrt[5] + x + x^2])/4766560 - (441*(1900 - 1629*Sqrt[5])*Log[2*Sqrt[5] + x + x^2])/4766560 + (243*(
970 - 513*Sqrt[5])*Log[2*Sqrt[5] + x + x^2])/2383280 - (81*(530 - 419*Sqrt[5])*Log[2*Sqrt[5] + x + x^2])/11916
40 - (351*(256 - 177*Sqrt[5])*Log[2*Sqrt[5] + x + x^2])/953312 + (63*(120 - 113*Sqrt[5])*Log[2*Sqrt[5] + x + x
^2])/238328 + (729*(220 - 47*Sqrt[5])*Log[2*Sqrt[5] + x + x^2])/1191640 - (9*(5048 + 1525*Sqrt[5])*Log[2*Sqrt[
5] + x + x^2])/1191640 + (11840*Defer[Int][Log[x]/(-20 + x^2 + 2*x^3 + x^4)^2, x])/31 + (15768*Defer[Int][(x*L
og[x])/(-20 + x^2 + 2*x^3 + x^4)^2, x])/31 + (2432*Defer[Int][(x^2*Log[x])/(-20 + x^2 + 2*x^3 + x^4)^2, x])/31
+ (784*Defer[Int][(x^3*Log[x])/(-20 + x^2 + 2*x^3 + x^4)^2, x])/31 + (352*Defer[Int][Log[x]/(-20 + x^2 + 2*x^
3 + x^4), x])/31 + (368*Defer[Int][(x*Log[x])/(-20 + x^2 + 2*x^3 + x^4), x])/31 + (15*Defer[Int][(x^2*Log[x])/
(-20 + x^2 + 2*x^3 + x^4), x])/31

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {720+720 x+364 x^2-588 x^3-48 x^4-52 x^5+24 x^6+16 x^7-4 x^8+\left (1200 x+1512 x^2-732 x^3+368 x^4+60 x^5-40 x^6-40 x^7+8 x^8\right ) \log (x)}{x \left (60-20 x-3 x^2-5 x^3-x^4+x^5\right )^2} \, dx\\ &=\int \left (\frac {720}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2}+\frac {720}{(-3+x)^2 x \left (-20+x^2+2 x^3+x^4\right )^2}+\frac {364 x}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2}-\frac {588 x^2}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2}-\frac {48 x^3}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2}-\frac {52 x^4}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2}+\frac {24 x^5}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2}+\frac {16 x^6}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2}-\frac {4 x^7}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2}+\frac {4 \left (300+378 x-183 x^2+92 x^3+15 x^4-10 x^5-10 x^6+2 x^7\right ) \log (x)}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {x^7}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2} \, dx\right )+4 \int \frac {\left (300+378 x-183 x^2+92 x^3+15 x^4-10 x^5-10 x^6+2 x^7\right ) \log (x)}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2} \, dx+16 \int \frac {x^6}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2} \, dx+24 \int \frac {x^5}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2} \, dx-48 \int \frac {x^3}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2} \, dx-52 \int \frac {x^4}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2} \, dx+364 \int \frac {x}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2} \, dx-588 \int \frac {x^2}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2} \, dx+720 \int \frac {1}{(-3+x)^2 \left (-20+x^2+2 x^3+x^4\right )^2} \, dx+720 \int \frac {1}{(-3+x)^2 x \left (-20+x^2+2 x^3+x^4\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 42, normalized size = 1.24 \begin {gather*} -\frac {4 \left (-3-4 x-3 x^2+x^3\right ) \log (x)}{60-20 x-3 x^2-5 x^3-x^4+x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(720 + 720*x + 364*x^2 - 588*x^3 - 48*x^4 - 52*x^5 + 24*x^6 + 16*x^7 - 4*x^8 + (1200*x + 1512*x^2 -
732*x^3 + 368*x^4 + 60*x^5 - 40*x^6 - 40*x^7 + 8*x^8)*Log[x])/(3600*x - 2400*x^2 + 40*x^3 - 480*x^4 + 89*x^5 +
190*x^6 - 9*x^7 + 4*x^8 - 9*x^9 - 2*x^10 + x^11),x]

[Out]

(-4*(-3 - 4*x - 3*x^2 + x^3)*Log[x])/(60 - 20*x - 3*x^2 - 5*x^3 - x^4 + x^5)

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fricas [A]  time = 0.72, size = 42, normalized size = 1.24 \begin {gather*} -\frac {4 \, {\left (x^{3} - 3 \, x^{2} - 4 \, x - 3\right )} \log \relax (x)}{x^{5} - x^{4} - 5 \, x^{3} - 3 \, x^{2} - 20 \, x + 60} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^8-40*x^7-40*x^6+60*x^5+368*x^4-732*x^3+1512*x^2+1200*x)*log(x)-4*x^8+16*x^7+24*x^6-52*x^5-48*x
^4-588*x^3+364*x^2+720*x+720)/(x^11-2*x^10-9*x^9+4*x^8-9*x^7+190*x^6+89*x^5-480*x^4+40*x^3-2400*x^2+3600*x),x,
algorithm="fricas")

[Out]

-4*(x^3 - 3*x^2 - 4*x - 3)*log(x)/(x^5 - x^4 - 5*x^3 - 3*x^2 - 20*x + 60)

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giac [A]  time = 0.18, size = 42, normalized size = 1.24 \begin {gather*} -\frac {4 \, {\left (x^{3} - 3 \, x^{2} - 4 \, x - 3\right )} \log \relax (x)}{x^{5} - x^{4} - 5 \, x^{3} - 3 \, x^{2} - 20 \, x + 60} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^8-40*x^7-40*x^6+60*x^5+368*x^4-732*x^3+1512*x^2+1200*x)*log(x)-4*x^8+16*x^7+24*x^6-52*x^5-48*x
^4-588*x^3+364*x^2+720*x+720)/(x^11-2*x^10-9*x^9+4*x^8-9*x^7+190*x^6+89*x^5-480*x^4+40*x^3-2400*x^2+3600*x),x,
algorithm="giac")

[Out]

-4*(x^3 - 3*x^2 - 4*x - 3)*log(x)/(x^5 - x^4 - 5*x^3 - 3*x^2 - 20*x + 60)

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maple [A]  time = 0.18, size = 43, normalized size = 1.26

method	result	size
risch	\(-\frac {4 \left (x^{3}-3 x^{2}-4 x -3\right ) \ln \relax (x )}{x^{5}-x^{4}-5 x^{3}-3 x^{2}-20 x +60}\)	\(43\)
norman	\(\frac {12 \ln \relax (x )-4 x^{3} \ln \relax (x )+12 x^{2} \ln \relax (x )+16 x \ln \relax (x )}{x^{5}-x^{4}-5 x^{3}-3 x^{2}-20 x +60}\)	\(51\)
default	\(\frac {\ln \relax (x )}{5}+\frac {103 i \left (-1+8 \sqrt {5}\right ) \arctan \left (\frac {2 x +1}{\sqrt {-1+8 \sqrt {5}}}\right )}{1240 \left (\frac {i \sqrt {-1+8 \sqrt {5}}}{4}+\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}+\frac {7 i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}} \ln \left (2 \sqrt {5}+x^{2}+x \right )}{620 \left (\frac {i \sqrt {-1+8 \sqrt {5}}}{4}+\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}-\frac {103 i \left (-1+8 \sqrt {5}\right ) \arctan \left (\frac {2 x +1}{\sqrt {-1+8 \sqrt {5}}}\right )}{1240 \left (-\frac {i \sqrt {-1+8 \sqrt {5}}}{4}-\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}+\frac {453 i \sqrt {-1+8 \sqrt {5}}\, \ln \left (2 \sqrt {5}+x^{2}+x \right )}{620 \left (-\frac {i \sqrt {-1+8 \sqrt {5}}}{4}-\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}-\frac {7 i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}} \ln \left (2 \sqrt {5}+x^{2}+x \right )}{620 \left (-\frac {i \sqrt {-1+8 \sqrt {5}}}{4}-\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}-\frac {453 i \sqrt {-1+8 \sqrt {5}}\, \ln \left (2 \sqrt {5}+x^{2}+x \right )}{620 \left (\frac {i \sqrt {-1+8 \sqrt {5}}}{4}+\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}-\frac {\ln \relax (x ) x \left (56 x^{3}+187 x^{2}+1051 x +1200\right )}{155 \left (x^{4}+2 x^{3}+x^{2}-20\right )}+\frac {2857 \ln \left (2 \sqrt {5}+x^{2}+x \right )}{2480 \left (\frac {i \sqrt {-1+8 \sqrt {5}}}{4}+\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}+\frac {2857 \ln \left (2 \sqrt {5}+x^{2}+x \right )}{2480 \left (-\frac {i \sqrt {-1+8 \sqrt {5}}}{4}-\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}+\frac {2857 \ln \left (x +\frac {1}{2}+\frac {\sqrt {1+8 \sqrt {5}}}{2}\right )}{1240 \left (\frac {\sqrt {1+8 \sqrt {5}}}{4}-\frac {\left (1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}+\frac {2857 \ln \left (x +\frac {1}{2}-\frac {\sqrt {1+8 \sqrt {5}}}{2}\right )}{1240 \left (-\frac {\sqrt {1+8 \sqrt {5}}}{4}+\frac {\left (1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}+\frac {5 \ln \relax (x ) x}{31 \left (x -3\right )}-\frac {453 \ln \left (x +\frac {1}{2}+\frac {\sqrt {1+8 \sqrt {5}}}{2}\right ) \sqrt {1+8 \sqrt {5}}}{310 \left (\frac {\sqrt {1+8 \sqrt {5}}}{4}-\frac {\left (1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}+\frac {103 \ln \left (x +\frac {1}{2}+\frac {\sqrt {1+8 \sqrt {5}}}{2}\right ) \left (1+8 \sqrt {5}\right )}{1240 \left (\frac {\sqrt {1+8 \sqrt {5}}}{4}-\frac {\left (1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}-\frac {7 \ln \left (x +\frac {1}{2}+\frac {\sqrt {1+8 \sqrt {5}}}{2}\right ) \left (1+8 \sqrt {5}\right )^{\frac {3}{2}}}{310 \left (\frac {\sqrt {1+8 \sqrt {5}}}{4}-\frac {\left (1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}+\frac {453 \ln \left (x +\frac {1}{2}-\frac {\sqrt {1+8 \sqrt {5}}}{2}\right ) \sqrt {1+8 \sqrt {5}}}{310 \left (-\frac {\sqrt {1+8 \sqrt {5}}}{4}+\frac {\left (1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}+\frac {103 \ln \left (x +\frac {1}{2}-\frac {\sqrt {1+8 \sqrt {5}}}{2}\right ) \left (1+8 \sqrt {5}\right )}{1240 \left (-\frac {\sqrt {1+8 \sqrt {5}}}{4}+\frac {\left (1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}+\frac {7 \ln \left (x +\frac {1}{2}-\frac {\sqrt {1+8 \sqrt {5}}}{2}\right ) \left (1+8 \sqrt {5}\right )^{\frac {3}{2}}}{310 \left (-\frac {\sqrt {1+8 \sqrt {5}}}{4}+\frac {\left (1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}+\textit {\_Z}^{2}-20\right )}{\sum }\frac {\left (56 \textit {\_R}^{3}+187 \textit {\_R}^{2}+1051 \textit {\_R} +1200\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}+3 \textit {\_R}^{2}+\textit {\_R}}\right )}{310}+\frac {7 \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}} \arctan \left (\frac {2 x +1}{\sqrt {-1+8 \sqrt {5}}}\right )}{310 \left (-\frac {i \sqrt {-1+8 \sqrt {5}}}{4}-\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}+\frac {7 \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}} \arctan \left (\frac {2 x +1}{\sqrt {-1+8 \sqrt {5}}}\right )}{310 \left (\frac {i \sqrt {-1+8 \sqrt {5}}}{4}+\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}-\frac {453 \sqrt {-1+8 \sqrt {5}}\, \arctan \left (\frac {2 x +1}{\sqrt {-1+8 \sqrt {5}}}\right )}{310 \left (\frac {i \sqrt {-1+8 \sqrt {5}}}{4}+\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}-\frac {2857 i \arctan \left (\frac {2 x +1}{\sqrt {-1+8 \sqrt {5}}}\right )}{1240 \left (\frac {i \sqrt {-1+8 \sqrt {5}}}{4}+\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}-\frac {453 \sqrt {-1+8 \sqrt {5}}\, \arctan \left (\frac {2 x +1}{\sqrt {-1+8 \sqrt {5}}}\right )}{310 \left (-\frac {i \sqrt {-1+8 \sqrt {5}}}{4}-\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}+\frac {2857 i \arctan \left (\frac {2 x +1}{\sqrt {-1+8 \sqrt {5}}}\right )}{1240 \left (-\frac {i \sqrt {-1+8 \sqrt {5}}}{4}-\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}-\frac {103 \left (-1+8 \sqrt {5}\right ) \ln \left (2 \sqrt {5}+x^{2}+x \right )}{2480 \left (\frac {i \sqrt {-1+8 \sqrt {5}}}{4}+\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}-\frac {103 \left (-1+8 \sqrt {5}\right ) \ln \left (2 \sqrt {5}+x^{2}+x \right )}{2480 \left (-\frac {i \sqrt {-1+8 \sqrt {5}}}{4}-\frac {i \left (-1+8 \sqrt {5}\right )^{\frac {3}{2}}}{4}\right )}\)	\(1272\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^8-40*x^7-40*x^6+60*x^5+368*x^4-732*x^3+1512*x^2+1200*x)*ln(x)-4*x^8+16*x^7+24*x^6-52*x^5-48*x^4-588*
x^3+364*x^2+720*x+720)/(x^11-2*x^10-9*x^9+4*x^8-9*x^7+190*x^6+89*x^5-480*x^4+40*x^3-2400*x^2+3600*x),x,method=
_RETURNVERBOSE)

[Out]

-4*(x^3-3*x^2-4*x-3)/(x^5-x^4-5*x^3-3*x^2-20*x+60)*ln(x)

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maxima [A]  time = 0.38, size = 42, normalized size = 1.24 \begin {gather*} -\frac {4 \, {\left (x^{3} - 3 \, x^{2} - 4 \, x - 3\right )} \log \relax (x)}{x^{5} - x^{4} - 5 \, x^{3} - 3 \, x^{2} - 20 \, x + 60} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^8-40*x^7-40*x^6+60*x^5+368*x^4-732*x^3+1512*x^2+1200*x)*log(x)-4*x^8+16*x^7+24*x^6-52*x^5-48*x
^4-588*x^3+364*x^2+720*x+720)/(x^11-2*x^10-9*x^9+4*x^8-9*x^7+190*x^6+89*x^5-480*x^4+40*x^3-2400*x^2+3600*x),x,
algorithm="maxima")

[Out]

-4*(x^3 - 3*x^2 - 4*x - 3)*log(x)/(x^5 - x^4 - 5*x^3 - 3*x^2 - 20*x + 60)

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mupad [B]  time = 4.42, size = 50, normalized size = 1.47 \begin {gather*} \frac {15\,\ln \relax (x)}{31\,\left (x-3\right )}-\frac {\ln \relax (x)\,\left (15\,x^3+199\,x^2+240\,x+224\right )}{31\,\left (x^4+2\,x^3+x^2-20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((720*x + 364*x^2 - 588*x^3 - 48*x^4 - 52*x^5 + 24*x^6 + 16*x^7 - 4*x^8 + log(x)*(1200*x + 1512*x^2 - 732*x
^3 + 368*x^4 + 60*x^5 - 40*x^6 - 40*x^7 + 8*x^8) + 720)/(3600*x - 2400*x^2 + 40*x^3 - 480*x^4 + 89*x^5 + 190*x
^6 - 9*x^7 + 4*x^8 - 9*x^9 - 2*x^10 + x^11),x)

[Out]

(15*log(x))/(31*(x - 3)) - (log(x)*(240*x + 199*x^2 + 15*x^3 + 224))/(31*(x^2 + 2*x^3 + x^4 - 20))

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sympy [A]  time = 0.29, size = 39, normalized size = 1.15 \begin {gather*} \frac {\left (- 4 x^{3} + 12 x^{2} + 16 x + 12\right ) \log {\relax (x )}}{x^{5} - x^{4} - 5 x^{3} - 3 x^{2} - 20 x + 60} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**8-40*x**7-40*x**6+60*x**5+368*x**4-732*x**3+1512*x**2+1200*x)*ln(x)-4*x**8+16*x**7+24*x**6-52
*x**5-48*x**4-588*x**3+364*x**2+720*x+720)/(x**11-2*x**10-9*x**9+4*x**8-9*x**7+190*x**6+89*x**5-480*x**4+40*x*
*3-2400*x**2+3600*x),x)

[Out]

(-4*x**3 + 12*x**2 + 16*x + 12)*log(x)/(x**5 - x**4 - 5*x**3 - 3*x**2 - 20*x + 60)

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