3.63.74 \(\int \frac {-732656250+46976265 x-1128063 x^2+12024 x^3-48 x^4+(-117206250+5632800 x-90120 x^2+480 x^3) \log (x)+(-7031625+225150 x-1800 x^2) \log ^2(x)+(-187500+3000 x) \log ^3(x)-1875 \log ^4(x)+(6250-2605 x+40 x^2+(250-100 x) \log (x)) \log (x^3)}{244140625 x-15656250 x^2+376001 x^3-4008 x^4+16 x^5+(39062500 x-1877500 x^2+30040 x^3-160 x^4) \log (x)+(2343750 x-75050 x^2+600 x^3) \log ^2(x)+(62500 x-1000 x^2) \log ^3(x)+625 x \log ^4(x)} \, dx\)

Optimal. Leaf size=27 \[ \left (-1+\frac {5}{x-(-2 x+5 (25+\log (x)))^2}\right ) \log \left (x^3\right ) \]

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Rubi [F]  time = 13.51, 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 {-732656250+46976265 x-1128063 x^2+12024 x^3-48 x^4+\left (-117206250+5632800 x-90120 x^2+480 x^3\right ) \log (x)+\left (-7031625+225150 x-1800 x^2\right ) \log ^2(x)+(-187500+3000 x) \log ^3(x)-1875 \log ^4(x)+\left (6250-2605 x+40 x^2+(250-100 x) \log (x)\right ) \log \left (x^3\right )}{244140625 x-15656250 x^2+376001 x^3-4008 x^4+16 x^5+\left (39062500 x-1877500 x^2+30040 x^3-160 x^4\right ) \log (x)+\left (2343750 x-75050 x^2+600 x^3\right ) \log ^2(x)+\left (62500 x-1000 x^2\right ) \log ^3(x)+625 x \log ^4(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-732656250 + 46976265*x - 1128063*x^2 + 12024*x^3 - 48*x^4 + (-117206250 + 5632800*x - 90120*x^2 + 480*x^
3)*Log[x] + (-7031625 + 225150*x - 1800*x^2)*Log[x]^2 + (-187500 + 3000*x)*Log[x]^3 - 1875*Log[x]^4 + (6250 -
2605*x + 40*x^2 + (250 - 100*x)*Log[x])*Log[x^3])/(244140625*x - 15656250*x^2 + 376001*x^3 - 4008*x^4 + 16*x^5
+ (39062500*x - 1877500*x^2 + 30040*x^3 - 160*x^4)*Log[x] + (2343750*x - 75050*x^2 + 600*x^3)*Log[x]^2 + (625
00*x - 1000*x^2)*Log[x]^3 + 625*x*Log[x]^4),x]

[Out]

-3*Log[x] - 15*Defer[Int][1/(x*(15625 - 501*x + 4*x^2 + 1250*Log[x] - 20*x*Log[x] + 25*Log[x]^2)), x] - 2605*D
efer[Int][Log[x^3]/(15625 - 501*x + 4*x^2 + 1250*Log[x] - 20*x*Log[x] + 25*Log[x]^2)^2, x] + 6250*Defer[Int][L
og[x^3]/(x*(15625 - 501*x + 4*x^2 + 1250*Log[x] - 20*x*Log[x] + 25*Log[x]^2)^2), x] + 40*Defer[Int][(x*Log[x^3
])/(15625 - 501*x + 4*x^2 + 1250*Log[x] - 20*x*Log[x] + 25*Log[x]^2)^2, x] - 100*Defer[Int][(Log[x]*Log[x^3])/
(15625 - 501*x + 4*x^2 + 1250*Log[x] - 20*x*Log[x] + 25*Log[x]^2)^2, x] + 250*Defer[Int][(Log[x]*Log[x^3])/(x*
(15625 - 501*x + 4*x^2 + 1250*Log[x] - 20*x*Log[x] + 25*Log[x]^2)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3 \left (244218750-15658755 x+376021 x^2-4008 x^3+16 x^4\right )-75 \left (93755-3002 x+24 x^2\right ) \log ^2(x)+1500 (-125+2 x) \log ^3(x)-1875 \log ^4(x)+5 \left (1250-521 x+8 x^2\right ) \log \left (x^3\right )+10 \log (x) \left (-11720625+563280 x-9012 x^2+48 x^3-5 (-5+2 x) \log \left (x^3\right )\right )}{x \left (15625-501 x+4 x^2+(1250-20 x) \log (x)+25 \log ^2(x)\right )^2} \, dx\\ &=\int \left (-\frac {3 \left (15625-501 x+4 x^2\right ) \left (15630-501 x+4 x^2\right )}{x \left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2}+\frac {5632800 \log (x)}{\left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2}-\frac {117206250 \log (x)}{x \left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2}-\frac {90120 x \log (x)}{\left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2}+\frac {480 x^2 \log (x)}{\left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2}-\frac {75 \left (93755-3002 x+24 x^2\right ) \log ^2(x)}{x \left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2}+\frac {1500 (-125+2 x) \log ^3(x)}{x \left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2}-\frac {1875 \log ^4(x)}{x \left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2}+\frac {5 \left (1250-521 x+8 x^2+50 \log (x)-20 x \log (x)\right ) \log \left (x^3\right )}{x \left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2}\right ) \, dx\\ &=-\left (3 \int \frac {\left (15625-501 x+4 x^2\right ) \left (15630-501 x+4 x^2\right )}{x \left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2} \, dx\right )+5 \int \frac {\left (1250-521 x+8 x^2+50 \log (x)-20 x \log (x)\right ) \log \left (x^3\right )}{x \left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2} \, dx-75 \int \frac {\left (93755-3002 x+24 x^2\right ) \log ^2(x)}{x \left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2} \, dx+480 \int \frac {x^2 \log (x)}{\left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2} \, dx+1500 \int \frac {(-125+2 x) \log ^3(x)}{x \left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2} \, dx-1875 \int \frac {\log ^4(x)}{x \left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2} \, dx-90120 \int \frac {x \log (x)}{\left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2} \, dx+5632800 \int \frac {\log (x)}{\left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2} \, dx-117206250 \int \frac {\log (x)}{x \left (15625-501 x+4 x^2+1250 \log (x)-20 x \log (x)+25 \log ^2(x)\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 37, normalized size = 1.37 \begin {gather*} -3 \log (x)-\frac {5 \log \left (x^3\right )}{15625-501 x+4 x^2+(1250-20 x) \log (x)+25 \log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-732656250 + 46976265*x - 1128063*x^2 + 12024*x^3 - 48*x^4 + (-117206250 + 5632800*x - 90120*x^2 +
480*x^3)*Log[x] + (-7031625 + 225150*x - 1800*x^2)*Log[x]^2 + (-187500 + 3000*x)*Log[x]^3 - 1875*Log[x]^4 + (6
250 - 2605*x + 40*x^2 + (250 - 100*x)*Log[x])*Log[x^3])/(244140625*x - 15656250*x^2 + 376001*x^3 - 4008*x^4 +
16*x^5 + (39062500*x - 1877500*x^2 + 30040*x^3 - 160*x^4)*Log[x] + (2343750*x - 75050*x^2 + 600*x^3)*Log[x]^2
+ (62500*x - 1000*x^2)*Log[x]^3 + 625*x*Log[x]^4),x]

[Out]

-3*Log[x] - (5*Log[x^3])/(15625 - 501*x + 4*x^2 + (1250 - 20*x)*Log[x] + 25*Log[x]^2)

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fricas [B]  time = 1.05, size = 61, normalized size = 2.26 \begin {gather*} \frac {3 \, {\left (10 \, {\left (2 \, x - 125\right )} \log \relax (x)^{2} - 25 \, \log \relax (x)^{3} - {\left (4 \, x^{2} - 501 \, x + 15630\right )} \log \relax (x)\right )}}{4 \, x^{2} - 10 \, {\left (2 \, x - 125\right )} \log \relax (x) + 25 \, \log \relax (x)^{2} - 501 \, x + 15625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-100*x+250)*log(x)+40*x^2-2605*x+6250)*log(x^3)-1875*log(x)^4+(3000*x-187500)*log(x)^3+(-1800*x^2
+225150*x-7031625)*log(x)^2+(480*x^3-90120*x^2+5632800*x-117206250)*log(x)-48*x^4+12024*x^3-1128063*x^2+469762
65*x-732656250)/(625*x*log(x)^4+(-1000*x^2+62500*x)*log(x)^3+(600*x^3-75050*x^2+2343750*x)*log(x)^2+(-160*x^4+
30040*x^3-1877500*x^2+39062500*x)*log(x)+16*x^5-4008*x^4+376001*x^3-15656250*x^2+244140625*x),x, algorithm="fr
icas")

[Out]

3*(10*(2*x - 125)*log(x)^2 - 25*log(x)^3 - (4*x^2 - 501*x + 15630)*log(x))/(4*x^2 - 10*(2*x - 125)*log(x) + 25
*log(x)^2 - 501*x + 15625)

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giac [A]  time = 0.14, size = 36, normalized size = 1.33 \begin {gather*} -\frac {15 \, \log \relax (x)}{4 \, x^{2} - 20 \, x \log \relax (x) + 25 \, \log \relax (x)^{2} - 501 \, x + 1250 \, \log \relax (x) + 15625} - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-100*x+250)*log(x)+40*x^2-2605*x+6250)*log(x^3)-1875*log(x)^4+(3000*x-187500)*log(x)^3+(-1800*x^2
+225150*x-7031625)*log(x)^2+(480*x^3-90120*x^2+5632800*x-117206250)*log(x)-48*x^4+12024*x^3-1128063*x^2+469762
65*x-732656250)/(625*x*log(x)^4+(-1000*x^2+62500*x)*log(x)^3+(600*x^3-75050*x^2+2343750*x)*log(x)^2+(-160*x^4+
30040*x^3-1877500*x^2+39062500*x)*log(x)+16*x^5-4008*x^4+376001*x^3-15656250*x^2+244140625*x),x, algorithm="gi
ac")

[Out]

-15*log(x)/(4*x^2 - 20*x*log(x) + 25*log(x)^2 - 501*x + 1250*log(x) + 15625) - 3*log(x)

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maple [C]  time = 0.14, size = 163, normalized size = 6.04

method	result	size
risch	\(-3 \ln \relax (x )-\frac {5 \left (6 \ln \relax (x )-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}-i \pi \mathrm {csgn}\left (i x^{3}\right )^{3}\right )}{2 \left (25 \ln \relax (x )^{2}-20 x \ln \relax (x )+4 x^{2}+1250 \ln \relax (x )-501 x +15625\right )}\)	\(163\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-100*x+250)*ln(x)+40*x^2-2605*x+6250)*ln(x^3)-1875*ln(x)^4+(3000*x-187500)*ln(x)^3+(-1800*x^2+225150*x-
7031625)*ln(x)^2+(480*x^3-90120*x^2+5632800*x-117206250)*ln(x)-48*x^4+12024*x^3-1128063*x^2+46976265*x-7326562
50)/(625*x*ln(x)^4+(-1000*x^2+62500*x)*ln(x)^3+(600*x^3-75050*x^2+2343750*x)*ln(x)^2+(-160*x^4+30040*x^3-18775
00*x^2+39062500*x)*ln(x)+16*x^5-4008*x^4+376001*x^3-15656250*x^2+244140625*x),x,method=_RETURNVERBOSE)

[Out]

-3*ln(x)-5/2*(6*ln(x)-I*Pi*csgn(I*x)^2*csgn(I*x^2)+2*I*Pi*csgn(I*x)*csgn(I*x^2)^2-I*Pi*csgn(I*x)*csgn(I*x^2)*c
sgn(I*x^3)+I*Pi*csgn(I*x)*csgn(I*x^3)^2-I*Pi*csgn(I*x^2)^3+I*Pi*csgn(I*x^2)*csgn(I*x^3)^2-I*Pi*csgn(I*x^3)^3)/
(25*ln(x)^2-20*x*ln(x)+4*x^2+1250*ln(x)-501*x+15625)

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maxima [A]  time = 0.40, size = 36, normalized size = 1.33 \begin {gather*} -\frac {15 \, \log \relax (x)}{4 \, x^{2} - 10 \, {\left (2 \, x - 125\right )} \log \relax (x) + 25 \, \log \relax (x)^{2} - 501 \, x + 15625} - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-100*x+250)*log(x)+40*x^2-2605*x+6250)*log(x^3)-1875*log(x)^4+(3000*x-187500)*log(x)^3+(-1800*x^2
+225150*x-7031625)*log(x)^2+(480*x^3-90120*x^2+5632800*x-117206250)*log(x)-48*x^4+12024*x^3-1128063*x^2+469762
65*x-732656250)/(625*x*log(x)^4+(-1000*x^2+62500*x)*log(x)^3+(600*x^3-75050*x^2+2343750*x)*log(x)^2+(-160*x^4+
30040*x^3-1877500*x^2+39062500*x)*log(x)+16*x^5-4008*x^4+376001*x^3-15656250*x^2+244140625*x),x, algorithm="ma
xima")

[Out]

-15*log(x)/(4*x^2 - 10*(2*x - 125)*log(x) + 25*log(x)^2 - 501*x + 15625) - 3*log(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\ln \relax (x)}^2\,\left (1800\,x^2-225150\,x+7031625\right )-46976265\,x+1875\,{\ln \relax (x)}^4+\ln \left (x^3\right )\,\left (2605\,x+\ln \relax (x)\,\left (100\,x-250\right )-40\,x^2-6250\right )+1128063\,x^2-12024\,x^3+48\,x^4-{\ln \relax (x)}^3\,\left (3000\,x-187500\right )-\ln \relax (x)\,\left (480\,x^3-90120\,x^2+5632800\,x-117206250\right )[phone]}{244140625\,x+{\ln \relax (x)}^3\,\left (62500\,x-1000\,x^2\right )+625\,x\,{\ln \relax (x)}^4+\ln \relax (x)\,\left (-160\,x^4+30040\,x^3-1877500\,x^2+39062500\,x\right )+{\ln \relax (x)}^2\,\left (600\,x^3-75050\,x^2+2343750\,x\right )-15656250\,x^2+376001\,x^3-4008\,x^4+16\,x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)^2*(1800*x^2 - 225150*x + 7031625) - 46976265*x + 1875*log(x)^4 + log(x^3)*(2605*x + log(x)*(100*x
- 250) - 40*x^2 - 6250) + 1128063*x^2 - 12024*x^3 + 48*x^4 - log(x)^3*(3000*x - 187500) - log(x)*(5632800*x -
90120*x^2 + 480*x^3 - 117206250) + 732656250)/(244140625*x + log(x)^3*(62500*x - 1000*x^2) + 625*x*log(x)^4 +
log(x)*(39062500*x - 1877500*x^2 + 30040*x^3 - 160*x^4) + log(x)^2*(2343750*x - 75050*x^2 + 600*x^3) - 156562
50*x^2 + 376001*x^3 - 4008*x^4 + 16*x^5),x)

[Out]

int(-(log(x)^2*(1800*x^2 - 225150*x + 7031625) - 46976265*x + 1875*log(x)^4 + log(x^3)*(2605*x + log(x)*(100*x
- 250) - 40*x^2 - 6250) + 1128063*x^2 - 12024*x^3 + 48*x^4 - log(x)^3*(3000*x - 187500) - log(x)*(5632800*x -
90120*x^2 + 480*x^3 - 117206250) + 732656250)/(244140625*x + log(x)^3*(62500*x - 1000*x^2) + 625*x*log(x)^4 +
log(x)*(39062500*x - 1877500*x^2 + 30040*x^3 - 160*x^4) + log(x)^2*(2343750*x - 75050*x^2 + 600*x^3) - 156562
50*x^2 + 376001*x^3 - 4008*x^4 + 16*x^5), x)

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sympy [A]  time = 0.36, size = 36, normalized size = 1.33 \begin {gather*} - 3 \log {\relax (x )} - \frac {15 \log {\relax (x )}}{4 x^{2} - 501 x + \left (1250 - 20 x\right ) \log {\relax (x )} + 25 \log {\relax (x )}^{2} + 15625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-100*x+250)*ln(x)+40*x**2-2605*x+6250)*ln(x**3)-1875*ln(x)**4+(3000*x-187500)*ln(x)**3+(-1800*x**
2+225150*x-7031625)*ln(x)**2+(480*x**3-90120*x**2+5632800*x-117206250)*ln(x)-48*x**4+12024*x**3-1128063*x**2+4
6976265*x-732656250)/(625*x*ln(x)**4+(-1000*x**2+62500*x)*ln(x)**3+(600*x**3-75050*x**2+2343750*x)*ln(x)**2+(-
160*x**4+30040*x**3-1877500*x**2+39062500*x)*ln(x)+16*x**5-4008*x**4+376001*x**3-15656250*x**2+244140625*x),x)

[Out]

-3*log(x) - 15*log(x)/(4*x**2 - 501*x + (1250 - 20*x)*log(x) + 25*log(x)**2 + 15625)

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