3.10.66 \(\int \frac {-600000-588000 x-232860 x^2-46572 x^3-4704 x^4-192 x^5+(300000+441000 x+265560 x^2+82968 x^3+14244 x^4+1284 x^5+48 x^6) \log (x)+(-147000 x-146400 x^2-58920 x^3-11904 x^4-1200 x^5-48 x^6) \log ^2(x)+(75000 x+112500 x^2+67500 x^3+21000 x^4+3600 x^5+324 x^6+12 x^7) \log ^3(x)}{(3125 x+3125 x^2+1250 x^3+250 x^4+25 x^5+x^6) \log ^3(x)} \, dx\)

Optimal. Leaf size=23 \[ 6 \left (2+x-\frac {4-\frac {x}{(5+x)^2}}{\log (x)}\right )^2 \]

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Rubi [F]  time = 1.12, 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 {-600000-588000 x-232860 x^2-46572 x^3-4704 x^4-192 x^5+\left (300000+441000 x+265560 x^2+82968 x^3+14244 x^4+1284 x^5+48 x^6\right ) \log (x)+\left (-147000 x-146400 x^2-58920 x^3-11904 x^4-1200 x^5-48 x^6\right ) \log ^2(x)+\left (75000 x+112500 x^2+67500 x^3+21000 x^4+3600 x^5+324 x^6+12 x^7\right ) \log ^3(x)}{\left (3125 x+3125 x^2+1250 x^3+250 x^4+25 x^5+x^6\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-600000 - 588000*x - 232860*x^2 - 46572*x^3 - 4704*x^4 - 192*x^5 + (300000 + 441000*x + 265560*x^2 + 8296
8*x^3 + 14244*x^4 + 1284*x^5 + 48*x^6)*Log[x] + (-147000*x - 146400*x^2 - 58920*x^3 - 11904*x^4 - 1200*x^5 - 4
8*x^6)*Log[x]^2 + (75000*x + 112500*x^2 + 67500*x^3 + 21000*x^4 + 3600*x^5 + 324*x^6 + 12*x^7)*Log[x]^3)/((312
5*x + 3125*x^2 + 1250*x^3 + 250*x^4 + 25*x^5 + x^6)*Log[x]^3),x]

[Out]

24*x + 6*x^2 - 12*Defer[Int][(100 + 39*x + 4*x^2)^2/(x*(5 + x)^4*Log[x]^3), x] + 12*Defer[Int][((100 + 39*x +
4*x^2)*(250 + 270*x + 106*x^2 + 17*x^3 + x^4))/(x*(5 + x)^5*Log[x]^2), x] - 24*Defer[Int][(245 + 146*x + 30*x^
2 + 2*x^3)/((5 + x)^3*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {12 \left (-\left ((5+x) \left (100+39 x+4 x^2\right )^2\right )+\left (25000+36750 x+22130 x^2+6914 x^3+1187 x^4+107 x^5+4 x^6\right ) \log (x)-2 x (5+x)^2 \left (245+146 x+30 x^2+2 x^3\right ) \log ^2(x)+x (2+x) (5+x)^5 \log ^3(x)\right )}{x (5+x)^5 \log ^3(x)} \, dx\\ &=12 \int \frac {-\left ((5+x) \left (100+39 x+4 x^2\right )^2\right )+\left (25000+36750 x+22130 x^2+6914 x^3+1187 x^4+107 x^5+4 x^6\right ) \log (x)-2 x (5+x)^2 \left (245+146 x+30 x^2+2 x^3\right ) \log ^2(x)+x (2+x) (5+x)^5 \log ^3(x)}{x (5+x)^5 \log ^3(x)} \, dx\\ &=12 \int \left (2+x-\frac {\left (100+39 x+4 x^2\right )^2}{x (5+x)^4 \log ^3(x)}+\frac {\left (100+39 x+4 x^2\right ) \left (250+270 x+106 x^2+17 x^3+x^4\right )}{x (5+x)^5 \log ^2(x)}-\frac {2 \left (245+146 x+30 x^2+2 x^3\right )}{(5+x)^3 \log (x)}\right ) \, dx\\ &=24 x+6 x^2-12 \int \frac {\left (100+39 x+4 x^2\right )^2}{x (5+x)^4 \log ^3(x)} \, dx+12 \int \frac {\left (100+39 x+4 x^2\right ) \left (250+270 x+106 x^2+17 x^3+x^4\right )}{x (5+x)^5 \log ^2(x)} \, dx-24 \int \frac {245+146 x+30 x^2+2 x^3}{(5+x)^3 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.08, size = 63, normalized size = 2.74 \begin {gather*} 12 \left (2 x+\frac {x^2}{2}+\frac {\left (100+39 x+4 x^2\right )^2}{2 (5+x)^4 \log ^2(x)}+\frac {-200-178 x-47 x^2-4 x^3}{(5+x)^2 \log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-600000 - 588000*x - 232860*x^2 - 46572*x^3 - 4704*x^4 - 192*x^5 + (300000 + 441000*x + 265560*x^2
+ 82968*x^3 + 14244*x^4 + 1284*x^5 + 48*x^6)*Log[x] + (-147000*x - 146400*x^2 - 58920*x^3 - 11904*x^4 - 1200*x
^5 - 48*x^6)*Log[x]^2 + (75000*x + 112500*x^2 + 67500*x^3 + 21000*x^4 + 3600*x^5 + 324*x^6 + 12*x^7)*Log[x]^3)
/((3125*x + 3125*x^2 + 1250*x^3 + 250*x^4 + 25*x^5 + x^6)*Log[x]^3),x]

[Out]

12*(2*x + x^2/2 + (100 + 39*x + 4*x^2)^2/(2*(5 + x)^4*Log[x]^2) + (-200 - 178*x - 47*x^2 - 4*x^3)/((5 + x)^2*L
og[x]))

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fricas [B]  time = 0.56, size = 107, normalized size = 4.65 \begin {gather*} \frac {6 \, {\left (16 \, x^{4} + 312 \, x^{3} + {\left (x^{6} + 24 \, x^{5} + 230 \, x^{4} + 1100 \, x^{3} + 2625 \, x^{2} + 2500 \, x\right )} \log \relax (x)^{2} + 2321 \, x^{2} - 2 \, {\left (4 \, x^{5} + 87 \, x^{4} + 748 \, x^{3} + 3155 \, x^{2} + 6450 \, x + 5000\right )} \log \relax (x) + 7800 \, x + 10000\right )}}{{\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^7+324*x^6+3600*x^5+21000*x^4+67500*x^3+112500*x^2+75000*x)*log(x)^3+(-48*x^6-1200*x^5-11904*x
^4-58920*x^3-146400*x^2-147000*x)*log(x)^2+(48*x^6+1284*x^5+14244*x^4+82968*x^3+265560*x^2+441000*x+300000)*lo
g(x)-192*x^5-4704*x^4-46572*x^3-232860*x^2-588000*x-600000)/(x^6+25*x^5+250*x^4+1250*x^3+3125*x^2+3125*x)/log(
x)^3,x, algorithm="fricas")

[Out]

6*(16*x^4 + 312*x^3 + (x^6 + 24*x^5 + 230*x^4 + 1100*x^3 + 2625*x^2 + 2500*x)*log(x)^2 + 2321*x^2 - 2*(4*x^5 +
87*x^4 + 748*x^3 + 3155*x^2 + 6450*x + 5000)*log(x) + 7800*x + 10000)/((x^4 + 20*x^3 + 150*x^2 + 500*x + 625)
*log(x)^2)

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giac [B]  time = 0.42, size = 110, normalized size = 4.78 \begin {gather*} 6 \, x^{2} + 24 \, x - \frac {6 \, {\left (8 \, x^{5} \log \relax (x) + 174 \, x^{4} \log \relax (x) - 16 \, x^{4} + 1496 \, x^{3} \log \relax (x) - 312 \, x^{3} + 6310 \, x^{2} \log \relax (x) - 2321 \, x^{2} + 12900 \, x \log \relax (x) - 7800 \, x + 10000 \, \log \relax (x) - 10000\right )}}{x^{4} \log \relax (x)^{2} + 20 \, x^{3} \log \relax (x)^{2} + 150 \, x^{2} \log \relax (x)^{2} + 500 \, x \log \relax (x)^{2} + 625 \, \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^7+324*x^6+3600*x^5+21000*x^4+67500*x^3+112500*x^2+75000*x)*log(x)^3+(-48*x^6-1200*x^5-11904*x
^4-58920*x^3-146400*x^2-147000*x)*log(x)^2+(48*x^6+1284*x^5+14244*x^4+82968*x^3+265560*x^2+441000*x+300000)*lo
g(x)-192*x^5-4704*x^4-46572*x^3-232860*x^2-588000*x-600000)/(x^6+25*x^5+250*x^4+1250*x^3+3125*x^2+3125*x)/log(
x)^3,x, algorithm="giac")

[Out]

6*x^2 + 24*x - 6*(8*x^5*log(x) + 174*x^4*log(x) - 16*x^4 + 1496*x^3*log(x) - 312*x^3 + 6310*x^2*log(x) - 2321*
x^2 + 12900*x*log(x) - 7800*x + 10000*log(x) - 10000)/(x^4*log(x)^2 + 20*x^3*log(x)^2 + 150*x^2*log(x)^2 + 500
*x*log(x)^2 + 625*log(x)^2)

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maple [B]  time = 0.03, size = 93, normalized size = 4.04

method	result	size
risch	\(6 x^{2}+24 x -\frac {6 \left (8 x^{5} \ln \relax (x )+174 x^{4} \ln \relax (x )-16 x^{4}+1496 x^{3} \ln \relax (x )-312 x^{3}+6310 x^{2} \ln \relax (x )-2321 x^{2}+12900 x \ln \relax (x )-7800 x +10000 \ln \relax (x )-10000\right )}{\left (x^{4}+20 x^{3}+150 x^{2}+500 x +625\right ) \ln \relax (x )^{2}}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x^7+324*x^6+3600*x^5+21000*x^4+67500*x^3+112500*x^2+75000*x)*ln(x)^3+(-48*x^6-1200*x^5-11904*x^4-5892
0*x^3-146400*x^2-147000*x)*ln(x)^2+(48*x^6+1284*x^5+14244*x^4+82968*x^3+265560*x^2+441000*x+300000)*ln(x)-192*
x^5-4704*x^4-46572*x^3-232860*x^2-588000*x-600000)/(x^6+25*x^5+250*x^4+1250*x^3+3125*x^2+3125*x)/ln(x)^3,x,met
hod=_RETURNVERBOSE)

[Out]

6*x^2+24*x-6*(8*x^5*ln(x)+174*x^4*ln(x)-16*x^4+1496*x^3*ln(x)-312*x^3+6310*x^2*ln(x)-2321*x^2+12900*x*ln(x)-78
00*x+10000*ln(x)-10000)/(x^4+20*x^3+150*x^2+500*x+625)/ln(x)^2

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maxima [B]  time = 0.55, size = 107, normalized size = 4.65 \begin {gather*} \frac {6 \, {\left (16 \, x^{4} + 312 \, x^{3} + {\left (x^{6} + 24 \, x^{5} + 230 \, x^{4} + 1100 \, x^{3} + 2625 \, x^{2} + 2500 \, x\right )} \log \relax (x)^{2} + 2321 \, x^{2} - 2 \, {\left (4 \, x^{5} + 87 \, x^{4} + 748 \, x^{3} + 3155 \, x^{2} + 6450 \, x + 5000\right )} \log \relax (x) + 7800 \, x + 10000\right )}}{{\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^7+324*x^6+3600*x^5+21000*x^4+67500*x^3+112500*x^2+75000*x)*log(x)^3+(-48*x^6-1200*x^5-11904*x
^4-58920*x^3-146400*x^2-147000*x)*log(x)^2+(48*x^6+1284*x^5+14244*x^4+82968*x^3+265560*x^2+441000*x+300000)*lo
g(x)-192*x^5-4704*x^4-46572*x^3-232860*x^2-588000*x-600000)/(x^6+25*x^5+250*x^4+1250*x^3+3125*x^2+3125*x)/log(
x)^3,x, algorithm="maxima")

[Out]

6*(16*x^4 + 312*x^3 + (x^6 + 24*x^5 + 230*x^4 + 1100*x^3 + 2625*x^2 + 2500*x)*log(x)^2 + 2321*x^2 - 2*(4*x^5 +
87*x^4 + 748*x^3 + 3155*x^2 + 6450*x + 5000)*log(x) + 7800*x + 10000)/((x^4 + 20*x^3 + 150*x^2 + 500*x + 625)
*log(x)^2)

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mupad [B]  time = 1.05, size = 314, normalized size = 13.65 \begin {gather*} \frac {\frac {6\,{\left (4\,x^2+39\,x+100\right )}^2}{{\left (x+5\right )}^4}-\frac {6\,\ln \relax (x)\,\left (4\,x^6+107\,x^5+1187\,x^4+6914\,x^3+22130\,x^2+36750\,x+25000\right )}{{\left (x+5\right )}^5}+\frac {12\,x\,{\ln \relax (x)}^2\,\left (2\,x^3+30\,x^2+146\,x+245\right )}{{\left (x+5\right )}^3}}{{\ln \relax (x)}^2}-480\,\ln \relax (x)-24\,x+\frac {\frac {12\,x\,{\ln \relax (x)}^2\,\left (2\,x^4+40\,x^3+304\,x^2+970\,x+1225\right )}{{\left (x+5\right )}^4}-\frac {6\,\left (4\,x^6+107\,x^5+1179\,x^4+6876\,x^3+22320\,x^2+37750\,x+25000\right )}{{\left (x+5\right )}^5}+\frac {12\,x\,\ln \relax (x)\,\left (2\,x^6+60\,x^5+748\,x^4+4914\,x^3+18090\,x^2+36100\,x+31875\right )}{{\left (x+5\right )}^6}}{\ln \relax (x)}+6\,x^2+\frac {72\,x^5+1812\,x^4+12420\,x^3+27300\,x^2}{x^6+30\,x^5+375\,x^4+2500\,x^3+9375\,x^2+18750\,x+15625}+\frac {\ln \relax (x)\,\left (-24\,x^5+5952\,x^3+60360\,x^2+225300\,x+300000\right )}{x^4+20\,x^3+150\,x^2+500\,x+625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(588000*x - log(x)^3*(75000*x + 112500*x^2 + 67500*x^3 + 21000*x^4 + 3600*x^5 + 324*x^6 + 12*x^7) - log(x
)*(441000*x + 265560*x^2 + 82968*x^3 + 14244*x^4 + 1284*x^5 + 48*x^6 + 300000) + 232860*x^2 + 46572*x^3 + 4704
*x^4 + 192*x^5 + log(x)^2*(147000*x + 146400*x^2 + 58920*x^3 + 11904*x^4 + 1200*x^5 + 48*x^6) + 600000)/(log(x
)^3*(3125*x + 3125*x^2 + 1250*x^3 + 250*x^4 + 25*x^5 + x^6)),x)

[Out]

((6*(39*x + 4*x^2 + 100)^2)/(x + 5)^4 - (6*log(x)*(36750*x + 22130*x^2 + 6914*x^3 + 1187*x^4 + 107*x^5 + 4*x^6
+ 25000))/(x + 5)^5 + (12*x*log(x)^2*(146*x + 30*x^2 + 2*x^3 + 245))/(x + 5)^3)/log(x)^2 - 480*log(x) - 24*x
+ ((12*x*log(x)^2*(970*x + 304*x^2 + 40*x^3 + 2*x^4 + 1225))/(x + 5)^4 - (6*(37750*x + 22320*x^2 + 6876*x^3 +
1179*x^4 + 107*x^5 + 4*x^6 + 25000))/(x + 5)^5 + (12*x*log(x)*(36100*x + 18090*x^2 + 4914*x^3 + 748*x^4 + 60*x
^5 + 2*x^6 + 31875))/(x + 5)^6)/log(x) + 6*x^2 + (27300*x^2 + 12420*x^3 + 1812*x^4 + 72*x^5)/(18750*x + 9375*x
^2 + 2500*x^3 + 375*x^4 + 30*x^5 + x^6 + 15625) + (log(x)*(225300*x + 60360*x^2 + 5952*x^3 - 24*x^5 + 300000))
/(500*x + 150*x^2 + 20*x^3 + x^4 + 625)

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sympy [B]  time = 0.30, size = 82, normalized size = 3.57 \begin {gather*} 6 x^{2} + 24 x + \frac {96 x^{4} + 1872 x^{3} + 13926 x^{2} + 46800 x + \left (- 48 x^{5} - 1044 x^{4} - 8976 x^{3} - 37860 x^{2} - 77400 x - 60000\right ) \log {\relax (x )} + 60000}{\left (x^{4} + 20 x^{3} + 150 x^{2} + 500 x + 625\right ) \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x**7+324*x**6+3600*x**5+21000*x**4+67500*x**3+112500*x**2+75000*x)*ln(x)**3+(-48*x**6-1200*x**5
-11904*x**4-58920*x**3-146400*x**2-147000*x)*ln(x)**2+(48*x**6+1284*x**5+14244*x**4+82968*x**3+265560*x**2+441
000*x+300000)*ln(x)-192*x**5-4704*x**4-46572*x**3-232860*x**2-588000*x-600000)/(x**6+25*x**5+250*x**4+1250*x**
3+3125*x**2+3125*x)/ln(x)**3,x)

[Out]

6*x**2 + 24*x + (96*x**4 + 1872*x**3 + 13926*x**2 + 46800*x + (-48*x**5 - 1044*x**4 - 8976*x**3 - 37860*x**2 -
77400*x - 60000)*log(x) + 60000)/((x**4 + 20*x**3 + 150*x**2 + 500*x + 625)*log(x)**2)

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