3.81.92 \(\int \frac {-305156281250-73279658750 x-7328903800 x^2-390814950 x^3-11721300 x^4-187500 x^5-1250 x^6+(-1464781300-234497450 x-14070000 x^2-375100 x^3-3750 x^4) \log (x)+(-2343700-187600 x-3750 x^2) \log ^2(x)-1250 \log ^3(x)}{3813964890624 x^3+916142488128 x^4+91662930072 x^5+4889659851 x^6+146670075 x^7+2345625 x^8+15625 x^9+(18308203200 x^3+2931843600 x^4+175965075 x^5+4691250 x^6+46875 x^7) \log (x)+(29295000 x^3+2345625 x^4+46875 x^5) \log ^2(x)+15625 x^3 \log ^3(x)} \, dx\)

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

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Rubi [F]  time = 2.06, 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 {-305156281250-73279658750 x-7328903800 x^2-390814950 x^3-11721300 x^4-187500 x^5-1250 x^6+\left (-1464781300-234497450 x-14070000 x^2-375100 x^3-3750 x^4\right ) \log (x)+\left (-2343700-187600 x-3750 x^2\right ) \log ^2(x)-1250 \log ^3(x)}{3813964890624 x^3+916142488128 x^4+91662930072 x^5+4889659851 x^6+146670075 x^7+2345625 x^8+15625 x^9+\left (18308203200 x^3+2931843600 x^4+175965075 x^5+4691250 x^6+46875 x^7\right ) \log (x)+\left (29295000 x^3+2345625 x^4+46875 x^5\right ) \log ^2(x)+15625 x^3 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-305156281250 - 73279658750*x - 7328903800*x^2 - 390814950*x^3 - 11721300*x^4 - 187500*x^5 - 1250*x^6 + (
-1464781300 - 234497450*x - 14070000*x^2 - 375100*x^3 - 3750*x^4)*Log[x] + (-2343700 - 187600*x - 3750*x^2)*Lo
g[x]^2 - 1250*Log[x]^3)/(3813964890624*x^3 + 916142488128*x^4 + 91662930072*x^5 + 4889659851*x^6 + 146670075*x
^7 + 2345625*x^8 + 15625*x^9 + (18308203200*x^3 + 2931843600*x^4 + 175965075*x^5 + 4691250*x^6 + 46875*x^7)*Lo
g[x] + (29295000*x^3 + 2345625*x^4 + 46875*x^5)*Log[x]^2 + 15625*x^3*Log[x]^3),x]

[Out]

1/(25*x^2) - (2302*Defer[Int][(15624 + 1251*x + 25*x^2 + 25*Log[x])^(-3), x])/25 - 2*Defer[Int][1/(x^3*(15624
+ 1251*x + 25*x^2 + 25*Log[x])^3), x] - (2402*Defer[Int][1/(x^2*(15624 + 1251*x + 25*x^2 + 25*Log[x])^3), x])/
25 + (4854*Defer[Int][1/(x*(15624 + 1251*x + 25*x^2 + 25*Log[x])^3), x])/25 - 4*Defer[Int][x/(15624 + 1251*x +
25*x^2 + 25*Log[x])^3, x] + 4*Defer[Int][(15624 + 1251*x + 25*x^2 + 25*Log[x])^(-2), x] - (52*Defer[Int][1/(x
^3*(15624 + 1251*x + 25*x^2 + 25*Log[x])^2), x])/25 - 98*Defer[Int][1/(x^2*(15624 + 1251*x + 25*x^2 + 25*Log[x
])^2), x] + (2402*Defer[Int][1/(x*(15624 + 1251*x + 25*x^2 + 25*Log[x])^2), x])/25 - (4*Defer[Int][1/(x^3*(156
24 + 1251*x + 25*x^2 + 25*Log[x])), x])/25 + (2*Defer[Int][1/(x^2*(15624 + 1251*x + 25*x^2 + 25*Log[x])), x])/
25

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {50 \left (-(25+x)^2 \left (9765001+1563749 x+93801 x^2+2500 x^3+25 x^4\right )-\left (29295626+4689949 x+281400 x^2+7502 x^3+75 x^4\right ) \log (x)-\left (46874+3752 x+75 x^2\right ) \log ^2(x)-25 \log ^3(x)\right )}{x^3 \left (15624+1251 x+25 x^2+25 \log (x)\right )^3} \, dx\\ &=50 \int \frac {-(25+x)^2 \left (9765001+1563749 x+93801 x^2+2500 x^3+25 x^4\right )-\left (29295626+4689949 x+281400 x^2+7502 x^3+75 x^4\right ) \log (x)-\left (46874+3752 x+75 x^2\right ) \log ^2(x)-25 \log ^3(x)}{x^3 \left (15624+1251 x+25 x^2+25 \log (x)\right )^3} \, dx\\ &=50 \int \left (-\frac {1}{625 x^3}-\frac {(-1+x)^2 \left (25+1251 x+50 x^2\right )}{625 x^3 \left (15624+1251 x+25 x^2+25 \log (x)\right )^3}+\frac {-26-1225 x+1201 x^2+50 x^3}{625 x^3 \left (15624+1251 x+25 x^2+25 \log (x)\right )^2}+\frac {-2+x}{625 x^3 \left (15624+1251 x+25 x^2+25 \log (x)\right )}\right ) \, dx\\ &=\frac {1}{25 x^2}-\frac {2}{25} \int \frac {(-1+x)^2 \left (25+1251 x+50 x^2\right )}{x^3 \left (15624+1251 x+25 x^2+25 \log (x)\right )^3} \, dx+\frac {2}{25} \int \frac {-26-1225 x+1201 x^2+50 x^3}{x^3 \left (15624+1251 x+25 x^2+25 \log (x)\right )^2} \, dx+\frac {2}{25} \int \frac {-2+x}{x^3 \left (15624+1251 x+25 x^2+25 \log (x)\right )} \, dx\\ &=\frac {1}{25 x^2}-\frac {2}{25} \int \left (\frac {1151}{\left (15624+1251 x+25 x^2+25 \log (x)\right )^3}+\frac {25}{x^3 \left (15624+1251 x+25 x^2+25 \log (x)\right )^3}+\frac {1201}{x^2 \left (15624+1251 x+25 x^2+25 \log (x)\right )^3}-\frac {2427}{x \left (15624+1251 x+25 x^2+25 \log (x)\right )^3}+\frac {50 x}{\left (15624+1251 x+25 x^2+25 \log (x)\right )^3}\right ) \, dx+\frac {2}{25} \int \left (\frac {50}{\left (15624+1251 x+25 x^2+25 \log (x)\right )^2}-\frac {26}{x^3 \left (15624+1251 x+25 x^2+25 \log (x)\right )^2}-\frac {1225}{x^2 \left (15624+1251 x+25 x^2+25 \log (x)\right )^2}+\frac {1201}{x \left (15624+1251 x+25 x^2+25 \log (x)\right )^2}\right ) \, dx+\frac {2}{25} \int \left (-\frac {2}{x^3 \left (15624+1251 x+25 x^2+25 \log (x)\right )}+\frac {1}{x^2 \left (15624+1251 x+25 x^2+25 \log (x)\right )}\right ) \, dx\\ &=\frac {1}{25 x^2}+\frac {2}{25} \int \frac {1}{x^2 \left (15624+1251 x+25 x^2+25 \log (x)\right )} \, dx-\frac {4}{25} \int \frac {1}{x^3 \left (15624+1251 x+25 x^2+25 \log (x)\right )} \, dx-2 \int \frac {1}{x^3 \left (15624+1251 x+25 x^2+25 \log (x)\right )^3} \, dx-\frac {52}{25} \int \frac {1}{x^3 \left (15624+1251 x+25 x^2+25 \log (x)\right )^2} \, dx-4 \int \frac {x}{\left (15624+1251 x+25 x^2+25 \log (x)\right )^3} \, dx+4 \int \frac {1}{\left (15624+1251 x+25 x^2+25 \log (x)\right )^2} \, dx-\frac {2302}{25} \int \frac {1}{\left (15624+1251 x+25 x^2+25 \log (x)\right )^3} \, dx-\frac {2402}{25} \int \frac {1}{x^2 \left (15624+1251 x+25 x^2+25 \log (x)\right )^3} \, dx+\frac {2402}{25} \int \frac {1}{x \left (15624+1251 x+25 x^2+25 \log (x)\right )^2} \, dx-98 \int \frac {1}{x^2 \left (15624+1251 x+25 x^2+25 \log (x)\right )^2} \, dx+\frac {4854}{25} \int \frac {1}{x \left (15624+1251 x+25 x^2+25 \log (x)\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 31, normalized size = 1.24 \begin {gather*} \frac {25 \left ((25+x)^2+\log (x)\right )^2}{x^2 \left (15624+1251 x+25 x^2+25 \log (x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-305156281250 - 73279658750*x - 7328903800*x^2 - 390814950*x^3 - 11721300*x^4 - 187500*x^5 - 1250*x
^6 + (-1464781300 - 234497450*x - 14070000*x^2 - 375100*x^3 - 3750*x^4)*Log[x] + (-2343700 - 187600*x - 3750*x
^2)*Log[x]^2 - 1250*Log[x]^3)/(3813964890624*x^3 + 916142488128*x^4 + 91662930072*x^5 + 4889659851*x^6 + 14667
0075*x^7 + 2345625*x^8 + 15625*x^9 + (18308203200*x^3 + 2931843600*x^4 + 175965075*x^5 + 4691250*x^6 + 46875*x
^7)*Log[x] + (29295000*x^3 + 2345625*x^4 + 46875*x^5)*Log[x]^2 + 15625*x^3*Log[x]^3),x]

[Out]

(25*((25 + x)^2 + Log[x])^2)/(x^2*(15624 + 1251*x + 25*x^2 + 25*Log[x])^2)

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fricas [B]  time = 0.53, size = 93, normalized size = 3.72 \begin {gather*} \frac {25 \, {\left (x^{4} + 100 \, x^{3} + 3750 \, x^{2} + 2 \, {\left (x^{2} + 50 \, x + 625\right )} \log \relax (x) + \log \relax (x)^{2} + 62500 \, x + 390625\right )}}{625 \, x^{6} + 62550 \, x^{5} + 2346201 \, x^{4} + 625 \, x^{2} \log \relax (x)^{2} + 39091248 \, x^{3} + 244109376 \, x^{2} + 50 \, {\left (25 \, x^{4} + 1251 \, x^{3} + 15624 \, x^{2}\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1250*log(x)^3+(-3750*x^2-187600*x-2343700)*log(x)^2+(-3750*x^4-375100*x^3-14070000*x^2-234497450*x
-1464781300)*log(x)-1250*x^6-187500*x^5-11721300*x^4-390814950*x^3-7328903800*x^2-73279658750*x-305156281250)/
(15625*x^3*log(x)^3+(46875*x^5+2345625*x^4+29295000*x^3)*log(x)^2+(46875*x^7+4691250*x^6+175965075*x^5+2931843
600*x^4+18308203200*x^3)*log(x)+15625*x^9+2345625*x^8+146670075*x^7+4889659851*x^6+91662930072*x^5+91614248812
8*x^4+3813964890624*x^3),x, algorithm="fricas")

[Out]

25*(x^4 + 100*x^3 + 3750*x^2 + 2*(x^2 + 50*x + 625)*log(x) + log(x)^2 + 62500*x + 390625)/(625*x^6 + 62550*x^5
+ 2346201*x^4 + 625*x^2*log(x)^2 + 39091248*x^3 + 244109376*x^2 + 50*(25*x^4 + 1251*x^3 + 15624*x^2)*log(x))

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giac [B]  time = 0.19, size = 90, normalized size = 3.60 \begin {gather*} -\frac {50 \, x^{3} + 2451 \, x^{2} + 50 \, x \log \relax (x) + 28748 \, x - 50 \, \log \relax (x) - 31249}{25 \, {\left (625 \, x^{6} + 62550 \, x^{5} + 1250 \, x^{4} \log \relax (x) + 2346201 \, x^{4} + 62550 \, x^{3} \log \relax (x) + 625 \, x^{2} \log \relax (x)^{2} + 39091248 \, x^{3} + 781200 \, x^{2} \log \relax (x) + 244109376 \, x^{2}\right )}} + \frac {1}{25 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1250*log(x)^3+(-3750*x^2-187600*x-2343700)*log(x)^2+(-3750*x^4-375100*x^3-14070000*x^2-234497450*x
-1464781300)*log(x)-1250*x^6-187500*x^5-11721300*x^4-390814950*x^3-7328903800*x^2-73279658750*x-305156281250)/
(15625*x^3*log(x)^3+(46875*x^5+2345625*x^4+29295000*x^3)*log(x)^2+(46875*x^7+4691250*x^6+175965075*x^5+2931843
600*x^4+18308203200*x^3)*log(x)+15625*x^9+2345625*x^8+146670075*x^7+4889659851*x^6+91662930072*x^5+91614248812
8*x^4+3813964890624*x^3),x, algorithm="giac")

[Out]

-1/25*(50*x^3 + 2451*x^2 + 50*x*log(x) + 28748*x - 50*log(x) - 31249)/(625*x^6 + 62550*x^5 + 1250*x^4*log(x) +
2346201*x^4 + 62550*x^3*log(x) + 625*x^2*log(x)^2 + 39091248*x^3 + 781200*x^2*log(x) + 244109376*x^2) + 1/25/
x^2

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maple [B]  time = 0.03, size = 52, normalized size = 2.08

method	result	size
risch	\(\frac {1}{25 x^{2}}-\frac {50 x^{3}+2451 x^{2}+50 x \ln \relax (x )+28748 x -50 \ln \relax (x )-31249}{25 x^{2} \left (25 x^{2}+25 \ln \relax (x )+1251 x +15624\right )^{2}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1250*ln(x)^3+(-3750*x^2-187600*x-2343700)*ln(x)^2+(-3750*x^4-375100*x^3-14070000*x^2-234497450*x-1464781
300)*ln(x)-1250*x^6-187500*x^5-11721300*x^4-390814950*x^3-7328903800*x^2-73279658750*x-305156281250)/(15625*x^
3*ln(x)^3+(46875*x^5+2345625*x^4+29295000*x^3)*ln(x)^2+(46875*x^7+4691250*x^6+175965075*x^5+2931843600*x^4+183
08203200*x^3)*ln(x)+15625*x^9+2345625*x^8+146670075*x^7+4889659851*x^6+91662930072*x^5+916142488128*x^4+381396
4890624*x^3),x,method=_RETURNVERBOSE)

[Out]

1/25/x^2-1/25*(50*x^3+2451*x^2+50*x*ln(x)+28748*x-50*ln(x)-31249)/x^2/(25*x^2+25*ln(x)+1251*x+15624)^2

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maxima [B]  time = 0.66, size = 93, normalized size = 3.72 \begin {gather*} \frac {25 \, {\left (x^{4} + 100 \, x^{3} + 3750 \, x^{2} + 2 \, {\left (x^{2} + 50 \, x + 625\right )} \log \relax (x) + \log \relax (x)^{2} + 62500 \, x + 390625\right )}}{625 \, x^{6} + 62550 \, x^{5} + 2346201 \, x^{4} + 625 \, x^{2} \log \relax (x)^{2} + 39091248 \, x^{3} + 244109376 \, x^{2} + 50 \, {\left (25 \, x^{4} + 1251 \, x^{3} + 15624 \, x^{2}\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1250*log(x)^3+(-3750*x^2-187600*x-2343700)*log(x)^2+(-3750*x^4-375100*x^3-14070000*x^2-234497450*x
-1464781300)*log(x)-1250*x^6-187500*x^5-11721300*x^4-390814950*x^3-7328903800*x^2-73279658750*x-305156281250)/
(15625*x^3*log(x)^3+(46875*x^5+2345625*x^4+29295000*x^3)*log(x)^2+(46875*x^7+4691250*x^6+175965075*x^5+2931843
600*x^4+18308203200*x^3)*log(x)+15625*x^9+2345625*x^8+146670075*x^7+4889659851*x^6+91662930072*x^5+91614248812
8*x^4+3813964890624*x^3),x, algorithm="maxima")

[Out]

25*(x^4 + 100*x^3 + 3750*x^2 + 2*(x^2 + 50*x + 625)*log(x) + log(x)^2 + 62500*x + 390625)/(625*x^6 + 62550*x^5
+ 2346201*x^4 + 625*x^2*log(x)^2 + 39091248*x^3 + 244109376*x^2 + 50*(25*x^4 + 1251*x^3 + 15624*x^2)*log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {73279658750\,x+{\ln \relax (x)}^2\,\left (3750\,x^2+187600\,x+2343700\right )+1250\,{\ln \relax (x)}^3+\ln \relax (x)\,\left (3750\,x^4+375100\,x^3+14070000\,x^2+234497450\,x+1464781300\right )[phone]\,x^2+390814950\,x^3+11721300\,x^4+187500\,x^5+1250\,x^6+305156281250}{15625\,x^3\,{\ln \relax (x)}^3+{\ln \relax (x)}^2\,\left (46875\,x^5+2345625\,x^4+29295000\,x^3\right )+\ln \relax (x)\,\left (46875\,x^7+4691250\,x^6+175965075\,x^5+2931843600\,x^4+18308203200\,x^3\right )[phone]\,x^3+916142488128\,x^4+91662930072\,x^5+4889659851\,x^6+146670075\,x^7+2345625\,x^8+15625\,x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(73279658750*x + log(x)^2*(187600*x + 3750*x^2 + 2343700) + 1250*log(x)^3 + log(x)*(234497450*x + 1407000
0*x^2 + 375100*x^3 + 3750*x^4 + 1464781300) + 7328903800*x^2 + 390814950*x^3 + 11721300*x^4 + 187500*x^5 + 125
0*x^6 + 305156281250)/(15625*x^3*log(x)^3 + log(x)^2*(29295000*x^3 + 2345625*x^4 + 46875*x^5) + log(x)*(183082
03200*x^3 + 2931843600*x^4 + 175965075*x^5 + 4691250*x^6 + 46875*x^7) + 3813964890624*x^3 + 916142488128*x^4 +
91662930072*x^5 + 4889659851*x^6 + 146670075*x^7 + 2345625*x^8 + 15625*x^9),x)

[Out]

int(-(73279658750*x + log(x)^2*(187600*x + 3750*x^2 + 2343700) + 1250*log(x)^3 + log(x)*(234497450*x + 1407000
0*x^2 + 375100*x^3 + 3750*x^4 + 1464781300) + 7328903800*x^2 + 390814950*x^3 + 11721300*x^4 + 187500*x^5 + 125
0*x^6 + 305156281250)/(15625*x^3*log(x)^3 + log(x)^2*(29295000*x^3 + 2345625*x^4 + 46875*x^5) + log(x)*(183082
03200*x^3 + 2931843600*x^4 + 175965075*x^5 + 4691250*x^6 + 46875*x^7) + 3813964890624*x^3 + 916142488128*x^4 +
91662930072*x^5 + 4889659851*x^6 + 146670075*x^7 + 2345625*x^8 + 15625*x^9), x)

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sympy [B]  time = 0.28, size = 83, normalized size = 3.32 \begin {gather*} \frac {- 50 x^{3} - 2451 x^{2} - 28748 x + \left (50 - 50 x\right ) \log {\relax (x )} + 31249}{15625 x^{6} + 1563750 x^{5} + 58655025 x^{4} + 977281200 x^{3} + 15625 x^{2} \log {\relax (x )}^{2} + 6102734400 x^{2} + \left (31250 x^{4} + 1563750 x^{3} + 19530000 x^{2}\right ) \log {\relax (x )}} + \frac {1}{25 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1250*ln(x)**3+(-3750*x**2-187600*x-2343700)*ln(x)**2+(-3750*x**4-375100*x**3-14070000*x**2-2344974
50*x-1464781300)*ln(x)-1250*x**6-187500*x**5-11721300*x**4-390814950*x**3-7328903800*x**2-73279658750*x-305156
281250)/(15625*x**3*ln(x)**3+(46875*x**5+2345625*x**4+29295000*x**3)*ln(x)**2+(46875*x**7+4691250*x**6+1759650
75*x**5+2931843600*x**4+18308203200*x**3)*ln(x)+15625*x**9+2345625*x**8+146670075*x**7+4889659851*x**6+9166293
0072*x**5+916142488128*x**4+3813964890624*x**3),x)

[Out]

(-50*x**3 - 2451*x**2 - 28748*x + (50 - 50*x)*log(x) + 31249)/(15625*x**6 + 1563750*x**5 + 58655025*x**4 + 977
281200*x**3 + 15625*x**2*log(x)**2 + 6102734400*x**2 + (31250*x**4 + 1563750*x**3 + 19530000*x**2)*log(x)) + 1
/(25*x**2)

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