3.9.42 \(\int \frac {-175-120 x-1254 x^3-1001 x^4-300 x^5-40 x^6-2 x^7+(-60-40 x+8 x^2-1000 x^3-800 x^4-200 x^5-16 x^6) \log (x)+(-5+x^2-300 x^3-200 x^4-36 x^5) \log ^2(x)+(-40 x^3-16 x^4) \log ^3(x)-2 x^3 \log ^4(x)}{625 x^2+500 x^3+150 x^4+20 x^5+x^6+(500 x^2+400 x^3+100 x^4+8 x^5) \log (x)+(150 x^2+100 x^3+18 x^4) \log ^2(x)+(20 x^2+8 x^3) \log ^3(x)+x^2 \log ^4(x)} \, dx\)

Optimal. Leaf size=28 \[ \frac {\left (5+x^2\right ) \left (-x+\frac {1}{2 x \log (x)+(5+x+\log (x))^2}\right )}{x} \]

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Rubi [F]  time = 3.06, 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 {-175-120 x-1254 x^3-1001 x^4-300 x^5-40 x^6-2 x^7+\left (-60-40 x+8 x^2-1000 x^3-800 x^4-200 x^5-16 x^6\right ) \log (x)+\left (-5+x^2-300 x^3-200 x^4-36 x^5\right ) \log ^2(x)+\left (-40 x^3-16 x^4\right ) \log ^3(x)-2 x^3 \log ^4(x)}{625 x^2+500 x^3+150 x^4+20 x^5+x^6+\left (500 x^2+400 x^3+100 x^4+8 x^5\right ) \log (x)+\left (150 x^2+100 x^3+18 x^4\right ) \log ^2(x)+\left (20 x^2+8 x^3\right ) \log ^3(x)+x^2 \log ^4(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-175 - 120*x - 1254*x^3 - 1001*x^4 - 300*x^5 - 40*x^6 - 2*x^7 + (-60 - 40*x + 8*x^2 - 1000*x^3 - 800*x^4
- 200*x^5 - 16*x^6)*Log[x] + (-5 + x^2 - 300*x^3 - 200*x^4 - 36*x^5)*Log[x]^2 + (-40*x^3 - 16*x^4)*Log[x]^3 -
2*x^3*Log[x]^4)/(625*x^2 + 500*x^3 + 150*x^4 + 20*x^5 + x^6 + (500*x^2 + 400*x^3 + 100*x^4 + 8*x^5)*Log[x] + (
150*x^2 + 100*x^3 + 18*x^4)*Log[x]^2 + (20*x^2 + 8*x^3)*Log[x]^3 + x^2*Log[x]^4),x]

[Out]

-x^2 - 20*Defer[Int][(25 + 10*x + x^2 + 10*Log[x] + 4*x*Log[x] + Log[x]^2)^(-2), x] - 50*Defer[Int][1/(x^2*(25
+ 10*x + x^2 + 10*Log[x] + 4*x*Log[x] + Log[x]^2)^2), x] - 70*Defer[Int][1/(x*(25 + 10*x + x^2 + 10*Log[x] +
4*x*Log[x] + Log[x]^2)^2), x] - 14*Defer[Int][x/(25 + 10*x + x^2 + 10*Log[x] + 4*x*Log[x] + Log[x]^2)^2, x] -
2*Defer[Int][x^2/(25 + 10*x + x^2 + 10*Log[x] + 4*x*Log[x] + Log[x]^2)^2, x] - 2*Defer[Int][Log[x]/(25 + 10*x
+ x^2 + 10*Log[x] + 4*x*Log[x] + Log[x]^2)^2, x] - 10*Defer[Int][Log[x]/(x^2*(25 + 10*x + x^2 + 10*Log[x] + 4*
x*Log[x] + Log[x]^2)^2), x] - 20*Defer[Int][Log[x]/(x*(25 + 10*x + x^2 + 10*Log[x] + 4*x*Log[x] + Log[x]^2)^2)
, x] - 4*Defer[Int][(x*Log[x])/(25 + 10*x + x^2 + 10*Log[x] + 4*x*Log[x] + Log[x]^2)^2, x] + Defer[Int][(25 +
10*x + x^2 + 10*Log[x] + 4*x*Log[x] + Log[x]^2)^(-1), x] - 5*Defer[Int][1/(x^2*(25 + 10*x + x^2 + 10*Log[x] +
4*x*Log[x] + Log[x]^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-175-120 x-1254 x^3-1001 x^4-300 x^5-40 x^6-2 x^7-4 \left (15+10 x-2 x^2+250 x^3+200 x^4+50 x^5+4 x^6\right ) \log (x)-\left (5-x^2+300 x^3+200 x^4+36 x^5\right ) \log ^2(x)-8 x^3 (5+2 x) \log ^3(x)-2 x^3 \log ^4(x)}{x^2 \left ((5+x)^2+2 (5+2 x) \log (x)+\log ^2(x)\right )^2} \, dx\\ &=\int \left (-2 x-\frac {2 \left (5+x^2\right ) \left (5+7 x+x^2+\log (x)+2 x \log (x)\right )}{x^2 \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {-5+x^2}{x^2 \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )}\right ) \, dx\\ &=-x^2-2 \int \frac {\left (5+x^2\right ) \left (5+7 x+x^2+\log (x)+2 x \log (x)\right )}{x^2 \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {-5+x^2}{x^2 \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )} \, dx\\ &=-x^2-2 \int \left (\frac {5+7 x+x^2+\log (x)+2 x \log (x)}{\left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {5 \left (5+7 x+x^2+\log (x)+2 x \log (x)\right )}{x^2 \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}\right ) \, dx+\int \left (\frac {1}{25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)}-\frac {5}{x^2 \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )}\right ) \, dx\\ &=-x^2-2 \int \frac {5+7 x+x^2+\log (x)+2 x \log (x)}{\left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-5 \int \frac {1}{x^2 \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )} \, dx-10 \int \frac {5+7 x+x^2+\log (x)+2 x \log (x)}{x^2 \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {1}{25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)} \, dx\\ &=-x^2-2 \int \left (\frac {5}{\left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {7 x}{\left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {x^2}{\left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {\log (x)}{\left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {2 x \log (x)}{\left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}\right ) \, dx-5 \int \frac {1}{x^2 \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )} \, dx-10 \int \left (\frac {1}{\left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {5}{x^2 \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {7}{x \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {\log (x)}{x^2 \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {2 \log (x)}{x \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}\right ) \, dx+\int \frac {1}{25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)} \, dx\\ &=-x^2-2 \int \frac {x^2}{\left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-2 \int \frac {\log (x)}{\left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-4 \int \frac {x \log (x)}{\left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-5 \int \frac {1}{x^2 \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )} \, dx-2 \left (10 \int \frac {1}{\left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx\right )-10 \int \frac {\log (x)}{x^2 \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-14 \int \frac {x}{\left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-20 \int \frac {\log (x)}{x \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-50 \int \frac {1}{x^2 \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-70 \int \frac {1}{x \left (25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {1}{25+10 x+x^2+10 \log (x)+4 x \log (x)+\log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 36, normalized size = 1.29 \begin {gather*} -x^2+\frac {5+x^2}{x \left ((5+x)^2+2 (5+2 x) \log (x)+\log ^2(x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-175 - 120*x - 1254*x^3 - 1001*x^4 - 300*x^5 - 40*x^6 - 2*x^7 + (-60 - 40*x + 8*x^2 - 1000*x^3 - 80
0*x^4 - 200*x^5 - 16*x^6)*Log[x] + (-5 + x^2 - 300*x^3 - 200*x^4 - 36*x^5)*Log[x]^2 + (-40*x^3 - 16*x^4)*Log[x
]^3 - 2*x^3*Log[x]^4)/(625*x^2 + 500*x^3 + 150*x^4 + 20*x^5 + x^6 + (500*x^2 + 400*x^3 + 100*x^4 + 8*x^5)*Log[
x] + (150*x^2 + 100*x^3 + 18*x^4)*Log[x]^2 + (20*x^2 + 8*x^3)*Log[x]^3 + x^2*Log[x]^4),x]

[Out]

-x^2 + (5 + x^2)/(x*((5 + x)^2 + 2*(5 + 2*x)*Log[x] + Log[x]^2))

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fricas [B]  time = 0.73, size = 78, normalized size = 2.79 \begin {gather*} -\frac {x^{5} + x^{3} \log \relax (x)^{2} + 10 \, x^{4} + 25 \, x^{3} - x^{2} + 2 \, {\left (2 \, x^{4} + 5 \, x^{3}\right )} \log \relax (x) - 5}{x^{3} + x \log \relax (x)^{2} + 10 \, x^{2} + 2 \, {\left (2 \, x^{2} + 5 \, x\right )} \log \relax (x) + 25 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3*log(x)^4+(-16*x^4-40*x^3)*log(x)^3+(-36*x^5-200*x^4-300*x^3+x^2-5)*log(x)^2+(-16*x^6-200*x^5
-800*x^4-1000*x^3+8*x^2-40*x-60)*log(x)-2*x^7-40*x^6-300*x^5-1001*x^4-1254*x^3-120*x-175)/(x^2*log(x)^4+(8*x^3
+20*x^2)*log(x)^3+(18*x^4+100*x^3+150*x^2)*log(x)^2+(8*x^5+100*x^4+400*x^3+500*x^2)*log(x)+x^6+20*x^5+150*x^4+
500*x^3+625*x^2),x, algorithm="fricas")

[Out]

-(x^5 + x^3*log(x)^2 + 10*x^4 + 25*x^3 - x^2 + 2*(2*x^4 + 5*x^3)*log(x) - 5)/(x^3 + x*log(x)^2 + 10*x^2 + 2*(2
*x^2 + 5*x)*log(x) + 25*x)

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giac [A]  time = 0.62, size = 44, normalized size = 1.57 \begin {gather*} -x^{2} + \frac {x^{2} + 5}{x^{3} + 4 \, x^{2} \log \relax (x) + x \log \relax (x)^{2} + 10 \, x^{2} + 10 \, x \log \relax (x) + 25 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3*log(x)^4+(-16*x^4-40*x^3)*log(x)^3+(-36*x^5-200*x^4-300*x^3+x^2-5)*log(x)^2+(-16*x^6-200*x^5
-800*x^4-1000*x^3+8*x^2-40*x-60)*log(x)-2*x^7-40*x^6-300*x^5-1001*x^4-1254*x^3-120*x-175)/(x^2*log(x)^4+(8*x^3
+20*x^2)*log(x)^3+(18*x^4+100*x^3+150*x^2)*log(x)^2+(8*x^5+100*x^4+400*x^3+500*x^2)*log(x)+x^6+20*x^5+150*x^4+
500*x^3+625*x^2),x, algorithm="giac")

[Out]

-x^2 + (x^2 + 5)/(x^3 + 4*x^2*log(x) + x*log(x)^2 + 10*x^2 + 10*x*log(x) + 25*x)

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maple [A]  time = 0.04, size = 39, normalized size = 1.39

method	result	size
risch	\(-x^{2}+\frac {x^{2}+5}{x \left (\ln \relax (x )^{2}+4 x \ln \relax (x )+x^{2}+10 \ln \relax (x )+10 x +25\right )}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^3*ln(x)^4+(-16*x^4-40*x^3)*ln(x)^3+(-36*x^5-200*x^4-300*x^3+x^2-5)*ln(x)^2+(-16*x^6-200*x^5-800*x^4-
1000*x^3+8*x^2-40*x-60)*ln(x)-2*x^7-40*x^6-300*x^5-1001*x^4-1254*x^3-120*x-175)/(x^2*ln(x)^4+(8*x^3+20*x^2)*ln
(x)^3+(18*x^4+100*x^3+150*x^2)*ln(x)^2+(8*x^5+100*x^4+400*x^3+500*x^2)*ln(x)+x^6+20*x^5+150*x^4+500*x^3+625*x^
2),x,method=_RETURNVERBOSE)

[Out]

-x^2+(x^2+5)/x/(ln(x)^2+4*x*ln(x)+x^2+10*ln(x)+10*x+25)

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maxima [B]  time = 0.47, size = 78, normalized size = 2.79 \begin {gather*} -\frac {x^{5} + x^{3} \log \relax (x)^{2} + 10 \, x^{4} + 25 \, x^{3} - x^{2} + 2 \, {\left (2 \, x^{4} + 5 \, x^{3}\right )} \log \relax (x) - 5}{x^{3} + x \log \relax (x)^{2} + 10 \, x^{2} + 2 \, {\left (2 \, x^{2} + 5 \, x\right )} \log \relax (x) + 25 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3*log(x)^4+(-16*x^4-40*x^3)*log(x)^3+(-36*x^5-200*x^4-300*x^3+x^2-5)*log(x)^2+(-16*x^6-200*x^5
-800*x^4-1000*x^3+8*x^2-40*x-60)*log(x)-2*x^7-40*x^6-300*x^5-1001*x^4-1254*x^3-120*x-175)/(x^2*log(x)^4+(8*x^3
+20*x^2)*log(x)^3+(18*x^4+100*x^3+150*x^2)*log(x)^2+(8*x^5+100*x^4+400*x^3+500*x^2)*log(x)+x^6+20*x^5+150*x^4+
500*x^3+625*x^2),x, algorithm="maxima")

[Out]

-(x^5 + x^3*log(x)^2 + 10*x^4 + 25*x^3 - x^2 + 2*(2*x^4 + 5*x^3)*log(x) - 5)/(x^3 + x*log(x)^2 + 10*x^2 + 2*(2
*x^2 + 5*x)*log(x) + 25*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {120\,x+{\ln \relax (x)}^3\,\left (16\,x^4+40\,x^3\right )+2\,x^3\,{\ln \relax (x)}^4+\ln \relax (x)\,\left (16\,x^6+200\,x^5+800\,x^4+1000\,x^3-8\,x^2+40\,x+60\right )+1254\,x^3+1001\,x^4+300\,x^5+40\,x^6+2\,x^7+{\ln \relax (x)}^2\,\left (36\,x^5+200\,x^4+300\,x^3-x^2+5\right )+175}{{\ln \relax (x)}^3\,\left (8\,x^3+20\,x^2\right )+x^2\,{\ln \relax (x)}^4+\ln \relax (x)\,\left (8\,x^5+100\,x^4+400\,x^3+500\,x^2\right )+{\ln \relax (x)}^2\,\left (18\,x^4+100\,x^3+150\,x^2\right )+625\,x^2+500\,x^3+150\,x^4+20\,x^5+x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(120*x + log(x)^3*(40*x^3 + 16*x^4) + 2*x^3*log(x)^4 + log(x)*(40*x - 8*x^2 + 1000*x^3 + 800*x^4 + 200*x^
5 + 16*x^6 + 60) + 1254*x^3 + 1001*x^4 + 300*x^5 + 40*x^6 + 2*x^7 + log(x)^2*(300*x^3 - x^2 + 200*x^4 + 36*x^5
+ 5) + 175)/(log(x)^3*(20*x^2 + 8*x^3) + x^2*log(x)^4 + log(x)*(500*x^2 + 400*x^3 + 100*x^4 + 8*x^5) + log(x)
^2*(150*x^2 + 100*x^3 + 18*x^4) + 625*x^2 + 500*x^3 + 150*x^4 + 20*x^5 + x^6),x)

[Out]

int(-(120*x + log(x)^3*(40*x^3 + 16*x^4) + 2*x^3*log(x)^4 + log(x)*(40*x - 8*x^2 + 1000*x^3 + 800*x^4 + 200*x^
5 + 16*x^6 + 60) + 1254*x^3 + 1001*x^4 + 300*x^5 + 40*x^6 + 2*x^7 + log(x)^2*(300*x^3 - x^2 + 200*x^4 + 36*x^5
+ 5) + 175)/(log(x)^3*(20*x^2 + 8*x^3) + x^2*log(x)^4 + log(x)*(500*x^2 + 400*x^3 + 100*x^4 + 8*x^5) + log(x)
^2*(150*x^2 + 100*x^3 + 18*x^4) + 625*x^2 + 500*x^3 + 150*x^4 + 20*x^5 + x^6), x)

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sympy [A]  time = 0.23, size = 37, normalized size = 1.32 \begin {gather*} - x^{2} + \frac {x^{2} + 5}{x^{3} + 10 x^{2} + x \log {\relax (x )}^{2} + 25 x + \left (4 x^{2} + 10 x\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**3*ln(x)**4+(-16*x**4-40*x**3)*ln(x)**3+(-36*x**5-200*x**4-300*x**3+x**2-5)*ln(x)**2+(-16*x**6
-200*x**5-800*x**4-1000*x**3+8*x**2-40*x-60)*ln(x)-2*x**7-40*x**6-300*x**5-1001*x**4-1254*x**3-120*x-175)/(x**
2*ln(x)**4+(8*x**3+20*x**2)*ln(x)**3+(18*x**4+100*x**3+150*x**2)*ln(x)**2+(8*x**5+100*x**4+400*x**3+500*x**2)*
ln(x)+x**6+20*x**5+150*x**4+500*x**3+625*x**2),x)

[Out]

-x**2 + (x**2 + 5)/(x**3 + 10*x**2 + x*log(x)**2 + 25*x + (4*x**2 + 10*x)*log(x))

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