3.52.19 \(\int \frac {-200-280 x+18 x^2+122 x^3+34 x^4+(20 x^2-2 x^4) \log (x)}{100-140 x+9 x^2+68 x^3-24 x^4-8 x^5+4 x^6+(20 x^2-14 x^3-4 x^4+4 x^5) \log (x)+x^4 \log ^2(x)} \, dx\)

Optimal. Leaf size=33 \[ -3+\frac {5+x}{3-x-\frac {5+\frac {x^2 \log (x)}{2+x}}{2 x}} \]

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Rubi [F]  time = 1.15, 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 {-200-280 x+18 x^2+122 x^3+34 x^4+\left (20 x^2-2 x^4\right ) \log (x)}{100-140 x+9 x^2+68 x^3-24 x^4-8 x^5+4 x^6+\left (20 x^2-14 x^3-4 x^4+4 x^5\right ) \log (x)+x^4 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-200 - 280*x + 18*x^2 + 122*x^3 + 34*x^4 + (20*x^2 - 2*x^4)*Log[x])/(100 - 140*x + 9*x^2 + 68*x^3 - 24*x^
4 - 8*x^5 + 4*x^6 + (20*x^2 - 14*x^3 - 4*x^4 + 4*x^5)*Log[x] + x^4*Log[x]^2),x]

[Out]

-400*Defer[Int][(10 - 7*x - 2*x^2 + 2*x^3 + x^2*Log[x])^(-2), x] - 140*Defer[Int][x/(10 - 7*x - 2*x^2 + 2*x^3
+ x^2*Log[x])^2, x] + 78*Defer[Int][x^2/(10 - 7*x - 2*x^2 + 2*x^3 + x^2*Log[x])^2, x] + 68*Defer[Int][x^3/(10
- 7*x - 2*x^2 + 2*x^3 + x^2*Log[x])^2, x] + 30*Defer[Int][x^4/(10 - 7*x - 2*x^2 + 2*x^3 + x^2*Log[x])^2, x] +
4*Defer[Int][x^5/(10 - 7*x - 2*x^2 + 2*x^3 + x^2*Log[x])^2, x] + 20*Defer[Int][(10 - 7*x - 2*x^2 + 2*x^3 + x^2
*Log[x])^(-1), x] - 2*Defer[Int][x^2/(10 - 7*x - 2*x^2 + 2*x^3 + x^2*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-100-140 x+9 x^2+61 x^3+17 x^4-x^2 \left (-10+x^2\right ) \log (x)\right )}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2} \, dx\\ &=2 \int \frac {-100-140 x+9 x^2+61 x^3+17 x^4-x^2 \left (-10+x^2\right ) \log (x)}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2} \, dx\\ &=2 \int \left (\frac {-200-70 x+39 x^2+34 x^3+15 x^4+2 x^5}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2}+\frac {10-x^2}{10-7 x-2 x^2+2 x^3+x^2 \log (x)}\right ) \, dx\\ &=2 \int \frac {-200-70 x+39 x^2+34 x^3+15 x^4+2 x^5}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2} \, dx+2 \int \frac {10-x^2}{10-7 x-2 x^2+2 x^3+x^2 \log (x)} \, dx\\ &=2 \int \left (-\frac {200}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2}-\frac {70 x}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2}+\frac {39 x^2}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2}+\frac {34 x^3}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2}+\frac {15 x^4}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2}+\frac {2 x^5}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2}\right ) \, dx+2 \int \left (\frac {10}{10-7 x-2 x^2+2 x^3+x^2 \log (x)}-\frac {x^2}{10-7 x-2 x^2+2 x^3+x^2 \log (x)}\right ) \, dx\\ &=-\left (2 \int \frac {x^2}{10-7 x-2 x^2+2 x^3+x^2 \log (x)} \, dx\right )+4 \int \frac {x^5}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2} \, dx+20 \int \frac {1}{10-7 x-2 x^2+2 x^3+x^2 \log (x)} \, dx+30 \int \frac {x^4}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2} \, dx+68 \int \frac {x^3}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2} \, dx+78 \int \frac {x^2}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2} \, dx-140 \int \frac {x}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2} \, dx-400 \int \frac {1}{\left (10-7 x-2 x^2+2 x^3+x^2 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.55, size = 34, normalized size = 1.03 \begin {gather*} -\frac {2 x \left (10+7 x+x^2\right )}{10-7 x-2 x^2+2 x^3+x^2 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-200 - 280*x + 18*x^2 + 122*x^3 + 34*x^4 + (20*x^2 - 2*x^4)*Log[x])/(100 - 140*x + 9*x^2 + 68*x^3 -
24*x^4 - 8*x^5 + 4*x^6 + (20*x^2 - 14*x^3 - 4*x^4 + 4*x^5)*Log[x] + x^4*Log[x]^2),x]

[Out]

(-2*x*(10 + 7*x + x^2))/(10 - 7*x - 2*x^2 + 2*x^3 + x^2*Log[x])

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fricas [A]  time = 0.51, size = 37, normalized size = 1.12 \begin {gather*} -\frac {2 \, {\left (x^{3} + 7 \, x^{2} + 10 \, x\right )}}{2 \, x^{3} + x^{2} \log \relax (x) - 2 \, x^{2} - 7 \, x + 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4+20*x^2)*log(x)+34*x^4+122*x^3+18*x^2-280*x-200)/(x^4*log(x)^2+(4*x^5-4*x^4-14*x^3+20*x^2)*l
og(x)+4*x^6-8*x^5-24*x^4+68*x^3+9*x^2-140*x+100),x, algorithm="fricas")

[Out]

-2*(x^3 + 7*x^2 + 10*x)/(2*x^3 + x^2*log(x) - 2*x^2 - 7*x + 10)

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giac [A]  time = 0.17, size = 37, normalized size = 1.12 \begin {gather*} -\frac {2 \, {\left (x^{3} + 7 \, x^{2} + 10 \, x\right )}}{2 \, x^{3} + x^{2} \log \relax (x) - 2 \, x^{2} - 7 \, x + 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4+20*x^2)*log(x)+34*x^4+122*x^3+18*x^2-280*x-200)/(x^4*log(x)^2+(4*x^5-4*x^4-14*x^3+20*x^2)*l
og(x)+4*x^6-8*x^5-24*x^4+68*x^3+9*x^2-140*x+100),x, algorithm="giac")

[Out]

-2*(x^3 + 7*x^2 + 10*x)/(2*x^3 + x^2*log(x) - 2*x^2 - 7*x + 10)

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maple [A]  time = 0.07, size = 35, normalized size = 1.06

method	result	size
risch	\(-\frac {2 \left (x^{2}+7 x +10\right ) x}{x^{2} \ln \relax (x )+2 x^{3}-2 x^{2}-7 x +10}\)	\(35\)
norman	\(\frac {-2 x^{3}-14 x^{2}-20 x}{x^{2} \ln \relax (x )+2 x^{3}-2 x^{2}-7 x +10}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^4+20*x^2)*ln(x)+34*x^4+122*x^3+18*x^2-280*x-200)/(x^4*ln(x)^2+(4*x^5-4*x^4-14*x^3+20*x^2)*ln(x)+4*x
^6-8*x^5-24*x^4+68*x^3+9*x^2-140*x+100),x,method=_RETURNVERBOSE)

[Out]

-2*(x^2+7*x+10)*x/(x^2*ln(x)+2*x^3-2*x^2-7*x+10)

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maxima [A]  time = 0.38, size = 37, normalized size = 1.12 \begin {gather*} -\frac {2 \, {\left (x^{3} + 7 \, x^{2} + 10 \, x\right )}}{2 \, x^{3} + x^{2} \log \relax (x) - 2 \, x^{2} - 7 \, x + 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4+20*x^2)*log(x)+34*x^4+122*x^3+18*x^2-280*x-200)/(x^4*log(x)^2+(4*x^5-4*x^4-14*x^3+20*x^2)*l
og(x)+4*x^6-8*x^5-24*x^4+68*x^3+9*x^2-140*x+100),x, algorithm="maxima")

[Out]

-2*(x^3 + 7*x^2 + 10*x)/(2*x^3 + x^2*log(x) - 2*x^2 - 7*x + 10)

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mupad [B]  time = 3.37, size = 39, normalized size = 1.18 \begin {gather*} -\frac {2\,x^3+14\,x^2+20\,x}{x^2\,\ln \relax (x)-7\,x-2\,x^2+2\,x^3+10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(20*x^2 - 2*x^4) - 280*x + 18*x^2 + 122*x^3 + 34*x^4 - 200)/(x^4*log(x)^2 - 140*x + log(x)*(20*x^2
- 14*x^3 - 4*x^4 + 4*x^5) + 9*x^2 + 68*x^3 - 24*x^4 - 8*x^5 + 4*x^6 + 100),x)

[Out]

-(20*x + 14*x^2 + 2*x^3)/(x^2*log(x) - 7*x - 2*x^2 + 2*x^3 + 10)

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sympy [A]  time = 0.18, size = 36, normalized size = 1.09 \begin {gather*} \frac {- 2 x^{3} - 14 x^{2} - 20 x}{2 x^{3} + x^{2} \log {\relax (x )} - 2 x^{2} - 7 x + 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**4+20*x**2)*ln(x)+34*x**4+122*x**3+18*x**2-280*x-200)/(x**4*ln(x)**2+(4*x**5-4*x**4-14*x**3+2
0*x**2)*ln(x)+4*x**6-8*x**5-24*x**4+68*x**3+9*x**2-140*x+100),x)

[Out]

(-2*x**3 - 14*x**2 - 20*x)/(2*x**3 + x**2*log(x) - 2*x**2 - 7*x + 10)

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