3.62.4 \(\int \frac {1250 x+1253 x^2+1004 x^3+3000 x^4+300 x^5+1500 x^6+40 x^7+280 x^8+2 x^9+18 x^{10}+(-2506 x-8 x^2-4000 x^3-1800 x^5-320 x^7-20 x^9) \log (x)+(3+4 x) \log ^2(x)}{2 x^2-4 x \log (x)+2 \log ^2(x)} \, dx\)

Optimal. Leaf size=27 \[ -1+x \left (\frac {3}{2}+x-\frac {x \left (5+x^2\right )^4}{-x+\log (x)}\right ) \]

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Rubi [F]  time = 1.39, 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 {1250 x+1253 x^2+1004 x^3+3000 x^4+300 x^5+1500 x^6+40 x^7+280 x^8+2 x^9+18 x^{10}+\left (-2506 x-8 x^2-4000 x^3-1800 x^5-320 x^7-20 x^9\right ) \log (x)+(3+4 x) \log ^2(x)}{2 x^2-4 x \log (x)+2 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1250*x + 1253*x^2 + 1004*x^3 + 3000*x^4 + 300*x^5 + 1500*x^6 + 40*x^7 + 280*x^8 + 2*x^9 + 18*x^10 + (-250
6*x - 8*x^2 - 4000*x^3 - 1800*x^5 - 320*x^7 - 20*x^9)*Log[x] + (3 + 4*x)*Log[x]^2)/(2*x^2 - 4*x*Log[x] + 2*Log
[x]^2),x]

[Out]

(3*x)/2 + x^2 + 625*Defer[Int][x/(x - Log[x])^2, x] - 625*Defer[Int][x^2/(x - Log[x])^2, x] + 500*Defer[Int][x
^3/(x - Log[x])^2, x] - 500*Defer[Int][x^4/(x - Log[x])^2, x] + 150*Defer[Int][x^5/(x - Log[x])^2, x] - 150*De
fer[Int][x^6/(x - Log[x])^2, x] + 20*Defer[Int][x^7/(x - Log[x])^2, x] - 20*Defer[Int][x^8/(x - Log[x])^2, x]
+ Defer[Int][x^9/(x - Log[x])^2, x] - Defer[Int][x^10/(x - Log[x])^2, x] + 1250*Defer[Int][x/(x - Log[x]), x]
+ 2000*Defer[Int][x^3/(x - Log[x]), x] + 900*Defer[Int][x^5/(x - Log[x]), x] + 160*Defer[Int][x^7/(x - Log[x])
, x] + 10*Defer[Int][x^9/(x - Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1250 x+1253 x^2+1004 x^3+3000 x^4+300 x^5+1500 x^6+40 x^7+280 x^8+2 x^9+18 x^{10}+\left (-2506 x-8 x^2-4000 x^3-1800 x^5-320 x^7-20 x^9\right ) \log (x)+(3+4 x) \log ^2(x)}{2 (x-\log (x))^2} \, dx\\ &=\frac {1}{2} \int \frac {1250 x+1253 x^2+1004 x^3+3000 x^4+300 x^5+1500 x^6+40 x^7+280 x^8+2 x^9+18 x^{10}+\left (-2506 x-8 x^2-4000 x^3-1800 x^5-320 x^7-20 x^9\right ) \log (x)+(3+4 x) \log ^2(x)}{(x-\log (x))^2} \, dx\\ &=\frac {1}{2} \int \left (3+4 x-\frac {2 (-1+x) x \left (5+x^2\right )^4}{(x-\log (x))^2}+\frac {20 x \left (1+x^2\right ) \left (5+x^2\right )^3}{x-\log (x)}\right ) \, dx\\ &=\frac {3 x}{2}+x^2+10 \int \frac {x \left (1+x^2\right ) \left (5+x^2\right )^3}{x-\log (x)} \, dx-\int \frac {(-1+x) x \left (5+x^2\right )^4}{(x-\log (x))^2} \, dx\\ &=\frac {3 x}{2}+x^2+10 \int \left (\frac {125 x}{x-\log (x)}+\frac {200 x^3}{x-\log (x)}+\frac {90 x^5}{x-\log (x)}+\frac {16 x^7}{x-\log (x)}+\frac {x^9}{x-\log (x)}\right ) \, dx-\int \left (-\frac {625 x}{(x-\log (x))^2}+\frac {625 x^2}{(x-\log (x))^2}-\frac {500 x^3}{(x-\log (x))^2}+\frac {500 x^4}{(x-\log (x))^2}-\frac {150 x^5}{(x-\log (x))^2}+\frac {150 x^6}{(x-\log (x))^2}-\frac {20 x^7}{(x-\log (x))^2}+\frac {20 x^8}{(x-\log (x))^2}-\frac {x^9}{(x-\log (x))^2}+\frac {x^{10}}{(x-\log (x))^2}\right ) \, dx\\ &=\frac {3 x}{2}+x^2+10 \int \frac {x^9}{x-\log (x)} \, dx+20 \int \frac {x^7}{(x-\log (x))^2} \, dx-20 \int \frac {x^8}{(x-\log (x))^2} \, dx+150 \int \frac {x^5}{(x-\log (x))^2} \, dx-150 \int \frac {x^6}{(x-\log (x))^2} \, dx+160 \int \frac {x^7}{x-\log (x)} \, dx+500 \int \frac {x^3}{(x-\log (x))^2} \, dx-500 \int \frac {x^4}{(x-\log (x))^2} \, dx+625 \int \frac {x}{(x-\log (x))^2} \, dx-625 \int \frac {x^2}{(x-\log (x))^2} \, dx+900 \int \frac {x^5}{x-\log (x)} \, dx+1250 \int \frac {x}{x-\log (x)} \, dx+2000 \int \frac {x^3}{x-\log (x)} \, dx+\int \frac {x^9}{(x-\log (x))^2} \, dx-\int \frac {x^{10}}{(x-\log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 28, normalized size = 1.04 \begin {gather*} \frac {1}{2} x \left (3+2 x+\frac {2 x \left (5+x^2\right )^4}{x-\log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1250*x + 1253*x^2 + 1004*x^3 + 3000*x^4 + 300*x^5 + 1500*x^6 + 40*x^7 + 280*x^8 + 2*x^9 + 18*x^10 +
(-2506*x - 8*x^2 - 4000*x^3 - 1800*x^5 - 320*x^7 - 20*x^9)*Log[x] + (3 + 4*x)*Log[x]^2)/(2*x^2 - 4*x*Log[x] +
2*Log[x]^2),x]

[Out]

(x*(3 + 2*x + (2*x*(5 + x^2)^4)/(x - Log[x])))/2

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fricas [A]  time = 0.64, size = 54, normalized size = 2.00 \begin {gather*} \frac {2 \, x^{10} + 40 \, x^{8} + 300 \, x^{6} + 1000 \, x^{4} + 2 \, x^{3} + 1253 \, x^{2} - {\left (2 \, x^{2} + 3 \, x\right )} \log \relax (x)}{2 \, {\left (x - \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+4*x)*log(x)^2+(-20*x^9-320*x^7-1800*x^5-4000*x^3-8*x^2-2506*x)*log(x)+18*x^10+2*x^9+280*x^8+40*x
^7+1500*x^6+300*x^5+3000*x^4+1004*x^3+1253*x^2+1250*x)/(2*log(x)^2-4*x*log(x)+2*x^2),x, algorithm="fricas")

[Out]

1/2*(2*x^10 + 40*x^8 + 300*x^6 + 1000*x^4 + 2*x^3 + 1253*x^2 - (2*x^2 + 3*x)*log(x))/(x - log(x))

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giac [A]  time = 0.17, size = 40, normalized size = 1.48 \begin {gather*} x^{2} + \frac {3}{2} \, x + \frac {x^{10} + 20 \, x^{8} + 150 \, x^{6} + 500 \, x^{4} + 625 \, x^{2}}{x - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+4*x)*log(x)^2+(-20*x^9-320*x^7-1800*x^5-4000*x^3-8*x^2-2506*x)*log(x)+18*x^10+2*x^9+280*x^8+40*x
^7+1500*x^6+300*x^5+3000*x^4+1004*x^3+1253*x^2+1250*x)/(2*log(x)^2-4*x*log(x)+2*x^2),x, algorithm="giac")

[Out]

x^2 + 3/2*x + (x^10 + 20*x^8 + 150*x^6 + 500*x^4 + 625*x^2)/(x - log(x))

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maple [A]  time = 0.03, size = 40, normalized size = 1.48

method	result	size
risch	\(x^{2}+\frac {3 x}{2}+\frac {\left (x^{8}+20 x^{6}+150 x^{4}+500 x^{2}+625\right ) x^{2}}{x -\ln \relax (x )}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3+4*x)*ln(x)^2+(-20*x^9-320*x^7-1800*x^5-4000*x^3-8*x^2-2506*x)*ln(x)+18*x^10+2*x^9+280*x^8+40*x^7+1500*
x^6+300*x^5+3000*x^4+1004*x^3+1253*x^2+1250*x)/(2*ln(x)^2-4*x*ln(x)+2*x^2),x,method=_RETURNVERBOSE)

[Out]

x^2+3/2*x+(x^8+20*x^6+150*x^4+500*x^2+625)*x^2/(x-ln(x))

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maxima [A]  time = 0.37, size = 54, normalized size = 2.00 \begin {gather*} \frac {2 \, x^{10} + 40 \, x^{8} + 300 \, x^{6} + 1000 \, x^{4} + 2 \, x^{3} + 1253 \, x^{2} - {\left (2 \, x^{2} + 3 \, x\right )} \log \relax (x)}{2 \, {\left (x - \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+4*x)*log(x)^2+(-20*x^9-320*x^7-1800*x^5-4000*x^3-8*x^2-2506*x)*log(x)+18*x^10+2*x^9+280*x^8+40*x
^7+1500*x^6+300*x^5+3000*x^4+1004*x^3+1253*x^2+1250*x)/(2*log(x)^2-4*x*log(x)+2*x^2),x, algorithm="maxima")

[Out]

1/2*(2*x^10 + 40*x^8 + 300*x^6 + 1000*x^4 + 2*x^3 + 1253*x^2 - (2*x^2 + 3*x)*log(x))/(x - log(x))

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mupad [B]  time = 4.84, size = 61, normalized size = 2.26 \begin {gather*} \frac {x\,\left (2\,x+3\right )}{2}+\frac {\frac {x\,\left (2\,x^9+40\,x^7+300\,x^5+1000\,x^3+2\,x^2+1253\,x\right )}{2}-\frac {x^2\,\left (2\,x+3\right )}{2}}{x-\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1250*x - log(x)*(2506*x + 8*x^2 + 4000*x^3 + 1800*x^5 + 320*x^7 + 20*x^9) + 1253*x^2 + 1004*x^3 + 3000*x^
4 + 300*x^5 + 1500*x^6 + 40*x^7 + 280*x^8 + 2*x^9 + 18*x^10 + log(x)^2*(4*x + 3))/(2*log(x)^2 - 4*x*log(x) + 2
*x^2),x)

[Out]

(x*(2*x + 3))/2 + ((x*(1253*x + 2*x^2 + 1000*x^3 + 300*x^5 + 40*x^7 + 2*x^9))/2 - (x^2*(2*x + 3))/2)/(x - log(
x))

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sympy [A]  time = 0.14, size = 37, normalized size = 1.37 \begin {gather*} x^{2} + \frac {3 x}{2} + \frac {- x^{10} - 20 x^{8} - 150 x^{6} - 500 x^{4} - 625 x^{2}}{- x + \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3+4*x)*ln(x)**2+(-20*x**9-320*x**7-1800*x**5-4000*x**3-8*x**2-2506*x)*ln(x)+18*x**10+2*x**9+280*x*
*8+40*x**7+1500*x**6+300*x**5+3000*x**4+1004*x**3+1253*x**2+1250*x)/(2*ln(x)**2-4*x*ln(x)+2*x**2),x)

[Out]

x**2 + 3*x/2 + (-x**10 - 20*x**8 - 150*x**6 - 500*x**4 - 625*x**2)/(-x + log(x))

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