3.95.28 \(\int \frac {8 x^2+9 x^3+6 x^4+2 x^5+(120 x+4 x^2-50 x^3-29 x^4-4 x^5) \log (x)+(-300-120 x+88 x^2+90 x^3+24 x^4+2 x^5) \log ^2(x)}{2 x^4+(-20 x^3-4 x^4) \log (x)+(50 x^2+20 x^3+2 x^4) \log ^2(x)} \, dx\)

Optimal. Leaf size=33 \[ \frac {(4+x) \left (3-x \left (-x+\frac {x}{-x+(5+x) \log (x)}\right )\right )}{2 x} \]

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Rubi [F]  time = 0.96, 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 {8 x^2+9 x^3+6 x^4+2 x^5+\left (120 x+4 x^2-50 x^3-29 x^4-4 x^5\right ) \log (x)+\left (-300-120 x+88 x^2+90 x^3+24 x^4+2 x^5\right ) \log ^2(x)}{2 x^4+\left (-20 x^3-4 x^4\right ) \log (x)+\left (50 x^2+20 x^3+2 x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8*x^2 + 9*x^3 + 6*x^4 + 2*x^5 + (120*x + 4*x^2 - 50*x^3 - 29*x^4 - 4*x^5)*Log[x] + (-300 - 120*x + 88*x^2
+ 90*x^3 + 24*x^4 + 2*x^5)*Log[x]^2)/(2*x^4 + (-20*x^3 - 4*x^4)*Log[x] + (50*x^2 + 20*x^3 + 2*x^4)*Log[x]^2),
x]

[Out]

6/x + 2*x + x^2/2 + (25*Defer[Int][(-x + 5*Log[x] + x*Log[x])^(-2), x])/2 + 2*Defer[Int][x/(-x + 5*Log[x] + x*
Log[x])^2, x] + Defer[Int][x^2/(-x + 5*Log[x] + x*Log[x])^2, x]/2 - (25*Defer[Int][1/((5 + x)*(-x + 5*Log[x] +
x*Log[x])^2), x])/2 - (5*Defer[Int][(-x + 5*Log[x] + x*Log[x])^(-1), x])/2 - Defer[Int][x/(-x + 5*Log[x] + x*
Log[x]), x]/2 + (5*Defer[Int][1/((5 + x)*(-x + 5*Log[x] + x*Log[x])), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \left (8+9 x+6 x^2+2 x^3\right )+x \left (120+4 x-50 x^2-29 x^3-4 x^4\right ) \log (x)+2 (5+x)^2 \left (-6+2 x^2+x^3\right ) \log ^2(x)}{2 x^2 (x-(5+x) \log (x))^2} \, dx\\ &=\frac {1}{2} \int \frac {x^2 \left (8+9 x+6 x^2+2 x^3\right )+x \left (120+4 x-50 x^2-29 x^3-4 x^4\right ) \log (x)+2 (5+x)^2 \left (-6+2 x^2+x^3\right ) \log ^2(x)}{x^2 (x-(5+x) \log (x))^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {2 \left (-6+2 x^2+x^3\right )}{x^2}+\frac {100+45 x+9 x^2+x^3}{(5+x) (-x+5 \log (x)+x \log (x))^2}+\frac {-20-10 x-x^2}{(5+x) (-x+5 \log (x)+x \log (x))}\right ) \, dx\\ &=\frac {1}{2} \int \frac {100+45 x+9 x^2+x^3}{(5+x) (-x+5 \log (x)+x \log (x))^2} \, dx+\frac {1}{2} \int \frac {-20-10 x-x^2}{(5+x) (-x+5 \log (x)+x \log (x))} \, dx+\int \frac {-6+2 x^2+x^3}{x^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {25}{(-x+5 \log (x)+x \log (x))^2}+\frac {4 x}{(-x+5 \log (x)+x \log (x))^2}+\frac {x^2}{(-x+5 \log (x)+x \log (x))^2}-\frac {25}{(5+x) (-x+5 \log (x)+x \log (x))^2}\right ) \, dx+\frac {1}{2} \int \left (-\frac {5}{-x+5 \log (x)+x \log (x)}-\frac {x}{-x+5 \log (x)+x \log (x)}+\frac {5}{(5+x) (-x+5 \log (x)+x \log (x))}\right ) \, dx+\int \left (2-\frac {6}{x^2}+x\right ) \, dx\\ &=\frac {6}{x}+2 x+\frac {x^2}{2}+\frac {1}{2} \int \frac {x^2}{(-x+5 \log (x)+x \log (x))^2} \, dx-\frac {1}{2} \int \frac {x}{-x+5 \log (x)+x \log (x)} \, dx+2 \int \frac {x}{(-x+5 \log (x)+x \log (x))^2} \, dx-\frac {5}{2} \int \frac {1}{-x+5 \log (x)+x \log (x)} \, dx+\frac {5}{2} \int \frac {1}{(5+x) (-x+5 \log (x)+x \log (x))} \, dx+\frac {25}{2} \int \frac {1}{(-x+5 \log (x)+x \log (x))^2} \, dx-\frac {25}{2} \int \frac {1}{(5+x) (-x+5 \log (x)+x \log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 36, normalized size = 1.09 \begin {gather*} \frac {1}{2} \left (\frac {12}{x}+4 x+x^2-\frac {x (4+x)}{-x+5 \log (x)+x \log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*x^2 + 9*x^3 + 6*x^4 + 2*x^5 + (120*x + 4*x^2 - 50*x^3 - 29*x^4 - 4*x^5)*Log[x] + (-300 - 120*x +
88*x^2 + 90*x^3 + 24*x^4 + 2*x^5)*Log[x]^2)/(2*x^4 + (-20*x^3 - 4*x^4)*Log[x] + (50*x^2 + 20*x^3 + 2*x^4)*Log[
x]^2),x]

[Out]

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

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fricas [A]  time = 0.79, size = 58, normalized size = 1.76 \begin {gather*} \frac {x^{4} + 5 \, x^{3} + 4 \, x^{2} - {\left (x^{4} + 9 \, x^{3} + 20 \, x^{2} + 12 \, x + 60\right )} \log \relax (x) + 12 \, x}{2 \, {\left (x^{2} - {\left (x^{2} + 5 \, x\right )} \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5+24*x^4+90*x^3+88*x^2-120*x-300)*log(x)^2+(-4*x^5-29*x^4-50*x^3+4*x^2+120*x)*log(x)+2*x^5+6*x
^4+9*x^3+8*x^2)/((2*x^4+20*x^3+50*x^2)*log(x)^2+(-4*x^4-20*x^3)*log(x)+2*x^4),x, algorithm="fricas")

[Out]

1/2*(x^4 + 5*x^3 + 4*x^2 - (x^4 + 9*x^3 + 20*x^2 + 12*x + 60)*log(x) + 12*x)/(x^2 - (x^2 + 5*x)*log(x))

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giac [A]  time = 0.20, size = 37, normalized size = 1.12 \begin {gather*} \frac {1}{2} \, x^{2} + 2 \, x - \frac {x^{2} + 4 \, x}{2 \, {\left (x \log \relax (x) - x + 5 \, \log \relax (x)\right )}} + \frac {6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5+24*x^4+90*x^3+88*x^2-120*x-300)*log(x)^2+(-4*x^5-29*x^4-50*x^3+4*x^2+120*x)*log(x)+2*x^5+6*x
^4+9*x^3+8*x^2)/((2*x^4+20*x^3+50*x^2)*log(x)^2+(-4*x^4-20*x^3)*log(x)+2*x^4),x, algorithm="giac")

[Out]

1/2*x^2 + 2*x - 1/2*(x^2 + 4*x)/(x*log(x) - x + 5*log(x)) + 6/x

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maple [A]  time = 0.06, size = 37, normalized size = 1.12

method	result	size
risch	\(\frac {x^{3}+4 x^{2}+12}{2 x}-\frac {\left (4+x \right ) x}{2 \left (x \ln \relax (x )+5 \ln \relax (x )-x \right )}\)	\(37\)
norman	\(\frac {-4 x \ln \relax (x )+8 x^{2} \ln \relax (x )-6 x -\frac {5 x^{3}}{2}-\frac {x^{4}}{2}+\frac {9 x^{3} \ln \relax (x )}{2}+\frac {x^{4} \ln \relax (x )}{2}+30 \ln \relax (x )}{x \left (x \ln \relax (x )+5 \ln \relax (x )-x \right )}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^5+24*x^4+90*x^3+88*x^2-120*x-300)*ln(x)^2+(-4*x^5-29*x^4-50*x^3+4*x^2+120*x)*ln(x)+2*x^5+6*x^4+9*x^3
+8*x^2)/((2*x^4+20*x^3+50*x^2)*ln(x)^2+(-4*x^4-20*x^3)*ln(x)+2*x^4),x,method=_RETURNVERBOSE)

[Out]

1/2*(x^3+4*x^2+12)/x-1/2*(4+x)*x/(x*ln(x)+5*ln(x)-x)

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maxima [A]  time = 0.40, size = 58, normalized size = 1.76 \begin {gather*} \frac {x^{4} + 5 \, x^{3} + 4 \, x^{2} - {\left (x^{4} + 9 \, x^{3} + 20 \, x^{2} + 12 \, x + 60\right )} \log \relax (x) + 12 \, x}{2 \, {\left (x^{2} - {\left (x^{2} + 5 \, x\right )} \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5+24*x^4+90*x^3+88*x^2-120*x-300)*log(x)^2+(-4*x^5-29*x^4-50*x^3+4*x^2+120*x)*log(x)+2*x^5+6*x
^4+9*x^3+8*x^2)/((2*x^4+20*x^3+50*x^2)*log(x)^2+(-4*x^4-20*x^3)*log(x)+2*x^4),x, algorithm="maxima")

[Out]

1/2*(x^4 + 5*x^3 + 4*x^2 - (x^4 + 9*x^3 + 20*x^2 + 12*x + 60)*log(x) + 12*x)/(x^2 - (x^2 + 5*x)*log(x))

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mupad [B]  time = 9.33, size = 60, normalized size = 1.82 \begin {gather*} 2\,x+\frac {6}{x}+\frac {x^2}{2}+\frac {x^5+9\,x^4+45\,x^3+100\,x^2}{2\,\left (x-\ln \relax (x)\,\left (x+5\right )\right )\,\left (x^3+5\,x^2+25\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2*(88*x^2 - 120*x + 90*x^3 + 24*x^4 + 2*x^5 - 300) - log(x)*(50*x^3 - 4*x^2 - 120*x + 29*x^4 + 4*x
^5) + 8*x^2 + 9*x^3 + 6*x^4 + 2*x^5)/(log(x)^2*(50*x^2 + 20*x^3 + 2*x^4) - log(x)*(20*x^3 + 4*x^4) + 2*x^4),x)

[Out]

2*x + 6/x + x^2/2 + (100*x^2 + 45*x^3 + 9*x^4 + x^5)/(2*(x - log(x)*(x + 5))*(25*x + 5*x^2 + x^3))

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sympy [A]  time = 0.22, size = 31, normalized size = 0.94 \begin {gather*} \frac {x^{2}}{2} + 2 x + \frac {- x^{2} - 4 x}{- 2 x + \left (2 x + 10\right ) \log {\relax (x )}} + \frac {6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**5+24*x**4+90*x**3+88*x**2-120*x-300)*ln(x)**2+(-4*x**5-29*x**4-50*x**3+4*x**2+120*x)*ln(x)+2*
x**5+6*x**4+9*x**3+8*x**2)/((2*x**4+20*x**3+50*x**2)*ln(x)**2+(-4*x**4-20*x**3)*ln(x)+2*x**4),x)

[Out]

x**2/2 + 2*x + (-x**2 - 4*x)/(-2*x + (2*x + 10)*log(x)) + 6/x

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