3.75.45 \(\int \frac {-2 x+x^4-2 x^5+(4 x+10 x^2-20 x^3) \log (x)+(25-50 x) \log ^2(x)}{x^4+10 x^2 \log (x)+25 \log ^2(x)} \, dx\)

Optimal. Leaf size=22 \[ 3+x-x^2-\frac {2}{5+\frac {x^2}{\log (x)}} \]

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Rubi [F]  time = 0.42, 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 {-2 x+x^4-2 x^5+\left (4 x+10 x^2-20 x^3\right ) \log (x)+(25-50 x) \log ^2(x)}{x^4+10 x^2 \log (x)+25 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x + x^4 - 2*x^5 + (4*x + 10*x^2 - 20*x^3)*Log[x] + (25 - 50*x)*Log[x]^2)/(x^4 + 10*x^2*Log[x] + 25*Log
[x]^2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 24, normalized size = 1.09 \begin {gather*} x-x^2+\frac {2 x^2}{5 \left (x^2+5 \log (x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x + x^4 - 2*x^5 + (4*x + 10*x^2 - 20*x^3)*Log[x] + (25 - 50*x)*Log[x]^2)/(x^4 + 10*x^2*Log[x] +
25*Log[x]^2),x]

[Out]

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

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fricas [A]  time = 1.05, size = 39, normalized size = 1.77 \begin {gather*} -\frac {5 \, x^{4} - 5 \, x^{3} - 2 \, x^{2} + 25 \, {\left (x^{2} - x\right )} \log \relax (x)}{5 \, {\left (x^{2} + 5 \, \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*x+25)*log(x)^2+(-20*x^3+10*x^2+4*x)*log(x)-2*x^5+x^4-2*x)/(25*log(x)^2+10*x^2*log(x)+x^4),x, a
lgorithm="fricas")

[Out]

-1/5*(5*x^4 - 5*x^3 - 2*x^2 + 25*(x^2 - x)*log(x))/(x^2 + 5*log(x))

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giac [A]  time = 0.35, size = 22, normalized size = 1.00 \begin {gather*} -x^{2} + x + \frac {2 \, x^{2}}{5 \, {\left (x^{2} + 5 \, \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*x+25)*log(x)^2+(-20*x^3+10*x^2+4*x)*log(x)-2*x^5+x^4-2*x)/(25*log(x)^2+10*x^2*log(x)+x^4),x, a
lgorithm="giac")

[Out]

-x^2 + x + 2/5*x^2/(x^2 + 5*log(x))

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maple [A]  time = 0.03, size = 23, normalized size = 1.05

method	result	size
risch	\(-x^{2}+x +\frac {2 x^{2}}{5 \left (x^{2}+5 \ln \relax (x )\right )}\)	\(23\)
norman	\(\frac {x^{3}-2 \ln \relax (x )-x^{4}+5 x \ln \relax (x )-5 x^{2} \ln \relax (x )}{x^{2}+5 \ln \relax (x )}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-50*x+25)*ln(x)^2+(-20*x^3+10*x^2+4*x)*ln(x)-2*x^5+x^4-2*x)/(25*ln(x)^2+10*x^2*ln(x)+x^4),x,method=_RETU
RNVERBOSE)

[Out]

-x^2+x+2/5*x^2/(x^2+5*ln(x))

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maxima [A]  time = 0.40, size = 39, normalized size = 1.77 \begin {gather*} -\frac {5 \, x^{4} - 5 \, x^{3} - 2 \, x^{2} + 25 \, {\left (x^{2} - x\right )} \log \relax (x)}{5 \, {\left (x^{2} + 5 \, \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*x+25)*log(x)^2+(-20*x^3+10*x^2+4*x)*log(x)-2*x^5+x^4-2*x)/(25*log(x)^2+10*x^2*log(x)+x^4),x, a
lgorithm="maxima")

[Out]

-1/5*(5*x^4 - 5*x^3 - 2*x^2 + 25*(x^2 - x)*log(x))/(x^2 + 5*log(x))

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mupad [B]  time = 5.38, size = 24, normalized size = 1.09 \begin {gather*} x+\frac {2\,x^2}{5\,\left (5\,\ln \relax (x)+x^2\right )}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - x^4 + 2*x^5 + log(x)^2*(50*x - 25) - log(x)*(4*x + 10*x^2 - 20*x^3))/(10*x^2*log(x) + 25*log(x)^2
+ x^4),x)

[Out]

x + (2*x^2)/(5*(5*log(x) + x^2)) - x^2

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sympy [A]  time = 0.11, size = 19, normalized size = 0.86 \begin {gather*} - x^{2} + \frac {2 x^{2}}{5 x^{2} + 25 \log {\relax (x )}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*x+25)*ln(x)**2+(-20*x**3+10*x**2+4*x)*ln(x)-2*x**5+x**4-2*x)/(25*ln(x)**2+10*x**2*ln(x)+x**4),
x)

[Out]

-x**2 + 2*x**2/(5*x**2 + 25*log(x)) + x

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