3.43.52 \(\int \frac {10 x-20 x \log (x)+(45+110 x+27 x^2) \log ^2(x)}{-10 x^2 \log (x)+(45 x+55 x^2+9 x^3) \log ^2(x)} \, dx\)

Optimal. Leaf size=30 \[ \log \left (\frac {1}{3} x \left ((-3-x)^2-x \left (-5-\frac {4 x}{5}+\frac {2}{\log (x)}\right )\right )\right ) \]

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Rubi [F]  time = 2.05, 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 {10 x-20 x \log (x)+\left (45+110 x+27 x^2\right ) \log ^2(x)}{-10 x^2 \log (x)+\left (45 x+55 x^2+9 x^3\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(10*x - 20*x*Log[x] + (45 + 110*x + 27*x^2)*Log[x]^2)/(-10*x^2*Log[x] + (45*x + 55*x^2 + 9*x^3)*Log[x]^2),
x]

[Out]

Log[x] + Log[45 + 55*x + 9*x^2] - Log[Log[x]] + 65*Defer[Int][(-10*x + 45*Log[x] + 55*x*Log[x] + 9*x^2*Log[x])
^(-1), x] + 3240*Sqrt[5/281]*Defer[Int][1/((-55 + Sqrt[1405] - 18*x)*(-10*x + 45*Log[x] + 55*x*Log[x] + 9*x^2*
Log[x])), x] + 45*Defer[Int][1/(x*(-10*x + 45*Log[x] + 55*x*Log[x] + 9*x^2*Log[x])), x] + 9*Defer[Int][x/(-10*
x + 45*Log[x] + 55*x*Log[x] + 9*x^2*Log[x]), x] - (550*(281 - 11*Sqrt[1405])*Defer[Int][1/((55 - Sqrt[1405] +
18*x)*(-10*x + 45*Log[x] + 55*x*Log[x] + 9*x^2*Log[x])), x])/281 + 3240*Sqrt[5/281]*Defer[Int][1/((55 + Sqrt[1
405] + 18*x)*(-10*x + 45*Log[x] + 55*x*Log[x] + 9*x^2*Log[x])), x] - (550*(281 + 11*Sqrt[1405])*Defer[Int][1/(
(55 + Sqrt[1405] + 18*x)*(-10*x + 45*Log[x] + 55*x*Log[x] + 9*x^2*Log[x])), x])/281

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-10 x+20 x \log (x)-\left (45+110 x+27 x^2\right ) \log ^2(x)}{x \log (x) \left (10 x-45 \log (x)-55 x \log (x)-9 x^2 \log (x)\right )} \, dx\\ &=\int \left (\frac {45+110 x+27 x^2}{x \left (45+55 x+9 x^2\right )}-\frac {1}{x \log (x)}+\frac {90 \left (-5+x^2\right )}{\left (45+55 x+9 x^2\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )}+\frac {45+55 x+9 x^2}{x \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )}\right ) \, dx\\ &=90 \int \frac {-5+x^2}{\left (45+55 x+9 x^2\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )} \, dx+\int \frac {45+110 x+27 x^2}{x \left (45+55 x+9 x^2\right )} \, dx-\int \frac {1}{x \log (x)} \, dx+\int \frac {45+55 x+9 x^2}{x \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )} \, dx\\ &=90 \int \left (\frac {1}{9 \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )}-\frac {5 (18+11 x)}{9 \left (45+55 x+9 x^2\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )}\right ) \, dx+\int \left (\frac {1}{x}+\frac {55+18 x}{45+55 x+9 x^2}\right ) \, dx+\int \left (\frac {55}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)}+\frac {45}{x \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )}+\frac {9 x}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)}\right ) \, dx-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=\log (x)-\log (\log (x))+9 \int \frac {x}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)} \, dx+10 \int \frac {1}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)} \, dx+45 \int \frac {1}{x \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )} \, dx-50 \int \frac {18+11 x}{\left (45+55 x+9 x^2\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )} \, dx+55 \int \frac {1}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)} \, dx+\int \frac {55+18 x}{45+55 x+9 x^2} \, dx\\ &=\log (x)+\log \left (45+55 x+9 x^2\right )-\log (\log (x))+9 \int \frac {x}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)} \, dx+10 \int \frac {1}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)} \, dx+45 \int \frac {1}{x \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )} \, dx-50 \int \left (\frac {18}{\left (45+55 x+9 x^2\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )}+\frac {11 x}{\left (45+55 x+9 x^2\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )}\right ) \, dx+55 \int \frac {1}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)} \, dx\\ &=\log (x)+\log \left (45+55 x+9 x^2\right )-\log (\log (x))+9 \int \frac {x}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)} \, dx+10 \int \frac {1}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)} \, dx+45 \int \frac {1}{x \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )} \, dx+55 \int \frac {1}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)} \, dx-550 \int \frac {x}{\left (45+55 x+9 x^2\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )} \, dx-900 \int \frac {1}{\left (45+55 x+9 x^2\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )} \, dx\\ &=\log (x)+\log \left (45+55 x+9 x^2\right )-\log (\log (x))+9 \int \frac {x}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)} \, dx+10 \int \frac {1}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)} \, dx+45 \int \frac {1}{x \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )} \, dx+55 \int \frac {1}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)} \, dx-550 \int \left (\frac {1-11 \sqrt {\frac {5}{281}}}{\left (55-\sqrt {1405}+18 x\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )}+\frac {1+11 \sqrt {\frac {5}{281}}}{\left (55+\sqrt {1405}+18 x\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )}\right ) \, dx-900 \int \left (-\frac {18}{\sqrt {1405} \left (-55+\sqrt {1405}-18 x\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )}-\frac {18}{\sqrt {1405} \left (55+\sqrt {1405}+18 x\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )}\right ) \, dx\\ &=\log (x)+\log \left (45+55 x+9 x^2\right )-\log (\log (x))+9 \int \frac {x}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)} \, dx+10 \int \frac {1}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)} \, dx+45 \int \frac {1}{x \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )} \, dx+55 \int \frac {1}{-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)} \, dx+\left (3240 \sqrt {\frac {5}{281}}\right ) \int \frac {1}{\left (-55+\sqrt {1405}-18 x\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )} \, dx+\left (3240 \sqrt {\frac {5}{281}}\right ) \int \frac {1}{\left (55+\sqrt {1405}+18 x\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )} \, dx-\frac {1}{281} \left (550 \left (281-11 \sqrt {1405}\right )\right ) \int \frac {1}{\left (55-\sqrt {1405}+18 x\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )} \, dx-\frac {1}{281} \left (550 \left (281+11 \sqrt {1405}\right )\right ) \int \frac {1}{\left (55+\sqrt {1405}+18 x\right ) \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.44, size = 29, normalized size = 0.97 \begin {gather*} \log (x)-\log (\log (x))+\log \left (-10 x+45 \log (x)+55 x \log (x)+9 x^2 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10*x - 20*x*Log[x] + (45 + 110*x + 27*x^2)*Log[x]^2)/(-10*x^2*Log[x] + (45*x + 55*x^2 + 9*x^3)*Log[
x]^2),x]

[Out]

Log[x] - Log[Log[x]] + Log[-10*x + 45*Log[x] + 55*x*Log[x] + 9*x^2*Log[x]]

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fricas [B]  time = 0.61, size = 52, normalized size = 1.73 \begin {gather*} \log \left (9 \, x^{3} + 55 \, x^{2} + 45 \, x\right ) + \log \left (\frac {{\left (9 \, x^{2} + 55 \, x + 45\right )} \log \relax (x) - 10 \, x}{9 \, x^{2} + 55 \, x + 45}\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^2+110*x+45)*log(x)^2-20*x*log(x)+10*x)/((9*x^3+55*x^2+45*x)*log(x)^2-10*x^2*log(x)),x, algori
thm="fricas")

[Out]

log(9*x^3 + 55*x^2 + 45*x) + log(((9*x^2 + 55*x + 45)*log(x) - 10*x)/(9*x^2 + 55*x + 45)) - log(log(x))

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giac [A]  time = 0.14, size = 29, normalized size = 0.97 \begin {gather*} \log \left (9 \, x^{2} \log \relax (x) + 55 \, x \log \relax (x) - 10 \, x + 45 \, \log \relax (x)\right ) + \log \relax (x) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^2+110*x+45)*log(x)^2-20*x*log(x)+10*x)/((9*x^3+55*x^2+45*x)*log(x)^2-10*x^2*log(x)),x, algori
thm="giac")

[Out]

log(9*x^2*log(x) + 55*x*log(x) - 10*x + 45*log(x)) + log(x) - log(log(x))

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maple [A]  time = 0.04, size = 30, normalized size = 1.00

method	result	size
norman	\(\ln \relax (x )-\ln \left (\ln \relax (x )\right )+\ln \left (9 x^{2} \ln \relax (x )+55 x \ln \relax (x )+45 \ln \relax (x )-10 x \right )\)	\(30\)
risch	\(\ln \left (9 x^{3}+55 x^{2}+45 x \right )+\ln \left (\ln \relax (x )-\frac {10 x}{9 x^{2}+55 x +45}\right )-\ln \left (\ln \relax (x )\right )\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((27*x^2+110*x+45)*ln(x)^2-20*x*ln(x)+10*x)/((9*x^3+55*x^2+45*x)*ln(x)^2-10*x^2*ln(x)),x,method=_RETURNVER
BOSE)

[Out]

ln(x)-ln(ln(x))+ln(9*x^2*ln(x)+55*x*ln(x)+45*ln(x)-10*x)

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maxima [A]  time = 0.37, size = 50, normalized size = 1.67 \begin {gather*} \log \left (9 \, x^{2} + 55 \, x + 45\right ) + \log \relax (x) + \log \left (\frac {{\left (9 \, x^{2} + 55 \, x + 45\right )} \log \relax (x) - 10 \, x}{9 \, x^{2} + 55 \, x + 45}\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^2+110*x+45)*log(x)^2-20*x*log(x)+10*x)/((9*x^3+55*x^2+45*x)*log(x)^2-10*x^2*log(x)),x, algori
thm="maxima")

[Out]

log(9*x^2 + 55*x + 45) + log(x) + log(((9*x^2 + 55*x + 45)*log(x) - 10*x)/(9*x^2 + 55*x + 45)) - log(log(x))

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mupad [B]  time = 3.10, size = 29, normalized size = 0.97 \begin {gather*} \ln \left (45\,\ln \relax (x)-10\,x+9\,x^2\,\ln \relax (x)+55\,x\,\ln \relax (x)\right )-\ln \left (\ln \relax (x)\right )+\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(10*x + log(x)^2*(110*x + 27*x^2 + 45) - 20*x*log(x))/(10*x^2*log(x) - log(x)^2*(45*x + 55*x^2 + 9*x^3)),
x)

[Out]

log(45*log(x) - 10*x + 9*x^2*log(x) + 55*x*log(x)) - log(log(x)) + log(x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x**2+110*x+45)*ln(x)**2-20*x*ln(x)+10*x)/((9*x**3+55*x**2+45*x)*ln(x)**2-10*x**2*ln(x)),x)

[Out]

Exception raised: PolynomialError

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