3.11.31 \(\int \frac {-6-12 x-11 x^2+2 x^3+2 x^2 \log (x)}{25 x-20 x^2+4 x^3+(-10 x+4 x^2) \log (x)+x \log ^2(x)} \, dx\)

Optimal. Leaf size=15 \[ \frac {6+x^2}{-5+2 x+\log (x)} \]

________________________________________________________________________________________

Rubi [F]  time = 0.62, 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 {-6-12 x-11 x^2+2 x^3+2 x^2 \log (x)}{25 x-20 x^2+4 x^3+\left (-10 x+4 x^2\right ) \log (x)+x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6 - 12*x - 11*x^2 + 2*x^3 + 2*x^2*Log[x])/(25*x - 20*x^2 + 4*x^3 + (-10*x + 4*x^2)*Log[x] + x*Log[x]^2),
x]

[Out]

-12*Defer[Int][(-5 + 2*x + Log[x])^(-2), x] - 6*Defer[Int][1/(x*(-5 + 2*x + Log[x])^2), x] - Defer[Int][x/(-5
+ 2*x + Log[x])^2, x] - 2*Defer[Int][x^2/(-5 + 2*x + Log[x])^2, x] + 2*Defer[Int][x/(-5 + 2*x + Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6-12 x-11 x^2+2 x^3+2 x^2 \log (x)}{x (5-2 x-\log (x))^2} \, dx\\ &=\int \left (\frac {-6-12 x-x^2-2 x^3}{x (-5+2 x+\log (x))^2}+\frac {2 x}{-5+2 x+\log (x)}\right ) \, dx\\ &=2 \int \frac {x}{-5+2 x+\log (x)} \, dx+\int \frac {-6-12 x-x^2-2 x^3}{x (-5+2 x+\log (x))^2} \, dx\\ &=2 \int \frac {x}{-5+2 x+\log (x)} \, dx+\int \left (-\frac {12}{(-5+2 x+\log (x))^2}-\frac {6}{x (-5+2 x+\log (x))^2}-\frac {x}{(-5+2 x+\log (x))^2}-\frac {2 x^2}{(-5+2 x+\log (x))^2}\right ) \, dx\\ &=-\left (2 \int \frac {x^2}{(-5+2 x+\log (x))^2} \, dx\right )+2 \int \frac {x}{-5+2 x+\log (x)} \, dx-6 \int \frac {1}{x (-5+2 x+\log (x))^2} \, dx-12 \int \frac {1}{(-5+2 x+\log (x))^2} \, dx-\int \frac {x}{(-5+2 x+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.23, size = 15, normalized size = 1.00 \begin {gather*} \frac {6+x^2}{-5+2 x+\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 - 12*x - 11*x^2 + 2*x^3 + 2*x^2*Log[x])/(25*x - 20*x^2 + 4*x^3 + (-10*x + 4*x^2)*Log[x] + x*Log[
x]^2),x]

[Out]

(6 + x^2)/(-5 + 2*x + Log[x])

________________________________________________________________________________________

fricas [A]  time = 0.94, size = 15, normalized size = 1.00 \begin {gather*} \frac {x^{2} + 6}{2 \, x + \log \relax (x) - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*log(x)+2*x^3-11*x^2-12*x-6)/(x*log(x)^2+(4*x^2-10*x)*log(x)+4*x^3-20*x^2+25*x),x, algorithm="
fricas")

[Out]

(x^2 + 6)/(2*x + log(x) - 5)

________________________________________________________________________________________

giac [A]  time = 0.67, size = 15, normalized size = 1.00 \begin {gather*} \frac {x^{2} + 6}{2 \, x + \log \relax (x) - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*log(x)+2*x^3-11*x^2-12*x-6)/(x*log(x)^2+(4*x^2-10*x)*log(x)+4*x^3-20*x^2+25*x),x, algorithm="
giac")

[Out]

(x^2 + 6)/(2*x + log(x) - 5)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 16, normalized size = 1.07

method	result	size
norman	\(\frac {x^{2}+6}{2 x +\ln \relax (x )-5}\)	\(16\)
risch	\(\frac {x^{2}+6}{2 x +\ln \relax (x )-5}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2*ln(x)+2*x^3-11*x^2-12*x-6)/(x*ln(x)^2+(4*x^2-10*x)*ln(x)+4*x^3-20*x^2+25*x),x,method=_RETURNVERBOSE
)

[Out]

(x^2+6)/(2*x+ln(x)-5)

________________________________________________________________________________________

maxima [A]  time = 0.53, size = 15, normalized size = 1.00 \begin {gather*} \frac {x^{2} + 6}{2 \, x + \log \relax (x) - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*log(x)+2*x^3-11*x^2-12*x-6)/(x*log(x)^2+(4*x^2-10*x)*log(x)+4*x^3-20*x^2+25*x),x, algorithm="
maxima")

[Out]

(x^2 + 6)/(2*x + log(x) - 5)

________________________________________________________________________________________

mupad [B]  time = 0.85, size = 15, normalized size = 1.00 \begin {gather*} \frac {x^2+6}{2\,x+\ln \relax (x)-5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(12*x - 2*x^2*log(x) + 11*x^2 - 2*x^3 + 6)/(25*x + x*log(x)^2 - log(x)*(10*x - 4*x^2) - 20*x^2 + 4*x^3),x
)

[Out]

(x^2 + 6)/(2*x + log(x) - 5)

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 12, normalized size = 0.80 \begin {gather*} \frac {x^{2} + 6}{2 x + \log {\relax (x )} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2*ln(x)+2*x**3-11*x**2-12*x-6)/(x*ln(x)**2+(4*x**2-10*x)*ln(x)+4*x**3-20*x**2+25*x),x)

[Out]

(x**2 + 6)/(2*x + log(x) - 5)

________________________________________________________________________________________