3.66 \(\int \frac{\log (x)}{x \log ^2(\frac{a+b x}{(b c-a d) x})} \, dx\)

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\log (x)}{x \log ^2\left (\frac{a+b x}{x (b c-a d)}\right )},x\right ) \]

[Out]

Unintegrable[Log[x]/(x*Log[(a + b*x)/((b*c - a*d)*x)]^2), x]

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Rubi [A]  time = 0.0219153, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\log (x)}{x \log ^2\left (\frac{a+b x}{(b c-a d) x}\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Log[x]/(x*Log[(a + b*x)/((b*c - a*d)*x)]^2),x]

[Out]

Defer[Int][Log[x]/(x*Log[(a + b*x)/((b*c - a*d)*x)]^2), x]

Rubi steps

\begin{align*} \int \frac{\log (x)}{x \log ^2\left (\frac{a+b x}{(b c-a d) x}\right )} \, dx &=\int \frac{\log (x)}{x \log ^2\left (\frac{a+b x}{(b c-a d) x}\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 33.1028, size = 0, normalized size = 0. \[ \int \frac{\log (x)}{x \log ^2\left (\frac{a+b x}{(b c-a d) x}\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Log[x]/(x*Log[(a + b*x)/((b*c - a*d)*x)]^2),x]

[Out]

Integrate[Log[x]/(x*Log[(a + b*x)/((b*c - a*d)*x)]^2), x]

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Maple [A]  time = 0.327, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( x \right ) }{x} \left ( \ln \left ({\frac{bx+a}{ \left ( -ad+bc \right ) x}} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x)/x/ln((b*x+a)/(-a*d+b*c)/x)^2,x)

[Out]

int(ln(x)/x/ln((b*x+a)/(-a*d+b*c)/x)^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (b x + a\right )} \log \left (x\right )}{a \log \left (b c - a d\right ) - a \log \left (b x + a\right ) + a \log \left (x\right )} - \int -\frac{b x \log \left (x\right ) + b x + a}{a x \log \left (b c - a d\right ) - a x \log \left (b x + a\right ) + a x \log \left (x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)/x/log((b*x+a)/(-a*d+b*c)/x)^2,x, algorithm="maxima")

[Out]

-(b*x + a)*log(x)/(a*log(b*c - a*d) - a*log(b*x + a) + a*log(x)) - integrate(-(b*x*log(x) + b*x + a)/(a*x*log(
b*c - a*d) - a*x*log(b*x + a) + a*x*log(x)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (x\right )}{x \log \left (\frac{b x + a}{{\left (b c - a d\right )} x}\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)/x/log((b*x+a)/(-a*d+b*c)/x)^2,x, algorithm="fricas")

[Out]

integral(log(x)/(x*log((b*x + a)/((b*c - a*d)*x))^2), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a \log{\left (x \right )} + b x \log{\left (x \right )}}{a \log{\left (\frac{a + b x}{x \left (- a d + b c\right )} \right )}} - \frac{\int \frac{b}{\log{\left (\frac{a}{- a d x + b c x} + \frac{b x}{- a d x + b c x} \right )}}\, dx + \int \frac{a}{x \log{\left (\frac{a}{- a d x + b c x} + \frac{b x}{- a d x + b c x} \right )}}\, dx + \int \frac{b \log{\left (x \right )}}{\log{\left (\frac{a}{- a d x + b c x} + \frac{b x}{- a d x + b c x} \right )}}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(x)/x/ln((b*x+a)/(-a*d+b*c)/x)**2,x)

[Out]

(a*log(x) + b*x*log(x))/(a*log((a + b*x)/(x*(-a*d + b*c)))) - (Integral(b/log(a/(-a*d*x + b*c*x) + b*x/(-a*d*x
+ b*c*x)), x) + Integral(a/(x*log(a/(-a*d*x + b*c*x) + b*x/(-a*d*x + b*c*x))), x) + Integral(b*log(x)/log(a/(
-a*d*x + b*c*x) + b*x/(-a*d*x + b*c*x)), x))/a

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (x\right )}{x \log \left (\frac{b x + a}{{\left (b c - a d\right )} x}\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)/x/log((b*x+a)/(-a*d+b*c)/x)^2,x, algorithm="giac")

[Out]

integrate(log(x)/(x*log((b*x + a)/((b*c - a*d)*x))^2), x)