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

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\log (x)}{x \log ^2\left (\frac {a+b x}{x (b c-a d)}\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, 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 = {} \[ \int \frac {\log (x)}{x \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )} \, dx \]

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*}

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Mathematica [A]  time = 33.66, size = 0, normalized size = 0.00 \[ \int \frac {\log (x)}{x \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )} \, dx \]

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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fricas [A]  time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \relax (x)}{x \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \relax (x)}{x \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )^{2}}\,{d x} \]

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)

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maple [A]  time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\ln \relax (x )}{x \ln \left (\frac {b x +a}{\left (-a d +b c \right ) x}\right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (b x + a\right )} \log \relax (x)}{a \log \left (b c - a d\right ) - a \log \left (b x + a\right ) + a \log \relax (x)} - \int -\frac {b x \log \relax (x) + b x + a}{a x \log \left (b c - a d\right ) - a x \log \left (b x + a\right ) + a x \log \relax (x)}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\ln \relax (x)}{x\,{\ln \left (-\frac {a+b\,x}{x\,\left (a\,d-b\,c\right )}\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a \log {\relax (x )} + b x \log {\relax (x )}}{a \log {\left (\frac {a + b x}{x \left (- a d + b c\right )} \right )}} - \frac {\int \frac {b}{\log {\left (\frac {a}{- a d x + b c x} + \frac {b x}{- a d x + b c x} \right )}}\, dx + \int \frac {a}{x \log {\left (\frac {a}{- a d x + b c x} + \frac {b x}{- a d x + b c x} \right )}}\, dx + \int \frac {b \log {\relax (x )}}{\log {\left (\frac {a}{- a d x + b c x} + \frac {b x}{- a d x + b c x} \right )}}\, dx}{a} \]

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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