3.62 \(\int \frac{\log (x) \log ^3(\frac{a+b x}{(b c-a d) x})}{x} \, dx\)

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\log (x) \log ^3\left (\frac{a+b x}{x (b c-a d)}\right )}{x},x\right ) \]

[Out]

Unintegrable[(Log[x]*Log[(a + b*x)/((b*c - a*d)*x)]^3)/x, x]

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Rubi [A]  time = 0.021354, 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 = {} \[ \int \frac{\log (x) \log ^3\left (\frac{a+b x}{(b c-a d) x}\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Log[x]*Log[(a + b*x)/((b*c - a*d)*x)]^3)/x,x]

[Out]

Defer[Int][(Log[x]*Log[(a + b*x)/((b*c - a*d)*x)]^3)/x, x]

Rubi steps

\begin{align*} \int \frac{\log (x) \log ^3\left (\frac{a+b x}{(b c-a d) x}\right )}{x} \, dx &=\int \frac{\log (x) \log ^3\left (\frac{a+b x}{(b c-a d) x}\right )}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 5.09395, size = 0, normalized size = 0. \[ \int \frac{\log (x) \log ^3\left (\frac{a+b x}{(b c-a d) x}\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Log[x]*Log[(a + b*x)/((b*c - a*d)*x)]^3)/x,x]

[Out]

Integrate[(Log[x]*Log[(a + b*x)/((b*c - a*d)*x)]^3)/x, x]

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Maple [A]  time = 0.385, 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 ) ^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x)*ln((b*x+a)/(-a*d+b*c)/x)^3/x,x)

[Out]

int(ln(x)*ln((b*x+a)/(-a*d+b*c)/x)^3/x,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, \log \left (b x + a\right )^{3} \log \left (x\right )^{2} - \int \frac{2 \,{\left (b x + a\right )} \log \left (x\right )^{4} + 6 \,{\left (b x \log \left (b c - a d\right ) + a \log \left (b c - a d\right )\right )} \log \left (x\right )^{3} + 3 \,{\left ({\left (3 \, b x + 2 \, a\right )} \log \left (x\right )^{2} + 2 \,{\left (b x \log \left (b c - a d\right ) + a \log \left (b c - a d\right )\right )} \log \left (x\right )\right )} \log \left (b x + a\right )^{2} + 6 \,{\left (b x \log \left (b c - a d\right )^{2} + a \log \left (b c - a d\right )^{2}\right )} \log \left (x\right )^{2} - 6 \,{\left ({\left (b x + a\right )} \log \left (x\right )^{3} + 2 \,{\left (b x \log \left (b c - a d\right ) + a \log \left (b c - a d\right )\right )} \log \left (x\right )^{2} +{\left (b x \log \left (b c - a d\right )^{2} + a \log \left (b c - a d\right )^{2}\right )} \log \left (x\right )\right )} \log \left (b x + a\right ) + 2 \,{\left (b x \log \left (b c - a d\right )^{3} + a \log \left (b c - a d\right )^{3}\right )} \log \left (x\right )}{2 \,{\left (b x^{2} + a x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)*log((b*x+a)/(-a*d+b*c)/x)^3/x,x, algorithm="maxima")

[Out]

1/2*log(b*x + a)^3*log(x)^2 - integrate(1/2*(2*(b*x + a)*log(x)^4 + 6*(b*x*log(b*c - a*d) + a*log(b*c - a*d))*
log(x)^3 + 3*((3*b*x + 2*a)*log(x)^2 + 2*(b*x*log(b*c - a*d) + a*log(b*c - a*d))*log(x))*log(b*x + a)^2 + 6*(b
*x*log(b*c - a*d)^2 + a*log(b*c - a*d)^2)*log(x)^2 - 6*((b*x + a)*log(x)^3 + 2*(b*x*log(b*c - a*d) + a*log(b*c
- a*d))*log(x)^2 + (b*x*log(b*c - a*d)^2 + a*log(b*c - a*d)^2)*log(x))*log(b*x + a) + 2*(b*x*log(b*c - a*d)^3
+ a*log(b*c - a*d)^3)*log(x))/(b*x^2 + a*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 ) \log \left (\frac{b x + a}{{\left (b c - a d\right )} x}\right )^{3}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)*log((b*x+a)/(-a*d+b*c)/x)^3/x,x, algorithm="fricas")

[Out]

integral(log(x)*log((b*x + a)/((b*c - a*d)*x))^3/x, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3 a \int \frac{\log{\left (x \right )}^{2} \log{\left (\frac{a}{- a d x + b c x} + \frac{b x}{- a d x + b c x} \right )}^{2}}{a x + b x^{2}}\, dx}{2} + \frac{\log{\left (x \right )}^{2} \log{\left (\frac{a + b x}{x \left (- a d + b c\right )} \right )}^{3}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(x)*ln((b*x+a)/(-a*d+b*c)/x)**3/x,x)

[Out]

3*a*Integral(log(x)**2*log(a/(-a*d*x + b*c*x) + b*x/(-a*d*x + b*c*x))**2/(a*x + b*x**2), x)/2 + log(x)**2*log(
(a + b*x)/(x*(-a*d + b*c)))**3/2

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (x\right ) \log \left (\frac{b x + a}{{\left (b c - a d\right )} x}\right )^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)*log((b*x+a)/(-a*d+b*c)/x)^3/x,x, algorithm="giac")

[Out]

integrate(log(x)*log((b*x + a)/((b*c - a*d)*x))^3/x, x)