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

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\log (x) \log ^3\left (\frac {a+b x}{x (b c-a d)}\right )}{x},x\right ) \]

[Out]

Unintegrable(ln(x)*ln((b*x+a)/(-a*d+b*c)/x)^3/x,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 = {} \[ \int \frac {\log (x) \log ^3\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx \]

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*}

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Mathematica [A]  time = 5.10, size = 0, normalized size = 0.00 \[ \int \frac {\log (x) \log ^3\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx \]

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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fricas [A]  time = 1.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \relax (x) \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )^{3}}{x}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \relax (x) \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )^{3}}{x}\,{d x} \]

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)

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maple [A]  time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\ln \relax (x ) \ln \left (\frac {b x +a}{\left (-a d +b c \right ) x}\right )^{3}}{x}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, \log \left (b x + a\right )^{3} \log \relax (x)^{2} - \int \frac {2 \, {\left (b x + a\right )} \log \relax (x)^{4} + 6 \, {\left (b x \log \left (b c - a d\right ) + a \log \left (b c - a d\right )\right )} \log \relax (x)^{3} + 3 \, {\left ({\left (3 \, b x + 2 \, a\right )} \log \relax (x)^{2} + 2 \, {\left (b x \log \left (b c - a d\right ) + a \log \left (b c - a d\right )\right )} \log \relax (x)\right )} \log \left (b x + a\right )^{2} + 6 \, {\left (b x \log \left (b c - a d\right )^{2} + a \log \left (b c - a d\right )^{2}\right )} \log \relax (x)^{2} - 6 \, {\left ({\left (b x + a\right )} \log \relax (x)^{3} + 2 \, {\left (b x \log \left (b c - a d\right ) + a \log \left (b c - a d\right )\right )} \log \relax (x)^{2} + {\left (b x \log \left (b c - a d\right )^{2} + a \log \left (b c - a d\right )^{2}\right )} \log \relax (x)\right )} \log \left (b x + a\right ) + 2 \, {\left (b x \log \left (b c - a d\right )^{3} + a \log \left (b c - a d\right )^{3}\right )} \log \relax (x)}{2 \, {\left (b x^{2} + a x\right )}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\ln \left (-\frac {a+b\,x}{x\,\left (a\,d-b\,c\right )}\right )}^3\,\ln \relax (x)}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {3 a \int \frac {\log {\relax (x )}^{2} \log {\left (\frac {a}{- a d x + b c x} + \frac {b x}{- a d x + b c x} \right )}^{2}}{a x + b x^{2}}\, dx}{2} + \frac {\log {\relax (x )}^{2} \log {\left (\frac {a + b x}{x \left (- a d + b c\right )} \right )}^{3}}{2} \]

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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