\(\int \frac {x}{(d+e x) (a+b \log (c x^n))} \, dx\) [158]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {x}{(d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Int}\left (\frac {x}{(d+e x) \left (a+b \log \left (c x^n\right )\right )},x\right ) \]

[Out]

Unintegrable(x/(e*x+d)/(a+b*ln(c*x^n)),x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x}{(d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {x}{(d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {x}{(d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {x}{(d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {x}{(d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

\[\int \frac {x}{\left (e x +d \right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}d x\]

[In]

int(x/(e*x+d)/(a+b*ln(c*x^n)),x)

[Out]

int(x/(e*x+d)/(a+b*ln(c*x^n)),x)

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {x}{(d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {x}{{\left (e x + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

integral(x/(a*e*x + a*d + (b*e*x + b*d)*log(c*x^n)), x)

Sympy [N/A]

Not integrable

Time = 0.85 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {x}{(d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {x}{\left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e x\right )}\, dx \]

[In]

integrate(x/(e*x+d)/(a+b*ln(c*x**n)),x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {x}{(d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {x}{{\left (e x + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

integrate(x/((e*x + d)*(b*log(c*x^n) + a)), x)

Giac [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {x}{(d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {x}{{\left (e x + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

integrate(x/((e*x + d)*(b*log(c*x^n) + a)), x)

Mupad [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {x}{(d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {x}{\left (a+b\,\ln \left (c\,x^n\right )\right )\,\left (d+e\,x\right )} \,d x \]

[In]

int(x/((a + b*log(c*x^n))*(d + e*x)),x)

[Out]

int(x/((a + b*log(c*x^n))*(d + e*x)), x)