\(\int \frac {1}{x (d+e x) (a+b \log (c x^n))} \, dx\) [160]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {1}{x (d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Int}\left (\frac {1}{x (d+e x) \left (a+b \log \left (c x^n\right )\right )},x\right ) \]

[Out]

Unintegrable(1/x/(e*x+d)/(a+b*ln(c*x^n)),x)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {1}{x (d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x (d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx \]

[In]

Int[1/(x*(d + e*x)*(a + b*Log[c*x^n])),x]

[Out]

Defer[Int][1/(x*(d + e*x)*(a + b*Log[c*x^n])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x (d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.89 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x (d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x (d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx \]

[In]

Integrate[1/(x*(d + e*x)*(a + b*Log[c*x^n])),x]

[Out]

Integrate[1/(x*(d + e*x)*(a + b*Log[c*x^n])), x]

Maple [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

\[\int \frac {1}{x \left (e x +d \right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}d x\]

[In]

int(1/x/(e*x+d)/(a+b*ln(c*x^n)),x)

[Out]

int(1/x/(e*x+d)/(a+b*ln(c*x^n)),x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x (d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{{\left (e x + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )} x} \,d x } \]

[In]

integrate(1/x/(e*x+d)/(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

integral(1/(a*e*x^2 + a*d*x + (b*e*x^2 + b*d*x)*log(c*x^n)), x)

Sympy [N/A]

Not integrable

Time = 1.44 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x (d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x \left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e x\right )}\, dx \]

[In]

integrate(1/x/(e*x+d)/(a+b*ln(c*x**n)),x)

[Out]

Integral(1/(x*(a + b*log(c*x**n))*(d + e*x)), x)

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x (d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{{\left (e x + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )} x} \,d x } \]

[In]

integrate(1/x/(e*x+d)/(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

integrate(1/((e*x + d)*(b*log(c*x^n) + a)*x), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x (d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{{\left (e x + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )} x} \,d x } \]

[In]

integrate(1/x/(e*x+d)/(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

integrate(1/((e*x + d)*(b*log(c*x^n) + a)*x), x)

Mupad [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x (d+e x) \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x\,\left (a+b\,\ln \left (c\,x^n\right )\right )\,\left (d+e\,x\right )} \,d x \]

[In]

int(1/(x*(a + b*log(c*x^n))*(d + e*x)),x)

[Out]

int(1/(x*(a + b*log(c*x^n))*(d + e*x)), x)