\(\int \frac {d x^m+e \log ^{-1+q}(c x^n)}{x (a x^m+b \log ^q(c x^n))} \, dx\) [35]

Optimal result

Integrand size = 40, antiderivative size = 40 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\frac {e \log \left (a x^m+b \log ^q\left (c x^n\right )\right )}{b n q}+\left (d-\frac {a e m}{b n q}\right ) \text {Int}\left (\frac {x^{-1+m}}{a x^m+b \log ^q\left (c x^n\right )},x\right ) \]

[Out]

(d-a*e*m/b/n/q)*CannotIntegrate(x^(-1+m)/(a*x^m+b*ln(c*x^n)^q),x)+e*ln(a*x^m+b*ln(c*x^n)^q)/b/n/q

Rubi [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 40, 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 = {} \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 6.71 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\int { \frac {d x^{m} + e \log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} x} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.38 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\int { \frac {d x^{m} + e \log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} x} \,d x } \]

[In]

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

[Out]

e*log(log(c) + log(x^n))/(b*n) + integrate((b*d*x^m*log(x^n) + (b*d*log(c) - a*e)*x^m)/(a*b*x*x^m*log(c) + a*b
*x*x^m*log(x^n) + (b^2*x*log(c) + b^2*x*log(x^n))*(log(c) + log(x^n))^q), x)

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\int { \frac {d x^{m} + e \log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.64 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\int \frac {d\,x^m+e\,{\ln \left (c\,x^n\right )}^{q-1}}{x\,\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )} \,d x \]

[In]

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

[Out]

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