### 3.35 $$\int \frac{d x^m+e \log ^{-1+q}(c x^n)}{x (a x^m+b \log ^q(c x^n))} \, dx$$

Optimal. Leaf size=72 $\left (d-\frac{a e m}{b n q}\right ) \text{CannotIntegrate}\left (\frac{x^{m-1}}{a x^m+b \log ^q\left (c x^n\right )},x\right )+\frac{e \log \left (a x^m+b \log ^q\left (c x^n\right )\right )}{b n q}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.267419, 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 = {} $\int \frac{d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx$

Veriﬁcation is Not applicable to the result.

[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*} \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 ) \int \frac{x^{-1+m}}{a x^m+b \log ^q\left (c x^n\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 4.74691, size = 0, normalized size = 0. $\int \frac{d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx$

Veriﬁcation is Not applicable to the result.

[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 [A]  time = 35.223, size = 0, normalized size = 0. \begin{align*} \int{\frac{d{x}^{m}+e \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{-1+q}}{x \left ( a{x}^{m}+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{q} \right ) }}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{d x^{m} + e \log \left (c x^{n}\right )^{q - 1}}{a x x^{m} + b x \log \left (c x^{n}\right )^{q}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)