### 3.1 $$\int \frac{\log ^{-1+q}(c x^n) (a x^m+b \log ^q(c x^n))^p}{x} \, dx$$

Optimal. Leaf size=75 $\frac{\left (a x^m+b \log ^q\left (c x^n\right )\right )^{p+1}}{b n (p+1) q}-\frac{a m \text{CannotIntegrate}\left (x^{m-1} \left (a x^m+b \log ^q\left (c x^n\right )\right )^p,x\right )}{b n q}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.24677, 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 = {} $\int \frac{\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.18046, size = 0, normalized size = 0. $\int \frac{\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.48, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{-1+q} \left ( a{x}^{m}+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{q} \right ) ^{p}}{x}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError