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

Optimal. Leaf size=68 \[ \frac {\log \left (a x^m+b \log ^q\left (c x^n\right )\right )}{b n q}-\frac {a m \text {Int}\left (\frac {x^{m-1}}{a x^m+b \log ^q\left (c x^n\right )},x\right )}{b n q} \]

[Out]

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

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Rubi [A]  time = 0.24, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (c x^{n}\right )^{q - 1}}{a x x^{m} + b x \log \left (c x^{n}\right )^{q}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 29.52, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \,x^{n}\right )^{q -1}}{\left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right ) x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -a \int \frac {x^{m}}{a b x x^{m} \log \relax (c) + a b x x^{m} \log \left (x^{n}\right ) + {\left (b^{2} x \log \relax (c) + b^{2} x \log \left (x^{n}\right )\right )} {\left (\log \relax (c) + \log \left (x^{n}\right )\right )}^{q}}\,{d x} + \frac {\log \left (\log \relax (c) + \log \left (x^{n}\right )\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*integrate(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) + log(log(c) + log(x^n))/(b*n)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\ln \left (c\,x^n\right )}^{q-1}}{x\,\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (c x^{n} \right )}^{q - 1}}{x \left (a x^{m} + b \log {\left (c x^{n} \right )}^{q}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(c*x**n)**(q - 1)/(x*(a*x**m + b*log(c*x**n)**q)), x)

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