3.7 \(\int \frac {\log ^{-1+q}(c x^n)}{x (a x^m+b \log ^q(c x^n))^2} \, dx\)

Optimal. Leaf size=70 \[ -\frac {a m \text {Int}\left (\frac {x^{m-1}}{\left (a x^m+b \log ^q\left (c x^n\right )\right )^2},x\right )}{b n q}-\frac {1}{b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )} \]

[Out]

-a*m*CannotIntegrate(x^(-1+m)/(a*x^m+b*ln(c*x^n)^q)^2,x)/b/n/q-1/b/n/q/(a*x^m+b*ln(c*x^n)^q)

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Rubi [A]  time = 0.29, 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 = {} \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx \]

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)^2),x]

[Out]

-(1/(b*n*q*(a*x^m + b*Log[c*x^n]^q))) - (a*m*Defer[Int][x^(-1 + m)/(a*x^m + b*Log[c*x^n]^q)^2, 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 )^2} \, dx &=-\frac {1}{b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )}-\frac {(a m) \int \frac {x^{-1+m}}{\left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx}{b n q}\\ \end {align*}

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Mathematica [A]  time = 0.80, size = 0, normalized size = 0.00 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx \]

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)^2),x]

[Out]

Integrate[Log[c*x^n]^(-1 + q)/(x*(a*x^m + b*Log[c*x^n]^q)^2), x]

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fricas [A]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (c x^{n}\right )^{q - 1}}{2 \, a b x x^{m} \log \left (c x^{n}\right )^{q} + a^{2} x x^{2 \, m} + b^{2} x \log \left (c x^{n}\right )^{2 \, q}}, x\right ) \]

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)^2,x, algorithm="fricas")

[Out]

integral(log(c*x^n)^(q - 1)/(2*a*b*x*x^m*log(c*x^n)^q + a^2*x*x^(2*m) + b^2*x*log(c*x^n)^(2*q)), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{2} x}\,{d x} \]

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)^2,x, algorithm="giac")

[Out]

integrate(log(c*x^n)^(q - 1)/((a*x^m + b*log(c*x^n)^q)^2*x), x)

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maple [A]  time = 50.17, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \,x^{n}\right )^{q -1}}{\left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{2} x}\, dx \]

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)^2,x)

[Out]

int(ln(c*x^n)^(q-1)/x/(a*x^m+b*ln(c*x^n)^q)^2,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{a b m x^{m} \log \left (x^{n}\right ) - {\left (n q - m \log \relax (c)\right )} a b x^{m} + {\left (b^{2} m \log \left (x^{n}\right ) - {\left (n q - m \log \relax (c)\right )} b^{2}\right )} {\left (\log \relax (c) + \log \left (x^{n}\right )\right )}^{q}} + \int -\frac {m n {\left (q - 1\right )} - m^{2} \log \relax (c) - m^{2} \log \left (x^{n}\right )}{a b m^{2} x x^{m} \log \left (x^{n}\right )^{2} - 2 \, {\left (m n q - m^{2} \log \relax (c)\right )} a b x x^{m} \log \left (x^{n}\right ) + {\left (n^{2} q^{2} - 2 \, m n q \log \relax (c) + m^{2} \log \relax (c)^{2}\right )} a b x x^{m} + {\left (b^{2} m^{2} x \log \left (x^{n}\right )^{2} - 2 \, {\left (m n q - m^{2} \log \relax (c)\right )} b^{2} x \log \left (x^{n}\right ) + {\left (n^{2} q^{2} - 2 \, m n q \log \relax (c) + m^{2} \log \relax (c)^{2}\right )} b^{2} x\right )} {\left (\log \relax (c) + \log \left (x^{n}\right )\right )}^{q}}\,{d x} \]

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)^2,x, algorithm="maxima")

[Out]

1/(a*b*m*x^m*log(x^n) - (n*q - m*log(c))*a*b*x^m + (b^2*m*log(x^n) - (n*q - m*log(c))*b^2)*(log(c) + log(x^n))
^q) + integrate(-(m*n*(q - 1) - m^2*log(c) - m^2*log(x^n))/(a*b*m^2*x*x^m*log(x^n)^2 - 2*(m*n*q - m^2*log(c))*
a*b*x*x^m*log(x^n) + (n^2*q^2 - 2*m*n*q*log(c) + m^2*log(c)^2)*a*b*x*x^m + (b^2*m^2*x*log(x^n)^2 - 2*(m*n*q -
m^2*log(c))*b^2*x*log(x^n) + (n^2*q^2 - 2*m*n*q*log(c) + m^2*log(c)^2)*b^2*x)*(log(c) + log(x^n))^q), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\ln \left (c\,x^n\right )}^{q-1}}{x\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^2} \,d x \]

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)^2),x)

[Out]

int(log(c*x^n)^(q - 1)/(x*(a*x^m + b*log(c*x^n)^q)^2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**2,x)

[Out]

Timed out

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