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=73 \[ \left (d-\frac {a e m}{b n q}\right ) \text {Int}\left (\frac {x^{m-1}}{a x^m+b \log ^q\left (c x^n\right )},x\right )+\frac {e \log \left (a x^m+b \log ^q\left (c x^n\right )\right )}{b n q} \]

[Out]

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

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Rubi [A]  time = 0.27, 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 = {} \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx \]

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*}

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Mathematica [A]  time = 5.11, size = 0, normalized size = 0.00 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx \]

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]

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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d x^{m} + e \log \left (c x^{n}\right )^{q - 1}}{a x x^{m} + b x \log \left (c x^{n}\right )^{q}}, x\right ) \]

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)

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

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)

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maple [A]  time = 114.84, size = 0, normalized size = 0.00 \[ \int \frac {d \,x^{m}+e \ln \left (c \,x^{n}\right )^{q -1}}{\left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right ) x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((d*x^m+e*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 \[ \frac {e \log \left (\log \relax (c) + \log \left (x^{n}\right )\right )}{b n} + \int \frac {b d x^{m} \log \left (x^{n}\right ) + {\left (b d \log \relax (c) - a e\right )} x^{m}}{a b x x^{m} \log \relax (c) + a b x x^{m} \log \left (x^{n}\right ) + {\left (b^{2} x \log \relax (c) + b^{2} x \log \left (x^{n}\right )\right )} {\left (\log \relax (c) + \log \left (x^{n}\right )\right )}^{q}}\,{d x} \]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d*x**m + e*log(c*x**n)**q/log(c*x**n))/(x*(a*x**m + b*log(c*x**n)**q)), x)

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