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

Optimal. Leaf size=17 \[ \log \left (a x^m+b \log ^q\left (c x^n\right )\right ) \]

[Out]

Log[a*x^m + b*Log[c*x^n]^q]

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Rubi [A]  time = 0.18538, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.023, Rules used = {2541} \[ \log \left (a x^m+b \log ^q\left (c x^n\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a*m*x^m + b*n*q*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]

Rule 2541

Int[(Log[(c_.)*(x_)^(n_.)]^(r_.)*(e_.) + (d_.)*(x_)^(m_.))/((x_)*(Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_
)^(m_.))), x_Symbol] :> Simp[(e*Log[a*x^m + b*Log[c*x^n]^q])/(b*n*q), x] /; FreeQ[{a, b, c, d, e, m, n, q, r},
x] && EqQ[r, q - 1] && EqQ[a*e*m - b*d*n*q, 0]

Rubi steps

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

Mathematica [A]  time = 0.233164, size = 17, normalized size = 1. \[ \log \left (a x^m+b \log ^q\left (c x^n\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a*m*x^m + b*n*q*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]

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Maple [C]  time = 0.164, size = 213, normalized size = 12.5 \begin{align*} q\ln \left ( \ln \left ({x}^{n} \right ) -{\frac{i}{2}} \left ( \pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ) -\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,i\ln \left ( c \right ) \right ) \right ) -q\ln \left ( \ln \left ( c \right ) +\ln \left ({x}^{n} \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) \right ) +\ln \left ( \left ( \ln \left ( c \right ) +\ln \left ({x}^{n} \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) \right ) ^{q}+{\frac{a{x}^{m}}{b}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

q*ln(ln(x^n)-1/2*I*(Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-Pi*csgn(I*c)*csgn(I*c*x^n)^2-Pi*csgn(I*x^n)*csgn(I*
c*x^n)^2+Pi*csgn(I*c*x^n)^3+2*I*ln(c)))-q*ln(ln(c)+ln(x^n)-1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*c*x^n)+csgn(I*c))*(
-csgn(I*c*x^n)+csgn(I*x^n)))+ln((ln(c)+ln(x^n)-1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*c*x^n)+csgn(I*c))*(-csgn(I*c*x^
n)+csgn(I*x^n)))^q+1/b*a*x^m)

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Maxima [A]  time = 1.55993, size = 30, normalized size = 1.76 \begin{align*} \log \left (\frac{a x^{m} + b{\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{q}}{b}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((a*x^m + b*(log(c) + log(x^n))^q)/b)

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Fricas [A]  time = 1.82366, size = 51, normalized size = 3. \begin{align*} \log \left ({\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b + a x^{m}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((n*log(x) + log(c))^q*b + a*x^m)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b n q \log \left (c x^{n}\right )^{q - 1} + a m x^{m}}{{\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((a*m*x^m+b*n*q*log(c*x^n)^(-1+q))/x/(a*x^m+b*log(c*x^n)^q),x, algorithm="giac")

[Out]

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