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

Optimal. Leaf size=20 \[ -\frac{1}{a x^m+b \log ^q\left (c x^n\right )} \]

[Out]

-(a*x^m + b*Log[c*x^n]^q)^(-1)

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Rubi [A]  time = 0.181982, antiderivative size = 20, 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 = {2544} \[ -\frac{1}{a x^m+b \log ^q\left (c x^n\right )} \]

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)^2),x]

[Out]

-(a*x^m + b*Log[c*x^n]^q)^(-1)

Rule 2544

Int[((Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_)^(m_.))^(p_.)*(Log[(c_.)*(x_)^(n_.)]^(r_.)*(e_.) + (d_.)*(x
_)^(m_.)))/(x_), x_Symbol] :> Simp[(e*(a*x^m + b*Log[c*x^n]^q)^(p + 1))/(b*n*q*(p + 1)), x] /; FreeQ[{a, b, c,
d, e, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -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 )^2} \, dx &=-\frac{1}{a x^m+b \log ^q\left (c x^n\right )}\\ \end{align*}

Mathematica [A]  time = 0.0401056, size = 20, normalized size = 1. \[ -\frac{1}{a x^m+b \log ^q\left (c x^n\right )} \]

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)^2),x]

[Out]

-(a*x^m + b*Log[c*x^n]^q)^(-1)

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Maple [C]  time = 0.129, size = 68, normalized size = 3.4 \begin{align*} - \left ( a{x}^{m}+b \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} \right ) ^{-1} \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)^2,x)

[Out]

-1/(a*x^m+b*(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)

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Maxima [A]  time = 1.71604, size = 28, normalized size = 1.4 \begin{align*} -\frac{1}{a x^{m} + b{\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{q}} \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)^2,x, algorithm="maxima")

[Out]

-1/(a*x^m + b*(log(c) + log(x^n))^q)

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Fricas [A]  time = 1.99025, size = 51, normalized size = 2.55 \begin{align*} -\frac{1}{{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b + a x^{m}} \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)^2,x, algorithm="fricas")

[Out]

-1/((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)**2,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 )}^{2} 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)^2,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)^2*x), x)