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

Optimal. Leaf size=26 \[ \frac{d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )} \]

[Out]

(d*Log[c*x^n])/(a*x^m + b*Log[c*x^n]^q)

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Rubi [A]  time = 0.245221, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 60, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.017, Rules used = {2546} \[ \frac{d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Log[c*x^n])/(a*x^m + b*Log[c*x^n]^q)

Rule 2546

Int[(Log[(c_.)*(x_)^(n_.)]^(q_.)*(f_.) + (d_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]*(e_.)*(x_)^(m_.))/((x_)*(Log
[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_)^(m_.))^2), x_Symbol] :> Simp[(d*Log[c*x^n])/(a*n*(a*x^m + b*Log[c*x
^n]^q)), x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[e*n + d*m, 0] && EqQ[a*f + b*d*(q - 1), 0]

Rubi steps

\begin{align*} \int \frac{a d n x^m-a d m x^m \log \left (c x^n\right )-b d n (-1+q) \log ^q\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx &=\frac{d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )}\\ \end{align*}

Mathematica [A]  time = 0.356638, size = 26, normalized size = 1. \[ \frac{d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Log[c*x^n])/(a*x^m + b*Log[c*x^n]^q)

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Maple [C]  time = 0.128, size = 158, normalized size = 6.1 \begin{align*}{\frac{d \left ( 2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) -i\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ) +i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3} \right ) }{2\,a{x}^{m}+2\,b \left ( \ln \left ( c \right ) +\ln \left ({x}^{n} \right ) -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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*ln(c)+2*ln(x^n)-I*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*Pi*csgn(I*x^n
)*csgn(I*c*x^n)^2-I*Pi*csgn(I*c*x^n)^3)*d/(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 = 1.67436, size = 42, normalized size = 1.62 \begin{align*} \frac{d \log \left (c\right ) + d \log \left (x^{n}\right )}{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*d*n*x^m-a*d*m*x^m*log(c*x^n)-b*d*n*(-1+q)*log(c*x^n)^q)/x/(a*x^m+b*log(c*x^n)^q)^2,x, algorithm="
maxima")

[Out]

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

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Fricas [A]  time = 1.93464, size = 80, normalized size = 3.08 \begin{align*} \frac{d n \log \left (x\right ) + d \log \left (c\right )}{{\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*d*n*x^m-a*d*m*x^m*log(c*x^n)-b*d*n*(-1+q)*log(c*x^n)^q)/x/(a*x^m+b*log(c*x^n)^q)^2,x, algorithm="
fricas")

[Out]

(d*n*log(x) + d*log(c))/((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*d*n*x**m-a*d*m*x**m*ln(c*x**n)-b*d*n*(-1+q)*ln(c*x**n)**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 d n{\left (q - 1\right )} \log \left (c x^{n}\right )^{q} + a d m x^{m} \log \left (c x^{n}\right ) - a d n 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*d*n*x^m-a*d*m*x^m*log(c*x^n)-b*d*n*(-1+q)*log(c*x^n)^q)/x/(a*x^m+b*log(c*x^n)^q)^2,x, algorithm="
giac")

[Out]

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