\(\int \frac {a d n x^m-a d m x^m \log (c x^n)-b d n (-1+q) \log ^q(c x^n)}{x (a x^m+b \log ^q(c x^n))^2} \, dx\) [38]

Optimal result

Integrand size = 60, antiderivative size = 26 \[ \int \frac {a d n x^m-a d m x^m \log \left (c x^n\right )-b d n (-1+q) \log ^q\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx=\frac {d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {2626} \[ \int \frac {a d n x^m-a d m x^m \log \left (c x^n\right )-b d n (-1+q) \log ^q\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx=\frac {d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )} \]

[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 2626

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*} \text {integral}& = \frac {d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {a d n x^m-a d m x^m \log \left (c x^n\right )-b d n (-1+q) \log ^q\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx=\frac {d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )} \]

[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)

Maple [A] (verified)

Time = 10.79 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04

method	result	size
parallelrisch	\(\frac {d \ln \left (c \,x^{n}\right )}{a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}}\)	\(27\)
risch	\(\frac {\left (-i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i x^{n}\right )-i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )+2 \ln \left (c \right )+2 \ln \left (x^{n}\right )\right ) d}{2 a \,x^{m}+2 b {\left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i x^{n}\right )\right )}{2}\right )}^{q}}\)	\(158\)

[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,method=_RETURNVERBOSE
)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {a d n x^m-a d m x^m \log \left (c x^n\right )-b d n (-1+q) \log ^q\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx=\frac {d n \log \left (x\right ) + d \log \left (c\right )}{{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b + a x^{m}} \]

[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)

Sympy [A] (verification not implemented)

Time = 22.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {a d n x^m-a d m x^m \log \left (c x^n\right )-b d n (-1+q) \log ^q\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx=\frac {d \log {\left (c x^{n} \right )}}{a x^{m} + b \log {\left (c x^{n} \right )}^{q}} \]

[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]

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

Maxima [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {a d n x^m-a d m x^m \log \left (c x^n\right )-b d n (-1+q) \log ^q\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx=\frac {d \log \left (c\right ) + d \log \left (x^{n}\right )}{a x^{m} + b {\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{q}} \]

[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)

Giac [F]

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

[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)

Mupad [B] (verification not implemented)

Time = 1.52 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {a d n x^m-a d m x^m \log \left (c x^n\right )-b d n (-1+q) \log ^q\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx=\frac {d\,\ln \left (c\,x^n\right )}{a\,x^m+b\,{\ln \left (c\,x^n\right )}^q} \]

[In]

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