\(\int \frac {a (-1+m) x^{-1+m}+b n q \log ^{-1+q}(c x^n)}{a x^m+b x \log ^q(c x^n)} \, dx\) [29]

Optimal result

Integrand size = 45, antiderivative size = 19 \[ \int \frac {a (-1+m) x^{-1+m}+b n q \log ^{-1+q}\left (c x^n\right )}{a x^m+b x \log ^q\left (c x^n\right )} \, dx=\log \left (a x^{-1+m}+b \log ^q\left (c x^n\right )\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {2641, 2621} \[ \int \frac {a (-1+m) x^{-1+m}+b n q \log ^{-1+q}\left (c x^n\right )}{a x^m+b x \log ^q\left (c x^n\right )} \, dx=\log \left (a x^{m-1}+b \log ^q\left (c x^n\right )\right ) \]

[In]

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

[Out]

Log[a*x^(-1 + m) + b*Log[c*x^n]^q]

Rule 2621

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]

Rule 2641

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

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

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {a (-1+m) x^{-1+m}+b n q \log ^{-1+q}\left (c x^n\right )}{a x^m+b x \log ^q\left (c x^n\right )} \, dx=-\log (x)+\log \left (a x^m+b x \log ^q\left (c x^n\right )\right ) \]

[In]

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

[Out]

-Log[x] + Log[a*x^m + b*x*Log[c*x^n]^q]

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 24.97 (sec) , antiderivative size = 216, normalized size of antiderivative = 11.37

method	result	size
risch	\(q \ln \left (\ln \left (x^{n}\right )-\frac {i \left (\pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )-\pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-\pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+\pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 i \ln \left (c \right )\right )}{2}\right )-q \ln \left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i x^{n}\right )\right )}{2}\right )+\ln \left ({\left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i x^{n}\right )\right )}{2}\right )}^{q}+\frac {a \,x^{m}}{x b}\right )\)	\(216\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {a (-1+m) x^{-1+m}+b n q \log ^{-1+q}\left (c x^n\right )}{a x^m+b x \log ^q\left (c x^n\right )} \, dx=\log \left (\frac {{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b x + a x^{m}}{x}\right ) \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37 \[ \int \frac {a (-1+m) x^{-1+m}+b n q \log ^{-1+q}\left (c x^n\right )}{a x^m+b x \log ^q\left (c x^n\right )} \, dx=\log \left (\frac {b x {\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{q} + a x^{m}}{b x}\right ) \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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