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

Optimal. Leaf size=19 \[ \log \left (a x^{-1+m}+b \log ^q\left (c x^n\right )\right ) \]

[Out]

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

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Rubi [A]
time = 0.23, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {2641, 2621} \begin {gather*} \log \left (a x^{m-1}+b \log ^q\left (c x^n\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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*} \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 (-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*}

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Mathematica [A]
time = 0.21, size = 23, normalized size = 1.21 \begin {gather*} -\log (x)+\log \left (a x^m+b x \log ^q\left (c x^n\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.13, size = 215, normalized size = 11.32

method	result	size
risch	\(q \ln \left (-\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\ln \left (c \right )+\ln \left (x^{n}\right )\right )-q \ln \left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i x^{n}\right )\right )}{2}\right )+\ln \left (\left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i x^{n}\right )\right )}{2}\right )^{q}+\frac {a \,x^{m}}{x b}\right )\)	\(215\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.41, size = 26, normalized size = 1.37 \begin {gather*} \log \left (\frac {b x {\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{q} + a x^{m}}{b x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Fricas [A]
time = 0.38, size = 23, normalized size = 1.21 \begin {gather*} \log \left (\frac {{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b x + a x^{m}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Exception raised: SystemError >> excessive stack use: stack is 3435 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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