\(\int \frac {(d x)^m}{a+b \log (c x^n)} \, dx\) [153]

Optimal result

Integrand size = 18, antiderivative size = 66 \[ \int \frac {(d x)^m}{a+b \log \left (c x^n\right )} \, dx=\frac {e^{-\frac {a (1+m)}{b n}} (d x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \operatorname {ExpIntegralEi}\left (\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b d n} \]

[Out]

(d*x)^(1+m)*Ei((1+m)*(a+b*ln(c*x^n))/b/n)/b/d/exp(a*(1+m)/b/n)/n/((c*x^n)^((1+m)/n))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2347, 2209} \[ \int \frac {(d x)^m}{a+b \log \left (c x^n\right )} \, dx=\frac {(d x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b d n} \]

[In]

Int[(d*x)^m/(a + b*Log[c*x^n]),x]

[Out]

((d*x)^(1 + m)*ExpIntegralEi[((1 + m)*(a + b*Log[c*x^n]))/(b*n)])/(b*d*E^((a*(1 + m))/(b*n))*n*(c*x^n)^((1 + m
)/n))

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2347

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.02 \[ \int \frac {(d x)^m}{a+b \log \left (c x^n\right )} \, dx=\frac {e^{-\frac {(1+m) \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{b n}} x^{-m} (d x)^m \operatorname {ExpIntegralEi}\left (\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]

[In]

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

[Out]

((d*x)^m*ExpIntegralEi[((1 + m)*(a + b*Log[c*x^n]))/(b*n)])/(b*E^(((1 + m)*(a + b*(-(n*Log[x]) + Log[c*x^n])))
/(b*n))*n*x^m)

Maple [F]

\[\int \frac {\left (d x \right )^{m}}{a +b \ln \left (c \,x^{n}\right )}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03 \[ \int \frac {(d x)^m}{a+b \log \left (c x^n\right )} \, dx=\frac {{\rm Ei}\left (\frac {{\left (b m + b\right )} n \log \left (x\right ) + a m + {\left (b m + b\right )} \log \left (c\right ) + a}{b n}\right ) e^{\left (\frac {b m n \log \left (d\right ) - a m - {\left (b m + b\right )} \log \left (c\right ) - a}{b n}\right )}}{b n} \]

[In]

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

[Out]

Ei(((b*m + b)*n*log(x) + a*m + (b*m + b)*log(c) + a)/(b*n))*e^((b*m*n*log(d) - a*m - (b*m + b)*log(c) - a)/(b*
n))/(b*n)

Sympy [F]

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

[In]

integrate((d*x)**m/(a+b*ln(c*x**n)),x)

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((d*x)^m/(a + b*log(c*x^n)),x)

[Out]

int((d*x)^m/(a + b*log(c*x^n)), x)