\(\int \frac {a+b \log (c (d+e x^m)^n)}{x} \, dx\) [624]

Optimal result

Integrand size = 20, antiderivative size = 49 \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x} \, dx=\frac {\log \left (-\frac {e x^m}{d}\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{m}+\frac {b n \operatorname {PolyLog}\left (2,1+\frac {e x^m}{d}\right )}{m} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2504, 2441, 2352} \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x} \, dx=\frac {\log \left (-\frac {e x^m}{d}\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{m}+\frac {b n \operatorname {PolyLog}\left (2,\frac {e x^m}{d}+1\right )}{m} \]

[In]

Int[(a + b*Log[c*(d + e*x^m)^n])/x,x]

[Out]

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx,x,x^m\right )}{m} \\ & = \frac {\log \left (-\frac {e x^m}{d}\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{m}-\frac {(b e n) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^m\right )}{m} \\ & = \frac {\log \left (-\frac {e x^m}{d}\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{m}+\frac {b n \text {Li}_2\left (1+\frac {e x^m}{d}\right )}{m} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x} \, dx=a \log (x)+\frac {b \left (\log \left (-\frac {e x^m}{d}\right ) \log \left (c \left (d+e x^m\right )^n\right )+n \operatorname {PolyLog}\left (2,\frac {d+e x^m}{d}\right )\right )}{m} \]

[In]

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

[Out]

a*Log[x] + (b*(Log[-((e*x^m)/d)]*Log[c*(d + e*x^m)^n] + n*PolyLog[2, (d + e*x^m)/d]))/m

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.14 (sec) , antiderivative size = 180, normalized size of antiderivative = 3.67

method	result	size
risch	\(b \ln \left (x \right ) \ln \left (\left (d +e \,x^{m}\right )^{n}\right )+\left (\frac {i b \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{m}\right )^{n}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{m}\right )^{n}\right )}^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{m}\right )^{n}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{m}\right )^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{m}\right )^{n}\right )}^{3}}{2}+\frac {i b \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{m}\right )^{n}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \ln \left (x \right )-\frac {b n \operatorname {dilog}\left (\frac {d +e \,x^{m}}{d}\right )}{m}-b n \ln \left (x \right ) \ln \left (\frac {d +e \,x^{m}}{d}\right )\)	\(180\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.41 \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x} \, dx=\frac {b m n \log \left (e x^{m} + d\right ) \log \left (x\right ) - b m n \log \left (x\right ) \log \left (\frac {e x^{m} + d}{d}\right ) - b n {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) + {\left (b m \log \left (c\right ) + a m\right )} \log \left (x\right )}{m} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*log(c*(d + e*x^m)^n))/x,x)

[Out]

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