\(\int \frac {\log (c (a+\frac {b}{x})^p)}{d+e x} \, dx\) [201]

Optimal result

Integrand size = 20, antiderivative size = 113 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e}+\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{e}+\frac {p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2512, 266, 2463, 2441, 2352, 2440, 2438} \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=\frac {\log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{e}-\frac {p \log (d+e x) \log \left (-\frac {e (a x+b)}{a d-b e}\right )}{e}+\frac {p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e}+\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e} \]

[In]

Int[Log[c*(a + b/x)^p]/(d + e*x),x]

[Out]

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 2438

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

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 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 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2512

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[f +
g*x]*((a + b*Log[c*(d + e*x^n)^p])/g), x] - Dist[b*e*n*(p/g), Int[x^(n - 1)*(Log[f + g*x]/(d + e*x^n)), x], x]
/; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && RationalQ[n]

Rubi steps \begin{align*} \text {integral}& = \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e}+\frac {(b p) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x}\right ) x^2} \, dx}{e} \\ & = \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e}+\frac {(b p) \int \left (\frac {\log (d+e x)}{b x}-\frac {a \log (d+e x)}{b (b+a x)}\right ) \, dx}{e} \\ & = \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e}+\frac {p \int \frac {\log (d+e x)}{x} \, dx}{e}-\frac {(a p) \int \frac {\log (d+e x)}{b+a x} \, dx}{e} \\ & = \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e}+\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e}-p \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx+p \int \frac {\log \left (\frac {e (b+a x)}{-a d+b e}\right )}{d+e x} \, dx \\ & = \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e}+\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e}+\frac {p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e}+\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-a d+b e}\right )}{x} \, dx,x,d+e x\right )}{e} \\ & = \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e}+\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e}-\frac {p \text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )}{e}+\frac {p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.01 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e}+\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e}+\frac {p \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{e} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.58 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.21

method	result	size
parts	\(\frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right ) \ln \left (e x +d \right )}{e}+p b \left (\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{b e}-\frac {a \left (\frac {\operatorname {dilog}\left (\frac {-a d +a \left (e x +d \right )+b e}{-a d +b e}\right )}{a}+\frac {\ln \left (e x +d \right ) \ln \left (\frac {-a d +a \left (e x +d \right )+b e}{-a d +b e}\right )}{a}\right )}{b e}\right )\)	\(137\)

[In]

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

[Out]

ln(c*(a+b/x)^p)*ln(e*x+d)/e+p*b*(1/b/e*(dilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d))-a/b/e*(dilog((-a*d+a*(e*x+d)+b*e)/
(-a*d+b*e))/a+ln(e*x+d)*ln((-a*d+a*(e*x+d)+b*e)/(-a*d+b*e))/a))

Fricas [F]

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

[In]

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

[Out]

integral(log(c*((a*x + b)/x)^p)/(e*x + d), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.41 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=\frac {b p {\left (\frac {\log \left (e x + d\right ) \log \left (a + \frac {b}{x}\right )}{b} - \frac {\log \left (e x + d\right ) \log \left (-\frac {a e x + a d}{a d - b e} + 1\right ) + {\rm Li}_2\left (\frac {a e x + a d}{a d - b e}\right )}{b} + \frac {\log \left (e x + d\right ) \log \left (-\frac {e x + d}{d} + 1\right ) + {\rm Li}_2\left (\frac {e x + d}{d}\right )}{b}\right )}}{e} - \frac {p \log \left (e x + d\right ) \log \left (a + \frac {b}{x}\right )}{e} + \frac {\log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right ) \log \left (e x + d\right )}{e} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(log(c*(a + b/x)^p)/(d + e*x),x)

[Out]

int(log(c*(a + b/x)^p)/(d + e*x), x)