\(\int \frac {a+b \log (c x^n)}{d+\frac {e}{x}} \, dx\) [331]

Optimal result

Integrand size = 20, antiderivative size = 69 \[ \int \frac {a+b \log \left (c x^n\right )}{d+\frac {e}{x}} \, dx=\frac {a x}{d}-\frac {b n x}{d}+\frac {b x \log \left (c x^n\right )}{d}-\frac {e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{d^2}-\frac {b e n \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {a+b \log \left (c x^n\right )}{d+\frac {e}{x}} \, dx=\frac {a d x-b d n x-a e \log \left (1+\frac {d x}{e}\right )+b \log \left (c x^n\right ) \left (d x-e \log \left (1+\frac {d x}{e}\right )\right )-b e n \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2767, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2767

\(\displaystyle \int \left (\frac {a+b \log \left (c x^n\right )}{d}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d (d x+e)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {e \log \left (\frac {d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}+\frac {a x}{d}+\frac {b x \log \left (c x^n\right )}{d}-\frac {b e n \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^2}-\frac {b n x}{d}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2767

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^( 
q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (d + e*x 
^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x] 
&& IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))
 

Maple [C] (warning: unable to verify)

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

Time = 0.23 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.72

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right ) x}{d}-\frac {b \ln \left (x^{n}\right ) e \ln \left (d x +e \right )}{d^{2}}-\frac {b n x}{d}-\frac {b n e}{d^{2}}+\frac {b n e \ln \left (d x +e \right ) \ln \left (-\frac {d x}{e}\right )}{d^{2}}+\frac {b n e \operatorname {dilog}\left (-\frac {d x}{e}\right )}{d^{2}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x}{d}-\frac {e \ln \left (d x +e \right )}{d^{2}}\right )\)	\(188\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [A] (verification not implemented)

Time = 42.52 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.36 \[ \int \frac {a+b \log \left (c x^n\right )}{d+\frac {e}{x}} \, dx=- \frac {a e \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d} + \frac {a x}{d} + \frac {b e n \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (e \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (e \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (e \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (e \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{d} - \frac {b e \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d} - \frac {b n x}{d} + \frac {b x \log {\left (c x^{n} \right )}}{d} \] Input:

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

Output:

-a*e*Piecewise((x/e, Eq(d, 0)), (log(d*x + e)/d, True))/d + a*x/d + b*e*n* 
Piecewise((x/e, Eq(d, 0)), (Piecewise((-polylog(2, d*x*exp_polar(I*pi)/e), 
 (Abs(x) < 1) & (1/Abs(x) < 1)), (log(e)*log(x) - polylog(2, d*x*exp_polar 
(I*pi)/e), Abs(x) < 1), (-log(e)*log(1/x) - polylog(2, d*x*exp_polar(I*pi) 
/e), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(e) + meij 
erg(((1, 1), ()), ((), (0, 0)), x)*log(e) - polylog(2, d*x*exp_polar(I*pi) 
/e), True))/d, True))/d - b*e*Piecewise((x/e, Eq(d, 0)), (log(d*x + e)/d, 
True))*log(c*x**n)/d - b*n*x/d + b*x*log(c*x**n)/d
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{d+\frac {e}{x}} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{d \,x^{2}+e x}d x \right ) b \,e^{2} n -2 \,\mathrm {log}\left (d x +e \right ) a e n -\mathrm {log}\left (x^{n} c \right )^{2} b e +2 \,\mathrm {log}\left (x^{n} c \right ) b d n x +2 a d n x -2 b d \,n^{2} x}{2 d^{2} n} \] Input:

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

Output:

(2*int(log(x**n*c)/(d*x**2 + e*x),x)*b*e**2*n - 2*log(d*x + e)*a*e*n - log 
(x**n*c)**2*b*e + 2*log(x**n*c)*b*d*n*x + 2*a*d*n*x - 2*b*d*n**2*x)/(2*d** 
2*n)