\(\int \frac {a+b \log (c x^n)}{x^2 (d+e x)} \, dx\) [36]

Optimal result

Integrand size = 21, antiderivative size = 74 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=-\frac {b n}{d x}-\frac {a+b \log \left (c x^n\right )}{d x}+\frac {e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac {b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=-\frac {\frac {2 b d n}{x}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{b n}-2 e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-2 b e n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{2 d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2780, 2741, 2779, 2838}

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.

\(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx\)

\(\Big \downarrow \) 2780

\(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x^2}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}\)

\(\Big \downarrow \) 2741

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

\(\Big \downarrow \) 2779

\(\displaystyle \frac {-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}}{d}-\frac {e \left (\frac {b n \int \frac {\log \left (\frac {d}{e x}+1\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2741

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> 
Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( 
m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
 

rule 2779

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

rule 2780

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.))/((d_) + (e_.)* 
(x_)^(r_.)), x_Symbol] :> Simp[1/d   Int[x^m*(a + b*Log[c*x^n])^p, x], x] - 
 Simp[e/d   Int[(x^(m + r)*(a + b*Log[c*x^n])^p)/(d + e*x^r), x], x] /; Fre 
eQ[{a, b, c, d, e, m, n, r}, x] && IGtQ[p, 0] && IGtQ[r, 0] && ILtQ[m, -1]
 

rule 2838

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]
 

Maple [C] (warning: unable to verify)

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

Time = 0.47 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.99

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right ) e \ln \left (e x +d \right )}{d^{2}}-\frac {b \ln \left (x^{n}\right )}{d x}-\frac {b \ln \left (x^{n}\right ) e \ln \left (x \right )}{d^{2}}-\frac {b n e \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{2}}-\frac {b n e \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{2}}-\frac {b n}{d x}+\frac {b n e \ln \left (x \right )^{2}}{2 d^{2}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\frac {e \ln \left (e x +d \right )}{d^{2}}-\frac {1}{d x}-\frac {e \ln \left (x \right )}{d^{2}}\right )\)	\(221\)

Input:

int((a+b*ln(c*x^n))/x^2/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

b*ln(x^n)*e/d^2*ln(e*x+d)-b*ln(x^n)/d/x-b*ln(x^n)*e/d^2*ln(x)-b*n*e/d^2*ln 
(e*x+d)*ln(-e*x/d)-b*n*e/d^2*dilog(-e*x/d)-b*n/d/x+1/2*b*n*e/d^2*ln(x)^2+( 
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)*(e/d^2*ln(e*x+d)-1/d/x-e/d^2*ln(x))
 

Fricas [F]

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

integrate((a+b*log(c*x^n))/x^2/(e*x+d),x, algorithm="fricas")
 

Output:

integral((b*log(c*x^n) + a)/(e*x^3 + d*x^2), x)
 

Sympy [A] (verification not implemented)

Time = 40.15 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.92 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=- \frac {a}{d x} + \frac {a e^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{d^{2}} - \frac {a e \log {\left (x \right )}}{d^{2}} - \frac {b n}{d x} - \frac {b \log {\left (c x^{n} \right )}}{d x} - \frac {b e^{2} n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{d^{2}} + \frac {b e^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{2}} + \frac {b e n \log {\left (x \right )}^{2}}{2 d^{2}} - \frac {b e \log {\left (x \right )} \log {\left (c x^{n} \right )}}{d^{2}} \] Input:

integrate((a+b*ln(c*x**n))/x**2/(e*x+d),x)
 

Output:

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

Maxima [F]

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

integrate((a+b*log(c*x^n))/x^2/(e*x+d),x, algorithm="maxima")
 

Output:

a*(e*log(e*x + d)/d^2 - e*log(x)/d^2 - 1/(d*x)) + b*integrate((log(c) + lo 
g(x^n))/(e*x^3 + d*x^2), x)
 

Giac [F]

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

integrate((a+b*log(c*x^n))/x^2/(e*x+d),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=\frac {\left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e \,x^{3}+d \,x^{2}}d x \right ) b \,d^{2} x +\mathrm {log}\left (e x +d \right ) a e x -\mathrm {log}\left (x \right ) a e x -a d}{d^{2} x} \] Input:

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

Output:

(int(log(x**n*c)/(d*x**2 + e*x**3),x)*b*d**2*x + log(d + e*x)*a*e*x - log( 
x)*a*e*x - a*d)/(d**2*x)