\(\int \frac {a+b \log (c x^n)}{x^3 (d+e x^2)} \, dx\) [214]

Optimal result

Integrand size = 23, antiderivative size = 83 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )} \, dx=-\frac {b n}{4 d x^2}-\frac {a+b \log \left (c x^n\right )}{2 d x^2}+\frac {e \log \left (1+\frac {d}{e x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2}-\frac {b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x^2}\right )}{4 d^2} \] Output:

-1/4*b*n/d/x^2-1/2*(a+b*ln(c*x^n))/d/x^2+1/2*e*ln(1+d/e/x^2)*(a+b*ln(c*x^n 
))/d^2-1/4*b*e*n*polylog(2,-d/e/x^2)/d^2
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.89 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )} \, dx=\frac {-\frac {b d n}{x^2}-\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{b n}+2 e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )+2 e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )+2 b e n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )+2 b e n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{4 d^2} \] Input:

Integrate[(a + b*Log[c*x^n])/(x^3*(d + e*x^2)),x]
 

Output:

(-((b*d*n)/x^2) - (2*d*(a + b*Log[c*x^n]))/x^2 - (2*e*(a + b*Log[c*x^n])^2 
)/(b*n) + 2*e*(a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]] + 2*e*(a + 
b*Log[c*x^n])*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)] + 2*b*e*n*PolyLog[2, (Sqrt 
[e]*x)/Sqrt[-d]] + 2*b*e*n*PolyLog[2, (d*Sqrt[e]*x)/(-d)^(3/2)])/(4*d^2)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, 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.

\(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )} \, dx\)

\(\Big \downarrow \) 2780

\(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x^3}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{x \left (e x^2+d\right )}dx}{d}\)

\(\Big \downarrow \) 2741

\(\displaystyle \frac {-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{x \left (e x^2+d\right )}dx}{d}\)

\(\Big \downarrow \) 2779

\(\displaystyle \frac {-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}}{d}-\frac {e \left (\frac {b n \int \frac {\log \left (\frac {d}{e x^2}+1\right )}{x}dx}{2 d}-\frac {\log \left (\frac {d}{e x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d}\right )}{d}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}}{d}-\frac {e \left (\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x^2}\right )}{4 d}-\frac {\log \left (\frac {d}{e x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d}\right )}{d}\)

Input:

Int[(a + b*Log[c*x^n])/(x^3*(d + e*x^2)),x]
 

Output:

(-1/4*(b*n)/x^2 - (a + b*Log[c*x^n])/(2*x^2))/d - (e*(-1/2*(Log[1 + d/(e*x 
^2)]*(a + b*Log[c*x^n]))/d + (b*n*PolyLog[2, -(d/(e*x^2))])/(4*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 = 0.65 (sec) , antiderivative size = 317, normalized size of antiderivative = 3.82

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right ) e \ln \left (e \,x^{2}+d \right )}{2 d^{2}}-\frac {b \ln \left (x^{n}\right )}{2 d \,x^{2}}-\frac {b \ln \left (x^{n}\right ) e \ln \left (x \right )}{d^{2}}-\frac {b n e \ln \left (x \right ) \ln \left (e \,x^{2}+d \right )}{2 d^{2}}+\frac {b n e \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}+\frac {b n e \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}+\frac {b n e \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}+\frac {b n e \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}-\frac {b n}{4 d \,x^{2}}+\frac {b n e \ln \left (x \right )^{2}}{2 d^{2}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\frac {e \ln \left (e \,x^{2}+d \right )}{2 d^{2}}-\frac {1}{2 d \,x^{2}}-\frac {e \ln \left (x \right )}{d^{2}}\right )\)	\(317\)

Input:

int((a+b*ln(c*x^n))/x^3/(e*x^2+d),x,method=_RETURNVERBOSE)
 

Output:

1/2*b*ln(x^n)*e/d^2*ln(e*x^2+d)-1/2*b*ln(x^n)/d/x^2-b*ln(x^n)*e/d^2*ln(x)- 
1/2*b*n*e/d^2*ln(x)*ln(e*x^2+d)+1/2*b*n*e/d^2*ln(x)*ln((-e*x+(-d*e)^(1/2)) 
/(-d*e)^(1/2))+1/2*b*n*e/d^2*ln(x)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/2 
*b*n*e/d^2*dilog((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/2*b*n*e/d^2*dilog((e* 
x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/4*b*n/d/x^2+1/2*b*n*e/d^2*ln(x)^2+(1/2*I*P 
i*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)*(1/2*e/d^2*ln(e*x^2+d)-1/2/d/x^2-e/d^2*ln(x))
 

Fricas [F]

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

integrate((a+b*log(c*x^n))/x^3/(e*x^2+d),x, algorithm="fricas")
 

Output:

integral((b*log(c*x^n) + a)/(e*x^5 + d*x^3), x)
 

Sympy [F(-1)]

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

integrate((a+b*ln(c*x**n))/x**3/(e*x**2+d),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((a+b*log(c*x^n))/x^3/(e*x^2+d),x, algorithm="maxima")
 

Output:

1/2*a*(e*log(e*x^2 + d)/d^2 - 2*e*log(x)/d^2 - 1/(d*x^2)) + b*integrate((l 
og(c) + log(x^n))/(e*x^5 + d*x^3), x)
 

Giac [F]

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

integrate((a+b*log(c*x^n))/x^3/(e*x^2+d),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e \,x^{5}+d \,x^{3}}d x \right ) b \,d^{2} x^{2}+\mathrm {log}\left (e \,x^{2}+d \right ) a e \,x^{2}-2 \,\mathrm {log}\left (x \right ) a e \,x^{2}-a d}{2 d^{2} x^{2}} \] Input:

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

Output:

(2*int(log(x**n*c)/(d*x**3 + e*x**5),x)*b*d**2*x**2 + log(d + e*x**2)*a*e* 
x**2 - 2*log(x)*a*e*x**2 - a*d)/(2*d**2*x**2)