\(\int \frac {a+b \log (c x^n)}{x (d+e x^2)^3} \, dx\) [234]

Optimal result

Integrand size = 23, antiderivative size = 115 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^3} \, dx=\frac {a+b \log \left (c x^n\right )}{4 d \left (d+e x^2\right )^2}-\frac {\log \left (1+\frac {d}{e x^2}\right ) \left (4 a-3 b n+4 b \log \left (c x^n\right )\right )}{8 d^3}+\frac {4 a-b n+4 b \log \left (c x^n\right )}{8 d^2 \left (d+e x^2\right )}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x^2}\right )}{4 d^3} \] Output:

1/4*(a+b*ln(c*x^n))/d/(e*x^2+d)^2-1/8*ln(1+d/e/x^2)*(4*a-3*b*n+4*b*ln(c*x^ 
n))/d^3+1/8*(4*a-b*n+4*b*ln(c*x^n))/d^2/(e*x^2+d)+1/4*b*n*polylog(2,-d/e/x 
^2)/d^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.12 (sec) , antiderivative size = 396, normalized size of antiderivative = 3.44 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^3} \, dx=\frac {\frac {4 d^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2}+\frac {8 d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{d+e x^2}+16 \log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )-8 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+e x^2\right )-b n \left (\frac {d}{d-i \sqrt {d} \sqrt {e} x}+\frac {d}{d+i \sqrt {d} \sqrt {e} x}+2 \log (x)-\frac {d \log (x)}{\left (\sqrt {d}-i \sqrt {e} x\right )^2}-\frac {d \log (x)}{\left (\sqrt {d}+i \sqrt {e} x\right )^2}+\frac {5 \sqrt {e} x \log (x)}{-i \sqrt {d}+\sqrt {e} x}+\frac {5 \sqrt {e} x \log (x)}{i \sqrt {d}+\sqrt {e} x}-8 \log ^2(x)-6 \log \left (i \sqrt {d}-\sqrt {e} x\right )-6 \log \left (i \sqrt {d}+\sqrt {e} x\right )+8 \log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+8 \log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+8 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+8 \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{16 d^3} \] Input:

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

Output:

((4*d^2*(a - b*n*Log[x] + b*Log[c*x^n]))/(d + e*x^2)^2 + (8*d*(a - b*n*Log 
[x] + b*Log[c*x^n]))/(d + e*x^2) + 16*Log[x]*(a - b*n*Log[x] + b*Log[c*x^n 
]) - 8*(a - b*n*Log[x] + b*Log[c*x^n])*Log[d + e*x^2] - b*n*(d/(d - I*Sqrt 
[d]*Sqrt[e]*x) + d/(d + I*Sqrt[d]*Sqrt[e]*x) + 2*Log[x] - (d*Log[x])/(Sqrt 
[d] - I*Sqrt[e]*x)^2 - (d*Log[x])/(Sqrt[d] + I*Sqrt[e]*x)^2 + (5*Sqrt[e]*x 
*Log[x])/((-I)*Sqrt[d] + Sqrt[e]*x) + (5*Sqrt[e]*x*Log[x])/(I*Sqrt[d] + Sq 
rt[e]*x) - 8*Log[x]^2 - 6*Log[I*Sqrt[d] - Sqrt[e]*x] - 6*Log[I*Sqrt[d] + S 
qrt[e]*x] + 8*Log[x]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]] + 8*Log[x]*Log[1 + (I* 
Sqrt[e]*x)/Sqrt[d]] + 8*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]] + 8*PolyLog[2 
, (I*Sqrt[e]*x)/Sqrt[d]]))/(16*d^3)
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2785, 25, 2785, 27, 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.

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

\(\Big \downarrow \) 2785

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2785

\(\displaystyle \frac {\frac {4 a+4 b \log \left (c x^n\right )-b n}{2 d \left (d+e x^2\right )}-\frac {\int -\frac {2 \left (4 a-3 b n+4 b \log \left (c x^n\right )\right )}{x \left (e x^2+d\right )}dx}{2 d}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d \left (d+e x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {4 a-3 b n+4 b \log \left (c x^n\right )}{x \left (e x^2+d\right )}dx}{d}+\frac {4 a+4 b \log \left (c x^n\right )-b n}{2 d \left (d+e x^2\right )}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d \left (d+e x^2\right )^2}\)

\(\Big \downarrow \) 2779

\(\displaystyle \frac {\frac {\frac {2 b n \int \frac {\log \left (\frac {d}{e x^2}+1\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x^2}+1\right ) \left (4 a+4 b \log \left (c x^n\right )-3 b n\right )}{2 d}}{d}+\frac {4 a+4 b \log \left (c x^n\right )-b n}{2 d \left (d+e x^2\right )}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d \left (d+e x^2\right )^2}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x^2}\right )}{d}-\frac {\log \left (\frac {d}{e x^2}+1\right ) \left (4 a+4 b \log \left (c x^n\right )-3 b n\right )}{2 d}}{d}+\frac {4 a+4 b \log \left (c x^n\right )-b n}{2 d \left (d+e x^2\right )}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d \left (d+e x^2\right )^2}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 2785

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_)^2)^(q_.), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(q + 1)*((a + 
 b*Log[c*x^n])/(2*d*f*(q + 1))), x] + Simp[1/(2*d*(q + 1))   Int[(f*x)^m*(d 
 + e*x^2)^(q + 1)*(a*(m + 2*q + 3) + b*n + b*(m + 2*q + 3)*Log[c*x^n]), x], 
 x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && ILtQ[m, 0]
 

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 = 1.74 (sec) , antiderivative size = 390, normalized size of antiderivative = 3.39

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right )}{2 d^{2} \left (e \,x^{2}+d \right )}-\frac {b \ln \left (x^{n}\right ) \ln \left (e \,x^{2}+d \right )}{2 d^{3}}+\frac {b \ln \left (x^{n}\right )}{4 d \left (e \,x^{2}+d \right )^{2}}+\frac {b \ln \left (x^{n}\right ) \ln \left (x \right )}{d^{3}}-\frac {b n \ln \left (x \right )^{2}}{2 d^{3}}+\frac {b n \ln \left (x \right ) \ln \left (e \,x^{2}+d \right )}{2 d^{3}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{3}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{3}}-\frac {b n \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{3}}-\frac {b n \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{3}}-\frac {b n}{8 d^{2} \left (e \,x^{2}+d \right )}+\frac {3 b n \ln \left (e \,x^{2}+d \right )}{8 d^{3}}-\frac {3 b n \ln \left (x \right )}{4 d^{3}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {e \left (-\frac {d}{e \left (e \,x^{2}+d \right )}+\frac {\ln \left (e \,x^{2}+d \right )}{e}-\frac {d^{2}}{2 e \left (e \,x^{2}+d \right )^{2}}\right )}{2 d^{3}}+\frac {\ln \left (x \right )}{d^{3}}\right )\)	\(390\)

Input:

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

Output:

1/2*b*ln(x^n)/d^2/(e*x^2+d)-1/2*b*ln(x^n)/d^3*ln(e*x^2+d)+1/4*b*ln(x^n)/d/ 
(e*x^2+d)^2+b*ln(x^n)/d^3*ln(x)-1/2*b*n/d^3*ln(x)^2+1/2*b*n/d^3*ln(x)*ln(e 
*x^2+d)-1/2*b*n/d^3*ln(x)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/2*b*n/d^3 
*ln(x)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/2*b*n/d^3*dilog((-e*x+(-d*e)^ 
(1/2))/(-d*e)^(1/2))-1/2*b*n/d^3*dilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/ 
8*b*n/d^2/(e*x^2+d)+3/8*b*n/d^3*ln(e*x^2+d)-3/4*b*n*ln(x)/d^3+(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 
)*(-1/2*e/d^3*(-d/e/(e*x^2+d)+ln(e*x^2+d)/e-1/2*d^2/e/(e*x^2+d)^2)+1/d^3*l 
n(x))
 

Fricas [F]

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

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

Output:

integral((b*log(c*x^n) + a)/(e^3*x^7 + 3*d*e^2*x^5 + 3*d^2*e*x^3 + d^3*x), 
 x)
 

Sympy [A] (verification not implemented)

Time = 138.29 (sec) , antiderivative size = 403, normalized size of antiderivative = 3.50 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-a*e*Piecewise((x**2/(2*d**3), Eq(e, 0)), (-1/(4*e*(d + e*x**2)**2), True) 
)/d - a*e*Piecewise((x**2/(2*d**2), Eq(e, 0)), (-1/(2*d*e + 2*e**2*x**2), 
True))/d**2 + a*log(x)/d**3 - a*log(d + e*x**2)/(2*d**3) + b*e**2*n*Piecew 
ise((-1/(2*e**3*x**2), Eq(d, 0)), (-1/(4*d*e**2 + 4*e**3*x**2) - log(d + e 
*x**2)/(4*d*e**2), True))/(2*d**2) - b*e**2*Piecewise((1/(e**3*x**2), Eq(d 
, 0)), (-1/(2*d*(d/x**2 + e)**2), True))*log(c*x**n)/(2*d**2) - b*e*n*Piec 
ewise((-1/(2*e**2*x**2), Eq(d, 0)), (-log(d + e*x**2)/(2*d*e), True))/d**2 
 + b*e*Piecewise((1/(e**2*x**2), Eq(d, 0)), (-1/(d**2/x**2 + d*e), True))* 
log(c*x**n)/d**2 + b*n*Piecewise((-1/(2*e*x**2), Eq(d, 0)), (Piecewise((po 
lylog(2, d*exp_polar(I*pi)/(e*x**2))/2, (Abs(x) < 1) & (1/Abs(x) < 1)), (l 
og(e)*log(x) + polylog(2, d*exp_polar(I*pi)/(e*x**2))/2, Abs(x) < 1), (-lo 
g(e)*log(1/x) + polylog(2, d*exp_polar(I*pi)/(e*x**2))/2, 1/Abs(x) < 1), ( 
-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(e) + meijerg(((1, 1), ()), (() 
, (0, 0)), x)*log(e) + polylog(2, d*exp_polar(I*pi)/(e*x**2))/2, True))/d, 
 True))/(2*d**2) - b*Piecewise((1/(e*x**2), Eq(d, 0)), (log(d/x**2 + e)/d, 
 True))*log(c*x**n)/(2*d**2)
 

Maxima [F]

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

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

Output:

1/4*a*((2*e*x^2 + 3*d)/(d^2*e^2*x^4 + 2*d^3*e*x^2 + d^4) - 2*log(e*x^2 + d 
)/d^3 + 4*log(x)/d^3) + b*integrate((log(c) + log(x^n))/(e^3*x^7 + 3*d*e^2 
*x^5 + 3*d^2*e*x^3 + d^3*x), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^3} \, dx=\frac {4 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{7}+3 d \,e^{2} x^{5}+3 d^{2} e \,x^{3}+d^{3} x}d x \right ) b \,d^{5}+8 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{7}+3 d \,e^{2} x^{5}+3 d^{2} e \,x^{3}+d^{3} x}d x \right ) b \,d^{4} e \,x^{2}+4 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{7}+3 d \,e^{2} x^{5}+3 d^{2} e \,x^{3}+d^{3} x}d x \right ) b \,d^{3} e^{2} x^{4}-2 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,d^{2}-4 \,\mathrm {log}\left (e \,x^{2}+d \right ) a d e \,x^{2}-2 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,e^{2} x^{4}+4 \,\mathrm {log}\left (x \right ) a \,d^{2}+8 \,\mathrm {log}\left (x \right ) a d e \,x^{2}+4 \,\mathrm {log}\left (x \right ) a \,e^{2} x^{4}+2 a \,d^{2}-a \,e^{2} x^{4}}{4 d^{3} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

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

Output:

(4*int(log(x**n*c)/(d**3*x + 3*d**2*e*x**3 + 3*d*e**2*x**5 + e**3*x**7),x) 
*b*d**5 + 8*int(log(x**n*c)/(d**3*x + 3*d**2*e*x**3 + 3*d*e**2*x**5 + e**3 
*x**7),x)*b*d**4*e*x**2 + 4*int(log(x**n*c)/(d**3*x + 3*d**2*e*x**3 + 3*d* 
e**2*x**5 + e**3*x**7),x)*b*d**3*e**2*x**4 - 2*log(d + e*x**2)*a*d**2 - 4* 
log(d + e*x**2)*a*d*e*x**2 - 2*log(d + e*x**2)*a*e**2*x**4 + 4*log(x)*a*d* 
*2 + 8*log(x)*a*d*e*x**2 + 4*log(x)*a*e**2*x**4 + 2*a*d**2 - a*e**2*x**4)/ 
(4*d**3*(d**2 + 2*d*e*x**2 + e**2*x**4))