\(\int \frac {a+b \log (c x^n)}{x^4 (d+e x^2)^3} \, dx\) [240]

Optimal result

Integrand size = 23, antiderivative size = 259 \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx=-\frac {35 b n}{72 d^3 x^3}+\frac {35 b e n}{8 d^4 x}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}+\frac {e \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^4 x}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^{9/2}}-\frac {35 b e^{3/2} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{16 (-d)^{9/2}}+\frac {35 b e^{3/2} n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{16 (-d)^{9/2}} \] Output:

-35/72*b*n/d^3/x^3+35/8*b*e*n/d^4/x+1/4*(a+b*ln(c*x^n))/d/x^3/(e*x^2+d)^2+ 
1/8*(7*a-b*n+7*b*ln(c*x^n))/d^2/x^3/(e*x^2+d)-1/24*(35*a-12*b*n+35*b*ln(c* 
x^n))/d^3/x^3+1/8*e*(35*a-12*b*n+35*b*ln(c*x^n))/d^4/x+1/8*e^(3/2)*arctan( 
e^(1/2)*x/d^(1/2))*(35*a-12*b*n+35*b*ln(c*x^n))/d^(9/2)-35/16*b*e^(3/2)*n* 
polylog(2,-e^(1/2)*x/(-d)^(1/2))/(-d)^(9/2)+35/16*b*e^(3/2)*n*polylog(2,e^ 
(1/2)*x/(-d)^(1/2))/(-d)^(9/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(584\) vs. \(2(259)=518\).

Time = 1.95 (sec) , antiderivative size = 584, normalized size of antiderivative = 2.25 \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx=\frac {1}{144} \left (-\frac {16 b n}{d^3 x^3}+\frac {432 b e n}{d^4 x}-\frac {48 \left (a+b \log \left (c x^n\right )\right )}{d^3 x^3}+\frac {432 e \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac {9 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{(-d)^{7/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {99 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{d^4 \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {9 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{(-d)^{7/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {99 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{d^4 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {99 b e^{3/2} n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{9/2}}-\frac {99 b e^{3/2} n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{(-d)^{9/2}}-\frac {9 b e^{3/2} n \left (\frac {1}{\sqrt {-d} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {\log (x)}{d}+\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right )}{d}\right )}{(-d)^{7/2}}-\frac {315 e^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{9/2}}+\frac {9 b e^{3/2} n \left (\frac {1}{\sqrt {-d} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\log (x)}{d}+\frac {\log \left ((-d)^{3/2}+d \sqrt {e} x\right )}{d}\right )}{(-d)^{7/2}}+\frac {315 e^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{9/2}}+\frac {315 b e^{3/2} n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{9/2}}-\frac {315 b e^{3/2} n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{9/2}}\right ) \] Input:

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

Output:

((-16*b*n)/(d^3*x^3) + (432*b*e*n)/(d^4*x) - (48*(a + b*Log[c*x^n]))/(d^3* 
x^3) + (432*e*(a + b*Log[c*x^n]))/(d^4*x) - (9*e^(3/2)*(a + b*Log[c*x^n])) 
/((-d)^(7/2)*(Sqrt[-d] - Sqrt[e]*x)^2) - (99*e^(3/2)*(a + b*Log[c*x^n]))/( 
d^4*(Sqrt[-d] - Sqrt[e]*x)) + (9*e^(3/2)*(a + b*Log[c*x^n]))/((-d)^(7/2)*( 
Sqrt[-d] + Sqrt[e]*x)^2) + (99*e^(3/2)*(a + b*Log[c*x^n]))/(d^4*(Sqrt[-d] 
+ Sqrt[e]*x)) + (99*b*e^(3/2)*n*(Log[x] - Log[Sqrt[-d] - Sqrt[e]*x]))/(-d) 
^(9/2) - (99*b*e^(3/2)*n*(Log[x] - Log[Sqrt[-d] + Sqrt[e]*x]))/(-d)^(9/2) 
- (9*b*e^(3/2)*n*(1/(Sqrt[-d]*(Sqrt[-d] + Sqrt[e]*x)) - Log[x]/d + Log[Sqr 
t[-d] + Sqrt[e]*x]/d))/(-d)^(7/2) - (315*e^(3/2)*(a + b*Log[c*x^n])*Log[1 
+ (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(9/2) + (9*b*e^(3/2)*n*(1/(Sqrt[-d]*(Sqrt[-d 
] - Sqrt[e]*x)) - Log[x]/d + Log[(-d)^(3/2) + d*Sqrt[e]*x]/d))/(-d)^(7/2) 
+ (315*e^(3/2)*(a + b*Log[c*x^n])*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^ 
(9/2) + (315*b*e^(3/2)*n*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(9/2) - (3 
15*b*e^(3/2)*n*PolyLog[2, (d*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^(9/2))/144
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.21 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2785, 25, 2785, 25, 2780, 2741, 2780, 2741, 2761, 27, 5355, 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.

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

\(\Big \downarrow \) 2785

\(\displaystyle \frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}-\frac {\int -\frac {7 a-b n+7 b \log \left (c x^n\right )}{x^4 \left (e x^2+d\right )^2}dx}{4 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {7 a-b n+7 b \log \left (c x^n\right )}{x^4 \left (e x^2+d\right )^2}dx}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}\)

\(\Big \downarrow \) 2785

\(\displaystyle \frac {\frac {7 a+7 b \log \left (c x^n\right )-b n}{2 d x^3 \left (d+e x^2\right )}-\frac {\int -\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{x^4 \left (e x^2+d\right )}dx}{2 d}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {35 a-12 b n+35 b \log \left (c x^n\right )}{x^4 \left (e x^2+d\right )}dx}{2 d}+\frac {7 a+7 b \log \left (c x^n\right )-b n}{2 d x^3 \left (d+e x^2\right )}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}\)

\(\Big \downarrow \) 2780

\(\displaystyle \frac {\frac {\frac {\int \frac {35 a-12 b n+35 b \log \left (c x^n\right )}{x^4}dx}{d}-\frac {e \int \frac {35 a-12 b n+35 b \log \left (c x^n\right )}{x^2 \left (e x^2+d\right )}dx}{d}}{2 d}+\frac {7 a+7 b \log \left (c x^n\right )-b n}{2 d x^3 \left (d+e x^2\right )}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}\)

\(\Big \downarrow \) 2741

\(\displaystyle \frac {\frac {\frac {-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{3 x^3}-\frac {35 b n}{9 x^3}}{d}-\frac {e \int \frac {35 a-12 b n+35 b \log \left (c x^n\right )}{x^2 \left (e x^2+d\right )}dx}{d}}{2 d}+\frac {7 a+7 b \log \left (c x^n\right )-b n}{2 d x^3 \left (d+e x^2\right )}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}\)

\(\Big \downarrow \) 2780

\(\displaystyle \frac {\frac {\frac {-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{3 x^3}-\frac {35 b n}{9 x^3}}{d}-\frac {e \left (\frac {\int \frac {35 a-12 b n+35 b \log \left (c x^n\right )}{x^2}dx}{d}-\frac {e \int \frac {35 a-12 b n+35 b \log \left (c x^n\right )}{e x^2+d}dx}{d}\right )}{d}}{2 d}+\frac {7 a+7 b \log \left (c x^n\right )-b n}{2 d x^3 \left (d+e x^2\right )}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}\)

\(\Big \downarrow \) 2741

\(\displaystyle \frac {\frac {\frac {-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{3 x^3}-\frac {35 b n}{9 x^3}}{d}-\frac {e \left (\frac {-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{x}-\frac {35 b n}{x}}{d}-\frac {e \int \frac {35 a-12 b n+35 b \log \left (c x^n\right )}{e x^2+d}dx}{d}\right )}{d}}{2 d}+\frac {7 a+7 b \log \left (c x^n\right )-b n}{2 d x^3 \left (d+e x^2\right )}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}\)

\(\Big \downarrow \) 2761

\(\displaystyle \frac {\frac {\frac {-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{3 x^3}-\frac {35 b n}{9 x^3}}{d}-\frac {e \left (\frac {-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{x}-\frac {35 b n}{x}}{d}-\frac {e \left (\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a+35 b \log \left (c x^n\right )-12 b n\right )}{\sqrt {d} \sqrt {e}}-35 b n \int \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x}dx\right )}{d}\right )}{d}}{2 d}+\frac {7 a+7 b \log \left (c x^n\right )-b n}{2 d x^3 \left (d+e x^2\right )}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{3 x^3}-\frac {35 b n}{9 x^3}}{d}-\frac {e \left (\frac {-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{x}-\frac {35 b n}{x}}{d}-\frac {e \left (\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a+35 b \log \left (c x^n\right )-12 b n\right )}{\sqrt {d} \sqrt {e}}-\frac {35 b n \int \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x}dx}{\sqrt {d} \sqrt {e}}\right )}{d}\right )}{d}}{2 d}+\frac {7 a+7 b \log \left (c x^n\right )-b n}{2 d x^3 \left (d+e x^2\right )}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}\)

\(\Big \downarrow \) 5355

\(\displaystyle \frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {\frac {7 a+7 b \log \left (c x^n\right )-b n}{2 d x^3 \left (d+e x^2\right )}+\frac {\frac {-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{3 x^3}-\frac {35 b n}{9 x^3}}{d}-\frac {e \left (\frac {-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{x}-\frac {35 b n}{x}}{d}-\frac {e \left (\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a+35 b \log \left (c x^n\right )-12 b n\right )}{\sqrt {d} \sqrt {e}}-\frac {35 b n \left (\frac {1}{2} i \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x}dx-\frac {1}{2} i \int \frac {\log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right )}{x}dx\right )}{\sqrt {d} \sqrt {e}}\right )}{d}\right )}{d}}{2 d}}{4 d}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {\frac {7 a+7 b \log \left (c x^n\right )-b n}{2 d x^3 \left (d+e x^2\right )}+\frac {\frac {-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{3 x^3}-\frac {35 b n}{9 x^3}}{d}-\frac {e \left (\frac {-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{x}-\frac {35 b n}{x}}{d}-\frac {e \left (\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a+35 b \log \left (c x^n\right )-12 b n\right )}{\sqrt {d} \sqrt {e}}-\frac {35 b n \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {d} \sqrt {e}}\right )}{d}\right )}{d}}{2 d}}{4 d}\)

Input:

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

Output:

(a + b*Log[c*x^n])/(4*d*x^3*(d + e*x^2)^2) + ((7*a - b*n + 7*b*Log[c*x^n]) 
/(2*d*x^3*(d + e*x^2)) + (((-35*b*n)/(9*x^3) - (35*a - 12*b*n + 35*b*Log[c 
*x^n])/(3*x^3))/d - (e*(((-35*b*n)/x - (35*a - 12*b*n + 35*b*Log[c*x^n])/x 
)/d - (e*((ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(35*a - 12*b*n + 35*b*Log[c*x^n]))/ 
(Sqrt[d]*Sqrt[e]) - (35*b*n*((I/2)*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]] - 
(I/2)*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]]))/(Sqrt[d]*Sqrt[e])))/d))/d)/(2*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 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 2761

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/((d_) + (e_.)*(x_)^2), x_Symbol] 
:> With[{u = IntHide[1/(d + e*x^2), x]}, Simp[u*(a + b*Log[c*x^n]), x] - Si 
mp[b*n   Int[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n}, x]
 

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 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]
 

rule 5355

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] 
 + (Simp[I*(b/2)   Int[Log[1 - I*c*x]/x, x], x] - Simp[I*(b/2)   Int[Log[1 
+ I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]
 

Maple [C] (warning: unable to verify)

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

Time = 3.90 (sec) , antiderivative size = 1029, normalized size of antiderivative = 3.97

method	result	size
risch	\(\text {Expression too large to display}\)	\(1029\)

Input:

int((a+b*ln(c*x^n))/x^4/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/8*b*n*e^2/d^4*x/(e*x^2+d)+3/16*b*n*e^2/d^2*ln(x)/(e*x^2+d)^2/(-d*e)^(1/ 
2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-3/16*b*n*e^2/d^2*ln(x)/(e*x^2+d)^2 
/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/2*b*n*e^2/d^3*ln(x)/(e 
*x^2+d)/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/2*b*n*e^2/d^3* 
ln(x)/(e*x^2+d)/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+11/8*b/d^ 
4*e^3/(e*x^2+d)^2*x^3*ln(x^n)+35/8*b*e^2/d^4/(d*e)^(1/2)*arctan(x*e/(d*e)^ 
(1/2))*ln(x^n)-3/2*b*n*e^2/d^4/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+35/16*b 
*n*e^2/d^4/(-d*e)^(1/2)*dilog((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-35/16*b*n* 
e^2/d^4/(-d*e)^(1/2)*dilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+(1/2*I*Pi*b*cs 
gn(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/d^4*e^2*((11/8*e*x^3+13/8*d*x)/(e*x^2+d)^2+35/8/(d*e)^(1/2)*arctan(x*e/ 
(d*e)^(1/2)))-1/3/d^3/x^3+3/d^4*e/x)+3*b*ln(x^n)/d^4*e/x-35/8*b*e^2/d^4/(d 
*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*n*ln(x)+3/2*b*n/d^4*e^2*ln(x)/(-d*e)^(1/ 
2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-3/2*b*n/d^4*e^2*ln(x)/(-d*e)^(1/2) 
*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-b*n*e^2/d^3*ln(x)/(e*x^2+d)^2*x-b*n*e 
^3/d^4*ln(x)/(e*x^2+d)^2*x^3+13/8*b*ln(x^n)*e^2/d^3/(e*x^2+d)^2*x-3/8*b*n* 
e^3/d^3*ln(x)/(e*x^2+d)^2/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2)) 
*x^2+1/2*b*n*e^3/d^4*ln(x)/(e*x^2+d)/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/( 
-d*e)^(1/2))*x^2-1/2*b*n*e^3/d^4*ln(x)/(e*x^2+d)/(-d*e)^(1/2)*ln((e*x+(...
 

Fricas [F]

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

integrate((a+b*log(c*x^n))/x^4/(e*x^2+d)^3,x, algorithm="fricas")
 

Output:

integral((b*log(c*x^n) + a)/(e^3*x^10 + 3*d*e^2*x^8 + 3*d^2*e*x^6 + d^3*x^ 
4), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((a+b*ln(c*x**n))/x**4/(e*x**2+d)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*log(c*x^n))/x^4/(e*x^2+d)^3,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

integrate((a+b*log(c*x^n))/x^4/(e*x^2+d)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^4\,{\left (e\,x^2+d\right )}^3} \,d x \] Input:

int((a + b*log(c*x^n))/(x^4*(d + e*x^2)^3),x)
                                                                                    
                                                                                    
 

Output:

int((a + b*log(c*x^n))/(x^4*(d + e*x^2)^3), x)
 

Reduce [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx=\frac {105 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,d^{2} e \,x^{3}+210 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d \,e^{2} x^{5}+105 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,e^{3} x^{7}+24 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{10}+3 d \,e^{2} x^{8}+3 d^{2} e \,x^{6}+d^{3} x^{4}}d x \right ) b \,d^{7} x^{3}+48 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{10}+3 d \,e^{2} x^{8}+3 d^{2} e \,x^{6}+d^{3} x^{4}}d x \right ) b \,d^{6} e \,x^{5}+24 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{10}+3 d \,e^{2} x^{8}+3 d^{2} e \,x^{6}+d^{3} x^{4}}d x \right ) b \,d^{5} e^{2} x^{7}-8 a \,d^{4}+56 a \,d^{3} e \,x^{2}+175 a \,d^{2} e^{2} x^{4}+105 a d \,e^{3} x^{6}}{24 d^{5} x^{3} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

int((a+b*log(c*x^n))/x^4/(e*x^2+d)^3,x)
 

Output:

(105*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d**2*e*x**3 + 210*sqr 
t(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d*e**2*x**5 + 105*sqrt(e)*sqr 
t(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*e**3*x**7 + 24*int(log(x**n*c)/(d**3* 
x**4 + 3*d**2*e*x**6 + 3*d*e**2*x**8 + e**3*x**10),x)*b*d**7*x**3 + 48*int 
(log(x**n*c)/(d**3*x**4 + 3*d**2*e*x**6 + 3*d*e**2*x**8 + e**3*x**10),x)*b 
*d**6*e*x**5 + 24*int(log(x**n*c)/(d**3*x**4 + 3*d**2*e*x**6 + 3*d*e**2*x* 
*8 + e**3*x**10),x)*b*d**5*e**2*x**7 - 8*a*d**4 + 56*a*d**3*e*x**2 + 175*a 
*d**2*e**2*x**4 + 105*a*d*e**3*x**6)/(24*d**5*x**3*(d**2 + 2*d*e*x**2 + e* 
*2*x**4))