\(\int \frac {x^4 (a+b \log (c x^n))}{(d+e x^2)^2} \, dx\) [226]

Optimal result

Integrand size = 23, antiderivative size = 190 \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\frac {a x}{e^2}-\frac {b n x}{e^2}-\frac {b \sqrt {d} n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {b x \log \left (c x^n\right )}{e^2}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}-\frac {3 \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{5/2}}-\frac {3 b \sqrt {-d} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{4 e^{5/2}}+\frac {3 b \sqrt {-d} n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{4 e^{5/2}} \] Output:

a*x/e^2-b*n*x/e^2-1/2*b*d^(1/2)*n*arctan(e^(1/2)*x/d^(1/2))/e^(5/2)+b*x*ln 
(c*x^n)/e^2+1/2*d*x*(a+b*ln(c*x^n))/e^2/(e*x^2+d)-3/2*d^(1/2)*arctan(e^(1/ 
2)*x/d^(1/2))*(a+b*ln(c*x^n))/e^(5/2)-3/4*b*(-d)^(1/2)*n*polylog(2,-e^(1/2 
)*x/(-d)^(1/2))/e^(5/2)+3/4*b*(-d)^(1/2)*n*polylog(2,e^(1/2)*x/(-d)^(1/2)) 
/e^(5/2)
 

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.56 \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\frac {4 a \sqrt {e} x-4 b \sqrt {e} n x+4 b \sqrt {e} x \log \left (c x^n\right )-\frac {d \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-d}-\sqrt {e} x}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-d}+\sqrt {e} x}+\frac {b d n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{\sqrt {-d}}+b \sqrt {-d} n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )-3 \sqrt {-d} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )+3 \sqrt {-d} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )+3 b \sqrt {-d} n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )-3 b \sqrt {-d} n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{4 e^{5/2}} \] Input:

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

Output:

(4*a*Sqrt[e]*x - 4*b*Sqrt[e]*n*x + 4*b*Sqrt[e]*x*Log[c*x^n] - (d*(a + b*Lo 
g[c*x^n]))/(Sqrt[-d] - Sqrt[e]*x) + (d*(a + b*Log[c*x^n]))/(Sqrt[-d] + Sqr 
t[e]*x) + (b*d*n*(Log[x] - Log[Sqrt[-d] - Sqrt[e]*x]))/Sqrt[-d] + b*Sqrt[- 
d]*n*(Log[x] - Log[Sqrt[-d] + Sqrt[e]*x]) - 3*Sqrt[-d]*(a + b*Log[c*x^n])* 
Log[1 + (Sqrt[e]*x)/Sqrt[-d]] + 3*Sqrt[-d]*(a + b*Log[c*x^n])*Log[1 + (d*S 
qrt[e]*x)/(-d)^(3/2)] + 3*b*Sqrt[-d]*n*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]] - 
3*b*Sqrt[-d]*n*PolyLog[2, (d*Sqrt[e]*x)/(-d)^(3/2)])/(4*e^(5/2))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2793, 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.

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

\(\Big \downarrow \) 2793

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{5/2}}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \left (d+e x^2\right )}+\frac {a x}{e^2}-\frac {b \sqrt {d} n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{5/2}}+\frac {b x \log \left (c x^n\right )}{e^2}+\frac {3 i b \sqrt {d} n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {d} n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 e^{5/2}}-\frac {b n x}{e^2}\)

Input:

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

Output:

(a*x)/e^2 - (b*n*x)/e^2 - (b*Sqrt[d]*n*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*e^( 
5/2)) + (b*x*Log[c*x^n])/e^2 + (d*x*(a + b*Log[c*x^n]))/(2*e^2*(d + e*x^2) 
) - (3*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*e^(5/2)) 
 + (((3*I)/4)*b*Sqrt[d]*n*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]])/e^(5/2) - 
(((3*I)/4)*b*Sqrt[d]*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/e^(5/2)
 

Defintions of rubi rules used

rule 2009

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

rule 2793

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

Maple [C] (warning: unable to verify)

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

Time = 0.71 (sec) , antiderivative size = 559, normalized size of antiderivative = 2.94

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right ) x}{e^{2}}+\frac {b \ln \left (x^{n}\right ) d x}{2 e^{2} \left (e \,x^{2}+d \right )}+\frac {3 b d \arctan \left (\frac {x e}{\sqrt {d e}}\right ) n \ln \left (x \right )}{2 e^{2} \sqrt {d e}}-\frac {3 b d \arctan \left (\frac {x e}{\sqrt {d e}}\right ) \ln \left (x^{n}\right )}{2 e^{2} \sqrt {d e}}-\frac {b n x}{e^{2}}-\frac {b n d \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{2 e^{2} \sqrt {d e}}+\frac {b n d \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2}}{4 e \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {b n d \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2}}{4 e \left (e \,x^{2}+d \right ) \sqrt {-d e}}+\frac {b n \,d^{2} \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 e^{2} \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {b n \,d^{2} \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 e^{2} \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {3 b n d \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 e^{2} \sqrt {-d e}}+\frac {3 b n d \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 e^{2} \sqrt {-d e}}-\frac {b n d \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{e^{2} \sqrt {-d e}}+\frac {b n d \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{e^{2} \sqrt {-d e}}+\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}{e^{2}}-\frac {d \left (-\frac {x}{2 \left (e \,x^{2}+d \right )}+\frac {3 \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{2 \sqrt {d e}}\right )}{e^{2}}\right )\)	\(559\)

Input:

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

Output:

b*ln(x^n)/e^2*x+1/2*b*ln(x^n)*d/e^2*x/(e*x^2+d)+3/2*b*d/e^2/(d*e)^(1/2)*ar 
ctan(x*e/(d*e)^(1/2))*n*ln(x)-3/2*b*d/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/ 
2))*ln(x^n)-b*n*x/e^2-1/2*b*n*d/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+1/ 
4*b*n*d/e*ln(x)/(e*x^2+d)/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2) 
)*x^2-1/4*b*n*d/e*ln(x)/(e*x^2+d)/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e 
)^(1/2))*x^2+1/4*b*n*d^2/e^2*ln(x)/(e*x^2+d)/(-d*e)^(1/2)*ln((-e*x+(-d*e)^ 
(1/2))/(-d*e)^(1/2))-1/4*b*n*d^2/e^2*ln(x)/(e*x^2+d)/(-d*e)^(1/2)*ln((e*x+ 
(-d*e)^(1/2))/(-d*e)^(1/2))-3/4*b*n*d/e^2/(-d*e)^(1/2)*dilog((-e*x+(-d*e)^ 
(1/2))/(-d*e)^(1/2))+3/4*b*n*d/e^2/(-d*e)^(1/2)*dilog((e*x+(-d*e)^(1/2))/( 
-d*e)^(1/2))-b*n/e^2*d*ln(x)/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1 
/2))+b*n/e^2*d*ln(x)/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+(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)*c 
sgn(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/e^2-1/e^2*d*(-1/2*x/(e*x^2+d)+3/2/(d*e)^(1/2)*arctan(x*e/(d*e 
)^(1/2))))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\frac {-3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d -3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a e \,x^{2}-4 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b d n -4 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b e n \,x^{2}-6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b \,d^{3} e -6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b \,d^{2} e^{2} x^{2}+6 \,\mathrm {log}\left (x^{n} c \right ) b d e x +2 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{2} x^{3}+3 a d e x +2 a \,e^{2} x^{3}-2 b d e n x -2 b \,e^{2} n \,x^{3}}{2 e^{3} \left (e \,x^{2}+d \right )} \] Input:

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

Output:

( - 3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d - 3*sqrt(e)*sqrt(d 
)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*e*x**2 - 4*sqrt(e)*sqrt(d)*atan((e*x)/(s 
qrt(e)*sqrt(d)))*b*d*n - 4*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*b 
*e*n*x**2 - 6*int(log(x**n*c)/(d**2 + 2*d*e*x**2 + e**2*x**4),x)*b*d**3*e 
- 6*int(log(x**n*c)/(d**2 + 2*d*e*x**2 + e**2*x**4),x)*b*d**2*e**2*x**2 + 
6*log(x**n*c)*b*d*e*x + 2*log(x**n*c)*b*e**2*x**3 + 3*a*d*e*x + 2*a*e**2*x 
**3 - 2*b*d*e*n*x - 2*b*e**2*n*x**3)/(2*e**3*(d + e*x**2))