\(\int \frac {x^5 (a+b \log (c x^n))}{(d+e x^2)^3} \, dx\) [231]

Optimal result

Integrand size = 23, antiderivative size = 152 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {b d n}{8 e^3 \left (d+e x^2\right )}+\frac {b n \log (x)}{4 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \left (d+e x^2\right )^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}+\frac {3 b n \log \left (d+e x^2\right )}{8 e^3}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^3}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{4 e^3} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.62 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.28 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {-2 d^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+8 d \left (d+e x^2\right ) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+4 \left (d+e x^2\right )^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+e x^2\right )+b n \left (d^2+d e x^2-4 d e x^2 \log (x)-6 e^2 x^4 \log (x)+3 d^2 \log \left (i \sqrt {d}-\sqrt {e} x\right )+6 d e x^2 \log \left (i \sqrt {d}-\sqrt {e} x\right )+3 e^2 x^4 \log \left (i \sqrt {d}-\sqrt {e} x\right )+3 d^2 \log \left (i \sqrt {d}+\sqrt {e} x\right )+6 d e x^2 \log \left (i \sqrt {d}+\sqrt {e} x\right )+3 e^2 x^4 \log \left (i \sqrt {d}+\sqrt {e} x\right )+4 d^2 \log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+8 d e x^2 \log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+4 e^2 x^4 \log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+4 d^2 \log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+8 d e x^2 \log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+4 e^2 x^4 \log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+4 \left (d+e x^2\right )^2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+4 \left (d+e x^2\right )^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{8 e^3 \left (d+e x^2\right )^2} \] Input:

Integrate[(x^5*(a + b*Log[c*x^n]))/(d + e*x^2)^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, 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.

\(\displaystyle \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx\)

\(\Big \downarrow \) 2793

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac {\log \left (\frac {e x^2}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{4 e^3}+\frac {b d n}{8 e^3 \left (d+e x^2\right )}+\frac {3 b n \log \left (d+e x^2\right )}{8 e^3}+\frac {b n \log (x)}{4 e^3}\)

Input:

Int[(x^5*(a + b*Log[c*x^n]))/(d + e*x^2)^3,x]
 

Output:

(b*d*n)/(8*e^3*(d + e*x^2)) + (b*n*Log[x])/(4*e^3) - (d^2*(a + b*Log[c*x^n 
]))/(4*e^3*(d + e*x^2)^2) - (x^2*(a + b*Log[c*x^n]))/(e^2*(d + e*x^2)) + ( 
3*b*n*Log[d + e*x^2])/(8*e^3) + ((a + b*Log[c*x^n])*Log[1 + (e*x^2)/d])/(2 
*e^3) + (b*n*PolyLog[2, -((e*x^2)/d)])/(4*e^3)
 

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 = 1.33 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.36

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right ) d}{e^{3} \left (e \,x^{2}+d \right )}+\frac {b \ln \left (x^{n}\right ) \ln \left (e \,x^{2}+d \right )}{2 e^{3}}-\frac {b \ln \left (x^{n}\right ) d^{2}}{4 e^{3} \left (e \,x^{2}+d \right )^{2}}+\frac {b d n}{8 e^{3} \left (e \,x^{2}+d \right )}+\frac {3 b n \ln \left (e \,x^{2}+d \right )}{8 e^{3}}-\frac {3 b n \ln \left (x \right )}{4 e^{3}}-\frac {b n \ln \left (x \right ) \ln \left (e \,x^{2}+d \right )}{2 e^{3}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{3}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{3}}+\frac {b n \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{3}}+\frac {b n \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{3}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\frac {d}{e^{3} \left (e \,x^{2}+d \right )}+\frac {\ln \left (e \,x^{2}+d \right )}{2 e^{3}}-\frac {d^{2}}{4 e^{3} \left (e \,x^{2}+d \right )^{2}}\right )\)	\(359\)

Input:

int(x^5*(a+b*ln(c*x^n))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

integrate(x^5*(a+b*log(c*x^n))/(e*x^2+d)^3,x, algorithm="fricas")
 

Output:

integral((b*x^5*log(c*x^n) + a*x^5)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + 
 d^3), x)
 

Sympy [A] (verification not implemented)

Time = 69.82 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.65 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(x**5*(a+b*ln(c*x**n))/(e*x**2+d)**3,x)
 

Output:

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

Maxima [F]

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

integrate(x^5*(a+b*log(c*x^n))/(e*x^2+d)^3,x, algorithm="maxima")
 

Output:

1/4*a*((4*d*e*x^2 + 3*d^2)/(e^5*x^4 + 2*d*e^4*x^2 + d^2*e^3) + 2*log(e*x^2 
 + d)/e^3) + b*integrate((x^5*log(c) + x^5*log(x^n))/(e^3*x^6 + 3*d*e^2*x^ 
4 + 3*d^2*e*x^2 + d^3), x)
 

Giac [F]

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

integrate(x^5*(a+b*log(c*x^n))/(e*x^2+d)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {-4 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{7}+3 d \,e^{2} x^{5}+3 d^{2} e \,x^{3}+d^{3} x}d x \right ) b \,d^{5} n -8 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{7}+3 d \,e^{2} x^{5}+3 d^{2} e \,x^{3}+d^{3} x}d x \right ) b \,d^{4} e n \,x^{2}-4 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{7}+3 d \,e^{2} x^{5}+3 d^{2} e \,x^{3}+d^{3} x}d x \right ) b \,d^{3} e^{2} n \,x^{4}+2 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,d^{2} n +4 \,\mathrm {log}\left (e \,x^{2}+d \right ) a d e n \,x^{2}+2 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,e^{2} n \,x^{4}+3 \,\mathrm {log}\left (e \,x^{2}+d \right ) b \,d^{2} n^{2}+6 \,\mathrm {log}\left (e \,x^{2}+d \right ) b d e \,n^{2} x^{2}+3 \,\mathrm {log}\left (e \,x^{2}+d \right ) b \,e^{2} n^{2} x^{4}+2 \mathrm {log}\left (x^{n} c \right )^{2} b \,d^{2}+4 \mathrm {log}\left (x^{n} c \right )^{2} b d e \,x^{2}+2 \mathrm {log}\left (x^{n} c \right )^{2} b \,e^{2} x^{4}+3 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} n -3 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{2} n \,x^{4}-3 \,\mathrm {log}\left (x \right ) b \,d^{2} n^{2}-6 \,\mathrm {log}\left (x \right ) b d e \,n^{2} x^{2}-3 \,\mathrm {log}\left (x \right ) b \,e^{2} n^{2} x^{4}+a \,d^{2} n -2 a \,e^{2} n \,x^{4}}{4 e^{3} n \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

int(x^5*(a+b*log(c*x^n))/(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*n - 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*n*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*n*x**4 + 2*log(d + e*x**2)*a* 
d**2*n + 4*log(d + e*x**2)*a*d*e*n*x**2 + 2*log(d + e*x**2)*a*e**2*n*x**4 
+ 3*log(d + e*x**2)*b*d**2*n**2 + 6*log(d + e*x**2)*b*d*e*n**2*x**2 + 3*lo 
g(d + e*x**2)*b*e**2*n**2*x**4 + 2*log(x**n*c)**2*b*d**2 + 4*log(x**n*c)** 
2*b*d*e*x**2 + 2*log(x**n*c)**2*b*e**2*x**4 + 3*log(x**n*c)*b*d**2*n - 3*l 
og(x**n*c)*b*e**2*n*x**4 - 3*log(x)*b*d**2*n**2 - 6*log(x)*b*d*e*n**2*x**2 
 - 3*log(x)*b*e**2*n**2*x**4 + a*d**2*n - 2*a*e**2*n*x**4)/(4*e**3*n*(d**2 
 + 2*d*e*x**2 + e**2*x**4))