\(\int \frac {x^3 (a+b \log (c x^n))}{(d+e x^2)^2} \, dx\) [222]

Optimal result

Integrand size = 23, antiderivative size = 95 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e \left (d+e x^2\right )}+\frac {b n \log \left (d+e x^2\right )}{4 e^2}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^2}{d}\right )}{2 e^2}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x^2}{d}\right )}{4 e^2} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 321, normalized size of antiderivative = 3.38 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx=\frac {\frac {2 d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{d+e x^2}+2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+e x^2\right )+\frac {b n \left (-2 e x^2 \log (x)+d \log \left (i \sqrt {d}-\sqrt {e} x\right )+e x^2 \log \left (i \sqrt {d}-\sqrt {e} x\right )+d \log \left (i \sqrt {d}+\sqrt {e} x\right )+e x^2 \log \left (i \sqrt {d}+\sqrt {e} x\right )+2 d \log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+2 e x^2 \log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+2 d \log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+2 e x^2 \log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+2 \left (d+e x^2\right ) \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+2 \left (d+e x^2\right ) \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{d+e x^2}}{4 e^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 95, 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.

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

\(\Big \downarrow \) 2793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*(x^2*(a + b*Log[c*x^n]))/(e*(d + e*x^2)) + (b*n*Log[d + e*x^2])/(4*e^ 
2) + ((a + b*Log[c*x^n])*Log[1 + (e*x^2)/d])/(2*e^2) + (b*n*PolyLog[2, -(( 
e*x^2)/d)])/(4*e^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 = 0.68 (sec) , antiderivative size = 305, normalized size of antiderivative = 3.21

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right ) d}{2 e^{2} \left (e \,x^{2}+d \right )}+\frac {b \ln \left (x^{n}\right ) \ln \left (e \,x^{2}+d \right )}{2 e^{2}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}-\frac {b n \ln \left (x \right ) \ln \left (e \,x^{2}+d \right )}{2 e^{2}}+\frac {b n \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}+\frac {b n \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e^{2}}+\frac {b n \ln \left (e \,x^{2}+d \right )}{4 e^{2}}-\frac {b n \ln \left (x \right )}{2 e^{2}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\frac {d}{2 e^{2} \left (e \,x^{2}+d \right )}+\frac {\ln \left (e \,x^{2}+d \right )}{2 e^{2}}\right )\)	\(305\)

Input:

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

Output:

1/2*b*ln(x^n)*d/e^2/(e*x^2+d)+1/2*b*ln(x^n)/e^2*ln(e*x^2+d)+1/2*b*n/e^2*ln 
(x)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/2*b*n/e^2*ln(x)*ln((e*x+(-d*e)^ 
(1/2))/(-d*e)^(1/2))-1/2*b*n/e^2*ln(x)*ln(e*x^2+d)+1/2*b*n/e^2*dilog((-e*x 
+(-d*e)^(1/2))/(-d*e)^(1/2))+1/2*b*n/e^2*dilog((e*x+(-d*e)^(1/2))/(-d*e)^( 
1/2))+1/4*b*n*ln(e*x^2+d)/e^2-1/2*b*n/e^2*ln(x)+(1/2*I*Pi*b*csgn(I*x^n)*cs 
gn(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*cs 
gn(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*d/e^2/( 
e*x^2+d)+1/2/e^2*ln(e*x^2+d))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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