\(\int x^2 (a+b \log (c x^n)) \log (d (\frac {1}{d}+f x^2)) \, dx\) [34]

Optimal result

Integrand size = 26, antiderivative size = 241 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=-\frac {8 b n x}{9 d f}+\frac {4}{27} b n x^3+\frac {2 b n \arctan \left (\sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}-\frac {1}{9} b n x^3 \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {i b n \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}-\frac {i b n \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}} \] Output:

-8/9*b*n*x/d/f+4/27*b*n*x^3+2/9*b*n*arctan(d^(1/2)*f^(1/2)*x)/d^(3/2)/f^(3 
/2)+2/3*x*(a+b*ln(c*x^n))/d/f-2/9*x^3*(a+b*ln(c*x^n))-2/3*arctan(d^(1/2)*f 
^(1/2)*x)*(a+b*ln(c*x^n))/d^(3/2)/f^(3/2)-1/9*b*n*x^3*ln(d*f*x^2+1)+1/3*x^ 
3*(a+b*ln(c*x^n))*ln(d*f*x^2+1)+1/3*I*b*n*polylog(2,-I*d^(1/2)*f^(1/2)*x)/ 
d^(3/2)/f^(3/2)-1/3*I*b*n*polylog(2,I*d^(1/2)*f^(1/2)*x)/d^(3/2)/f^(3/2)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.51 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\frac {2 a x}{3 d f}-\frac {2 a x^3}{9}-\frac {2 a \arctan \left (\sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}+\frac {2 b x \left (-n+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{9 d f}-\frac {2}{27} b x^3 \left (-n+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right )-\frac {2 b \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (-n+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{9 d^{3/2} f^{3/2}}+\frac {1}{3} a x^3 \log \left (1+d f x^2\right )+\frac {1}{9} b x^3 \left (-n+3 n \log (x)+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \log \left (1+d f x^2\right )-\frac {2}{3} b d f n \left (-\frac {x (-1+\log (x))}{d^2 f^2}+\frac {-\frac {x^3}{9}+\frac {1}{3} x^3 \log (x)}{d f}-\frac {i \left (\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )\right )}{2 d^{5/2} f^{5/2}}+\frac {i \left (\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )\right )}{2 d^{5/2} f^{5/2}}\right ) \] Input:

Integrate[x^2*(a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^2)],x]
 

Output:

(2*a*x)/(3*d*f) - (2*a*x^3)/9 - (2*a*ArcTan[Sqrt[d]*Sqrt[f]*x])/(3*d^(3/2) 
*f^(3/2)) + (2*b*x*(-n + 3*(-(n*Log[x]) + Log[c*x^n])))/(9*d*f) - (2*b*x^3 
*(-n + 3*(-(n*Log[x]) + Log[c*x^n])))/27 - (2*b*ArcTan[Sqrt[d]*Sqrt[f]*x]* 
(-n + 3*(-(n*Log[x]) + Log[c*x^n])))/(9*d^(3/2)*f^(3/2)) + (a*x^3*Log[1 + 
d*f*x^2])/3 + (b*x^3*(-n + 3*n*Log[x] + 3*(-(n*Log[x]) + Log[c*x^n]))*Log[ 
1 + d*f*x^2])/9 - (2*b*d*f*n*(-((x*(-1 + Log[x]))/(d^2*f^2)) + (-1/9*x^3 + 
 (x^3*Log[x])/3)/(d*f) - ((I/2)*(Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + Pol 
yLog[2, (-I)*Sqrt[d]*Sqrt[f]*x]))/(d^(5/2)*f^(5/2)) + ((I/2)*(Log[x]*Log[1 
 - I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]))/(d^(5/2)*f^(5/ 
2))))/3
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2823, 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.

\(\displaystyle \int x^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 2823

\(\displaystyle -b n \int \left (\frac {1}{3} \log \left (d f x^2+1\right ) x^2-\frac {2 x^2}{9}+\frac {2}{3 d f}-\frac {2 \arctan \left (\sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2} x}\right )dx-\frac {2 \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}+\frac {1}{3} x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}+\frac {1}{3} x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-b n \left (-\frac {2 \arctan \left (\sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}-\frac {i \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}+\frac {i \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )}{3 d^{3/2} f^{3/2}}+\frac {1}{9} x^3 \log \left (d f x^2+1\right )+\frac {8 x}{9 d f}-\frac {4 x^3}{27}\right )\)

Input:

Int[x^2*(a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^2)],x]
 

Output:

(2*x*(a + b*Log[c*x^n]))/(3*d*f) - (2*x^3*(a + b*Log[c*x^n]))/9 - (2*ArcTa 
n[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]))/(3*d^(3/2)*f^(3/2)) + (x^3*(a + b 
*Log[c*x^n])*Log[1 + d*f*x^2])/3 - b*n*((8*x)/(9*d*f) - (4*x^3)/27 - (2*Ar 
cTan[Sqrt[d]*Sqrt[f]*x])/(9*d^(3/2)*f^(3/2)) + (x^3*Log[1 + d*f*x^2])/9 - 
((I/3)*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x])/(d^(3/2)*f^(3/2)) + ((I/3)*Poly 
Log[2, I*Sqrt[d]*Sqrt[f]*x])/(d^(3/2)*f^(3/2)))
 

Defintions of rubi rules used

rule 2009

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

rule 2823

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* 
(e + f*x^m)^r], x]}, Simp[(a + b*Log[c*x^n])   u, x] - Simp[b*n   Int[1/x 
 u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q 
+ 1)/m] || (RationalQ[m] && RationalQ[q])) && NeQ[q, -1]
 

Maple [C] (warning: unable to verify)

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

Time = 32.18 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.75

method	result	size
risch	\(\frac {b \,x^{3} \ln \left (d f \,x^{2}+1\right ) \ln \left (x^{n}\right )}{3}-\frac {2 b \,x^{3} \ln \left (x^{n}\right )}{9}+\frac {2 b x \ln \left (x^{n}\right )}{3 d f}+\frac {2 b \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) n \ln \left (x \right )}{3 d f \sqrt {d f}}-\frac {2 b \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) \ln \left (x^{n}\right )}{3 d f \sqrt {d f}}-\frac {b n \,x^{3} \ln \left (d f \,x^{2}+1\right )}{9}+\frac {4 b n \,x^{3}}{27}-\frac {8 b n x}{9 d f}+\frac {2 b n \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{9 d f \sqrt {d f}}+\frac {b n \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1+x \sqrt {-d f}\right )}{3 d^{2} f^{2}}-\frac {b n \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1-x \sqrt {-d f}\right )}{3 d^{2} f^{2}}+\frac {b n \sqrt {-d f}\, \operatorname {dilog}\left (1+x \sqrt {-d f}\right )}{3 d^{2} f^{2}}-\frac {b n \sqrt {-d f}\, \operatorname {dilog}\left (1-x \sqrt {-d f}\right )}{3 d^{2} f^{2}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x^{3} \ln \left (d f \,x^{2}+1\right )}{3}-\frac {2 d f \left (\frac {\frac {1}{3} x^{3} d f -x}{d^{2} f^{2}}+\frac {\arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{d^{2} f^{2} \sqrt {d f}}\right )}{3}\right )\)	\(422\)

Input:

int(x^2*(a+b*ln(c*x^n))*ln(d*(1/d+f*x^2)),x,method=_RETURNVERBOSE)
 

Output:

1/3*b*x^3*ln(d*f*x^2+1)*ln(x^n)-2/9*b*x^3*ln(x^n)+2/3*b/d/f*x*ln(x^n)+2/3* 
b/d/f/(d*f)^(1/2)*arctan(x*d*f/(d*f)^(1/2))*n*ln(x)-2/3*b/d/f/(d*f)^(1/2)* 
arctan(x*d*f/(d*f)^(1/2))*ln(x^n)-1/9*b*n*x^3*ln(d*f*x^2+1)+4/27*b*n*x^3-8 
/9*b*n*x/d/f+2/9*b*n/d/f/(d*f)^(1/2)*arctan(x*d*f/(d*f)^(1/2))+1/3*b*n/d^2 
/f^2*(-d*f)^(1/2)*ln(x)*ln(1+x*(-d*f)^(1/2))-1/3*b*n/d^2/f^2*(-d*f)^(1/2)* 
ln(x)*ln(1-x*(-d*f)^(1/2))+1/3*b*n/d^2/f^2*(-d*f)^(1/2)*dilog(1+x*(-d*f)^( 
1/2))-1/3*b*n/d^2/f^2*(-d*f)^(1/2)*dilog(1-x*(-d*f)^(1/2))+(1/2*I*Pi*b*csg 
n(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)*( 
1/3*x^3*ln(d*f*x^2+1)-2/3*d*f*(1/d^2/f^2*(1/3*x^3*d*f-x)+1/d^2/f^2/(d*f)^( 
1/2)*arctan(x*d*f/(d*f)^(1/2))))
 

Fricas [F]

\[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

integrate(x^2*(a+b*log(c*x^n))*log(d*(1/d+f*x^2)),x, algorithm="fricas")
 

Output:

integral(b*x^2*log(d*f*x^2 + 1)*log(c*x^n) + a*x^2*log(d*f*x^2 + 1), x)
 

Sympy [F(-1)]

Timed out. \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\text {Timed out} \] Input:

integrate(x**2*(a+b*ln(c*x**n))*ln(d*(1/d+f*x**2)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

integrate(x^2*(a+b*log(c*x^n))*log(d*(1/d+f*x^2)),x, algorithm="maxima")
 

Output:

1/9*(3*b*x^3*log(x^n) - (b*(n - 3*log(c)) - 3*a)*x^3)*log(d*f*x^2 + 1) - i 
ntegrate(2/9*(3*b*d*f*x^4*log(x^n) + (3*a*d*f - (d*f*n - 3*d*f*log(c))*b)* 
x^4)/(d*f*x^2 + 1), x)
 

Giac [F]

\[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

integrate(x^2*(a+b*log(c*x^n))*log(d*(1/d+f*x^2)),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int x^2\,\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \] Input:

int(x^2*log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n)),x)
 

Output:

int(x^2*log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n)), x)
 

Reduce [F]

\[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\frac {-18 \sqrt {f}\, \sqrt {d}\, \mathit {atan} \left (\frac {d f x}{\sqrt {f}\, \sqrt {d}}\right ) a +6 \sqrt {f}\, \sqrt {d}\, \mathit {atan} \left (\frac {d f x}{\sqrt {f}\, \sqrt {d}}\right ) b n -18 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{d f \,x^{2}+1}d x \right ) b d f +9 \,\mathrm {log}\left (d f \,x^{2}+1\right ) \mathrm {log}\left (x^{n} c \right ) b \,d^{2} f^{2} x^{3}+9 \,\mathrm {log}\left (d f \,x^{2}+1\right ) a \,d^{2} f^{2} x^{3}-3 \,\mathrm {log}\left (d f \,x^{2}+1\right ) b \,d^{2} f^{2} n \,x^{3}-6 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} f^{2} x^{3}+18 \,\mathrm {log}\left (x^{n} c \right ) b d f x -6 a \,d^{2} f^{2} x^{3}+18 a d f x +4 b \,d^{2} f^{2} n \,x^{3}-24 b d f n x}{27 d^{2} f^{2}} \] Input:

int(x^2*(a+b*log(c*x^n))*log(d*(1/d+f*x^2)),x)
 

Output:

( - 18*sqrt(f)*sqrt(d)*atan((d*f*x)/(sqrt(f)*sqrt(d)))*a + 6*sqrt(f)*sqrt( 
d)*atan((d*f*x)/(sqrt(f)*sqrt(d)))*b*n - 18*int(log(x**n*c)/(d*f*x**2 + 1) 
,x)*b*d*f + 9*log(d*f*x**2 + 1)*log(x**n*c)*b*d**2*f**2*x**3 + 9*log(d*f*x 
**2 + 1)*a*d**2*f**2*x**3 - 3*log(d*f*x**2 + 1)*b*d**2*f**2*n*x**3 - 6*log 
(x**n*c)*b*d**2*f**2*x**3 + 18*log(x**n*c)*b*d*f*x - 6*a*d**2*f**2*x**3 + 
18*a*d*f*x + 4*b*d**2*f**2*n*x**3 - 24*b*d*f*n*x)/(27*d**2*f**2)