Integrand size = 28, antiderivative size = 612 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=-\frac {16 a b n x}{9 d f}+\frac {52 b^2 n^2 x}{27 d f}-\frac {4}{27} b^2 n^2 x^3-\frac {4 b^2 n^2 \arctan \left (\sqrt {d} \sqrt {f} x\right )}{27 d^{3/2} f^{3/2}}-\frac {16 b^2 n x \log \left (c x^n\right )}{9 d f}+\frac {8}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b n \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{3/2} f^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{3 d f}-\frac {2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {2}{27} b^2 n^2 x^3 \log \left (1+d f x^2\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}-\frac {2 i b^2 n^2 \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}+\frac {2 i b^2 n^2 \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{3/2}}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{3/2}} \]
-16/9*a*b*n*x/d/f+52/27*b^2*n^2*x/d/f-4/27*b^2*n^2*x^3-4/27*b^2*n^2*arctan (x*d^(1/2)*f^(1/2))/d^(3/2)/f^(3/2)-16/9*b^2*n*x*ln(c*x^n)/d/f+8/27*b*n*x^ 3*(a+b*ln(c*x^n))+4/9*b*n*arctan(x*d^(1/2)*f^(1/2))*(a+b*ln(c*x^n))/d^(3/2 )/f^(3/2)+2/3*x*(a+b*ln(c*x^n))^2/d/f-2/9*x^3*(a+b*ln(c*x^n))^2+2/27*b^2*n ^2*x^3*ln(d*f*x^2+1)-2/9*b*n*x^3*(a+b*ln(c*x^n))*ln(d*f*x^2+1)+1/3*x^3*(a+ b*ln(c*x^n))^2*ln(d*f*x^2+1)-1/3*(a+b*ln(c*x^n))^2*ln(1-x*(-d)^(1/2)*f^(1/ 2))/(-d)^(3/2)/f^(3/2)+1/3*(a+b*ln(c*x^n))^2*ln(1+x*(-d)^(1/2)*f^(1/2))/(- d)^(3/2)/f^(3/2)+2/3*b*n*(a+b*ln(c*x^n))*polylog(2,-x*(-d)^(1/2)*f^(1/2))/ (-d)^(3/2)/f^(3/2)-2/3*b*n*(a+b*ln(c*x^n))*polylog(2,x*(-d)^(1/2)*f^(1/2)) /(-d)^(3/2)/f^(3/2)-2/9*I*b^2*n^2*polylog(2,-I*x*d^(1/2)*f^(1/2))/d^(3/2)/ f^(3/2)+2/9*I*b^2*n^2*polylog(2,I*x*d^(1/2)*f^(1/2))/d^(3/2)/f^(3/2)-2/3*b ^2*n^2*polylog(3,-x*(-d)^(1/2)*f^(1/2))/(-d)^(3/2)/f^(3/2)+2/3*b^2*n^2*pol ylog(3,x*(-d)^(1/2)*f^(1/2))/(-d)^(3/2)/f^(3/2)
Time = 0.41 (sec) , antiderivative size = 703, normalized size of antiderivative = 1.15 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\frac {6 \sqrt {d} \sqrt {f} x \left (9 a^2-6 a b n+2 b^2 n^2+6 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+18 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+9 b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right )-2 d^{3/2} f^{3/2} x^3 \left (9 a^2-6 a b n+2 b^2 n^2+6 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+18 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+9 b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right )-6 \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (9 a^2-6 a b n+2 b^2 n^2+6 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+18 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+9 b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right )+3 d^{3/2} f^{3/2} x^3 \left (9 a^2-6 a b n+2 b^2 n^2-6 b (-3 a+b n) \log \left (c x^n\right )+9 b^2 \log ^2\left (c x^n\right )\right ) \log \left (1+d f x^2\right )-18 b n \left (3 a-b n-3 b n \log (x)+3 b \log \left (c x^n\right )\right ) \left (-2 \sqrt {d} \sqrt {f} x (-1+\log (x))+\frac {2}{9} d^{3/2} f^{3/2} x^3 (-1+3 \log (x))-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 )+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 )\right )+54 b^2 n^2 \left (\sqrt {d} \sqrt {f} x \left (2-2 \log (x)+\log ^2(x)\right )-\frac {1}{27} d^{3/2} f^{3/2} x^3 \left (2-6 \log (x)+9 \log ^2(x)\right )+\frac {1}{2} i \left (\log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )-2 \operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )\right )-\frac {1}{2} i \left (\log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )-2 \operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )\right )\right )}{81 d^{3/2} f^{3/2}} \]
(6*Sqrt[d]*Sqrt[f]*x*(9*a^2 - 6*a*b*n + 2*b^2*n^2 + 6*b^2*n*(n*Log[x] - Lo g[c*x^n]) + 18*a*b*(-(n*Log[x]) + Log[c*x^n]) + 9*b^2*(-(n*Log[x]) + Log[c *x^n])^2) - 2*d^(3/2)*f^(3/2)*x^3*(9*a^2 - 6*a*b*n + 2*b^2*n^2 + 6*b^2*n*( n*Log[x] - Log[c*x^n]) + 18*a*b*(-(n*Log[x]) + Log[c*x^n]) + 9*b^2*(-(n*Lo g[x]) + Log[c*x^n])^2) - 6*ArcTan[Sqrt[d]*Sqrt[f]*x]*(9*a^2 - 6*a*b*n + 2* b^2*n^2 + 6*b^2*n*(n*Log[x] - Log[c*x^n]) + 18*a*b*(-(n*Log[x]) + Log[c*x^ n]) + 9*b^2*(-(n*Log[x]) + Log[c*x^n])^2) + 3*d^(3/2)*f^(3/2)*x^3*(9*a^2 - 6*a*b*n + 2*b^2*n^2 - 6*b*(-3*a + b*n)*Log[c*x^n] + 9*b^2*Log[c*x^n]^2)*L og[1 + d*f*x^2] - 18*b*n*(3*a - b*n - 3*b*n*Log[x] + 3*b*Log[c*x^n])*(-2*S qrt[d]*Sqrt[f]*x*(-1 + Log[x]) + (2*d^(3/2)*f^(3/2)*x^3*(-1 + 3*Log[x]))/9 - I*(Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, (-I)*Sqrt[d]*Sqrt[f ]*x]) + I*(Log[x]*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, I*Sqrt[d]*Sqrt [f]*x])) + 54*b^2*n^2*(Sqrt[d]*Sqrt[f]*x*(2 - 2*Log[x] + Log[x]^2) - (d^(3 /2)*f^(3/2)*x^3*(2 - 6*Log[x] + 9*Log[x]^2))/27 + (I/2)*(Log[x]^2*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + 2*Log[x]*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - 2*Pol yLog[3, (-I)*Sqrt[d]*Sqrt[f]*x]) - (I/2)*(Log[x]^2*Log[1 - I*Sqrt[d]*Sqrt[ f]*x] + 2*Log[x]*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] - 2*PolyLog[3, I*Sqrt[d]* Sqrt[f]*x])))/(81*d^(3/2)*f^(3/2))
Time = 1.14 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2825, 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.
| \(\displaystyle \int x^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2825 |
| \(\displaystyle -2 f \int \left (\frac {d \left (a+b \log \left (c x^n\right )\right )^2 x^4}{3 \left (d f x^2+1\right )}-\frac {2 b d n \left (a+b \log \left (c x^n\right )\right ) x^4}{9 \left (d f x^2+1\right )}+\frac {2 b^2 d n^2 x^4}{27 \left (d f x^2+1\right )}\right )dx+\frac {1}{3} x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {2}{9} b n x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2}{27} b^2 n^2 x^3 \log \left (d f x^2+1\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 f \left (-\frac {2 b n \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{3/2} f^{5/2}}-\frac {b n \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 (-d)^{3/2} f^{5/2}}+\frac {b n \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 (-d)^{3/2} f^{5/2}}+\frac {\log \left (1-\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )^2}{6 (-d)^{3/2} f^{5/2}}-\frac {\log \left (\sqrt {-d} \sqrt {f} x+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{6 (-d)^{3/2} f^{5/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{3 d f^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{9 f}-\frac {4 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{27 f}+\frac {8 a b n x}{9 d f^2}+\frac {2 b^2 n^2 \arctan \left (\sqrt {d} \sqrt {f} x\right )}{27 d^{3/2} f^{5/2}}+\frac {8 b^2 n x \log \left (c x^n\right )}{9 d f^2}+\frac {i b^2 n^2 \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{5/2}}-\frac {i b^2 n^2 \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )}{9 d^{3/2} f^{5/2}}+\frac {b^2 n^2 \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{5/2}}-\frac {b^2 n^2 \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right )}{3 (-d)^{3/2} f^{5/2}}-\frac {26 b^2 n^2 x}{27 d f^2}+\frac {2 b^2 n^2 x^3}{27 f}\right )+\frac {1}{3} x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {2}{9} b n x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2}{27} b^2 n^2 x^3 \log \left (d f x^2+1\right )\) |
(2*b^2*n^2*x^3*Log[1 + d*f*x^2])/27 - (2*b*n*x^3*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/9 + (x^3*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/3 - 2*f*((8*a* b*n*x)/(9*d*f^2) - (26*b^2*n^2*x)/(27*d*f^2) + (2*b^2*n^2*x^3)/(27*f) + (2 *b^2*n^2*ArcTan[Sqrt[d]*Sqrt[f]*x])/(27*d^(3/2)*f^(5/2)) + (8*b^2*n*x*Log[ c*x^n])/(9*d*f^2) - (4*b*n*x^3*(a + b*Log[c*x^n]))/(27*f) - (2*b*n*ArcTan[ Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]))/(9*d^(3/2)*f^(5/2)) - (x*(a + b*Log [c*x^n])^2)/(3*d*f^2) + (x^3*(a + b*Log[c*x^n])^2)/(9*f) + ((a + b*Log[c*x ^n])^2*Log[1 - Sqrt[-d]*Sqrt[f]*x])/(6*(-d)^(3/2)*f^(5/2)) - ((a + b*Log[c *x^n])^2*Log[1 + Sqrt[-d]*Sqrt[f]*x])/(6*(-d)^(3/2)*f^(5/2)) - (b*n*(a + b *Log[c*x^n])*PolyLog[2, -(Sqrt[-d]*Sqrt[f]*x)])/(3*(-d)^(3/2)*f^(5/2)) + ( b*n*(a + b*Log[c*x^n])*PolyLog[2, Sqrt[-d]*Sqrt[f]*x])/(3*(-d)^(3/2)*f^(5/ 2)) + ((I/9)*b^2*n^2*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x])/(d^(3/2)*f^(5/2)) - ((I/9)*b^2*n^2*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])/(d^(3/2)*f^(5/2)) + (b^ 2*n^2*PolyLog[3, -(Sqrt[-d]*Sqrt[f]*x)])/(3*(-d)^(3/2)*f^(5/2)) - (b^2*n^2 *PolyLog[3, Sqrt[-d]*Sqrt[f]*x])/(3*(-d)^(3/2)*f^(5/2)))
3.1.36.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q* (a + b*Log[c*x^n])^p, x]}, Simp[Log[d*(e + f*x^m)^r] u, x] - Simp[f*m*r Int[x^(m - 1)/(e + f*x^m) u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m , n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q]
\[\int x^{2} {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )d x\]
\[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \]
integral(b^2*x^2*log(d*f*x^2 + 1)*log(c*x^n)^2 + 2*a*b*x^2*log(d*f*x^2 + 1 )*log(c*x^n) + a^2*x^2*log(d*f*x^2 + 1), x)
Timed out. \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\text {Timed out} \]
\[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \]
1/27*(9*b^2*x^3*log(x^n)^2 - 6*(b^2*(n - 3*log(c)) - 3*a*b)*x^3*log(x^n) + ((2*n^2 - 6*n*log(c) + 9*log(c)^2)*b^2 - 6*a*b*(n - 3*log(c)) + 9*a^2)*x^ 3)*log(d*f*x^2 + 1) - integrate(2/27*(9*b^2*d*f*x^4*log(x^n)^2 + 6*(3*a*b* d*f - (d*f*n - 3*d*f*log(c))*b^2)*x^4*log(x^n) + (9*a^2*d*f - 6*(d*f*n - 3 *d*f*log(c))*a*b + (2*d*f*n^2 - 6*d*f*n*log(c) + 9*d*f*log(c)^2)*b^2)*x^4) /(d*f*x^2 + 1), x)
\[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right ) \,d x } \]
Timed out. \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx=\int x^2\,\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]