Integrand size = 28, antiderivative size = 571 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=-\frac {52 b^2 f m n^2}{27 e x}-\frac {4 b^2 f^{3/2} m n^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 e^{3/2}}-\frac {16 b f m n \left (a+b \log \left (c x^n\right )\right )}{9 e x}-\frac {4 b f^{3/2} m n \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^{3/2}}-\frac {2 f m \left (a+b \log \left (c x^n\right )\right )^2}{3 e x}+\frac {f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac {2 b f^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}+\frac {2 b f^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}+\frac {2 i b^2 f^{3/2} m n^2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 e^{3/2}}-\frac {2 i b^2 f^{3/2} m n^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 e^{3/2}}+\frac {2 b^2 f^{3/2} m n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {2 b^2 f^{3/2} m n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}} \]
-52/27*b^2*f*m*n^2/e/x-4/27*b^2*f^(3/2)*m*n^2*arctan(x*f^(1/2)/e^(1/2))/e^ (3/2)-16/9*b*f*m*n*(a+b*ln(c*x^n))/e/x-4/9*b*f^(3/2)*m*n*arctan(x*f^(1/2)/ e^(1/2))*(a+b*ln(c*x^n))/e^(3/2)-2/3*f*m*(a+b*ln(c*x^n))^2/e/x-2/27*b^2*n^ 2*ln(d*(f*x^2+e)^m)/x^3-2/9*b*n*(a+b*ln(c*x^n))*ln(d*(f*x^2+e)^m)/x^3-1/3* (a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m)/x^3+1/3*f^(3/2)*m*(a+b*ln(c*x^n))^2*ln (1-x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-1/3*f^(3/2)*m*(a+b*ln(c*x^n))^2*ln(1+x *f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-2/3*b*f^(3/2)*m*n*(a+b*ln(c*x^n))*polylog( 2,-x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)+2/3*b*f^(3/2)*m*n*(a+b*ln(c*x^n))*poly log(2,x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-2/9*I*b^2*f^(3/2)*m*n^2*polylog(2,I *x*f^(1/2)/e^(1/2))/e^(3/2)+2/9*I*b^2*f^(3/2)*m*n^2*polylog(2,-I*x*f^(1/2) /e^(1/2))/e^(3/2)+2/3*b^2*f^(3/2)*m*n^2*polylog(3,-x*f^(1/2)/(-e)^(1/2))/( -e)^(3/2)-2/3*b^2*f^(3/2)*m*n^2*polylog(3,x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)
Time = 0.28 (sec) , antiderivative size = 1083, normalized size of antiderivative = 1.90 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\frac {-18 a^2 \sqrt {e} f m x^2-48 a b \sqrt {e} f m n x^2-52 b^2 \sqrt {e} f m n^2 x^2-18 a^2 f^{3/2} m x^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-12 a b f^{3/2} m n x^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-4 b^2 f^{3/2} m n^2 x^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )+36 a b f^{3/2} m n x^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)+12 b^2 f^{3/2} m n^2 x^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)-18 b^2 f^{3/2} m n^2 x^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2(x)-36 a b \sqrt {e} f m x^2 \log \left (c x^n\right )-48 b^2 \sqrt {e} f m n x^2 \log \left (c x^n\right )-36 a b f^{3/2} m x^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )-12 b^2 f^{3/2} m n x^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )+36 b^2 f^{3/2} m n x^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x) \log \left (c x^n\right )-18 b^2 \sqrt {e} f m x^2 \log ^2\left (c x^n\right )-18 b^2 f^{3/2} m x^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2\left (c x^n\right )-18 i a b f^{3/2} m n x^3 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-6 i b^2 f^{3/2} m n^2 x^3 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+9 i b^2 f^{3/2} m n^2 x^3 \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-18 i b^2 f^{3/2} m n x^3 \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+18 i a b f^{3/2} m n x^3 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+6 i b^2 f^{3/2} m n^2 x^3 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-9 i b^2 f^{3/2} m n^2 x^3 \log ^2(x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+18 i b^2 f^{3/2} m n x^3 \log (x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-9 a^2 e^{3/2} \log \left (d \left (e+f x^2\right )^m\right )-6 a b e^{3/2} n \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 e^{3/2} n^2 \log \left (d \left (e+f x^2\right )^m\right )-18 a b e^{3/2} \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-6 b^2 e^{3/2} n \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-9 b^2 e^{3/2} \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+6 i b f^{3/2} m n x^3 \left (3 a+b n+3 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-6 i b f^{3/2} m n x^3 \left (3 a+b n+3 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )-18 i b^2 f^{3/2} m n^2 x^3 \operatorname {PolyLog}\left (3,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+18 i b^2 f^{3/2} m n^2 x^3 \operatorname {PolyLog}\left (3,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{27 e^{3/2} x^3} \]
(-18*a^2*Sqrt[e]*f*m*x^2 - 48*a*b*Sqrt[e]*f*m*n*x^2 - 52*b^2*Sqrt[e]*f*m*n ^2*x^2 - 18*a^2*f^(3/2)*m*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 12*a*b*f^(3/2) *m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 4*b^2*f^(3/2)*m*n^2*x^3*ArcTan[(Sqr t[f]*x)/Sqrt[e]] + 36*a*b*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[ x] + 12*b^2*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 18*b^2* f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 - 36*a*b*Sqrt[e]*f* m*x^2*Log[c*x^n] - 48*b^2*Sqrt[e]*f*m*n*x^2*Log[c*x^n] - 36*a*b*f^(3/2)*m* x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] - 12*b^2*f^(3/2)*m*n*x^3*ArcTan [(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 36*b^2*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]* x)/Sqrt[e]]*Log[x]*Log[c*x^n] - 18*b^2*Sqrt[e]*f*m*x^2*Log[c*x^n]^2 - 18*b ^2*f^(3/2)*m*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 - (18*I)*a*b*f^( 3/2)*m*n*x^3*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (6*I)*b^2*f^(3/2)*m*n ^2*x^3*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (9*I)*b^2*f^(3/2)*m*n^2*x^3 *Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - (18*I)*b^2*f^(3/2)*m*n*x^3*Log[ x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (18*I)*a*b*f^(3/2)*m*n*x^3* Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + (6*I)*b^2*f^(3/2)*m*n^2*x^3*Log[x] *Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - (9*I)*b^2*f^(3/2)*m*n^2*x^3*Log[x]^2*Log [1 + (I*Sqrt[f]*x)/Sqrt[e]] + (18*I)*b^2*f^(3/2)*m*n*x^3*Log[x]*Log[c*x^n] *Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 9*a^2*e^(3/2)*Log[d*(e + f*x^2)^m] - 6*a *b*e^(3/2)*n*Log[d*(e + f*x^2)^m] - 2*b^2*e^(3/2)*n^2*Log[d*(e + f*x^2)...
Time = 0.98 (sec) , antiderivative size = 560, normalized size of antiderivative = 0.98, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 2825 |
| \(\displaystyle -2 f m \int \left (-\frac {2 b^2 n^2}{27 x^2 \left (f x^2+e\right )}-\frac {2 b \left (a+b \log \left (c x^n\right )\right ) n}{9 x^2 \left (f x^2+e\right )}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 x^2 \left (f x^2+e\right )}\right )dx-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 f m \left (\frac {2 b \sqrt {f} n \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^{3/2}}+\frac {b \sqrt {f} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 (-e)^{3/2}}-\frac {b \sqrt {f} n \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 (-e)^{3/2}}-\frac {\sqrt {f} \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{6 (-e)^{3/2}}+\frac {\sqrt {f} \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{6 (-e)^{3/2}}+\frac {8 b n \left (a+b \log \left (c x^n\right )\right )}{9 e x}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e x}+\frac {2 b^2 \sqrt {f} n^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 e^{3/2}}-\frac {i b^2 \sqrt {f} n^2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 e^{3/2}}+\frac {i b^2 \sqrt {f} n^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 e^{3/2}}-\frac {b^2 \sqrt {f} n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}+\frac {b^2 \sqrt {f} n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}+\frac {26 b^2 n^2}{27 e x}\right )-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac {2 b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}\) |
(-2*b^2*n^2*Log[d*(e + f*x^2)^m])/(27*x^3) - (2*b*n*(a + b*Log[c*x^n])*Log [d*(e + f*x^2)^m])/(9*x^3) - ((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/( 3*x^3) - 2*f*m*((26*b^2*n^2)/(27*e*x) + (2*b^2*Sqrt[f]*n^2*ArcTan[(Sqrt[f] *x)/Sqrt[e]])/(27*e^(3/2)) + (8*b*n*(a + b*Log[c*x^n]))/(9*e*x) + (2*b*Sqr t[f]*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(9*e^(3/2)) + (a + b*Log[c*x^n])^2/(3*e*x) - (Sqrt[f]*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x )/Sqrt[-e]])/(6*(-e)^(3/2)) + (Sqrt[f]*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[ f]*x)/Sqrt[-e]])/(6*(-e)^(3/2)) + (b*Sqrt[f]*n*(a + b*Log[c*x^n])*PolyLog[ 2, -((Sqrt[f]*x)/Sqrt[-e])])/(3*(-e)^(3/2)) - (b*Sqrt[f]*n*(a + b*Log[c*x^ n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) - ((I/9)*b^2*Sqrt[f]* n^2*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/e^(3/2) + ((I/9)*b^2*Sqrt[f]*n^2 *PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/e^(3/2) - (b^2*Sqrt[f]*n^2*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(3*(-e)^(3/2)) + (b^2*Sqrt[f]*n^2*PolyLog[3, (Sq rt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)))
3.2.7.3.1 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]
\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{x^{4}}d x\]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^4} \,d x \]