Integrand size = 28, antiderivative size = 356 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^5} \, dx=-\frac {7 b^2 f m n^2}{32 e x^2}-\frac {b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}+\frac {b f^2 m n \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}+\frac {f^2 m \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}+\frac {b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac {b^2 f^2 m n^2 \operatorname {PolyLog}\left (2,-\frac {e}{f x^2}\right )}{16 e^2}-\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e}{f x^2}\right )}{4 e^2}-\frac {b^2 f^2 m n^2 \operatorname {PolyLog}\left (3,-\frac {e}{f x^2}\right )}{8 e^2} \]
-7/32*b^2*f*m*n^2/e/x^2-1/16*b^2*f^2*m*n^2*ln(x)/e^2-3/8*b*f*m*n*(a+b*ln(c *x^n))/e/x^2+1/8*b*f^2*m*n*ln(1+e/f/x^2)*(a+b*ln(c*x^n))/e^2-1/4*f*m*(a+b* ln(c*x^n))^2/e/x^2+1/4*f^2*m*ln(1+e/f/x^2)*(a+b*ln(c*x^n))^2/e^2+1/32*b^2* f^2*m*n^2*ln(f*x^2+e)/e^2-1/32*b^2*n^2*ln(d*(f*x^2+e)^m)/x^4-1/8*b*n*(a+b* ln(c*x^n))*ln(d*(f*x^2+e)^m)/x^4-1/4*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m)/x ^4-1/16*b^2*f^2*m*n^2*polylog(2,-e/f/x^2)/e^2-1/4*b*f^2*m*n*(a+b*ln(c*x^n) )*polylog(2,-e/f/x^2)/e^2-1/8*b^2*f^2*m*n^2*polylog(3,-e/f/x^2)/e^2
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1111, normalized size of antiderivative = 3.12 \[ \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^5} \, dx=-\frac {24 a^2 e f m x^2+36 a b e f m n x^2+21 b^2 e f m n^2 x^2+48 a^2 f^2 m x^4 \log (x)+24 a b f^2 m n x^4 \log (x)+6 b^2 f^2 m n^2 x^4 \log (x)-48 a b f^2 m n x^4 \log ^2(x)-12 b^2 f^2 m n^2 x^4 \log ^2(x)+16 b^2 f^2 m n^2 x^4 \log ^3(x)+48 a b e f m x^2 \log \left (c x^n\right )+36 b^2 e f m n x^2 \log \left (c x^n\right )+96 a b f^2 m x^4 \log (x) \log \left (c x^n\right )+24 b^2 f^2 m n x^4 \log (x) \log \left (c x^n\right )-48 b^2 f^2 m n x^4 \log ^2(x) \log \left (c x^n\right )+24 b^2 e f m x^2 \log ^2\left (c x^n\right )+48 b^2 f^2 m x^4 \log (x) \log ^2\left (c x^n\right )-48 a b f^2 m n x^4 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 b^2 f^2 m n^2 x^4 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+24 b^2 f^2 m n^2 x^4 \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-48 b^2 f^2 m n x^4 \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-48 a b f^2 m n x^4 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 b^2 f^2 m n^2 x^4 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+24 b^2 f^2 m n^2 x^4 \log ^2(x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-48 b^2 f^2 m n x^4 \log (x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-24 a^2 f^2 m x^4 \log \left (e+f x^2\right )-12 a b f^2 m n x^4 \log \left (e+f x^2\right )-3 b^2 f^2 m n^2 x^4 \log \left (e+f x^2\right )+48 a b f^2 m n x^4 \log (x) \log \left (e+f x^2\right )+12 b^2 f^2 m n^2 x^4 \log (x) \log \left (e+f x^2\right )-24 b^2 f^2 m n^2 x^4 \log ^2(x) \log \left (e+f x^2\right )-48 a b f^2 m x^4 \log \left (c x^n\right ) \log \left (e+f x^2\right )-12 b^2 f^2 m n x^4 \log \left (c x^n\right ) \log \left (e+f x^2\right )+48 b^2 f^2 m n x^4 \log (x) \log \left (c x^n\right ) \log \left (e+f x^2\right )-24 b^2 f^2 m x^4 \log ^2\left (c x^n\right ) \log \left (e+f x^2\right )+24 a^2 e^2 \log \left (d \left (e+f x^2\right )^m\right )+12 a b e^2 n \log \left (d \left (e+f x^2\right )^m\right )+3 b^2 e^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )+48 a b e^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+12 b^2 e^2 n \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+24 b^2 e^2 \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-12 b f^2 m n x^4 \left (4 a+b n+4 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-12 b f^2 m n x^4 \left (4 a+b n+4 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )+48 b^2 f^2 m n^2 x^4 \operatorname {PolyLog}\left (3,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+48 b^2 f^2 m n^2 x^4 \operatorname {PolyLog}\left (3,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{96 e^2 x^4} \]
-1/96*(24*a^2*e*f*m*x^2 + 36*a*b*e*f*m*n*x^2 + 21*b^2*e*f*m*n^2*x^2 + 48*a ^2*f^2*m*x^4*Log[x] + 24*a*b*f^2*m*n*x^4*Log[x] + 6*b^2*f^2*m*n^2*x^4*Log[ x] - 48*a*b*f^2*m*n*x^4*Log[x]^2 - 12*b^2*f^2*m*n^2*x^4*Log[x]^2 + 16*b^2* f^2*m*n^2*x^4*Log[x]^3 + 48*a*b*e*f*m*x^2*Log[c*x^n] + 36*b^2*e*f*m*n*x^2* Log[c*x^n] + 96*a*b*f^2*m*x^4*Log[x]*Log[c*x^n] + 24*b^2*f^2*m*n*x^4*Log[x ]*Log[c*x^n] - 48*b^2*f^2*m*n*x^4*Log[x]^2*Log[c*x^n] + 24*b^2*e*f*m*x^2*L og[c*x^n]^2 + 48*b^2*f^2*m*x^4*Log[x]*Log[c*x^n]^2 - 48*a*b*f^2*m*n*x^4*Lo g[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 12*b^2*f^2*m*n^2*x^4*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 24*b^2*f^2*m*n^2*x^4*Log[x]^2*Log[1 - (I*Sqrt[f]* x)/Sqrt[e]] - 48*b^2*f^2*m*n*x^4*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/S qrt[e]] - 48*a*b*f^2*m*n*x^4*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 12*b^ 2*f^2*m*n^2*x^4*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 24*b^2*f^2*m*n^2*x ^4*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 48*b^2*f^2*m*n*x^4*Log[x]*Log [c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 24*a^2*f^2*m*x^4*Log[e + f*x^2] - 12*a*b*f^2*m*n*x^4*Log[e + f*x^2] - 3*b^2*f^2*m*n^2*x^4*Log[e + f*x^2] + 48*a*b*f^2*m*n*x^4*Log[x]*Log[e + f*x^2] + 12*b^2*f^2*m*n^2*x^4*Log[x]*Log [e + f*x^2] - 24*b^2*f^2*m*n^2*x^4*Log[x]^2*Log[e + f*x^2] - 48*a*b*f^2*m* x^4*Log[c*x^n]*Log[e + f*x^2] - 12*b^2*f^2*m*n*x^4*Log[c*x^n]*Log[e + f*x^ 2] + 48*b^2*f^2*m*n*x^4*Log[x]*Log[c*x^n]*Log[e + f*x^2] - 24*b^2*f^2*m*x^ 4*Log[c*x^n]^2*Log[e + f*x^2] + 24*a^2*e^2*Log[d*(e + f*x^2)^m] + 12*a*...
Time = 0.79 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.94, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^5} \, dx\) |
| \(\Big \downarrow \) 2825 |
| \(\displaystyle -2 f m \int \left (-\frac {b^2 n^2}{32 x^3 \left (f x^2+e\right )}-\frac {b \left (a+b \log \left (c x^n\right )\right ) n}{8 x^3 \left (f x^2+e\right )}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{4 x^3 \left (f x^2+e\right )}\right )dx-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 f m \left (\frac {b f n \operatorname {PolyLog}\left (2,-\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^2}-\frac {b f n \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac {f \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )}{16 e x^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{8 e x^2}+\frac {b^2 f n^2 \operatorname {PolyLog}\left (2,-\frac {e}{f x^2}\right )}{32 e^2}+\frac {b^2 f n^2 \operatorname {PolyLog}\left (3,-\frac {e}{f x^2}\right )}{16 e^2}-\frac {b^2 f n^2 \log \left (e+f x^2\right )}{64 e^2}+\frac {b^2 f n^2 \log (x)}{32 e^2}+\frac {7 b^2 n^2}{64 e x^2}\right )-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}\) |
-1/32*(b^2*n^2*Log[d*(e + f*x^2)^m])/x^4 - (b*n*(a + b*Log[c*x^n])*Log[d*( e + f*x^2)^m])/(8*x^4) - ((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/(4*x^ 4) - 2*f*m*((7*b^2*n^2)/(64*e*x^2) + (b^2*f*n^2*Log[x])/(32*e^2) + (3*b*n* (a + b*Log[c*x^n]))/(16*e*x^2) - (b*f*n*Log[1 + e/(f*x^2)]*(a + b*Log[c*x^ n]))/(16*e^2) + (a + b*Log[c*x^n])^2/(8*e*x^2) - (f*Log[1 + e/(f*x^2)]*(a + b*Log[c*x^n])^2)/(8*e^2) - (b^2*f*n^2*Log[e + f*x^2])/(64*e^2) + (b^2*f* n^2*PolyLog[2, -(e/(f*x^2))])/(32*e^2) + (b*f*n*(a + b*Log[c*x^n])*PolyLog [2, -(e/(f*x^2))])/(8*e^2) + (b^2*f*n^2*PolyLog[3, -(e/(f*x^2))])/(16*e^2) )
3.2.3.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 76.38 (sec) , antiderivative size = 6432, normalized size of antiderivative = 18.07
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^5} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^5} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^5} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{5}} \,d x } \]
-1/32*(8*b^2*log(x^n)^2 + (n^2 + 4*n*log(c) + 8*log(c)^2)*b^2 + 4*a*b*(n + 4*log(c)) + 8*a^2 + 4*(b^2*(n + 4*log(c)) + 4*a*b)*log(x^n))*log((f*x^2 + e)^m)/x^4 + integrate(1/16*(16*b^2*e*log(c)^2*log(d) + 32*a*b*e*log(c)*lo g(d) + 16*a^2*e*log(d) + (8*(f*m + 2*f*log(d))*a^2 + 4*(f*m*n + 4*(f*m + 2 *f*log(d))*log(c))*a*b + (f*m*n^2 + 4*f*m*n*log(c) + 8*(f*m + 2*f*log(d))* log(c)^2)*b^2)*x^2 + 8*((f*m + 2*f*log(d))*b^2*x^2 + 2*b^2*e*log(d))*log(x ^n)^2 + 4*(8*b^2*e*log(c)*log(d) + 8*a*b*e*log(d) + (4*(f*m + 2*f*log(d))* a*b + (f*m*n + 4*(f*m + 2*f*log(d))*log(c))*b^2)*x^2)*log(x^n))/(f*x^7 + e *x^5), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^5} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^5} \, dx=\int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^5} \,d x \]