Integrand size = 25, antiderivative size = 172 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx=-\frac {e f^2 p}{4 d x^2}-\frac {e^2 f^2 p \log (x)}{2 d^2}+\frac {2 e f g p \log (x)}{d}+\frac {e^2 f^2 p \log \left (d+e x^2\right )}{4 d^2}-\frac {e f g p \log \left (d+e x^2\right )}{d}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{x^2}+\frac {1}{2} g^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} g^2 p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right ) \]
-1/4*e*f^2*p/d/x^2-1/2*e^2*f^2*p*ln(x)/d^2+2*e*f*g*p*ln(x)/d+1/4*e^2*f^2*p *ln(e*x^2+d)/d^2-e*f*g*p*ln(e*x^2+d)/d-1/4*f^2*ln(c*(e*x^2+d)^p)/x^4-f*g*l n(c*(e*x^2+d)^p)/x^2+1/2*g^2*ln(-e*x^2/d)*ln(c*(e*x^2+d)^p)+1/2*g^2*p*poly log(2,1+e*x^2/d)
Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.88 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx=\frac {1}{4} \left (\frac {8 e f g p \log (x)}{d}-\frac {4 e f g p \log \left (d+e x^2\right )}{d}-\frac {e f^2 p \left (d+2 e x^2 \log (x)-e x^2 \log \left (d+e x^2\right )\right )}{d^2 x^2}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^4}-\frac {4 f g \log \left (c \left (d+e x^2\right )^p\right )}{x^2}+2 g^2 \left (\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right )\right )\right ) \]
((8*e*f*g*p*Log[x])/d - (4*e*f*g*p*Log[d + e*x^2])/d - (e*f^2*p*(d + 2*e*x ^2*Log[x] - e*x^2*Log[d + e*x^2]))/(d^2*x^2) - (f^2*Log[c*(d + e*x^2)^p])/ x^4 - (4*f*g*Log[c*(d + e*x^2)^p])/x^2 + 2*g^2*(Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p] + p*PolyLog[2, 1 + (e*x^2)/d]))/4
Time = 0.39 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2925, 2863, 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.
| \(\displaystyle \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx\) |
| \(\Big \downarrow \) 2925 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (g x^2+f\right )^2 \log \left (c \left (e x^2+d\right )^p\right )}{x^6}dx^2\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\log \left (c \left (e x^2+d\right )^p\right ) f^2}{x^6}+\frac {2 g \log \left (c \left (e x^2+d\right )^p\right ) f}{x^4}+\frac {g^2 \log \left (c \left (e x^2+d\right )^p\right )}{x^2}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 x^4}-\frac {2 f g \log \left (c \left (d+e x^2\right )^p\right )}{x^2}+g^2 \log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )-\frac {e^2 f^2 p \log \left (x^2\right )}{2 d^2}+\frac {e^2 f^2 p \log \left (d+e x^2\right )}{2 d^2}-\frac {e f^2 p}{2 d x^2}+\frac {2 e f g p \log \left (x^2\right )}{d}-\frac {2 e f g p \log \left (d+e x^2\right )}{d}+g^2 p \operatorname {PolyLog}\left (2,\frac {e x^2}{d}+1\right )\right )\) |
(-1/2*(e*f^2*p)/(d*x^2) - (e^2*f^2*p*Log[x^2])/(2*d^2) + (2*e*f*g*p*Log[x^ 2])/d + (e^2*f^2*p*Log[d + e*x^2])/(2*d^2) - (2*e*f*g*p*Log[d + e*x^2])/d - (f^2*Log[c*(d + e*x^2)^p])/(2*x^4) - (2*f*g*Log[c*(d + e*x^2)^p])/x^2 + g^2*Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p] + g^2*p*PolyLog[2, 1 + (e*x^2)/ d])/2
3.4.28.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) ^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c , d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Si mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 ] || IGtQ[q, 0])
Time = 1.77 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.21
| method | result | size |
| parts | \(\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) g^{2} \ln \left (x \right )-\frac {f g \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{x^{2}}-\frac {f^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{4 x^{4}}-\frac {p e \left (4 g^{2} \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e}\right )-f \left (-\frac {\left (4 d g -e f \right ) \ln \left (e \,x^{2}+d \right )}{2 d^{2}}-\frac {f}{2 d \,x^{2}}+\frac {\left (4 d g -e f \right ) \ln \left (x \right )}{d^{2}}\right )\right )}{2}\) | \(208\) |
| risch | \(\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) g^{2} \ln \left (x \right )-\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f g}{x^{2}}-\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f^{2}}{4 x^{4}}-p \,g^{2} \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p \,g^{2} \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p \,g^{2} \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-p \,g^{2} \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )-\frac {e f g p \ln \left (e \,x^{2}+d \right )}{d}+\frac {e^{2} f^{2} p \ln \left (e \,x^{2}+d \right )}{4 d^{2}}-\frac {e \,f^{2} p}{4 d \,x^{2}}+\frac {2 e f g p \ln \left (x \right )}{d}-\frac {e^{2} f^{2} p \ln \left (x \right )}{2 d^{2}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (g^{2} \ln \left (x \right )-\frac {f g}{x^{2}}-\frac {f^{2}}{4 x^{4}}\right )\) | \(370\) |
ln(c*(e*x^2+d)^p)*g^2*ln(x)-f*g*ln(c*(e*x^2+d)^p)/x^2-1/4*f^2*ln(c*(e*x^2+ d)^p)/x^4-1/2*p*e*(4*g^2*(1/2*ln(x)*(ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+ ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2)))/e+1/2*(dilog((-e*x+(-d*e)^(1/2))/(-d* e)^(1/2))+dilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2)))/e)-f*(-1/2*(4*d*g-e*f)/d ^2*ln(e*x^2+d)-1/2*f/d/x^2+(4*d*g-e*f)/d^2*ln(x)))
\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx=\int { \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{5}} \,d x } \]
\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx=\int \frac {\left (f + g x^{2}\right )^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{x^{5}}\, dx \]
\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx=\int { \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{5}} \,d x } \]
\[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx=\int { \frac {{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,{\left (g\,x^2+f\right )}^2}{x^5} \,d x \]