Integrand size = 27, antiderivative size = 327 \[ \int \frac {\left (f+g x^{3 n}\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {2 d^2 f g p x^n}{3 e^2 n}+\frac {d^5 g^2 p x^n}{6 e^5 n}+\frac {d f g p x^{2 n}}{3 e n}-\frac {d^4 g^2 p x^{2 n}}{12 e^4 n}-\frac {2 f g p x^{3 n}}{9 n}+\frac {d^3 g^2 p x^{3 n}}{18 e^3 n}-\frac {d^2 g^2 p x^{4 n}}{24 e^2 n}+\frac {d g^2 p x^{5 n}}{30 e n}-\frac {g^2 p x^{6 n}}{36 n}+\frac {2 d^3 f g p \log \left (d+e x^n\right )}{3 e^3 n}-\frac {d^6 g^2 p \log \left (d+e x^n\right )}{6 e^6 n}+\frac {2 f g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {g^2 x^{6 n} \log \left (c \left (d+e x^n\right )^p\right )}{6 n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{n} \]
-2/3*d^2*f*g*p*x^n/e^2/n+1/6*d^5*g^2*p*x^n/e^5/n+1/3*d*f*g*p*x^(2*n)/e/n-1 /12*d^4*g^2*p*x^(2*n)/e^4/n-2/9*f*g*p*x^(3*n)/n+1/18*d^3*g^2*p*x^(3*n)/e^3 /n-1/24*d^2*g^2*p*x^(4*n)/e^2/n+1/30*d*g^2*p*x^(5*n)/e/n-1/36*g^2*p*x^(6*n )/n+2/3*d^3*f*g*p*ln(d+e*x^n)/e^3/n-1/6*d^6*g^2*p*ln(d+e*x^n)/e^6/n+2/3*f* g*x^(3*n)*ln(c*(d+e*x^n)^p)/n+1/6*g^2*x^(6*n)*ln(c*(d+e*x^n)^p)/n+f^2*ln(- e*x^n/d)*ln(c*(d+e*x^n)^p)/n+f^2*p*polylog(2,1+e*x^n/d)/n
Time = 0.20 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.64 \[ \int \frac {\left (f+g x^{3 n}\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {-e g p x^n \left (-60 d^5 g+30 d^4 e g x^n-20 d^3 e^2 g x^{2 n}+10 e^5 x^{2 n} \left (8 f+g x^{3 n}\right )-12 d e^4 x^n \left (10 f+g x^{3 n}\right )+15 d^2 e^3 \left (16 f+g x^{3 n}\right )\right )-60 d^3 g \left (-4 e^3 f+d^3 g\right ) p \log \left (d+e x^n\right )+60 e^6 \left (g x^{3 n} \left (4 f+g x^{3 n}\right )+6 f^2 \log \left (-\frac {e x^n}{d}\right )\right ) \log \left (c \left (d+e x^n\right )^p\right )+360 e^6 f^2 p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{360 e^6 n} \]
(-(e*g*p*x^n*(-60*d^5*g + 30*d^4*e*g*x^n - 20*d^3*e^2*g*x^(2*n) + 10*e^5*x ^(2*n)*(8*f + g*x^(3*n)) - 12*d*e^4*x^n*(10*f + g*x^(3*n)) + 15*d^2*e^3*(1 6*f + g*x^(3*n)))) - 60*d^3*g*(-4*e^3*f + d^3*g)*p*Log[d + e*x^n] + 60*e^6 *(g*x^(3*n)*(4*f + g*x^(3*n)) + 6*f^2*Log[-((e*x^n)/d)])*Log[c*(d + e*x^n) ^p] + 360*e^6*f^2*p*PolyLog[2, 1 + (e*x^n)/d])/(360*e^6*n)
Time = 0.49 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.87, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
| \(\displaystyle \int \frac {\left (f+g x^{3 n}\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx\) |
| \(\Big \downarrow \) 2925 |
| \(\displaystyle \frac {\int x^{-n} \left (g x^{3 n}+f\right )^2 \log \left (c \left (e x^n+d\right )^p\right )dx^n}{n}\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {\int \left (f^2 \log \left (c \left (e x^n+d\right )^p\right ) x^{-n}+2 f g \log \left (c \left (e x^n+d\right )^p\right ) x^{2 n}+g^2 \log \left (c \left (e x^n+d\right )^p\right ) x^{5 n}\right )dx^n}{n}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+\frac {2}{3} f g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )+\frac {1}{6} g^2 x^{6 n} \log \left (c \left (d+e x^n\right )^p\right )-\frac {d^6 g^2 p \log \left (d+e x^n\right )}{6 e^6}+\frac {d^5 g^2 p x^n}{6 e^5}-\frac {d^4 g^2 p x^{2 n}}{12 e^4}+\frac {2 d^3 f g p \log \left (d+e x^n\right )}{3 e^3}+\frac {d^3 g^2 p x^{3 n}}{18 e^3}-\frac {2 d^2 f g p x^n}{3 e^2}-\frac {d^2 g^2 p x^{4 n}}{24 e^2}+f^2 p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )+\frac {d f g p x^{2 n}}{3 e}+\frac {d g^2 p x^{5 n}}{30 e}-\frac {2}{9} f g p x^{3 n}-\frac {1}{36} g^2 p x^{6 n}}{n}\) |
((-2*d^2*f*g*p*x^n)/(3*e^2) + (d^5*g^2*p*x^n)/(6*e^5) + (d*f*g*p*x^(2*n))/ (3*e) - (d^4*g^2*p*x^(2*n))/(12*e^4) - (2*f*g*p*x^(3*n))/9 + (d^3*g^2*p*x^ (3*n))/(18*e^3) - (d^2*g^2*p*x^(4*n))/(24*e^2) + (d*g^2*p*x^(5*n))/(30*e) - (g^2*p*x^(6*n))/36 + (2*d^3*f*g*p*Log[d + e*x^n])/(3*e^3) - (d^6*g^2*p*L og[d + e*x^n])/(6*e^6) + (2*f*g*x^(3*n)*Log[c*(d + e*x^n)^p])/3 + (g^2*x^( 6*n)*Log[c*(d + e*x^n)^p])/6 + f^2*Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p] + f^2*p*PolyLog[2, 1 + (e*x^n)/d])/n
3.4.65.3.1 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])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.08 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(\frac {\left (g^{2} x^{6 n}+4 f g \,x^{3 n}+6 f^{2} \ln \left (x \right ) n \right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{6 n}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {g^{2} x^{6 n}}{6}+\frac {2 f g \,x^{3 n}}{3}+f^{2} \ln \left (x^{n}\right )\right )}{n}-\frac {g^{2} p \,x^{6 n}}{36 n}+\frac {d \,g^{2} p \,x^{5 n}}{30 e n}-\frac {d^{2} g^{2} p \,x^{4 n}}{24 e^{2} n}+\frac {d^{3} g^{2} p \,x^{3 n}}{18 e^{3} n}-\frac {d^{4} g^{2} p \,x^{2 n}}{12 e^{4} n}+\frac {d^{5} g^{2} p \,x^{n}}{6 e^{5} n}-\frac {d^{6} g^{2} p \ln \left (d +e \,x^{n}\right )}{6 e^{6} n}-\frac {p \,f^{2} \operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{n}-p \,f^{2} \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )-\frac {2 f g p \,x^{3 n}}{9 n}+\frac {d f g p \,x^{2 n}}{3 e n}-\frac {2 d^{2} f g p \,x^{n}}{3 e^{2} n}+\frac {2 d^{3} f g p \ln \left (d +e \,x^{n}\right )}{3 e^{3} n}\) | \(436\) |
1/6*(g^2*(x^n)^6+4*f*g*(x^n)^3+6*f^2*ln(x)*n)/n*ln((d+e*x^n)^p)+(1/2*I*Pi* csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2-1/2*I*Pi*csgn(I*(d+e*x^n)^p)*c sgn(I*c*(d+e*x^n)^p)*csgn(I*c)-1/2*I*Pi*csgn(I*c*(d+e*x^n)^p)^3+1/2*I*Pi*c sgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)+ln(c))/n*(1/6*g^2*(x^n)^6+2/3*f*g*(x^n)^3 +f^2*ln(x^n))-1/36*p/n*g^2*(x^n)^6+1/30*p/e/n*g^2*d*(x^n)^5-1/24*p/e^2/n*g ^2*d^2*(x^n)^4+1/18*p/e^3/n*g^2*d^3*(x^n)^3-1/12*p/e^4/n*g^2*d^4*(x^n)^2+1 /6*d^5*g^2*p*x^n/e^5/n-1/6*d^6*g^2*p*ln(d+e*x^n)/e^6/n-p/n*f^2*dilog((d+e* x^n)/d)-p*f^2*ln(x)*ln((d+e*x^n)/d)-2/9*p/n*f*g*(x^n)^3+1/3*p/e/n*f*g*d*(x ^n)^2-2/3*d^2*f*g*p*x^n/e^2/n+2/3*d^3*f*g*p*ln(d+e*x^n)/e^3/n
Time = 0.38 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.89 \[ \int \frac {\left (f+g x^{3 n}\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {360 \, e^{6} f^{2} n p \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) - 360 \, e^{6} f^{2} n \log \left (c\right ) \log \left (x\right ) - 12 \, d e^{5} g^{2} p x^{5 \, n} + 15 \, d^{2} e^{4} g^{2} p x^{4 \, n} + 360 \, e^{6} f^{2} p {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) - 30 \, {\left (4 \, d e^{5} f g - d^{4} e^{2} g^{2}\right )} p x^{2 \, n} + 60 \, {\left (4 \, d^{2} e^{4} f g - d^{5} e g^{2}\right )} p x^{n} + 10 \, {\left (e^{6} g^{2} p - 6 \, e^{6} g^{2} \log \left (c\right )\right )} x^{6 \, n} - 20 \, {\left (12 \, e^{6} f g \log \left (c\right ) - {\left (4 \, e^{6} f g - d^{3} e^{3} g^{2}\right )} p\right )} x^{3 \, n} - 60 \, {\left (6 \, e^{6} f^{2} n p \log \left (x\right ) + e^{6} g^{2} p x^{6 \, n} + 4 \, e^{6} f g p x^{3 \, n} + {\left (4 \, d^{3} e^{3} f g - d^{6} g^{2}\right )} p\right )} \log \left (e x^{n} + d\right )}{360 \, e^{6} n} \]
-1/360*(360*e^6*f^2*n*p*log(x)*log((e*x^n + d)/d) - 360*e^6*f^2*n*log(c)*l og(x) - 12*d*e^5*g^2*p*x^(5*n) + 15*d^2*e^4*g^2*p*x^(4*n) + 360*e^6*f^2*p* dilog(-(e*x^n + d)/d + 1) - 30*(4*d*e^5*f*g - d^4*e^2*g^2)*p*x^(2*n) + 60* (4*d^2*e^4*f*g - d^5*e*g^2)*p*x^n + 10*(e^6*g^2*p - 6*e^6*g^2*log(c))*x^(6 *n) - 20*(12*e^6*f*g*log(c) - (4*e^6*f*g - d^3*e^3*g^2)*p)*x^(3*n) - 60*(6 *e^6*f^2*n*p*log(x) + e^6*g^2*p*x^(6*n) + 4*e^6*f*g*p*x^(3*n) + (4*d^3*e^3 *f*g - d^6*g^2)*p)*log(e*x^n + d))/(e^6*n)
\[ \int \frac {\left (f+g x^{3 n}\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\left (f + g x^{3 n}\right )^{2} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \]
\[ \int \frac {\left (f+g x^{3 n}\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{3 \, n} + f\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \]
-1/360*(180*e^6*f^2*n^2*p*log(x)^2 - 12*d*e^5*g^2*p*x^(5*n) + 15*d^2*e^4*g ^2*p*x^(4*n) + 10*(e^6*g^2*p - 6*e^6*g^2*log(c))*x^(6*n) + 20*(4*e^6*f*g*p - d^3*e^3*g^2*p - 12*e^6*f*g*log(c))*x^(3*n) - 30*(4*d*e^5*f*g*p - d^4*e^ 2*g^2*p)*x^(2*n) + 60*(4*d^2*e^4*f*g*p - d^5*e*g^2*p)*x^n - 60*(6*e^6*f^2* n*log(x) + e^6*g^2*x^(6*n) + 4*e^6*f*g*x^(3*n))*log((e*x^n + d)^p) - 60*(4 *d^3*e^3*f*g*n*p - d^6*g^2*n*p + 6*e^6*f^2*n*log(c))*log(x))/(e^6*n) + int egrate(1/6*(6*d*e^6*f^2*n*p*log(x) - 4*d^4*e^3*f*g*p + d^7*g^2*p)/(e^7*x*x ^n + d*e^6*x), x)
\[ \int \frac {\left (f+g x^{3 n}\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{3 \, n} + f\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\left (f+g x^{3 n}\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f+g\,x^{3\,n}\right )}^2}{x} \,d x \]