Integrand size = 25, antiderivative size = 176 \[ \int \frac {\left (f+g x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {2 f g p x^n}{n}+\frac {d g^2 p x^n}{2 e n}-\frac {g^2 p x^{2 n}}{4 n}-\frac {d^2 g^2 p \log \left (d+e x^n\right )}{2 e^2 n}+\frac {g^2 x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {2 f g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{n} \]
-2*f*g*p*x^n/n+1/2*d*g^2*p*x^n/e/n-1/4*g^2*p*x^(2*n)/n-1/2*d^2*g^2*p*ln(d+ e*x^n)/e^2/n+1/2*g^2*x^(2*n)*ln(c*(d+e*x^n)^p)/n+2*f*g*(d+e*x^n)*ln(c*(d+e *x^n)^p)/e/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 = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.70 \[ \int \frac {\left (f+g x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {-e g p x^n \left (8 e f-2 d g+e g x^n\right )-2 d^2 g^2 p \log \left (d+e x^n\right )+2 e \left (4 d f g+e g x^n \left (4 f+g x^n\right )+2 e f^2 \log \left (-\frac {e x^n}{d}\right )\right ) \log \left (c \left (d+e x^n\right )^p\right )+4 e^2 f^2 p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{4 e^2 n} \]
(-(e*g*p*x^n*(8*e*f - 2*d*g + e*g*x^n)) - 2*d^2*g^2*p*Log[d + e*x^n] + 2*e *(4*d*f*g + e*g*x^n*(4*f + g*x^n) + 2*e*f^2*Log[-((e*x^n)/d)])*Log[c*(d + e*x^n)^p] + 4*e^2*f^2*p*PolyLog[2, 1 + (e*x^n)/d])/(4*e^2*n)
Time = 0.36 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.89, 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.
| \(\displaystyle \int \frac {\left (f+g x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx\) |
| \(\Big \downarrow \) 2925 |
| \(\displaystyle \frac {\int x^{-n} \left (g x^n+f\right )^2 \log \left (c \left (e x^n+d\right )^p\right )dx^n}{n}\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {\int \left (f^2 \log \left (c \left (e x^n+d\right )^p\right ) x^{-n}+g^2 \log \left (c \left (e x^n+d\right )^p\right ) x^n+2 f g \log \left (c \left (e x^n+d\right )^p\right )\right )dx^n}{n}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+\frac {2 f g \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e}+\frac {1}{2} g^2 x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )-\frac {d^2 g^2 p \log \left (d+e x^n\right )}{2 e^2}+f^2 p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )+\frac {d g^2 p x^n}{2 e}-2 f g p x^n-\frac {1}{4} g^2 p x^{2 n}}{n}\) |
(-2*f*g*p*x^n + (d*g^2*p*x^n)/(2*e) - (g^2*p*x^(2*n))/4 - (d^2*g^2*p*Log[d + e*x^n])/(2*e^2) + (g^2*x^(2*n)*Log[c*(d + e*x^n)^p])/2 + (2*f*g*(d + e* x^n)*Log[c*(d + e*x^n)^p])/e + 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
3.4.67.3.1 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 = 3.21 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.78
| method | result | size |
| risch | \(\frac {\left (2 f^{2} \ln \left (x \right ) n +g^{2} x^{2 n}+4 f g \,x^{n}\right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{2 n}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {g^{2} x^{2 n}}{2}+2 f g \,x^{n}+f^{2} \ln \left (x^{n}\right )\right )}{n}-\frac {g^{2} p \,x^{2 n}}{4 n}+\frac {d \,g^{2} p \,x^{n}}{2 e n}-\frac {d^{2} g^{2} p \ln \left (d +e \,x^{n}\right )}{2 e^{2} n}-\frac {p \,f^{2} \operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{n}-p \,f^{2} \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )-\frac {2 f g p \,x^{n}}{n}+\frac {2 p f g d \ln \left (d +e \,x^{n}\right )}{e n}\) | \(314\) |
1/2*(2*f^2*ln(x)*n+g^2*(x^n)^2+4*f*g*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)*csgn( 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*csgn( I*c*(d+e*x^n)^p)^2*csgn(I*c)+ln(c))/n*(1/2*g^2*(x^n)^2+2*f*g*x^n+f^2*ln(x^ n))-1/4*p/n*g^2*(x^n)^2+1/2*d*g^2*p*x^n/e/n-1/2*d^2*g^2*p*ln(d+e*x^n)/e^2/ n-p/n*f^2*dilog((d+e*x^n)/d)-p*f^2*ln(x)*ln((d+e*x^n)/d)-2*f*g*p*x^n/n+2*p /e/n*f*g*d*ln(d+e*x^n)
Time = 0.35 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.09 \[ \int \frac {\left (f+g x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {4 \, e^{2} f^{2} n p \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) - 4 \, e^{2} f^{2} n \log \left (c\right ) \log \left (x\right ) + 4 \, e^{2} f^{2} p {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + {\left (e^{2} g^{2} p - 2 \, e^{2} g^{2} \log \left (c\right )\right )} x^{2 \, n} - 2 \, {\left (4 \, e^{2} f g \log \left (c\right ) - {\left (4 \, e^{2} f g - d e g^{2}\right )} p\right )} x^{n} - 2 \, {\left (2 \, e^{2} f^{2} n p \log \left (x\right ) + e^{2} g^{2} p x^{2 \, n} + 4 \, e^{2} f g p x^{n} + {\left (4 \, d e f g - d^{2} g^{2}\right )} p\right )} \log \left (e x^{n} + d\right )}{4 \, e^{2} n} \]
-1/4*(4*e^2*f^2*n*p*log(x)*log((e*x^n + d)/d) - 4*e^2*f^2*n*log(c)*log(x) + 4*e^2*f^2*p*dilog(-(e*x^n + d)/d + 1) + (e^2*g^2*p - 2*e^2*g^2*log(c))*x ^(2*n) - 2*(4*e^2*f*g*log(c) - (4*e^2*f*g - d*e*g^2)*p)*x^n - 2*(2*e^2*f^2 *n*p*log(x) + e^2*g^2*p*x^(2*n) + 4*e^2*f*g*p*x^n + (4*d*e*f*g - d^2*g^2)* p)*log(e*x^n + d))/(e^2*n)
\[ \int \frac {\left (f+g x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\left (f + g x^{n}\right )^{2} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \]
\[ \int \frac {\left (f+g x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{n} + f\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \]
-1/4*(2*e^2*f^2*n^2*p*log(x)^2 + (e^2*g^2*p - 2*e^2*g^2*log(c))*x^(2*n) + 2*(4*e^2*f*g*p - d*e*g^2*p - 4*e^2*f*g*log(c))*x^n - 2*(2*e^2*f^2*n*log(x) + e^2*g^2*x^(2*n) + 4*e^2*f*g*x^n)*log((e*x^n + d)^p) - 2*(4*d*e*f*g*n*p - d^2*g^2*n*p + 2*e^2*f^2*n*log(c))*log(x))/(e^2*n) + integrate(1/2*(2*d*e ^2*f^2*n*p*log(x) - 4*d^2*e*f*g*p + d^3*g^2*p)/(e^3*x*x^n + d*e^2*x), x)
\[ \int \frac {\left (f+g x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{n} + f\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\left (f+g x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f+g\,x^n\right )}^2}{x} \,d x \]