Integrand size = 24, antiderivative size = 372 \[ \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {2 d^2 p^2 x^{1-2 n} (f x)^{-1+3 n}}{e^2 n}-\frac {d p^2 x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^2}{2 e^3 n}+\frac {2 p^2 x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^3}{27 e^3 n}-\frac {d^3 p^2 x^{1-3 n} (f x)^{-1+3 n} \log ^2\left (d+e x^n\right )}{3 e^3 n}-\frac {2 d^2 p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}+\frac {d p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{e^3 n}-\frac {2 p x^{1-3 n} (f x)^{-1+3 n} \left (d+e x^n\right )^3 \log \left (c \left (d+e x^n\right )^p\right )}{9 e^3 n}+\frac {2 d^3 p x^{1-3 n} (f x)^{-1+3 n} \log \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{3 e^3 n}+\frac {x (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{3 n} \]
2*d^2*p^2*x^(1-2*n)*(f*x)^(-1+3*n)/e^2/n-1/2*d*p^2*x^(1-3*n)*(f*x)^(-1+3*n )*(d+e*x^n)^2/e^3/n+2/27*p^2*x^(1-3*n)*(f*x)^(-1+3*n)*(d+e*x^n)^3/e^3/n-1/ 3*d^3*p^2*x^(1-3*n)*(f*x)^(-1+3*n)*ln(d+e*x^n)^2/e^3/n-2*d^2*p*x^(1-3*n)*( f*x)^(-1+3*n)*(d+e*x^n)*ln(c*(d+e*x^n)^p)/e^3/n+d*p*x^(1-3*n)*(f*x)^(-1+3* n)*(d+e*x^n)^2*ln(c*(d+e*x^n)^p)/e^3/n-2/9*p*x^(1-3*n)*(f*x)^(-1+3*n)*(d+e *x^n)^3*ln(c*(d+e*x^n)^p)/e^3/n+2/3*d^3*p*x^(1-3*n)*(f*x)^(-1+3*n)*ln(d+e* x^n)*ln(c*(d+e*x^n)^p)/e^3/n+1/3*x*(f*x)^(-1+3*n)*ln(c*(d+e*x^n)^p)^2/n
Time = 0.17 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.46 \[ \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {x^{-3 n} (f x)^{3 n} \left (-18 d^3 p^2 \log ^2\left (d+e x^n\right )+6 d^3 p \log \left (d+e x^n\right ) \left (-11 p+6 \log \left (c \left (d+e x^n\right )^p\right )\right )+e x^n \left (p^2 \left (66 d^2-15 d e x^n+4 e^2 x^{2 n}\right )-6 p \left (6 d^2-3 d e x^n+2 e^2 x^{2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )+18 e^2 x^{2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )\right )\right )}{54 e^3 f n} \]
((f*x)^(3*n)*(-18*d^3*p^2*Log[d + e*x^n]^2 + 6*d^3*p*Log[d + e*x^n]*(-11*p + 6*Log[c*(d + e*x^n)^p]) + e*x^n*(p^2*(66*d^2 - 15*d*e*x^n + 4*e^2*x^(2* n)) - 6*p*(6*d^2 - 3*d*e*x^n + 2*e^2*x^(2*n))*Log[c*(d + e*x^n)^p] + 18*e^ 2*x^(2*n)*Log[c*(d + e*x^n)^p]^2)))/(54*e^3*f*n*x^(3*n))
Time = 0.49 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.53, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2906, 2904, 2845, 2858, 25, 27, 2772, 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.
\(\displaystyle \int (f x)^{3 n-1} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx\) |
\(\Big \downarrow \) 2906 |
\(\displaystyle x^{1-3 n} (f x)^{3 n-1} \int x^{3 n-1} \log ^2\left (c \left (e x^n+d\right )^p\right )dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle \frac {x^{1-3 n} (f x)^{3 n-1} \int x^{2 n} \log ^2\left (c \left (e x^n+d\right )^p\right )dx^n}{n}\) |
\(\Big \downarrow \) 2845 |
\(\displaystyle \frac {x^{1-3 n} (f x)^{3 n-1} \left (\frac {1}{3} x^{3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )-\frac {2}{3} e p \int \frac {x^{3 n} \log \left (c \left (e x^n+d\right )^p\right )}{e x^n+d}dx^n\right )}{n}\) |
\(\Big \downarrow \) 2858 |
\(\displaystyle \frac {x^{1-3 n} (f x)^{3 n-1} \left (\frac {1}{3} x^{3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )-\frac {2}{3} p \int x^{2 n} \log \left (c \left (e x^n+d\right )^p\right )d\left (e x^n+d\right )\right )}{n}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {x^{1-3 n} (f x)^{3 n-1} \left (\frac {2}{3} p \int -x^{2 n} \log \left (c \left (e x^n+d\right )^p\right )d\left (e x^n+d\right )+\frac {1}{3} x^{3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^{1-3 n} (f x)^{3 n-1} \left (\frac {2 p \int -e^3 x^{2 n} \log \left (c \left (e x^n+d\right )^p\right )d\left (e x^n+d\right )}{3 e^3}+\frac {1}{3} x^{3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\) |
\(\Big \downarrow \) 2772 |
\(\displaystyle \frac {x^{1-3 n} (f x)^{3 n-1} \left (\frac {2 p \left (-p \int \left (d^3 \log \left (e x^n+d\right ) x^{-n}-\frac {x^{2 n}}{3}-3 d^2+\frac {3}{2} d \left (e x^n+d\right )\right )d\left (e x^n+d\right )+d^3 \log \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )-3 d^2 \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )+\frac {3}{2} d x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )-\frac {1}{3} x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )\right )}{3 e^3}+\frac {1}{3} x^{3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^{1-3 n} (f x)^{3 n-1} \left (\frac {2 p \left (d^3 \log \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )-3 d^2 \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )+\frac {3}{2} d x^{2 n} \log \left (c \left (d+e x^n\right )^p\right )-\frac {1}{3} x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )-p \left (\frac {1}{2} d^3 \log ^2\left (d+e x^n\right )-3 d^2 \left (d+e x^n\right )+\frac {3}{4} d x^{2 n}-\frac {x^{3 n}}{9}\right )\right )}{3 e^3}+\frac {1}{3} x^{3 n} \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\) |
(x^(1 - 3*n)*(f*x)^(-1 + 3*n)*((x^(3*n)*Log[c*(d + e*x^n)^p]^2)/3 + (2*p*( -(p*((3*d*x^(2*n))/4 - x^(3*n)/9 - 3*d^2*(d + e*x^n) + (d^3*Log[d + e*x^n] ^2)/2)) + (3*d*x^(2*n)*Log[c*(d + e*x^n)^p])/2 - (x^(3*n)*Log[c*(d + e*x^n )^p])/3 - 3*d^2*(d + e*x^n)*Log[c*(d + e*x^n)^p] + d^3*Log[d + e*x^n]*Log[ c*(d + e*x^n)^p]))/(3*e^3)))/n
3.2.63.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ .))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[m, -1])
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1))) Int[(f + g*x)^(q + 1) *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & & !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_)*( x_))^(m_), x_Symbol] :> Simp[(f*x)^m/x^m Int[x^m*(a + b*Log[c*(d + e*x^n) ^p])^q, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && IntegerQ[Simp lify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])
\[\int \left (f x \right )^{-1+3 n} {\ln \left (c \left (d +e \,x^{n}\right )^{p}\right )}^{2}d x\]
Time = 0.36 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.72 \[ \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {2 \, {\left (2 \, e^{3} p^{2} - 6 \, e^{3} p \log \left (c\right ) + 9 \, e^{3} \log \left (c\right )^{2}\right )} f^{3 \, n - 1} x^{3 \, n} - 3 \, {\left (5 \, d e^{2} p^{2} - 6 \, d e^{2} p \log \left (c\right )\right )} f^{3 \, n - 1} x^{2 \, n} + 6 \, {\left (11 \, d^{2} e p^{2} - 6 \, d^{2} e p \log \left (c\right )\right )} f^{3 \, n - 1} x^{n} + 18 \, {\left (e^{3} f^{3 \, n - 1} p^{2} x^{3 \, n} + d^{3} f^{3 \, n - 1} p^{2}\right )} \log \left (e x^{n} + d\right )^{2} + 6 \, {\left (3 \, d e^{2} f^{3 \, n - 1} p^{2} x^{2 \, n} - 6 \, d^{2} e f^{3 \, n - 1} p^{2} x^{n} - 2 \, {\left (e^{3} p^{2} - 3 \, e^{3} p \log \left (c\right )\right )} f^{3 \, n - 1} x^{3 \, n} - {\left (11 \, d^{3} p^{2} - 6 \, d^{3} p \log \left (c\right )\right )} f^{3 \, n - 1}\right )} \log \left (e x^{n} + d\right )}{54 \, e^{3} n} \]
1/54*(2*(2*e^3*p^2 - 6*e^3*p*log(c) + 9*e^3*log(c)^2)*f^(3*n - 1)*x^(3*n) - 3*(5*d*e^2*p^2 - 6*d*e^2*p*log(c))*f^(3*n - 1)*x^(2*n) + 6*(11*d^2*e*p^2 - 6*d^2*e*p*log(c))*f^(3*n - 1)*x^n + 18*(e^3*f^(3*n - 1)*p^2*x^(3*n) + d ^3*f^(3*n - 1)*p^2)*log(e*x^n + d)^2 + 6*(3*d*e^2*f^(3*n - 1)*p^2*x^(2*n) - 6*d^2*e*f^(3*n - 1)*p^2*x^n - 2*(e^3*p^2 - 3*e^3*p*log(c))*f^(3*n - 1)*x ^(3*n) - (11*d^3*p^2 - 6*d^3*p*log(c))*f^(3*n - 1))*log(e*x^n + d))/(e^3*n )
\[ \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int \left (f x\right )^{3 n - 1} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}^{2}\, dx \]
Time = 0.21 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.64 \[ \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {e p {\left (\frac {6 \, d^{3} f^{3 \, n} \log \left (\frac {e x^{n} + d}{e}\right )}{e^{4} n} - \frac {2 \, e^{2} f^{3 \, n} x^{3 \, n} - 3 \, d e f^{3 \, n} x^{2 \, n} + 6 \, d^{2} f^{3 \, n} x^{n}}{e^{3} n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{9 \, f} + \frac {\left (f x\right )^{3 \, n} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2}}{3 \, f n} - \frac {{\left (18 \, d^{3} f^{3 \, n} \log \left (e x^{n} + d\right )^{2} - 4 \, e^{3} f^{3 \, n} x^{3 \, n} + 15 \, d e^{2} f^{3 \, n} x^{2 \, n} - 66 \, d^{2} e f^{3 \, n} x^{n} - 6 \, {\left (6 \, f^{3 \, n} \log \left (e\right ) - 11 \, f^{3 \, n}\right )} d^{3} \log \left (e x^{n} + d\right )\right )} p^{2}}{54 \, e^{3} f n} \]
1/9*e*p*(6*d^3*f^(3*n)*log((e*x^n + d)/e)/(e^4*n) - (2*e^2*f^(3*n)*x^(3*n) - 3*d*e*f^(3*n)*x^(2*n) + 6*d^2*f^(3*n)*x^n)/(e^3*n))*log((e*x^n + d)^p*c )/f + 1/3*(f*x)^(3*n)*log((e*x^n + d)^p*c)^2/(f*n) - 1/54*(18*d^3*f^(3*n)* log(e*x^n + d)^2 - 4*e^3*f^(3*n)*x^(3*n) + 15*d*e^2*f^(3*n)*x^(2*n) - 66*d ^2*e*f^(3*n)*x^n - 6*(6*f^(3*n)*log(e) - 11*f^(3*n))*d^3*log(e*x^n + d))*p ^2/(e^3*f*n)
\[ \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{3 \, n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2} \,d x } \]
Timed out. \[ \int (f x)^{-1+3 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^2\,{\left (f\,x\right )}^{3\,n-1} \,d x \]