Integrand size = 29, antiderivative size = 214 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3} \, dx=-\frac {b n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{d^2 m^2 \left (d+e x^m\right )}-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{2 e m \left (d+e x^m\right )^2}-\frac {b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-m}}{e}\right )}{d^2 e m^2}+\frac {b^2 n^2 x^{1-m} (f x)^{-1+m} \log \left (d+e x^m\right )}{d^2 e m^3}+\frac {b^2 n^2 x^{1-m} (f x)^{-1+m} \operatorname {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )}{d^2 e m^3} \]
-b*n*x*(f*x)^(-1+m)*(a+b*ln(c*x^n))/d^2/m^2/(d+e*x^m)-1/2*x^(1-m)*(f*x)^(- 1+m)*(a+b*ln(c*x^n))^2/e/m/(d+e*x^m)^2-b*n*x^(1-m)*(f*x)^(-1+m)*(a+b*ln(c* x^n))*ln(1+d/e/(x^m))/d^2/e/m^2+b^2*n^2*x^(1-m)*(f*x)^(-1+m)*ln(d+e*x^m)/d ^2/e/m^3+b^2*n^2*x^(1-m)*(f*x)^(-1+m)*polylog(2,-d/e/(x^m))/d^2/e/m^3
Time = 0.49 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.97 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3} \, dx=\frac {x^{-m} (f x)^m \left (\frac {2 b m n \left (a+b \log \left (c x^n\right )\right )}{d \left (d+e x^m\right )}-\frac {m^2 \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^2}-\frac {2 a b m n \log \left (d-d x^m\right )}{d^2}+\frac {2 b^2 n^2 \log \left (d-d x^m\right )}{d^2}+\frac {2 b^2 m n \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^m\right )}{d^2}+\frac {2 b^2 n^2 \left (\frac {1}{2} m^2 \log ^2(x)+\left (-m \log (x)+\log \left (-\frac {e x^m}{d}\right )\right ) \log \left (d+e x^m\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^m}{d}\right )\right )}{d^2}\right )}{2 e f m^3} \]
((f*x)^m*((2*b*m*n*(a + b*Log[c*x^n]))/(d*(d + e*x^m)) - (m^2*(a + b*Log[c *x^n])^2)/(d + e*x^m)^2 - (2*a*b*m*n*Log[d - d*x^m])/d^2 + (2*b^2*n^2*Log[ d - d*x^m])/d^2 + (2*b^2*m*n*(n*Log[x] - Log[c*x^n])*Log[d - d*x^m])/d^2 + (2*b^2*n^2*((m^2*Log[x]^2)/2 + (-(m*Log[x]) + Log[-((e*x^m)/d)])*Log[d + e*x^m] + PolyLog[2, 1 + (e*x^m)/d]))/d^2))/(2*e*f*m^3*x^m)
Time = 0.88 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.80, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2777, 2776, 2791, 2773, 792, 2779, 2838}
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.
| \(\displaystyle \int \frac {(f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3} \, dx\) |
| \(\Big \downarrow \) 2777 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \int \frac {x^{m-1} \left (a+b \log \left (c x^n\right )\right )^2}{\left (e x^m+d\right )^3}dx\) |
| \(\Big \downarrow \) 2776 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {b n \int \frac {a+b \log \left (c x^n\right )}{x \left (e x^m+d\right )^2}dx}{e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e m \left (d+e x^m\right )^2}\right )\) |
| \(\Big \downarrow \) 2791 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^m+d\right )}dx}{d}-\frac {e \int \frac {x^{m-1} \left (a+b \log \left (c x^n\right )\right )}{\left (e x^m+d\right )^2}dx}{d}\right )}{e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e m \left (d+e x^m\right )^2}\right )\) |
| \(\Big \downarrow \) 2773 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^m+d\right )}dx}{d}-\frac {e \left (\frac {x^m \left (a+b \log \left (c x^n\right )\right )}{d m \left (d+e x^m\right )}-\frac {b n \int \frac {x^{m-1}}{e x^m+d}dx}{d m}\right )}{d}\right )}{e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e m \left (d+e x^m\right )^2}\right )\) |
| \(\Big \downarrow \) 792 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (e x^m+d\right )}dx}{d}-\frac {e \left (\frac {x^m \left (a+b \log \left (c x^n\right )\right )}{d m \left (d+e x^m\right )}-\frac {b n \log \left (d+e x^m\right )}{d e m^2}\right )}{d}\right )}{e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e m \left (d+e x^m\right )^2}\right )\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {b n \left (\frac {\frac {b n \int \frac {\log \left (\frac {d x^{-m}}{e}+1\right )}{x}dx}{d m}-\frac {\log \left (\frac {d x^{-m}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d m}}{d}-\frac {e \left (\frac {x^m \left (a+b \log \left (c x^n\right )\right )}{d m \left (d+e x^m\right )}-\frac {b n \log \left (d+e x^m\right )}{d e m^2}\right )}{d}\right )}{e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e m \left (d+e x^m\right )^2}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle x^{1-m} (f x)^{m-1} \left (\frac {b n \left (\frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )}{d m^2}-\frac {\log \left (\frac {d x^{-m}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d m}}{d}-\frac {e \left (\frac {x^m \left (a+b \log \left (c x^n\right )\right )}{d m \left (d+e x^m\right )}-\frac {b n \log \left (d+e x^m\right )}{d e m^2}\right )}{d}\right )}{e m}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e m \left (d+e x^m\right )^2}\right )\) |
x^(1 - m)*(f*x)^(-1 + m)*(-1/2*(a + b*Log[c*x^n])^2/(e*m*(d + e*x^m)^2) + (b*n*(-((e*((x^m*(a + b*Log[c*x^n]))/(d*m*(d + e*x^m)) - (b*n*Log[d + e*x^ m])/(d*e*m^2)))/d) + (-(((a + b*Log[c*x^n])*Log[1 + d/(e*x^m)])/(d*m)) + ( b*n*PolyLog[2, -(d/(e*x^m))])/(d*m^2))/d))/(e*m))
3.4.65.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveConten t[a + b*x^n, x]]/(b*n), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/(d*f*(m + 1))), x] - Simp[b*(n/(d*(m + 1))) Int[(f*x)^m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && Eq Q[m + r*(q + 1) + 1, 0] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :> Simp[f^m*(d + e*x^r)^(q + 1)*((a + b*L og[c*x^n])^p/(e*r*(q + 1))), x] - Simp[b*f^m*n*(p/(e*r*(q + 1))) Int[(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d , e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || G tQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_)*(x_))^(m_.)*((d_) + ( e_.)*(x_)^(r_))^(q_.), x_Symbol] :> Simp[(f*x)^m/x^m Int[x^m*(d + e*x^r)^ q*(a + b*Log[c*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && !(IntegerQ[m] || GtQ[f, 0])
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^ (q_))/(x_), x_Symbol] :> Simp[1/d Int[(d + e*x^r)^(q + 1)*((a + b*Log[c*x ^n])^p/x), x], x] - Simp[e/d Int[x^(r - 1)*(d + e*x^r)^q*(a + b*Log[c*x^n ])^p, x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0] && ILtQ[q, -1 ]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
\[\int \frac {\left (f x \right )^{m -1} {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{\left (d +e \,x^{m}\right )^{3}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (211) = 422\).
Time = 0.30 (sec) , antiderivative size = 535, normalized size of antiderivative = 2.50 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3} \, dx=\frac {{\left (b^{2} e^{2} m^{2} n^{2} \log \left (x\right )^{2} + 2 \, {\left (b^{2} e^{2} m^{2} n \log \left (c\right ) + a b e^{2} m^{2} n - b^{2} e^{2} m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{2 \, m} + 2 \, {\left (b^{2} d e m^{2} n^{2} \log \left (x\right )^{2} + b^{2} d e m n \log \left (c\right ) + a b d e m n + {\left (2 \, b^{2} d e m^{2} n \log \left (c\right ) + 2 \, a b d e m^{2} n - b^{2} d e m n^{2}\right )} \log \left (x\right )\right )} f^{m - 1} x^{m} - {\left (b^{2} d^{2} m^{2} \log \left (c\right )^{2} + a^{2} d^{2} m^{2} - 2 \, a b d^{2} m n + 2 \, {\left (a b d^{2} m^{2} - b^{2} d^{2} m n\right )} \log \left (c\right )\right )} f^{m - 1} - 2 \, {\left (b^{2} e^{2} f^{m - 1} n^{2} x^{2 \, m} + 2 \, b^{2} d e f^{m - 1} n^{2} x^{m} + b^{2} d^{2} f^{m - 1} n^{2}\right )} {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) - 2 \, {\left ({\left (b^{2} e^{2} m n \log \left (c\right ) + a b e^{2} m n - b^{2} e^{2} n^{2}\right )} f^{m - 1} x^{2 \, m} + 2 \, {\left (b^{2} d e m n \log \left (c\right ) + a b d e m n - b^{2} d e n^{2}\right )} f^{m - 1} x^{m} + {\left (b^{2} d^{2} m n \log \left (c\right ) + a b d^{2} m n - b^{2} d^{2} n^{2}\right )} f^{m - 1}\right )} \log \left (e x^{m} + d\right ) - 2 \, {\left (b^{2} e^{2} f^{m - 1} m n^{2} x^{2 \, m} \log \left (x\right ) + 2 \, b^{2} d e f^{m - 1} m n^{2} x^{m} \log \left (x\right ) + b^{2} d^{2} f^{m - 1} m n^{2} \log \left (x\right )\right )} \log \left (\frac {e x^{m} + d}{d}\right )}{2 \, {\left (d^{2} e^{3} m^{3} x^{2 \, m} + 2 \, d^{3} e^{2} m^{3} x^{m} + d^{4} e m^{3}\right )}} \]
1/2*((b^2*e^2*m^2*n^2*log(x)^2 + 2*(b^2*e^2*m^2*n*log(c) + a*b*e^2*m^2*n - b^2*e^2*m*n^2)*log(x))*f^(m - 1)*x^(2*m) + 2*(b^2*d*e*m^2*n^2*log(x)^2 + b^2*d*e*m*n*log(c) + a*b*d*e*m*n + (2*b^2*d*e*m^2*n*log(c) + 2*a*b*d*e*m^2 *n - b^2*d*e*m*n^2)*log(x))*f^(m - 1)*x^m - (b^2*d^2*m^2*log(c)^2 + a^2*d^ 2*m^2 - 2*a*b*d^2*m*n + 2*(a*b*d^2*m^2 - b^2*d^2*m*n)*log(c))*f^(m - 1) - 2*(b^2*e^2*f^(m - 1)*n^2*x^(2*m) + 2*b^2*d*e*f^(m - 1)*n^2*x^m + b^2*d^2*f ^(m - 1)*n^2)*dilog(-(e*x^m + d)/d + 1) - 2*((b^2*e^2*m*n*log(c) + a*b*e^2 *m*n - b^2*e^2*n^2)*f^(m - 1)*x^(2*m) + 2*(b^2*d*e*m*n*log(c) + a*b*d*e*m* n - b^2*d*e*n^2)*f^(m - 1)*x^m + (b^2*d^2*m*n*log(c) + a*b*d^2*m*n - b^2*d ^2*n^2)*f^(m - 1))*log(e*x^m + d) - 2*(b^2*e^2*f^(m - 1)*m*n^2*x^(2*m)*log (x) + 2*b^2*d*e*f^(m - 1)*m*n^2*x^m*log(x) + b^2*d^2*f^(m - 1)*m*n^2*log(x ))*log((e*x^m + d)/d))/(d^2*e^3*m^3*x^(2*m) + 2*d^3*e^2*m^3*x^m + d^4*e*m^ 3)
Timed out. \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \left (f x\right )^{m - 1}}{{\left (e x^{m} + d\right )}^{3}} \,d x } \]
a*b*f^m*n*(1/((d*e^2*f*m*x^m + d^2*e*f*m)*m) + log(x)/(d^2*e*f*m) - log(e* x^m + d)/(d^2*e*f*m^2)) - 1/2*(f^m*log(x^n)^2/(e^3*f*m*x^(2*m) + 2*d*e^2*f *m*x^m + d^2*e*f*m) - 2*integrate((e*f^m*m*x^m*log(c)^2 + (d*f^m*n + (2*e* f^m*m*log(c) + e*f^m*n)*x^m)*log(x^n))/(e^4*f*m*x*x^(3*m) + 3*d*e^3*f*m*x* x^(2*m) + 3*d^2*e^2*f*m*x*x^m + d^3*e*f*m*x), x))*b^2 - a*b*f^m*log(c*x^n) /(e^3*f*m*x^(2*m) + 2*d*e^2*f*m*x^m + d^2*e*f*m) - 1/2*a^2*f^m/(e^3*f*m*x^ (2*m) + 2*d*e^2*f*m*x^m + d^2*e*f*m)
\[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \left (f x\right )^{m - 1}}{{\left (e x^{m} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3} \, dx=\int \frac {{\left (f\,x\right )}^{m-1}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x^m\right )}^3} \,d x \]