Integrand size = 21, antiderivative size = 210 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\frac {b^2 n^2}{3 d e^2 (d+e x)}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^2}+\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 d e^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{6 d^2 e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d^2 e^2}-\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{3 d^2 e^2} \]
1/3*b^2*n^2/d/e^2/(e*x+d)-1/3*b*n*(a+b*ln(c*x^n))/e^2/(e*x+d)^2+1/3*b*n*(a +b*ln(c*x^n))/d/e^2/(e*x+d)+1/6*(a+b*ln(c*x^n))^2/d^2/e^2+1/3*d*(a+b*ln(c* x^n))^2/e^2/(e*x+d)^3-1/2*(a+b*ln(c*x^n))^2/e^2/(e*x+d)^2-1/3*b*n*(a+b*ln( c*x^n))*ln(1+e*x/d)/d^2/e^2-1/3*b^2*n^2*polylog(2,-e*x/d)/d^2/e^2
Time = 0.16 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.34 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\frac {2 b^2 d^3 n^2+2 a b d^2 e n x+4 b^2 d^2 e n^2 x+3 a^2 d e^2 x^2+2 a b d e^2 n x^2+2 b^2 d e^2 n^2 x^2+a^2 e^3 x^3+b^2 e^2 x^2 (3 d+e x) \log ^2\left (c x^n\right )-2 a b d^3 n \log \left (1+\frac {e x}{d}\right )-6 a b d^2 e n x \log \left (1+\frac {e x}{d}\right )-6 a b d e^2 n x^2 \log \left (1+\frac {e x}{d}\right )-2 a b e^3 n x^3 \log \left (1+\frac {e x}{d}\right )-2 b \log \left (c x^n\right ) \left (-e x (b d n (d+e x)+a e x (3 d+e x))+b n (d+e x)^3 \log \left (1+\frac {e x}{d}\right )\right )-2 b^2 n^2 (d+e x)^3 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{6 d^2 e^2 (d+e x)^3} \]
(2*b^2*d^3*n^2 + 2*a*b*d^2*e*n*x + 4*b^2*d^2*e*n^2*x + 3*a^2*d*e^2*x^2 + 2 *a*b*d*e^2*n*x^2 + 2*b^2*d*e^2*n^2*x^2 + a^2*e^3*x^3 + b^2*e^2*x^2*(3*d + e*x)*Log[c*x^n]^2 - 2*a*b*d^3*n*Log[1 + (e*x)/d] - 6*a*b*d^2*e*n*x*Log[1 + (e*x)/d] - 6*a*b*d*e^2*n*x^2*Log[1 + (e*x)/d] - 2*a*b*e^3*n*x^3*Log[1 + ( e*x)/d] - 2*b*Log[c*x^n]*(-(e*x*(b*d*n*(d + e*x) + a*e*x*(3*d + e*x))) + b *n*(d + e*x)^3*Log[1 + (e*x)/d]) - 2*b^2*n^2*(d + e*x)^3*PolyLog[2, -((e*x )/d)])/(6*d^2*e^2*(d + e*x)^3)
Time = 0.72 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2783, 2773, 49, 2009, 2781, 2784, 2754, 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.
| \(\displaystyle \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 2783 |
| \(\displaystyle -\frac {2 b n \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3}dx}{3 d}+\frac {\int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3}dx}{3 d}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}\) |
| \(\Big \downarrow \) 2773 |
| \(\displaystyle \frac {\int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3}dx}{3 d}-\frac {2 b n \left (\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d (d+e x)^2}-\frac {b n \int \frac {x}{(d+e x)^2}dx}{2 d}\right )}{3 d}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3}dx}{3 d}-\frac {2 b n \left (\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d (d+e x)^2}-\frac {b n \int \left (\frac {1}{e (d+e x)}-\frac {d}{e (d+e x)^2}\right )dx}{2 d}\right )}{3 d}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3}dx}{3 d}-\frac {2 b n \left (\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d (d+e x)^2}-\frac {b n \left (\frac {d}{e^2 (d+e x)}+\frac {\log (d+e x)}{e^2}\right )}{2 d}\right )}{3 d}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}\) |
| \(\Big \downarrow \) 2781 |
| \(\displaystyle \frac {\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac {b n \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}dx}{d}}{3 d}-\frac {2 b n \left (\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d (d+e x)^2}-\frac {b n \left (\frac {d}{e^2 (d+e x)}+\frac {\log (d+e x)}{e^2}\right )}{2 d}\right )}{3 d}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac {b n \left (\frac {\int \frac {a+b n+b \log \left (c x^n\right )}{d+e x}dx}{e}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}\right )}{d}}{3 d}-\frac {2 b n \left (\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d (d+e x)^2}-\frac {b n \left (\frac {d}{e^2 (d+e x)}+\frac {\log (d+e x)}{e^2}\right )}{2 d}\right )}{3 d}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle \frac {\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac {b n \left (\frac {\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )+b n\right )}{e}-\frac {b n \int \frac {\log \left (\frac {e x}{d}+1\right )}{x}dx}{e}}{e}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}\right )}{d}}{3 d}-\frac {2 b n \left (\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d (d+e x)^2}-\frac {b n \left (\frac {d}{e^2 (d+e x)}+\frac {\log (d+e x)}{e^2}\right )}{2 d}\right )}{3 d}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {2 b n \left (\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d (d+e x)^2}-\frac {b n \left (\frac {d}{e^2 (d+e x)}+\frac {\log (d+e x)}{e^2}\right )}{2 d}\right )}{3 d}+\frac {\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac {b n \left (\frac {\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )+b n\right )}{e}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}}{e}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}\right )}{d}}{3 d}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}\) |
(x^2*(a + b*Log[c*x^n])^2)/(3*d*(d + e*x)^3) - (2*b*n*((x^2*(a + b*Log[c*x ^n]))/(2*d*(d + e*x)^2) - (b*n*(d/(e^2*(d + e*x)) + Log[d + e*x]/e^2))/(2* d)))/(3*d) + ((x^2*(a + b*Log[c*x^n])^2)/(2*d*(d + e*x)^2) - (b*n*(-((x*(a + b*Log[c*x^n]))/(e*(d + e*x))) + (((a + b*n + b*Log[c*x^n])*Log[1 + (e*x )/d])/e + (b*n*PolyLog[2, -((e*x)/d)])/e)/e))/d)/(3*d)
3.2.16.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
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_))^(q_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + Simp[b*n*(p/(d*(q + 1))) Int[(f*x) ^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + (Simp[(m + q + 2)/(d*(q + 1)) Int[ (f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p, x], x] + Simp[b*n*(p/(d*(q + 1))) Int[(f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[p, 0] && L tQ[q, -1] && GtQ[m, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] )/(e*(q + 1))), x] - Simp[f/(e*(q + 1)) Int[(f*x)^(m - 1)*(d + e*x)^(q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.56 (sec) , antiderivative size = 508, normalized size of antiderivative = 2.42
| method | result | size |
| risch | \(-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{2 e^{2} \left (e x +d \right )^{2}}+\frac {b^{2} \ln \left (x^{n}\right )^{2} d}{3 e^{2} \left (e x +d \right )^{3}}-\frac {b^{2} n \ln \left (x^{n}\right )}{3 e^{2} \left (e x +d \right )^{2}}-\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{3 e^{2} d^{2}}+\frac {b^{2} n \ln \left (x^{n}\right )}{3 e^{2} d \left (e x +d \right )}+\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )}{3 e^{2} d^{2}}-\frac {b^{2} n^{2} \ln \left (x \right )^{2}}{6 e^{2} d^{2}}+\frac {b^{2} n^{2}}{3 d \,e^{2} \left (e x +d \right )}+\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{3 e^{2} d^{2}}+\frac {b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{3 e^{2} d^{2}}+\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (-\frac {\ln \left (x^{n}\right )}{2 e^{2} \left (e x +d \right )^{2}}+\frac {\ln \left (x^{n}\right ) d}{3 e^{2} \left (e x +d \right )^{3}}-\frac {n \left (\frac {\ln \left (e x +d \right )}{d^{2}}-\frac {1}{d \left (e x +d \right )}+\frac {1}{\left (e x +d \right )^{2}}-\frac {\ln \left (x \right )}{d^{2}}\right )}{6 e^{2}}\right )+\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2} \left (-\frac {1}{2 e^{2} \left (e x +d \right )^{2}}+\frac {d}{3 e^{2} \left (e x +d \right )^{3}}\right )}{4}\) | \(508\) |
-1/2*b^2*ln(x^n)^2/e^2/(e*x+d)^2+1/3*b^2*ln(x^n)^2/e^2*d/(e*x+d)^3-1/3*b^2 *n*ln(x^n)/e^2/(e*x+d)^2-1/3*b^2*n*ln(x^n)/e^2/d^2*ln(e*x+d)+1/3*b^2*n*ln( x^n)/e^2/d/(e*x+d)+1/3*b^2*n*ln(x^n)/e^2/d^2*ln(x)-1/6*b^2*n^2/e^2/d^2*ln( x)^2+1/3*b^2*n^2/d/e^2/(e*x+d)+1/3*b^2*n^2/e^2/d^2*ln(e*x+d)*ln(-e*x/d)+1/ 3*b^2*n^2/e^2/d^2*dilog(-e*x/d)+(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^ n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b *Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)*b*(-1/2*ln(x^n)/e^2/(e*x+d)^2+1/3*ln(x^ n)/e^2*d/(e*x+d)^3-1/6*n/e^2*(1/d^2*ln(e*x+d)-1/d/(e*x+d)+1/(e*x+d)^2-1/d^ 2*ln(x)))+1/4*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c )*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^ 3+2*b*ln(c)+2*a)^2*(-1/2/e^2/(e*x+d)^2+1/3/e^2*d/(e*x+d)^3)
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{{\left (e x + d\right )}^{4}} \,d x } \]
integral((b^2*x*log(c*x^n)^2 + 2*a*b*x*log(c*x^n) + a^2*x)/(e^4*x^4 + 4*d* e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4), x)
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{4}}\, dx \]
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{{\left (e x + d\right )}^{4}} \,d x } \]
1/3*a*b*n*(x/(d*e^3*x^2 + 2*d^2*e^2*x + d^3*e) - log(e*x + d)/(d^2*e^2) + log(x)/(d^2*e^2)) - 1/6*((3*e*x + d)*log(x^n)^2/(e^5*x^3 + 3*d*e^4*x^2 + 3 *d^2*e^3*x + d^3*e^2) - 6*integrate(1/3*(3*e^2*x^2*log(c)^2 + (4*d*e*n*x + d^2*n + 3*(e^2*n + 2*e^2*log(c))*x^2)*log(x^n))/(e^6*x^5 + 4*d*e^5*x^4 + 6*d^2*e^4*x^3 + 4*d^3*e^3*x^2 + d^4*e^2*x), x))*b^2 - 1/3*(3*e*x + d)*a*b* log(c*x^n)/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2) - 1/6*(3*e*x + d)*a^2/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2)
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{{\left (e x + d\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int \frac {x\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^4} \,d x \]