Integrand size = 20, antiderivative size = 126 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}-\frac {b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}+\frac {b^2 n^2 \log (d+e x)}{d^2 e}+\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^2 e} \]
-b*n*x*(a+b*ln(c*x^n))/d^2/(e*x+d)-b*n*ln(1+d/e/x)*(a+b*ln(c*x^n))/d^2/e-1 /2*(a+b*ln(c*x^n))^2/e/(e*x+d)^2+b^2*n^2*ln(e*x+d)/d^2/e+b^2*n^2*polylog(2 ,-d/e/x)/d^2/e
Time = 0.07 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}+\frac {b n \left (\frac {a+b \log \left (c x^n\right )}{d (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^2 n}-\frac {b n \left (\frac {\log (x)}{d}-\frac {\log (d+e x)}{d}\right )}{d}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {d+e x}{d}\right )}{d^2}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2}\right )}{e} \]
-1/2*(a + b*Log[c*x^n])^2/(e*(d + e*x)^2) + (b*n*((a + b*Log[c*x^n])/(d*(d + e*x)) + (a + b*Log[c*x^n])^2/(2*b*d^2*n) - (b*n*(Log[x]/d - Log[d + e*x ]/d))/d - ((a + b*Log[c*x^n])*Log[(d + e*x)/d])/d^2 - (b*n*PolyLog[2, -((e *x)/d)])/d^2))/e
Time = 0.50 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2756, 2789, 2751, 16, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {b n \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2}dx}{d}\right )}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle \frac {b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \int \frac {1}{d+e x}dx}{d}\right )}{d}\right )}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}\right )}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {b n \left (\frac {\frac {b n \int \frac {\log \left (\frac {d}{e x}+1\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}\right )}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {b n \left (\frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}\right )}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\) |
-1/2*(a + b*Log[c*x^n])^2/(e*(d + e*x)^2) + (b*n*(-((e*((x*(a + b*Log[c*x^ n]))/(d*(d + e*x)) - (b*n*Log[d + e*x])/(d*e)))/d) + (-((Log[1 + d/(e*x)]* (a + b*Log[c*x^n]))/d) + (b*n*PolyLog[2, -(d/(e*x))])/d)/d))/e
3.2.10.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
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_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
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 = 0.48 (sec) , antiderivative size = 435, normalized size of antiderivative = 3.45
| method | result | size |
| risch | \(-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{2 e \left (e x +d \right )^{2}}-\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e \,d^{2}}+\frac {b^{2} n \ln \left (x^{n}\right )}{e d \left (e x +d \right )}+\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )}{e \,d^{2}}-\frac {b^{2} n^{2} \ln \left (x \right )^{2}}{2 e \,d^{2}}+\frac {b^{2} n^{2} \ln \left (e x +d \right )}{d^{2} e}-\frac {b^{2} n^{2} \ln \left (x \right )}{e \,d^{2}}+\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e \,d^{2}}+\frac {b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e \,d^{2}}+\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (-\frac {\ln \left (x^{n}\right )}{2 e \left (e x +d \right )^{2}}+\frac {n \left (-\frac {\ln \left (e x +d \right )}{d^{2}}+\frac {1}{d \left (e x +d \right )}+\frac {\ln \left (x \right )}{d^{2}}\right )}{2 e}\right )-\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2}}{8 \left (e x +d \right )^{2} e}\) | \(435\) |
-1/2*b^2*ln(x^n)^2/e/(e*x+d)^2-b^2/e*n*ln(x^n)/d^2*ln(e*x+d)+b^2*n*ln(x^n) /e/d/(e*x+d)+b^2/e*n*ln(x^n)/d^2*ln(x)-1/2*b^2/e*n^2/d^2*ln(x)^2+b^2*n^2*l n(e*x+d)/d^2/e-b^2/e*n^2/d^2*ln(x)+b^2/e*n^2/d^2*ln(e*x+d)*ln(-e*x/d)+b^2/ e*n^2/d^2*dilog(-e*x/d)+(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*P i*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/(e*x+d)^2+1/2/e*n*(-1/d^2*ln( e*x+d)+1/d/(e*x+d)+1/d^2*ln(x)))-1/8*(-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/(e*x+d)^2/e
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e^3*x^3 + 3*d*e^2*x^ 2 + 3*d^2*e*x + d^3), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
a*b*n*(1/(d*e^2*x + d^2*e) - log(e*x + d)/(d^2*e) + log(x)/(d^2*e)) - 1/2* b^2*(log(x^n)^2/(e^3*x^2 + 2*d*e^2*x + d^2*e) - 2*integrate((e*x*log(c)^2 + (d*n + (e*n + 2*e*log(c))*x)*log(x^n))/(e^4*x^4 + 3*d*e^3*x^3 + 3*d^2*e^ 2*x^2 + d^3*e*x), x)) - a*b*log(c*x^n)/(e^3*x^2 + 2*d*e^2*x + d^2*e) - 1/2 *a^2/(e^3*x^2 + 2*d*e^2*x + d^2*e)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]