Integrand size = 29, antiderivative size = 551 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 \left (f+g x^2\right )} \, dx=\frac {b^2 e^2 n^2 \log (x)}{d^2 f}-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f x}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f x^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2}-\frac {b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{d^2 f}+\frac {b^2 e^2 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e x}\right )}{d^2 f}+\frac {b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^2}+\frac {b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^2}-\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{f^2}-\frac {b^2 g n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^2}-\frac {b^2 g n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^2}+\frac {2 b^2 g n^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )}{f^2} \]
b^2*e^2*n^2*ln(x)/d^2/f-b*e*n*(e*x+d)*(a+b*ln(c*(e*x+d)^n))/d^2/f/x-1/2*(a +b*ln(c*(e*x+d)^n))^2/f/x^2-g*ln(-e*x/d)*(a+b*ln(c*(e*x+d)^n))^2/f^2-b*e^2 *n*(a+b*ln(c*(e*x+d)^n))*ln(1-d/(e*x+d))/d^2/f+1/2*g*(a+b*ln(c*(e*x+d)^n)) ^2*ln(e*((-f)^(1/2)-x*g^(1/2))/(e*(-f)^(1/2)+d*g^(1/2)))/f^2+1/2*g*(a+b*ln (c*(e*x+d)^n))^2*ln(e*((-f)^(1/2)+x*g^(1/2))/(e*(-f)^(1/2)-d*g^(1/2)))/f^2 +b^2*e^2*n^2*polylog(2,d/(e*x+d))/d^2/f-2*b*g*n*(a+b*ln(c*(e*x+d)^n))*poly log(2,1+e*x/d)/f^2+b*g*n*(a+b*ln(c*(e*x+d)^n))*polylog(2,-(e*x+d)*g^(1/2)/ (e*(-f)^(1/2)-d*g^(1/2)))/f^2+b*g*n*(a+b*ln(c*(e*x+d)^n))*polylog(2,(e*x+d )*g^(1/2)/(e*(-f)^(1/2)+d*g^(1/2)))/f^2+2*b^2*g*n^2*polylog(3,1+e*x/d)/f^2 -b^2*g*n^2*polylog(3,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2)-d*g^(1/2)))/f^2-b^2*g* n^2*polylog(3,(e*x+d)*g^(1/2)/(e*(-f)^(1/2)+d*g^(1/2)))/f^2
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 811, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 \left (f+g x^2\right )} \, dx=\frac {-d^2 f \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-2 d^2 g x^2 \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+d^2 g x^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (f+g x^2\right )-2 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (f \left (d e x+e^2 x^2 \log (x)+\left (d^2-e^2 x^2\right ) \log (d+e x)\right )-d^2 g x^2 \left (\log (d+e x) \log \left (\frac {e \left (\sqrt {f}+i \sqrt {g} x\right )}{e \sqrt {f}-i d \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}-i d \sqrt {g}}\right )\right )-d^2 g x^2 \left (\log (d+e x) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{e \sqrt {f}+i d \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}+i d \sqrt {g}}\right )\right )+2 d^2 g x^2 \left (\log \left (-\frac {e x}{d}\right ) \log (d+e x)+\operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )\right )\right )+b^2 n^2 \left (f \left (2 e^2 x^2 \log (x)-\log (d+e x) \left (2 e^2 x^2 \log \left (-\frac {e x}{d}\right )+(d+e x) (2 e x+(d-e x) \log (d+e x))\right )-2 e^2 x^2 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )\right )+d^2 g x^2 \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )-2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )\right )+d^2 g x^2 \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )-2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )\right )-2 d^2 g x^2 \left (\log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)+2 \log (d+e x) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )\right )\right )}{2 d^2 f^2 x^2} \]
(-(d^2*f*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2) - 2*d^2*g*x^2*Lo g[x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + d^2*g*x^2*(a - b*n* Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*Log[f + g*x^2] - 2*b*n*(a - b*n*Log [d + e*x] + b*Log[c*(d + e*x)^n])*(f*(d*e*x + e^2*x^2*Log[x] + (d^2 - e^2* x^2)*Log[d + e*x]) - d^2*g*x^2*(Log[d + e*x]*Log[(e*(Sqrt[f] + I*Sqrt[g]*x ))/(e*Sqrt[f] - I*d*Sqrt[g])] + PolyLog[2, ((-I)*Sqrt[g]*(d + e*x))/(e*Sqr t[f] - I*d*Sqrt[g])]) - d^2*g*x^2*(Log[d + e*x]*Log[(e*(Sqrt[f] - I*Sqrt[g ]*x))/(e*Sqrt[f] + I*d*Sqrt[g])] + PolyLog[2, (I*Sqrt[g]*(d + e*x))/(e*Sqr t[f] + I*d*Sqrt[g])]) + 2*d^2*g*x^2*(Log[-((e*x)/d)]*Log[d + e*x] + PolyLo g[2, 1 + (e*x)/d])) + b^2*n^2*(f*(2*e^2*x^2*Log[x] - Log[d + e*x]*(2*e^2*x ^2*Log[-((e*x)/d)] + (d + e*x)*(2*e*x + (d - e*x)*Log[d + e*x])) - 2*e^2*x ^2*PolyLog[2, 1 + (e*x)/d]) + d^2*g*x^2*(Log[d + e*x]^2*Log[1 - (Sqrt[g]*( d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])] + 2*Log[d + e*x]*PolyLog[2, (Sqrt[ g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])] - 2*PolyLog[3, (Sqrt[g]*(d + e *x))/((-I)*e*Sqrt[f] + d*Sqrt[g])]) + d^2*g*x^2*(Log[d + e*x]^2*Log[1 - (S qrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])] + 2*Log[d + e*x]*PolyLog[2, ( Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])] - 2*PolyLog[3, (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])]) - 2*d^2*g*x^2*(Log[-((e*x)/d)]*Log[d + e*x]^2 + 2*Log[d + e*x]*PolyLog[2, 1 + (e*x)/d] - 2*PolyLog[3, 1 + (e*x)/d ])))/(2*d^2*f^2*x^2)
Time = 1.19 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 \left (f+g x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {g^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 \left (f+g x^2\right )}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 x}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b e^2 n \log \left (1-\frac {d}{d+e x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f}-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f x}+\frac {b g n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac {b g n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}-\frac {2 b g n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}+\frac {g \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2}+\frac {g \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f x^2}+\frac {b^2 e^2 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e x}\right )}{d^2 f}+\frac {b^2 e^2 n^2 \log (x)}{d^2 f}-\frac {b^2 g n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^2}-\frac {b^2 g n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{f^2}+\frac {2 b^2 g n^2 \operatorname {PolyLog}\left (3,\frac {e x}{d}+1\right )}{f^2}\) |
(b^2*e^2*n^2*Log[x])/(d^2*f) - (b*e*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]) )/(d^2*f*x) - (a + b*Log[c*(d + e*x)^n])^2/(2*f*x^2) - (g*Log[-((e*x)/d)]* (a + b*Log[c*(d + e*x)^n])^2)/f^2 + (g*(a + b*Log[c*(d + e*x)^n])^2*Log[(e *(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2) + (g*(a + b*Lo g[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g] )])/(2*f^2) - (b*e^2*n*(a + b*Log[c*(d + e*x)^n])*Log[1 - d/(d + e*x)])/(d ^2*f) + (b^2*e^2*n^2*PolyLog[2, d/(d + e*x)])/(d^2*f) + (b*g*n*(a + b*Log[ c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g])) ])/f^2 + (b*g*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/ (e*Sqrt[-f] + d*Sqrt[g])])/f^2 - (2*b*g*n*(a + b*Log[c*(d + e*x)^n])*PolyL og[2, 1 + (e*x)/d])/f^2 - (b^2*g*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*S qrt[-f] - d*Sqrt[g]))])/f^2 - (b^2*g*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e *Sqrt[-f] + d*Sqrt[g])])/f^2 + (2*b^2*g*n^2*PolyLog[3, 1 + (e*x)/d])/f^2
3.4.14.3.1 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]
\[\int \frac {{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{x^{3} \left (g \,x^{2}+f \right )}d x\]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 \left (f+g x^2\right )} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x^{2} + f\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 \left (f+g x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 \left (f+g x^2\right )} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x^{2} + f\right )} x^{3}} \,d x } \]
1/2*a^2*(g*log(g*x^2 + f)/f^2 - 2*g*log(x)/f^2 - 1/(f*x^2)) + integrate((b ^2*log((e*x + d)^n)^2 + b^2*log(c)^2 + 2*a*b*log(c) + 2*(b^2*log(c) + a*b) *log((e*x + d)^n))/(g*x^5 + f*x^3), x)
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 \left (f+g x^2\right )} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x^{2} + f\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 \left (f+g x^2\right )} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x^3\,\left (g\,x^2+f\right )} \,d x \]