Integrand size = 29, antiderivative size = 694 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx=-\frac {b^2 e^2 n^2}{3 d^2 f x}-\frac {b^2 e^3 n^2 \log (x)}{d^3 f}+\frac {b^2 e^3 n^2 \log (d+e x)}{3 d^3 f}-\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d f x^2}+\frac {2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f x}-\frac {2 b e g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}+\frac {g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{5/2}}-\frac {g^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}+\frac {2 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{3 d^3 f}-\frac {2 b^2 e^3 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e x}\right )}{3 d^3 f}-\frac {b g^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{5/2}}+\frac {b g^{3/2} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{5/2}}-\frac {2 b^2 e g n^2 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{d f^2}+\frac {b^2 g^{3/2} n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{5/2}}-\frac {b^2 g^{3/2} n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{5/2}} \]
-1/3*b^2*e^2*n^2/d^2/f/x-b^2*e^3*n^2*ln(x)/d^3/f+1/3*b^2*e^3*n^2*ln(e*x+d) /d^3/f-1/3*b*e*n*(a+b*ln(c*(e*x+d)^n))/d/f/x^2+2/3*b*e^2*n*(e*x+d)*(a+b*ln (c*(e*x+d)^n))/d^3/f/x-2*b*e*g*n*ln(-e*x/d)*(a+b*ln(c*(e*x+d)^n))/d/f^2-1/ 3*(a+b*ln(c*(e*x+d)^n))^2/f/x^3+g*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/d/f^2/x+ 2/3*b*e^3*n*(a+b*ln(c*(e*x+d)^n))*ln(1-d/(e*x+d))/d^3/f+1/2*g^(3/2)*(a+b*l n(c*(e*x+d)^n))^2*ln(e*((-f)^(1/2)-x*g^(1/2))/(e*(-f)^(1/2)+d*g^(1/2)))/(- f)^(5/2)-1/2*g^(3/2)*(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)^(5/2)-2/3*b^2*e^3*n^2*polylog(2,d/(e*x+d))/d ^3/f-2*b^2*e*g*n^2*polylog(2,1+e*x/d)/d/f^2-b*g^(3/2)*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)^(5/2)+b*g^(3 /2)*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)^(5/2)+b^2*g^(3/2)*n^2*polylog(3,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2)- d*g^(1/2)))/(-f)^(5/2)-b^2*g^(3/2)*n^2*polylog(3,(e*x+d)*g^(1/2)/(e*(-f)^( 1/2)+d*g^(1/2)))/(-f)^(5/2)
Result contains complex when optimal does not.
Time = 0.86 (sec) , antiderivative size = 930, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx=\frac {-2 d^3 f^{3/2} \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+6 d^3 \sqrt {f} g x^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+6 d^3 g^{3/2} x^3 \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+2 i b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (6 i d^2 \sqrt {f} g x^2 (e x \log (x)-(d+e x) \log (d+e x))+i f^{3/2} \left (d e x (d-2 e x)-2 e^3 x^3 \log (x)+2 \left (d^3+e^3 x^3\right ) \log (d+e x)\right )-3 d^3 g^{3/2} x^3 \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 )+3 d^3 g^{3/2} x^3 \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 )\right )+i b^2 n^2 \left (6 i d^2 \sqrt {f} g x^2 \left (2 e x \log \left (-\frac {e x}{d}\right ) \log (d+e x)-(d+e x) \log ^2(d+e x)+2 e x \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )\right )+2 i f^{3/2} \left (d e^2 x^2+3 e^3 x^3 \log (x)+d^2 e x \log (d+e x)-2 d e^2 x^2 \log (d+e x)-3 e^3 x^3 \log (d+e x)-2 e^3 x^3 \log \left (-\frac {e x}{d}\right ) \log (d+e x)+d^3 \log ^2(d+e x)+e^3 x^3 \log ^2(d+e x)-2 e^3 x^3 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )\right )+3 d^3 g^{3/2} x^3 \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 )-3 d^3 g^{3/2} x^3 \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 )\right )}{6 d^3 f^{5/2} x^3} \]
(-2*d^3*f^(3/2)*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 6*d^3*Sq rt[f]*g*x^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 6*d^3*g^(3/2 )*x^3*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x )^n])^2 + (2*I)*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*((6*I)*d ^2*Sqrt[f]*g*x^2*(e*x*Log[x] - (d + e*x)*Log[d + e*x]) + I*f^(3/2)*(d*e*x* (d - 2*e*x) - 2*e^3*x^3*Log[x] + 2*(d^3 + e^3*x^3)*Log[d + e*x]) - 3*d^3*g ^(3/2)*x^3*(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*Sqrt[f] - I*d*Sqrt[g])] ) + 3*d^3*g^(3/2)*x^3*(Log[d + e*x]*Log[(e*(Sqrt[f] - I*Sqrt[g]*x))/(e*Sqr t[f] + I*d*Sqrt[g])] + PolyLog[2, (I*Sqrt[g]*(d + e*x))/(e*Sqrt[f] + I*d*S qrt[g])])) + I*b^2*n^2*((6*I)*d^2*Sqrt[f]*g*x^2*(2*e*x*Log[-((e*x)/d)]*Log [d + e*x] - (d + e*x)*Log[d + e*x]^2 + 2*e*x*PolyLog[2, 1 + (e*x)/d]) + (2 *I)*f^(3/2)*(d*e^2*x^2 + 3*e^3*x^3*Log[x] + d^2*e*x*Log[d + e*x] - 2*d*e^2 *x^2*Log[d + e*x] - 3*e^3*x^3*Log[d + e*x] - 2*e^3*x^3*Log[-((e*x)/d)]*Log [d + e*x] + d^3*Log[d + e*x]^2 + e^3*x^3*Log[d + e*x]^2 - 2*e^3*x^3*PolyLo g[2, 1 + (e*x)/d]) + 3*d^3*g^(3/2)*x^3*(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])]) - 3*d^3*g^(3/2)*x^3*(Log[d + e*x]^2*Log [1 - (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])] + 2*Log[d + e*x]*Po...
Time = 1.33 (sec) , antiderivative size = 694, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 \left (f+g x^2\right )}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 x^2}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b e^3 n \log \left (1-\frac {d}{d+e x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f}+\frac {2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 f x}-\frac {2 b e g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f^2}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f^2 x}-\frac {b g^{3/2} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{5/2}}+\frac {b g^{3/2} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{5/2}}+\frac {g^{3/2} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{5/2}}-\frac {g^{3/2} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{5/2}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 f x^3}-\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d f x^2}-\frac {2 b^2 e^3 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e x}\right )}{3 d^3 f}-\frac {b^2 e^3 n^2 \log (x)}{d^3 f}+\frac {b^2 e^3 n^2 \log (d+e x)}{3 d^3 f}-\frac {b^2 e^2 n^2}{3 d^2 f x}-\frac {2 b^2 e g n^2 \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d f^2}+\frac {b^2 g^{3/2} n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{5/2}}-\frac {b^2 g^{3/2} n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{(-f)^{5/2}}\) |
-1/3*(b^2*e^2*n^2)/(d^2*f*x) - (b^2*e^3*n^2*Log[x])/(d^3*f) + (b^2*e^3*n^2 *Log[d + e*x])/(3*d^3*f) - (b*e*n*(a + b*Log[c*(d + e*x)^n]))/(3*d*f*x^2) + (2*b*e^2*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/(3*d^3*f*x) - (2*b*e*g* n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/(d*f^2) - (a + b*Log[c*(d + e*x)^n])^2/(3*f*x^3) + (g*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(d*f^2*x ) + (g^(3/2)*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/( e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(5/2)) - (g^(3/2)*(a + b*Log[c*(d + e*x) ^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*(-f)^( 5/2)) + (2*b*e^3*n*(a + b*Log[c*(d + e*x)^n])*Log[1 - d/(d + e*x)])/(3*d^3 *f) - (2*b^2*e^3*n^2*PolyLog[2, d/(d + e*x)])/(3*d^3*f) - (b*g^(3/2)*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*S qrt[g]))])/(-f)^(5/2) + (b*g^(3/2)*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(-f)^(5/2) - (2*b^2*e*g*n^ 2*PolyLog[2, 1 + (e*x)/d])/(d*f^2) + (b^2*g^(3/2)*n^2*PolyLog[3, -((Sqrt[g ]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(-f)^(5/2) - (b^2*g^(3/2)*n^2*Pol yLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(-f)^(5/2)
3.4.19.3.1 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]
\[\int \frac {{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{x^{4} \left (g \,x^{2}+f \right )}d x\]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x^{2} + f\right )} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x^{2} + f\right )} x^{4}} \,d x } \]
1/3*a^2*(3*g^2*arctan(g*x/sqrt(f*g))/(sqrt(f*g)*f^2) + (3*g*x^2 - f)/(f^2* x^3)) + 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^6 + f*x^4), x)
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x^{2} + f\right )} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 \left (f+g x^2\right )} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x^4\,\left (g\,x^2+f\right )} \,d x \]