Integrand size = 29, antiderivative size = 936 \[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\frac {2 a b d n x}{e g^2}-\frac {2 b^2 d n^2 x}{e g^2}+\frac {b^2 n^2 (d+e x)^2}{4 e^2 g^2}+\frac {2 b^2 d n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g^2}-\frac {b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}+\frac {e^2 f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (e^2 f+d^2 g\right )}-\frac {d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}+\frac {(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (f+g x^2\right )}-\frac {b e f \left (e f+d \sqrt {-f} \sqrt {g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^3 \left (e^2 f+d^2 g\right )}-\frac {f \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 )}{g^3}-\frac {b e (-f)^{3/2} \left (e \sqrt {-f}+d \sqrt {g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^3 \left (e^2 f+d^2 g\right )}-\frac {f \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 )}{g^3}-\frac {b^2 e (-f)^{3/2} \left (e \sqrt {-f}+d \sqrt {g}\right ) n^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^3 \left (e^2 f+d^2 g\right )}-\frac {2 b f 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 )}{g^3}-\frac {b^2 e (-f)^{3/2} \left (e \sqrt {-f}-d \sqrt {g}\right ) n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^3 \left (e^2 f+d^2 g\right )}-\frac {2 b f 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 )}{g^3}+\frac {2 b^2 f n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}+\frac {2 b^2 f n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^3} \]
2*a*b*d*n*x/e/g^2-2*b^2*d*n^2*x/e/g^2+1/4*b^2*n^2*(e*x+d)^2/e^2/g^2+2*b^2* d*n*(e*x+d)*ln(c*(e*x+d)^n)/e^2/g^2-1/2*b*n*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n) )/e^2/g^2+1/2*e^2*f^2*(a+b*ln(c*(e*x+d)^n))^2/g^3/(d^2*g+e^2*f)-d*(e*x+d)* (a+b*ln(c*(e*x+d)^n))^2/e^2/g^2+1/2*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))^2/e^2/ g^2-1/2*f^2*(a+b*ln(c*(e*x+d)^n))^2/g^3/(g*x^2+f)-f*(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)))/g^3-f*(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)))/g^3-2*b* f*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)))/g^3-2*b*f*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)))/g^3+2*b^2*f*n^2*polylog(3,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2)- d*g^(1/2)))/g^3+2*b^2*f*n^2*polylog(3,(e*x+d)*g^(1/2)/(e*(-f)^(1/2)+d*g^(1 /2)))/g^3-1/2*b^2*e*(-f)^(3/2)*n^2*polylog(2,(e*x+d)*g^(1/2)/(e*(-f)^(1/2) +d*g^(1/2)))*(e*(-f)^(1/2)-d*g^(1/2))/g^3/(d^2*g+e^2*f)-1/2*b*e*(-f)^(3/2) *n*(a+b*ln(c*(e*x+d)^n))*ln(e*((-f)^(1/2)+x*g^(1/2))/(e*(-f)^(1/2)-d*g^(1/ 2)))*(e*(-f)^(1/2)+d*g^(1/2))/g^3/(d^2*g+e^2*f)-1/2*b^2*e*(-f)^(3/2)*n^2*p olylog(2,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2)-d*g^(1/2)))*(e*(-f)^(1/2)+d*g^(1/2 ))/g^3/(d^2*g+e^2*f)-1/2*b*e*f*n*(a+b*ln(c*(e*x+d)^n))*ln(e*((-f)^(1/2)-x* g^(1/2))/(e*(-f)^(1/2)+d*g^(1/2)))*(e*f+d*(-f)^(1/2)*g^(1/2))/g^3/(d^2*g+e ^2*f)
Result contains complex when optimal does not.
Time = 1.80 (sec) , antiderivative size = 1254, normalized size of antiderivative = 1.34 \[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\frac {2 g x^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-\frac {2 f^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2}-4 f \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 (\frac {g \left (e x (2 d-e x)-2 \left (d^2-e^2 x^2\right ) \log (d+e x)\right )}{e^2}+\frac {f^{3/2} \left (i \sqrt {g} (d+e x) \log (d+e x)-e \left (\sqrt {f}+i \sqrt {g} x\right ) \log \left (i \sqrt {f}-\sqrt {g} x\right )\right )}{\left (e \sqrt {f}-i d \sqrt {g}\right ) \left (\sqrt {f}+i \sqrt {g} x\right )}+\frac {i f^{3/2} \left (-\sqrt {g} (d+e x) \log (d+e x)+e \left (i \sqrt {f}+\sqrt {g} x\right ) \log \left (i \sqrt {f}+\sqrt {g} x\right )\right )}{\left (e \sqrt {f}+i d \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}-4 f \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 )-4 f \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 )+b^2 n^2 \left (\frac {g \left (e x (-6 d+e x)+\left (6 d^2+4 d e x-2 e^2 x^2\right ) \log (d+e x)-2 \left (d^2-e^2 x^2\right ) \log ^2(d+e x)\right )}{e^2}+\frac {i f^{3/2} \left (-\sqrt {g} (d+e x) \log ^2(d+e x)+2 e \left (i \sqrt {f}+\sqrt {g} x\right ) \log (d+e x) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{e \sqrt {f}+i d \sqrt {g}}\right )+2 e \left (i \sqrt {f}+\sqrt {g} x\right ) \operatorname {PolyLog}\left (2,\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}+i d \sqrt {g}}\right )\right )}{\left (e \sqrt {f}+i d \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}-\frac {f^{3/2} \left (\log (d+e x) \left (-i \sqrt {g} (d+e x) \log (d+e x)+2 e \left (\sqrt {f}+i \sqrt {g} x\right ) \log \left (\frac {e \left (\sqrt {f}+i \sqrt {g} x\right )}{e \sqrt {f}-i d \sqrt {g}}\right )\right )+2 e \left (\sqrt {f}+i \sqrt {g} x\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )\right )}{\left (e \sqrt {f}-i d \sqrt {g}\right ) \left (\sqrt {f}+i \sqrt {g} x\right )}-4 f \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 )-4 f \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 )}{4 g^3} \]
(2*g*x^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 - (2*f^2*(a - b*n *Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2) - 4*f*(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])*((g*(e*x*(2*d - e*x) - 2*(d^2 - e^2*x^2)*Log[d + e*x]))/e^2 + (f^(3/2)*(I*Sqrt[g]*(d + e*x)*Log[d + e*x] - e*(Sqrt[f] + I*Sqrt[g]*x)*Log[I*Sqrt[f] - Sqrt[g]*x]))/((e*Sqrt[f] - I*d*Sqrt[g])*(Sqrt [f] + I*Sqrt[g]*x)) + (I*f^(3/2)*(-(Sqrt[g]*(d + e*x)*Log[d + e*x]) + e*(I *Sqrt[f] + Sqrt[g]*x)*Log[I*Sqrt[f] + Sqrt[g]*x]))/((e*Sqrt[f] + I*d*Sqrt[ g])*(Sqrt[f] - I*Sqrt[g]*x)) - 4*f*(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])]) - 4*f*(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])])) + b^2*n^2*((g*(e*x*(-6*d + e*x) + (6*d^2 + 4*d*e*x - 2*e^2*x^2)*Log[d + e*x] - 2*(d^2 - e^2*x^2)*Log[d + e*x]^2))/e^2 + (I*f^(3 /2)*(-(Sqrt[g]*(d + e*x)*Log[d + e*x]^2) + 2*e*(I*Sqrt[f] + Sqrt[g]*x)*Log [d + e*x]*Log[(e*(Sqrt[f] - I*Sqrt[g]*x))/(e*Sqrt[f] + I*d*Sqrt[g])] + 2*e *(I*Sqrt[f] + Sqrt[g]*x)*PolyLog[2, (I*Sqrt[g]*(d + e*x))/(e*Sqrt[f] + I*d *Sqrt[g])]))/((e*Sqrt[f] + I*d*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)) - (f^(3/2 )*(Log[d + e*x]*((-I)*Sqrt[g]*(d + e*x)*Log[d + e*x] + 2*e*(Sqrt[f] + I*Sq rt[g]*x)*Log[(e*(Sqrt[f] + I*Sqrt[g]*x))/(e*Sqrt[f] - I*d*Sqrt[g])]) + ...
Time = 1.77 (sec) , antiderivative size = 936, 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.
| \(\displaystyle \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {f^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2 \left (f+g x^2\right )^2}-\frac {2 f x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2 \left (f+g x^2\right )}+\frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {n^2 (d+e x)^2 b^2}{4 e^2 g^2}-\frac {2 d n^2 x b^2}{e g^2}+\frac {2 d n (d+e x) \log \left (c (d+e x)^n\right ) b^2}{e^2 g^2}-\frac {e (-f)^{3/2} \left (\sqrt {g} d+e \sqrt {-f}\right ) n^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) b^2}{2 g^3 \left (g d^2+e^2 f\right )}-\frac {e (-f)^{3/2} \left (e \sqrt {-f}-d \sqrt {g}\right ) n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) b^2}{2 g^3 \left (g d^2+e^2 f\right )}+\frac {2 f n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) b^2}{g^3}+\frac {2 f n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) b^2}{g^3}+\frac {2 a d n x b}{e g^2}-\frac {n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) b}{2 e^2 g^2}-\frac {e f \left (\sqrt {-f} \sqrt {g} d+e f\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {g} d+e \sqrt {-f}}\right ) b}{2 g^3 \left (g d^2+e^2 f\right )}-\frac {e (-f)^{3/2} \left (\sqrt {g} d+e \sqrt {-f}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {g} x+\sqrt {-f}\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) b}{2 g^3 \left (g d^2+e^2 f\right )}-\frac {2 f 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 ) b}{g^3}-\frac {2 f n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) b}{g^3}+\frac {(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}-\frac {d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}+\frac {e^2 f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (g d^2+e^2 f\right )}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3 \left (g x^2+f\right )}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {g} d+e \sqrt {-f}}\right )}{g^3}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {g} x+\sqrt {-f}\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}\) |
(2*a*b*d*n*x)/(e*g^2) - (2*b^2*d*n^2*x)/(e*g^2) + (b^2*n^2*(d + e*x)^2)/(4 *e^2*g^2) + (2*b^2*d*n*(d + e*x)*Log[c*(d + e*x)^n])/(e^2*g^2) - (b*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^2*g^2) + (e^2*f^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*g^3*(e^2*f + d^2*g)) - (d*(d + e*x)*(a + b*Log[c*(d + e* x)^n])^2)/(e^2*g^2) + ((d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2*g^ 2) - (f^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*g^3*(f + g*x^2)) - (b*e*f*(e*f + d*Sqrt[-f]*Sqrt[g])*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt [g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^3*(e^2*f + d^2*g)) - (f*(a + b*Log [c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g]) ])/g^3 - (b*e*(-f)^(3/2)*(e*Sqrt[-f] + d*Sqrt[g])*n*(a + b*Log[c*(d + e*x) ^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^3*(e^2 *f + d^2*g)) - (f*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]* x))/(e*Sqrt[-f] - d*Sqrt[g])])/g^3 - (b^2*e*(-f)^(3/2)*(e*Sqrt[-f] + d*Sqr t[g])*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2* g^3*(e^2*f + d^2*g)) - (2*b*f*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((S qrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^3 - (b^2*e*(-f)^(3/2)*(e*S qrt[-f] - d*Sqrt[g])*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sq rt[g])])/(2*g^3*(e^2*f + d^2*g)) - (2*b*f*n*(a + b*Log[c*(d + e*x)^n])*Pol yLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^3 + (2*b^2*f*n^2* PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^3 + (2*b...
3.4.20.3.1 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]
\[\int \frac {x^{5} {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{\left (g \,x^{2}+f \right )^{2}}d x\]
\[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{5}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
integral((b^2*x^5*log((e*x + d)^n*c)^2 + 2*a*b*x^5*log((e*x + d)^n*c) + a^ 2*x^5)/(g^2*x^4 + 2*f*g*x^2 + f^2), x)
Timed out. \[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{5}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
-1/2*a^2*(f^2/(g^4*x^2 + f*g^3) - x^2/g^2 + 2*f*log(g*x^2 + f)/g^3) + inte grate((b^2*x^5*log((e*x + d)^n)^2 + 2*(b^2*log(c) + a*b)*x^5*log((e*x + d) ^n) + (b^2*log(c)^2 + 2*a*b*log(c))*x^5)/(g^2*x^4 + 2*f*g*x^2 + f^2), x)
\[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{5}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int \frac {x^5\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{{\left (g\,x^2+f\right )}^2} \,d x \]