Integrand size = 29, antiderivative size = 447 \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx=-\frac {2 a b n x}{g}+\frac {2 b^2 n^2 x}{g}-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e g}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^{3/2}}-\frac {b \sqrt {-f} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^{3/2}}+\frac {b \sqrt {-f} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^{3/2}}+\frac {b^2 \sqrt {-f} n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^{3/2}}-\frac {b^2 \sqrt {-f} n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^{3/2}} \]
-2*a*b*n*x/g+2*b^2*n^2*x/g-2*b^2*n*(e*x+d)*ln(c*(e*x+d)^n)/e/g+(e*x+d)*(a+ b*ln(c*(e*x+d)^n))^2/e/g+1/2*(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)^(1/2)/g^(3/2)-1/2*(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)^(1/2)/g^( 3/2)-b*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)^(1/2)/g^(3/2)+b*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)^(1/2)/g^(3/2)+b^2*n^2*polylog(3,-(e* x+d)*g^(1/2)/(e*(-f)^(1/2)-d*g^(1/2)))*(-f)^(1/2)/g^(3/2)-b^2*n^2*polylog( 3,(e*x+d)*g^(1/2)/(e*(-f)^(1/2)+d*g^(1/2)))*(-f)^(1/2)/g^(3/2)
Result contains complex when optimal does not.
Time = 0.69 (sec) , antiderivative size = 623, normalized size of antiderivative = 1.39 \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx=\frac {e \sqrt {g} x \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+i b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (-2 i \sqrt {g} (d+e x) (-1+\log (d+e x))-e \sqrt {f} \left (\log (d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )\right )+e \sqrt {f} \left (\log (d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )\right )\right )+b^2 n^2 \left (\sqrt {g} \left (2 e x-2 (d+e x) \log (d+e x)+(d+e x) \log ^2(d+e x)\right )-\frac {1}{2} i e \sqrt {f} \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )-2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )\right )+\frac {1}{2} i e \sqrt {f} \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )-2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )\right )\right )}{e g^{3/2}} \]
(e*Sqrt[g]*x*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 - e*Sqrt[f]*A rcTan[(Sqrt[g]*x)/Sqrt[f]]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + I*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*((-2*I)*Sqrt[g]*(d + e*x)*(-1 + Log[d + e*x]) - e*Sqrt[f]*(Log[d + e*x]*Log[1 - (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])] + PolyLog[2, (Sqrt[g]*(d + e*x))/((-I )*e*Sqrt[f] + d*Sqrt[g])]) + e*Sqrt[f]*(Log[d + e*x]*Log[1 - (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])] + PolyLog[2, (Sqrt[g]*(d + e*x))/(I*e*Sq rt[f] + d*Sqrt[g])])) + b^2*n^2*(Sqrt[g]*(2*e*x - 2*(d + e*x)*Log[d + e*x] + (d + e*x)*Log[d + e*x]^2) - (I/2)*e*Sqrt[f]*(Log[d + e*x]^2*Log[1 - (Sq rt[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])]) + (I/2)*e*Sqrt[f]*(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])])))/(e*g^(3/2))
Time = 0.85 (sec) , antiderivative size = 447, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g \left (f+g x^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b \sqrt {-f} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^{3/2}}+\frac {b \sqrt {-f} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^{3/2}}+\frac {\sqrt {-f} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^{3/2}}-\frac {\sqrt {-f} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^{3/2}}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}-\frac {2 a b n x}{g}-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e g}+\frac {b^2 \sqrt {-f} n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^{3/2}}-\frac {b^2 \sqrt {-f} n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{g^{3/2}}+\frac {2 b^2 n^2 x}{g}\) |
(-2*a*b*n*x)/g + (2*b^2*n^2*x)/g - (2*b^2*n*(d + e*x)*Log[c*(d + e*x)^n])/ (e*g) + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e*g) + (Sqrt[-f]*(a + b* Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[ g])])/(2*g^(3/2)) - (Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f ] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^(3/2)) - (b*Sqrt[-f]*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*S qrt[g]))])/g^(3/2) + (b*Sqrt[-f]*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, ( Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^(3/2) + (b^2*Sqrt[-f]*n^2* PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^(3/2) - (b^ 2*Sqrt[-f]*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g ^(3/2)
3.4.16.3.1 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]
\[\int \frac {x^{2} {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{g \,x^{2}+f}d x\]
\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{2}}{g x^{2} + f} \,d x } \]
integral((b^2*x^2*log((e*x + d)^n*c)^2 + 2*a*b*x^2*log((e*x + d)^n*c) + a^ 2*x^2)/(g*x^2 + f), x)
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{2}}{g x^{2} + f} \,d x } \]
-a^2*(f*arctan(g*x/sqrt(f*g))/(sqrt(f*g)*g) - x/g) + integrate((b^2*x^2*lo g((e*x + d)^n)^2 + 2*(b^2*log(c) + a*b)*x^2*log((e*x + d)^n) + (b^2*log(c) ^2 + 2*a*b*log(c))*x^2)/(g*x^2 + f), x)
\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{2}}{g x^{2} + f} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx=\int \frac {x^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{g\,x^2+f} \,d x \]