Integrand size = 29, antiderivative size = 831 \[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx=-\frac {2 a b d f n x}{e g^2}+\frac {2 b^2 d f n^2 x}{e g^2}-\frac {2 b^2 d^3 n^2 x}{e^3 g}-\frac {b^2 f n^2 (d+e x)^2}{4 e^2 g^2}+\frac {3 b^2 d^2 n^2 (d+e x)^2}{4 e^4 g}-\frac {2 b^2 d n^2 (d+e x)^3}{9 e^4 g}+\frac {b^2 n^2 (d+e x)^4}{32 e^4 g}+\frac {b^2 d^4 n^2 \log ^2(d+e x)}{4 e^4 g}-\frac {2 b^2 d f n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g^2}+\frac {2 b d^3 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^4 g}+\frac {b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}-\frac {3 b d^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^4 g}+\frac {2 b d n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^4 g}-\frac {b n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{8 e^4 g}-\frac {b d^4 n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^4 g}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac {d f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}-\frac {f (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 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^3}+\frac {f^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 g^3}+\frac {b f^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 )}{g^3}+\frac {b f^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 )}{g^3}-\frac {b^2 f^2 n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}-\frac {b^2 f^2 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*f*n*x/e/g^2+2*b^2*d*f*n^2*x/e/g^2-2*b^2*d^3*n^2*x/e^3/g-1/4*b^2*f *n^2*(e*x+d)^2/e^2/g^2+3/4*b^2*d^2*n^2*(e*x+d)^2/e^4/g-2/9*b^2*d*n^2*(e*x+ d)^3/e^4/g+1/32*b^2*n^2*(e*x+d)^4/e^4/g+1/4*b^2*d^4*n^2*ln(e*x+d)^2/e^4/g- 2*b^2*d*f*n*(e*x+d)*ln(c*(e*x+d)^n)/e^2/g^2+2*b*d^3*n*(e*x+d)*(a+b*ln(c*(e *x+d)^n))/e^4/g+1/2*b*f*n*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))/e^2/g^2-3/2*b*d^ 2*n*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))/e^4/g+2/3*b*d*n*(e*x+d)^3*(a+b*ln(c*(e *x+d)^n))/e^4/g-1/8*b*n*(e*x+d)^4*(a+b*ln(c*(e*x+d)^n))/e^4/g-1/2*b*d^4*n* ln(e*x+d)*(a+b*ln(c*(e*x+d)^n))/e^4/g+1/4*x^4*(a+b*ln(c*(e*x+d)^n))^2/g+d* f*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e^2/g^2-1/2*f*(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*ln(e*((-f)^(1/2)-x*g^(1/2) )/(e*(-f)^(1/2)+d*g^(1/2)))/g^3+1/2*f^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)))/g^3+b*f^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)))/g^3+b*f^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)))/g^3-b ^2*f^2*n^2*polylog(3,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2)-d*g^(1/2)))/g^3-b^2*f^ 2*n^2*polylog(3,(e*x+d)*g^(1/2)/(e*(-f)^(1/2)+d*g^(1/2)))/g^3
Result contains complex when optimal does not.
Time = 0.89 (sec) , antiderivative size = 862, normalized size of antiderivative = 1.04 \[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx=\frac {-144 e^4 f g x^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+72 e^4 g^2 x^4 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+144 e^4 f^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 )-12 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (12 e^2 f g \left (e x (2 d-e x)-2 \left (d^2-e^2 x^2\right ) \log (d+e x)\right )+g^2 \left (e x \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+12 \left (d^4-e^4 x^4\right ) \log (d+e x)\right )-24 e^4 f^2 \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 )-24 e^4 f^2 \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 (-72 e^2 f 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 )-g^2 \left (e x \left (300 d^3-78 d^2 e x+28 d e^2 x^2-9 e^3 x^3\right )-12 \left (25 d^4+12 d^3 e x-6 d^2 e^2 x^2+4 d e^3 x^3-3 e^4 x^4\right ) \log (d+e x)+72 \left (d^4-e^4 x^4\right ) \log ^2(d+e x)\right )+144 e^4 f^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 )+144 e^4 f^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 )\right )}{288 e^4 g^3} \]
(-144*e^4*f*g*x^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 72*e^4 *g^2*x^4*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 144*e^4*f^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*Log[f + g*x^2] - 12*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(12*e^2*f*g*(e*x*(2*d - e*x) - 2* (d^2 - e^2*x^2)*Log[d + e*x]) + g^2*(e*x*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^ 2 + 3*e^3*x^3) + 12*(d^4 - e^4*x^4)*Log[d + e*x]) - 24*e^4*f^2*(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])]) - 24*e^4*f^2*(Log[d + e *x]*Log[1 - (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])] + PolyLog[2, (S qrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])])) + b^2*n^2*(-72*e^2*f*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) - g^2*(e*x*(300*d^3 - 78*d^2*e*x + 28*d*e^2*x^2 - 9*e ^3*x^3) - 12*(25*d^4 + 12*d^3*e*x - 6*d^2*e^2*x^2 + 4*d*e^3*x^3 - 3*e^4*x^ 4)*Log[d + e*x] + 72*(d^4 - e^4*x^4)*Log[d + e*x]^2) + 144*e^4*f^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])]) + 144*e^4*f ^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])])))/(288*...
Time = 1.46 (sec) , antiderivative size = 831, 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.
| \(\displaystyle \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {f^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2 \left (f+g x^2\right )}-\frac {f x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^2 n^2 \log ^2(d+e x) d^4}{4 e^4 g}-\frac {b n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^4}{2 e^4 g}-\frac {2 b^2 n^2 x d^3}{e^3 g}+\frac {2 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^3}{e^4 g}+\frac {3 b^2 n^2 (d+e x)^2 d^2}{4 e^4 g}-\frac {3 b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) d^2}{2 e^4 g}-\frac {2 b^2 n^2 (d+e x)^3 d}{9 e^4 g}+\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 d}{e^2 g^2}+\frac {2 b^2 f n^2 x d}{e g^2}-\frac {2 a b f n x d}{e g^2}-\frac {2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right ) d}{e^2 g^2}+\frac {2 b n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) d}{3 e^4 g}+\frac {b^2 n^2 (d+e x)^4}{32 e^4 g}-\frac {b^2 f n^2 (d+e x)^2}{4 e^2 g^2}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}-\frac {f (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}-\frac {b n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{8 e^4 g}+\frac {b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}+\frac {f^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 )}{\sqrt {g} d+e \sqrt {-f}}\right )}{2 g^3}+\frac {f^2 \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 )}{2 g^3}+\frac {b f^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 )}{g^3}+\frac {b f^2 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 )}{g^3}-\frac {b^2 f^2 n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}-\frac {b^2 f^2 n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{g^3}\) |
(-2*a*b*d*f*n*x)/(e*g^2) + (2*b^2*d*f*n^2*x)/(e*g^2) - (2*b^2*d^3*n^2*x)/( e^3*g) - (b^2*f*n^2*(d + e*x)^2)/(4*e^2*g^2) + (3*b^2*d^2*n^2*(d + e*x)^2) /(4*e^4*g) - (2*b^2*d*n^2*(d + e*x)^3)/(9*e^4*g) + (b^2*n^2*(d + e*x)^4)/( 32*e^4*g) + (b^2*d^4*n^2*Log[d + e*x]^2)/(4*e^4*g) - (2*b^2*d*f*n*(d + e*x )*Log[c*(d + e*x)^n])/(e^2*g^2) + (2*b*d^3*n*(d + e*x)*(a + b*Log[c*(d + e *x)^n]))/(e^4*g) + (b*f*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^2*g ^2) - (3*b*d^2*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^4*g) + (2*b* d*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(3*e^4*g) - (b*n*(d + e*x)^4*( a + b*Log[c*(d + e*x)^n]))/(8*e^4*g) - (b*d^4*n*Log[d + e*x]*(a + b*Log[c* (d + e*x)^n]))/(2*e^4*g) + (x^4*(a + b*Log[c*(d + e*x)^n])^2)/(4*g) + (d*f *(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e^2*g^2) - (f*(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*Lo g[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^3) + (f^2*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*S qrt[g])])/(2*g^3) + (b*f^2*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt [g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^3 + (b*f^2*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^3 - (b^2*f^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))]) /g^3 - (b^2*f^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g] )])/g^3
3.4.10.3.1 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]
\[\int \frac {x^{5} {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{g \,x^{2}+f}d x\]
\[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{5}}{g x^{2} + f} \,d x } \]
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*x^2 + f), x)
Timed out. \[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx=\text {Timed out} \]
\[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{5}}{g x^{2} + f} \,d x } \]
1/4*a^2*(2*f^2*log(g*x^2 + f)/g^3 + (g*x^4 - 2*f*x^2)/g^2) + integrate((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*x^2 + f), x)
\[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{5}}{g x^{2} + f} \,d x } \]
Timed out. \[ \int \frac {x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx=\int \frac {x^5\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{g\,x^2+f} \,d x \]