Integrand size = 27, antiderivative size = 430 \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (e^2 f+d^2 g\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}-\frac {b e \left (e f+d \sqrt {-f} \sqrt {g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f g \left (e^2 f+d^2 g\right )}-\frac {b e \left (e f-d \sqrt {-f} \sqrt {g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f g \left (e^2 f+d^2 g\right )}-\frac {b^2 e \left (e \sqrt {-f}+d \sqrt {g}\right ) n^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} g \left (e^2 f+d^2 g\right )}-\frac {b^2 e \left (e f+d \sqrt {-f} \sqrt {g}\right ) n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f g \left (e^2 f+d^2 g\right )} \]
1/2*e^2*(a+b*ln(c*(e*x+d)^n))^2/g/(d^2*g+e^2*f)-1/2*(a+b*ln(c*(e*x+d)^n))^ 2/g/(g*x^2+f)-1/2*b^2*e*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/(d^2*g+e^2*f)/(-f)^(1/2)-1/2*b*e*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))/f/g/(d^2*g+e^2*f)-1/2*b*e*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))/f/g/(d^2*g+e^2*f)-1/2*b^2*e*n^2*polylog(2,(e*x+d)*g^(1/2)/(e*(-f)^( 1/2)+d*g^(1/2)))*(e*f+d*(-f)^(1/2)*g^(1/2))/f/g/(d^2*g+e^2*f)
Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.37 \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\frac {-\frac {2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2}+\frac {2 b n \left (-a+b n \log (d+e x)-b \log \left (c (d+e x)^n\right )\right ) \left (2 \sqrt {f} g \left (d^2-e^2 x^2\right ) \log (d+e x)+e \left (f+g x^2\right ) \left (\left (e \sqrt {f}+i d \sqrt {g}\right ) \log \left (i \sqrt {f}-\sqrt {g} x\right )+\left (e \sqrt {f}-i d \sqrt {g}\right ) \log \left (i \sqrt {f}+\sqrt {g} x\right )\right )\right )}{\sqrt {f} \left (e^2 f+d^2 g\right ) \left (f+g x^2\right )}+\frac {i b^2 n^2 \left (\frac {-\sqrt {g} (d+e x) \log ^2(d+e x)+2 e \left (i \sqrt {f}+\sqrt {g} x\right ) \log (d+e x) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{e \sqrt {f}+i d \sqrt {g}}\right )+2 e \left (i \sqrt {f}+\sqrt {g} x\right ) \operatorname {PolyLog}\left (2,\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}+i d \sqrt {g}}\right )}{\left (e \sqrt {f}+i d \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+\frac {\log (d+e x) \left (\sqrt {g} (d+e x) \log (d+e x)+2 i e \left (\sqrt {f}+i \sqrt {g} x\right ) \log \left (\frac {e \left (\sqrt {f}+i \sqrt {g} x\right )}{e \sqrt {f}-i d \sqrt {g}}\right )\right )+2 i e \left (\sqrt {f}+i \sqrt {g} x\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )}{\left (e \sqrt {f}-i d \sqrt {g}\right ) \left (\sqrt {f}+i \sqrt {g} x\right )}\right )}{\sqrt {f}}}{4 g} \]
((-2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2) + (2*b*n *(-a + b*n*Log[d + e*x] - b*Log[c*(d + e*x)^n])*(2*Sqrt[f]*g*(d^2 - e^2*x^ 2)*Log[d + e*x] + e*(f + g*x^2)*((e*Sqrt[f] + I*d*Sqrt[g])*Log[I*Sqrt[f] - Sqrt[g]*x] + (e*Sqrt[f] - I*d*Sqrt[g])*Log[I*Sqrt[f] + Sqrt[g]*x])))/(Sqr t[f]*(e^2*f + d^2*g)*(f + g*x^2)) + (I*b^2*n^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)*PolyL og[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)) + (Log[d + e*x]*(Sqrt[g]*(d + e*x)*Log[d + e*x] + (2*I)*e*(Sqrt[f] + I*Sqrt[g]*x)*Log[(e*(Sqrt[f] + I*Sqrt[g]*x))/ (e*Sqrt[f] - I*d*Sqrt[g])]) + (2*I)*e*(Sqrt[f] + I*Sqrt[g]*x)*PolyLog[2, ( Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])])/((e*Sqrt[f] - I*d*Sqrt[g])* (Sqrt[f] + I*Sqrt[g]*x))))/Sqrt[f])/(4*g)
Time = 0.77 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2860, 2865, 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.
| \(\displaystyle \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2860 |
| \(\displaystyle \frac {b e n \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (g x^2+f\right )}dx}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}\) |
| \(\Big \downarrow \) 2865 |
| \(\displaystyle \frac {b e n \int \left (\frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (g d^2+e^2 f\right ) (d+e x)}-\frac {g (e x-d) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (g d^2+e^2 f\right ) \left (g x^2+f\right )}\right )dx}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b e n \left (\frac {e \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b n \left (d^2 g+e^2 f\right )}-\frac {\left (d \sqrt {-f} \sqrt {g}+e f\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (d^2 g+e^2 f\right )}-\frac {\left (e f-d \sqrt {-f} \sqrt {g}\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (d^2 g+e^2 f\right )}-\frac {b n \left (e f-d \sqrt {-f} \sqrt {g}\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f \left (d^2 g+e^2 f\right )}-\frac {b n \left (d \sqrt {-f} \sqrt {g}+e f\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{2 f \left (d^2 g+e^2 f\right )}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}\) |
-1/2*(a + b*Log[c*(d + e*x)^n])^2/(g*(f + g*x^2)) + (b*e*n*((e*(a + b*Log[ c*(d + e*x)^n])^2)/(2*b*(e^2*f + d^2*g)*n) - ((e*f + d*Sqrt[-f]*Sqrt[g])*( a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*S qrt[g])])/(2*f*(e^2*f + d^2*g)) - ((e*f - d*Sqrt[-f]*Sqrt[g])*(a + b*Log[c *(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/( 2*f*(e^2*f + d^2*g)) - (b*(e*f - d*Sqrt[-f]*Sqrt[g])*n*PolyLog[2, -((Sqrt[ g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*f*(e^2*f + d^2*g)) - (b*(e*f + d*Sqrt[-f]*Sqrt[g])*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqr t[g])])/(2*f*(e^2*f + d^2*g))))/g
3.4.22.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(x_)^(m_.)*( (f_.) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Simp[(f + g*x^r)^(q + 1)*((a + b*Log[c*(d + e*x)^n])^p/(g*r*(q + 1))), x] - Simp[b*e*n*(p/(g*r*(q + 1))) Int[(f + g*x^r)^(q + 1)*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, q, r}, x] && EqQ[m, r - 1] && N eQ[q, -1] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy mbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[ RFx, x] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.54 (sec) , antiderivative size = 1231, normalized size of antiderivative = 2.86
-1/2*b^2*ln((e*x+d)^n)^2/g/(g*x^2+f)-1/2*b^2/g*n^2*e^2/(d^2*g+e^2*f)*ln(e* x+d)^2+b^2/g*n*e^2/(d^2*g+e^2*f)*ln(e*x+d)*ln((e*x+d)^n)+1/2*b^2/g*n^2*e^2 /(d^2*g+e^2*f)*ln(g*(e*x+d)^2-2*(e*x+d)*d*g+d^2*g+f*e^2)*ln(e*x+d)-1/2*b^2 /g*n*e^2/(d^2*g+e^2*f)*ln(g*(e*x+d)^2-2*(e*x+d)*d*g+d^2*g+f*e^2)*ln((e*x+d )^n)-b^2*n^2*e/(d^2*g+e^2*f)*d/(f*g)^(1/2)*arctan(1/2*(2*g*(e*x+d)-2*d*g)/ e/(f*g)^(1/2))*ln(e*x+d)+b^2*n*e/(d^2*g+e^2*f)*d/(f*g)^(1/2)*arctan(1/2*(2 *g*(e*x+d)-2*d*g)/e/(f*g)^(1/2))*ln((e*x+d)^n)-1/2*b^2/g*n^2*e^2/(d^2*g+e^ 2*f)*ln(e*x+d)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))-1/2 *b^2/g*n^2*e^2/(d^2*g+e^2*f)*ln(e*x+d)*ln((e*(-f*g)^(1/2)+g*(e*x+d)-d*g)/( e*(-f*g)^(1/2)-d*g))+1/2*b^2*n^2*e/(d^2*g+e^2*f)*ln(e*x+d)/(-f*g)^(1/2)*ln ((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))*d-1/2*b^2*n^2*e/(d^2 *g+e^2*f)*ln(e*x+d)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f* g)^(1/2)-d*g))*d-1/2*b^2/g*n^2*e^2/(d^2*g+e^2*f)*dilog((e*(-f*g)^(1/2)-g*( e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))-1/2*b^2/g*n^2*e^2/(d^2*g+e^2*f)*dilog((e *(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))+1/2*b^2*n^2*e/(d^2*g+e^ 2*f)/(-f*g)^(1/2)*dilog((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g ))*d-1/2*b^2*n^2*e/(d^2*g+e^2*f)/(-f*g)^(1/2)*dilog((e*(-f*g)^(1/2)+g*(e*x +d)-d*g)/(e*(-f*g)^(1/2)-d*g))*d+(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*cs gn(I*(e*x+d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d) ^n)*csgn(I*c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*b*ln(c)+2*a)...
\[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
integral((b^2*x*log((e*x + d)^n*c)^2 + 2*a*b*x*log((e*x + d)^n*c) + a^2*x) /(g^2*x^4 + 2*f*g*x^2 + f^2), x)
Timed out. \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
-1/2*a*b*e*n*(e*log(g*x^2 + f)/(e^2*f*g + d^2*g^2) - 2*e*log(e*x + d)/(e^2 *f*g + d^2*g^2) - 2*d*arctan(g*x/sqrt(f*g))/((e^2*f + d^2*g)*sqrt(f*g))) - 1/2*b^2*(log((e*x + d)^n)^2/(g^2*x^2 + f*g) - 2*integrate((e*g*x^2*log(c) ^2 + d*g*x*log(c)^2 + (2*d*g*x*log(c) + e*f*n + (e*g*n + 2*e*g*log(c))*x^2 )*log((e*x + d)^n))/(e*g^3*x^5 + d*g^3*x^4 + 2*e*f*g^2*x^3 + 2*d*f*g^2*x^2 + e*f^2*g*x + d*f^2*g), x)) - a*b*log((e*x + d)^n*c)/(g^2*x^2 + f*g) - 1/ 2*a^2/(g^2*x^2 + f*g)
\[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\int \frac {x\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{{\left (g\,x^2+f\right )}^2} \,d x \]