Integrand size = 32, antiderivative size = 550 \[ \int \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=-\frac {n (a+b x) \log (a+b x)}{b g}+\frac {n (c+d x) \log (c+d x)}{d g}+\frac {x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{g}-\frac {\sqrt {f} \text {arctanh}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{g^{3/2}}-\frac {\sqrt {f} n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )}{2 g^{3/2}}+\frac {\sqrt {f} n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )}{2 g^{3/2}}+\frac {\sqrt {f} n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^{3/2}}+\frac {\sqrt {f} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^{3/2}}+\frac {\sqrt {f} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right )}{2 g^{3/2}} \]
-n*(b*x+a)*ln(b*x+a)/b/g+n*(d*x+c)*ln(d*x+c)/d/g+x*(n*ln(b*x+a)-ln(e*((b*x +a)/(d*x+c))^n)-n*ln(d*x+c))/g-arctanh(x*g^(1/2)/f^(1/2))*(n*ln(b*x+a)-ln( e*((b*x+a)/(d*x+c))^n)-n*ln(d*x+c))*f^(1/2)/g^(3/2)-1/2*n*ln(b*x+a)*ln(b*( f^(1/2)-x*g^(1/2))/(b*f^(1/2)+a*g^(1/2)))*f^(1/2)/g^(3/2)+1/2*n*ln(d*x+c)* ln(d*(f^(1/2)-x*g^(1/2))/(d*f^(1/2)+c*g^(1/2)))*f^(1/2)/g^(3/2)+1/2*n*ln(b *x+a)*ln(b*(f^(1/2)+x*g^(1/2))/(b*f^(1/2)-a*g^(1/2)))*f^(1/2)/g^(3/2)-1/2* n*ln(d*x+c)*ln(d*(f^(1/2)+x*g^(1/2))/(d*f^(1/2)-c*g^(1/2)))*f^(1/2)/g^(3/2 )+1/2*n*polylog(2,-(b*x+a)*g^(1/2)/(b*f^(1/2)-a*g^(1/2)))*f^(1/2)/g^(3/2)- 1/2*n*polylog(2,(b*x+a)*g^(1/2)/(b*f^(1/2)+a*g^(1/2)))*f^(1/2)/g^(3/2)-1/2 *n*polylog(2,-(d*x+c)*g^(1/2)/(d*f^(1/2)-c*g^(1/2)))*f^(1/2)/g^(3/2)+1/2*n *polylog(2,(d*x+c)*g^(1/2)/(d*f^(1/2)+c*g^(1/2)))*f^(1/2)/g^(3/2)
Time = 0.18 (sec) , antiderivative size = 467, normalized size of antiderivative = 0.85 \[ \int \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\frac {-\frac {2 \sqrt {g} (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}+\frac {2 (b c-a d) \sqrt {g} n \log (c+d x)}{b d}-\sqrt {f} \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )+\sqrt {f} \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )+\sqrt {f} n \left (\left (\log \left (\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right )-\log \left (\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right )\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )-\operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )\right )-\sqrt {f} n \left (\left (\log \left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )-\log \left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )-\operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )\right )}{2 g^{3/2}} \]
((-2*Sqrt[g]*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/b + (2*(b*c - a*d)* Sqrt[g]*n*Log[c + d*x])/(b*d) - Sqrt[f]*Log[e*((a + b*x)/(c + d*x))^n]*Log [Sqrt[f] - Sqrt[g]*x] + Sqrt[f]*Log[e*((a + b*x)/(c + d*x))^n]*Log[Sqrt[f] + Sqrt[g]*x] + Sqrt[f]*n*((Log[(Sqrt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g] )] - Log[(Sqrt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])])*Log[Sqrt[f] - Sqrt[ g]*x] + PolyLog[2, (b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[g])] - Po lyLog[2, (d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])]) - Sqrt[f]*n*( (Log[-((Sqrt[g]*(a + b*x))/(b*Sqrt[f] - a*Sqrt[g]))] - Log[-((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))])*Log[Sqrt[f] + Sqrt[g]*x] + PolyLog[2, (b *(Sqrt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])] - PolyLog[2, (d*(Sqrt[f] + Sqrt[g]*x))/(d*Sqrt[f] - c*Sqrt[g])]))/(2*g^(3/2))
Time = 0.90 (sec) , antiderivative size = 523, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2993, 262, 221, 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.
| \(\displaystyle \int \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx\) |
| \(\Big \downarrow \) 2993 |
| \(\displaystyle -\left (\left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right ) \int \frac {x^2}{f-g x^2}dx\right )+n \int \frac {x^2 \log (a+b x)}{f-g x^2}dx-n \int \frac {x^2 \log (c+d x)}{f-g x^2}dx\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\left (\left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right ) \left (\frac {f \int \frac {1}{f-g x^2}dx}{g}-\frac {x}{g}\right )\right )+n \int \frac {x^2 \log (a+b x)}{f-g x^2}dx-n \int \frac {x^2 \log (c+d x)}{f-g x^2}dx\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle n \int \frac {x^2 \log (a+b x)}{f-g x^2}dx-n \int \frac {x^2 \log (c+d x)}{f-g x^2}dx-\left (\left (\frac {\sqrt {f} \text {arctanh}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{g^{3/2}}-\frac {x}{g}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )\right )\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle n \int \left (\frac {f \log (a+b x)}{g \left (f-g x^2\right )}-\frac {\log (a+b x)}{g}\right )dx-n \int \left (\frac {f \log (c+d x)}{g \left (f-g x^2\right )}-\frac {\log (c+d x)}{g}\right )dx-\left (\left (\frac {\sqrt {f} \text {arctanh}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{g^{3/2}}-\frac {x}{g}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\left (\left (\frac {\sqrt {f} \text {arctanh}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{g^{3/2}}-\frac {x}{g}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )\right )+n \left (\frac {\sqrt {f} \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (a+b x)}{\sqrt {g} a+b \sqrt {f}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{a \sqrt {g}+b \sqrt {f}}\right )}{2 g^{3/2}}+\frac {\sqrt {f} \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^{3/2}}-\frac {(a+b x) \log (a+b x)}{b g}+\frac {x}{g}\right )-n \left (\frac {\sqrt {f} \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (c+d x)}{\sqrt {g} c+d \sqrt {f}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{c \sqrt {g}+d \sqrt {f}}\right )}{2 g^{3/2}}+\frac {\sqrt {f} \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^{3/2}}-\frac {(c+d x) \log (c+d x)}{d g}+\frac {x}{g}\right )\) |
-((-(x/g) + (Sqrt[f]*ArcTanh[(Sqrt[g]*x)/Sqrt[f]])/g^(3/2))*(n*Log[a + b*x ] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])) + n*(x/g - ((a + b*x )*Log[a + b*x])/(b*g) - (Sqrt[f]*Log[a + b*x]*Log[(b*(Sqrt[f] - Sqrt[g]*x) )/(b*Sqrt[f] + a*Sqrt[g])])/(2*g^(3/2)) + (Sqrt[f]*Log[a + b*x]*Log[(b*(Sq rt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])])/(2*g^(3/2)) + (Sqrt[f]*PolyL og[2, -((Sqrt[g]*(a + b*x))/(b*Sqrt[f] - a*Sqrt[g]))])/(2*g^(3/2)) - (Sqrt [f]*PolyLog[2, (Sqrt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*g^(3/2))) - n*(x/g - ((c + d*x)*Log[c + d*x])/(d*g) - (Sqrt[f]*Log[c + d*x]*Log[(d*( Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*g^(3/2)) + (Sqrt[f]*Log [c + d*x]*Log[(d*(Sqrt[f] + Sqrt[g]*x))/(d*Sqrt[f] - c*Sqrt[g])])/(2*g^(3/ 2)) + (Sqrt[f]*PolyLog[2, -((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))]) /(2*g^(3/2)) - (Sqrt[f]*PolyLog[2, (Sqrt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt [g])])/(2*g^(3/2)))
3.1.77.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
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[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) ^(r_.)]*(RFx_.), x_Symbol] :> Simp[p*r Int[RFx*Log[a + b*x], x], x] + (Si mp[q*r Int[RFx*Log[c + d*x], x], x] - Simp[(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]) Int[RFx, x], x]) /; FreeQ[ {a, b, c, d, e, f, p, q, r}, x] && RationalFunctionQ[RFx, x] && NeQ[b*c - a *d, 0] && !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; IntegersQ[ m, n]]
\[\int \frac {x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{-g \,x^{2}+f}d x\]
\[ \int \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\int { -\frac {x^{2} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{g x^{2} - f} \,d x } \]
Timed out. \[ \int \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1047 vs. \(2 (438) = 876\).
Time = 0.40 (sec) , antiderivative size = 1047, normalized size of antiderivative = 1.90 \[ \int \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\text {Too large to display} \]
-1/2*(2*b*c*(c^2/((b*c*d^3 - a*d^4)*g*x + (b*c^2*d^2 - a*c*d^3)*g) + a^2*l og(b*x + a)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g) + (b*c^2 - 2*a*c*d)*lo g(d*x + c)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*g))*d - 2*(c^3/((b*c*d^4 - a*d^5)*g*x + (b*c^2*d^3 - a*c*d^4)*g) + a^3*log(b*x + a)/((b^4*c^2 - 2* a*b^3*c*d + a^2*b^2*d^2)*g) + (2*b*c^3 - 3*a*c^2*d)*log(d*x + c)/((b^2*c^2 *d^3 - 2*a*b*c*d^4 + a^2*d^5)*g) - x/(b*d^2*g))*b*d^2 + 2*a*(c^2/((b*c*d^3 - a*d^4)*g*x + (b*c^2*d^2 - a*c*d^3)*g) + a^2*log(b*x + a)/((b^3*c^2 - 2* a*b^2*c*d + a^2*b*d^2)*g) + (b*c^2 - 2*a*c*d)*log(d*x + c)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*g))*d^2 - 2*a*c*d*(c/((b*c*d^2 - a*d^3)*g*x + (b*c ^2*d - a*c*d^2)*g) + a*log(b*x + a)/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*g) - a*log(d*x + c)/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*g)) - 2*b*d*(a^2*log(b*x + a)/((b^3*c - a*b^2*d)*g) - c^2*log(d*x + c)/((b*c*d^2 - a*d^3)*g) + x/(b* d*g)) + 2*b*c*(a*log(b*x + a)/((b^2*c - a*b*d)*g) - c*log(d*x + c)/((b*c*d - a*d^2)*g)) - (log(sqrt(g)*x - sqrt(f))*log((b*sqrt(g)*x - b*sqrt(f))/(b *sqrt(f) + a*sqrt(g)) + 1) + dilog(-(b*sqrt(g)*x - b*sqrt(f))/(b*sqrt(f) + a*sqrt(g))))*sqrt(f)/g^(3/2) + (log(sqrt(g)*x + sqrt(f))*log(-(b*sqrt(g)* x + b*sqrt(f))/(b*sqrt(f) - a*sqrt(g)) + 1) + dilog((b*sqrt(g)*x + b*sqrt( f))/(b*sqrt(f) - a*sqrt(g))))*sqrt(f)/g^(3/2) + (log(sqrt(g)*x - sqrt(f))* log((d*sqrt(g)*x - d*sqrt(f))/(d*sqrt(f) + c*sqrt(g)) + 1) + dilog(-(d*sqr t(g)*x - d*sqrt(f))/(d*sqrt(f) + c*sqrt(g))))*sqrt(f)/g^(3/2) - (log(sq...
\[ \int \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\int { -\frac {x^{2} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{g x^{2} - f} \,d x } \]
Timed out. \[ \int \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\int \frac {x^2\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{f-g\,x^2} \,d x \]