Integrand size = 27, antiderivative size = 491 \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {b e n \log (d+e x)}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{3/2}}-\frac {b e n \log (d+e x)}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{3/2}}+\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{3/2}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}-\frac {b e n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{3/2}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}+\frac {b n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}} \]
1/4*(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)))/g^(3/2)/(-f)^(1/2)-1/4*(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)))/g^(3/2)/(-f)^(1/2)-1/4*b*n*polylog(2,-(e*x+ d)*g^(1/2)/(e*(-f)^(1/2)-d*g^(1/2)))/g^(3/2)/(-f)^(1/2)+1/4*b*n*polylog(2, (e*x+d)*g^(1/2)/(e*(-f)^(1/2)+d*g^(1/2)))/g^(3/2)/(-f)^(1/2)+1/4*b*e*n*ln( e*x+d)/g^(3/2)/(e*(-f)^(1/2)-d*g^(1/2))-1/4*b*e*n*ln((-f)^(1/2)+x*g^(1/2)) /g^(3/2)/(e*(-f)^(1/2)-d*g^(1/2))-1/4*b*e*n*ln(e*x+d)/g^(3/2)/(e*(-f)^(1/2 )+d*g^(1/2))+1/4*b*e*n*ln((-f)^(1/2)-x*g^(1/2))/g^(3/2)/(e*(-f)^(1/2)+d*g^ (1/2))+1/4*(a+b*ln(c*(e*x+d)^n))/g^(3/2)/((-f)^(1/2)-x*g^(1/2))+1/4*(-a-b* ln(c*(e*x+d)^n))/g^(3/2)/((-f)^(1/2)+x*g^(1/2))
Time = 0.37 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.78 \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {\frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}-\sqrt {g} x}-\frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}+\sqrt {g} x}+\frac {b e n \left (-\log (d+e x)+\log \left (\sqrt {-f}-\sqrt {g} x\right )\right )}{e \sqrt {-f}+d \sqrt {g}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{\sqrt {-f}}+\frac {b e n \left (\log (d+e x)-\log \left (\sqrt {-f}+\sqrt {g} x\right )\right )}{e \sqrt {-f}-d \sqrt {g}}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2}}+\frac {b f n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2}}+\frac {b n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{\sqrt {-f}}}{4 g^{3/2}} \]
((a + b*Log[c*(d + e*x)^n])/(Sqrt[-f] - Sqrt[g]*x) - (a + b*Log[c*(d + e*x )^n])/(Sqrt[-f] + Sqrt[g]*x) + (b*e*n*(-Log[d + e*x] + Log[Sqrt[-f] - Sqrt [g]*x]))/(e*Sqrt[-f] + d*Sqrt[g]) + ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sq rt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/Sqrt[-f] + (b*e*n*(Log[d + e*x] - Log[Sqrt[-f] + Sqrt[g]*x]))/(e*Sqrt[-f] - d*Sqrt[g]) + (f*(a + b*L og[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g]) ])/(-f)^(3/2) + (b*f*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sq rt[g]))])/(-f)^(3/2) + (b*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d *Sqrt[g])])/Sqrt[-f])/(4*g^(3/2))
Time = 1.00 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {a+b \log \left (c (d+e x)^n\right )}{g \left (f+g x^2\right )}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \left (f+g x^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 \sqrt {-f} g^{3/2}}-\frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 \sqrt {-f} g^{3/2}}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}+\frac {b n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{4 \sqrt {-f} g^{3/2}}+\frac {b e n \log (d+e x)}{4 g^{3/2} \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {b e n \log (d+e x)}{4 g^{3/2} \left (d \sqrt {g}+e \sqrt {-f}\right )}+\frac {b e n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 g^{3/2} \left (d \sqrt {g}+e \sqrt {-f}\right )}-\frac {b e n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 g^{3/2} \left (e \sqrt {-f}-d \sqrt {g}\right )}\) |
(b*e*n*Log[d + e*x])/(4*(e*Sqrt[-f] - d*Sqrt[g])*g^(3/2)) - (b*e*n*Log[d + e*x])/(4*(e*Sqrt[-f] + d*Sqrt[g])*g^(3/2)) + (a + b*Log[c*(d + e*x)^n])/( 4*g^(3/2)*(Sqrt[-f] - Sqrt[g]*x)) - (a + b*Log[c*(d + e*x)^n])/(4*g^(3/2)* (Sqrt[-f] + Sqrt[g]*x)) + (b*e*n*Log[Sqrt[-f] - Sqrt[g]*x])/(4*(e*Sqrt[-f] + d*Sqrt[g])*g^(3/2)) + ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sq rt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*Sqrt[-f]*g^(3/2)) - (b*e*n*Log[Sqr t[-f] + Sqrt[g]*x])/(4*(e*Sqrt[-f] - d*Sqrt[g])*g^(3/2)) - ((a + b*Log[c*( d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(4* Sqrt[-f]*g^(3/2)) - (b*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d* Sqrt[g]))])/(4*Sqrt[-f]*g^(3/2)) + (b*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e* Sqrt[-f] + d*Sqrt[g])])/(4*Sqrt[-f]*g^(3/2))
3.3.72.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.31 (sec) , antiderivative size = 1521, normalized size of antiderivative = 3.10
1/2*b*e^2/(e^2*g*x^2+e^2*f)/g*x*n*ln(e*x+d)-1/2*b*e^2/(e^2*g*x^2+e^2*f)/g* x*ln((e*x+d)^n)-1/2*b/g/(f*g)^(1/2)*arctan(1/2*(2*g*(e*x+d)-2*d*g)/e/(f*g) ^(1/2))*n*ln(e*x+d)+1/2*b/g/(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)+1/4*b*e*n/g/(d^2*g+e^2*f)*d*ln(g*(e*x+d)^2-2*(e* x+d)*d*g+d^2*g+f*e^2)+1/2*b*e^2*n*f/g/(d^2*g+e^2*f)/(f*g)^(1/2)*arctan(1/2 *(2*g*(e*x+d)-2*d*g)/e/(f*g)^(1/2))-1/4*b*e^2*n*g*ln(e*x+d)/(d^2*g+e^2*f)/ (e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g) ^(1/2)+d*g))*x^2*d^2-1/4*b*e^4*n*f*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2* f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))*x^ 2+1/4*b*e^2*n*g*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln( (e*(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))*x^2*d^2+1/4*b*e^4*n*f *ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2) +g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))*x^2-1/2*b*e^3*n*ln(e*x+d)/(d^2*g+e^2 *f)/(e^2*g*x^2+e^2*f)*x^2*d-1/4*b*e^2*n*f*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x ^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d *g))*d^2-1/4*b*e^4*n*f^2/g*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g )^(1/2)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))+1/4*b*e^2* n*f*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1 /2)+g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))*d^2+1/4*b*e^4*n*f^2/g*ln(e*x+d)/( d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)+g*(e*x+d...
\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{2}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{2}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
-1/2*a*(x/(g^2*x^2 + f*g) - arctan(g*x/sqrt(f*g))/(sqrt(f*g)*g)) + b*integ rate((x^2*log((e*x + d)^n) + x^2*log(c))/(g^2*x^4 + 2*f*g*x^2 + f^2), x)
\[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{2}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{{\left (g\,x^2+f\right )}^2} \,d x \]