Integrand size = 27, antiderivative size = 534 \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {a x}{g^2}-\frac {b n x}{g^2}-\frac {b e f n \log (d+e x)}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{5/2}}+\frac {b e f n \log (d+e x)}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{5/2}}+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2} \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e f n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{5/2}}+\frac {3 \sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 g^{5/2}}+\frac {b e f n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{5/2}}-\frac {3 \sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 g^{5/2}}-\frac {3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 g^{5/2}}+\frac {3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 g^{5/2}} \]
a*x/g^2-b*n*x/g^2+b*(e*x+d)*ln(c*(e*x+d)^n)/e/g^2+3/4*(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)))*(-f)^(1/2)/g^(5/2) -3/4*(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)))*(-f)^(1/2)/g^(5/2)-3/4*b*n*polylog(2,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2) -d*g^(1/2)))*(-f)^(1/2)/g^(5/2)+3/4*b*n*polylog(2,(e*x+d)*g^(1/2)/(e*(-f)^ (1/2)+d*g^(1/2)))*(-f)^(1/2)/g^(5/2)-1/4*b*e*f*n*ln(e*x+d)/g^(5/2)/(e*(-f) ^(1/2)-d*g^(1/2))+1/4*b*e*f*n*ln((-f)^(1/2)+x*g^(1/2))/g^(5/2)/(e*(-f)^(1/ 2)-d*g^(1/2))+1/4*b*e*f*n*ln(e*x+d)/g^(5/2)/(e*(-f)^(1/2)+d*g^(1/2))-1/4*b *e*f*n*ln((-f)^(1/2)-x*g^(1/2))/g^(5/2)/(e*(-f)^(1/2)+d*g^(1/2))-1/4*f*(a+ b*ln(c*(e*x+d)^n))/g^(5/2)/((-f)^(1/2)-x*g^(1/2))+1/4*f*(a+b*ln(c*(e*x+d)^ n))/g^(5/2)/((-f)^(1/2)+x*g^(1/2))
Time = 0.47 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.81 \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {4 a \sqrt {g} x-4 b \sqrt {g} n x+\frac {4 b \sqrt {g} (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {-f}-\sqrt {g} x}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {-f}+\sqrt {g} x}+\frac {b e f n \left (\log (d+e x)-\log \left (\sqrt {-f}-\sqrt {g} x\right )\right )}{e \sqrt {-f}+d \sqrt {g}}+3 \sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )+\frac {b e f n \left (\log (d+e x)-\log \left (\sqrt {-f}+\sqrt {g} x\right )\right )}{-e \sqrt {-f}+d \sqrt {g}}-3 \sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )-3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )+3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 g^{5/2}} \]
(4*a*Sqrt[g]*x - 4*b*Sqrt[g]*n*x + (4*b*Sqrt[g]*(d + e*x)*Log[c*(d + e*x)^ n])/e - (f*(a + b*Log[c*(d + e*x)^n]))/(Sqrt[-f] - Sqrt[g]*x) + (f*(a + b* Log[c*(d + e*x)^n]))/(Sqrt[-f] + Sqrt[g]*x) + (b*e*f*n*(Log[d + e*x] - Log [Sqrt[-f] - Sqrt[g]*x]))/(e*Sqrt[-f] + d*Sqrt[g]) + 3*Sqrt[-f]*(a + b*Log[ c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])] + (b*e*f*n*(Log[d + e*x] - Log[Sqrt[-f] + Sqrt[g]*x]))/(-(e*Sqrt[-f]) + d*S qrt[g]) - 3*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g] *x))/(e*Sqrt[-f] - d*Sqrt[g])] - 3*b*Sqrt[-f]*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))] + 3*b*Sqrt[-f]*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*g^(5/2))
Time = 1.12 (sec) , antiderivative size = 534, 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.
| \(\displaystyle \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 \left (f+g x^2\right )^2}-\frac {2 f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 \left (f+g x^2\right )}+\frac {a+b \log \left (c (d+e x)^n\right )}{g^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {3 \sqrt {-f} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2}}-\frac {3 \sqrt {-f} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2}}+\frac {a x}{g^2}+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 g^{5/2}}+\frac {3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{4 g^{5/2}}-\frac {b e f n \log (d+e x)}{4 g^{5/2} \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {b e f n \log (d+e x)}{4 g^{5/2} \left (d \sqrt {g}+e \sqrt {-f}\right )}-\frac {b e f n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 g^{5/2} \left (d \sqrt {g}+e \sqrt {-f}\right )}+\frac {b e f n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 g^{5/2} \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {b n x}{g^2}\) |
(a*x)/g^2 - (b*n*x)/g^2 - (b*e*f*n*Log[d + e*x])/(4*(e*Sqrt[-f] - d*Sqrt[g ])*g^(5/2)) + (b*e*f*n*Log[d + e*x])/(4*(e*Sqrt[-f] + d*Sqrt[g])*g^(5/2)) + (b*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) - (f*(a + b*Log[c*(d + e*x)^n]) )/(4*g^(5/2)*(Sqrt[-f] - Sqrt[g]*x)) + (f*(a + b*Log[c*(d + e*x)^n]))/(4*g ^(5/2)*(Sqrt[-f] + Sqrt[g]*x)) - (b*e*f*n*Log[Sqrt[-f] - Sqrt[g]*x])/(4*(e *Sqrt[-f] + d*Sqrt[g])*g^(5/2)) + (3*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])*L og[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*g^(5/2)) + (b* e*f*n*Log[Sqrt[-f] + Sqrt[g]*x])/(4*(e*Sqrt[-f] - d*Sqrt[g])*g^(5/2)) - (3 *Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqr t[-f] - d*Sqrt[g])])/(4*g^(5/2)) - (3*b*Sqrt[-f]*n*PolyLog[2, -((Sqrt[g]*( d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(4*g^(5/2)) + (3*b*Sqrt[-f]*n*PolyLo g[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*g^(5/2))
3.3.71.3.1 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 = 1.32 (sec) , antiderivative size = 1619, normalized size of antiderivative = 3.03
-b*d*n/e/g^2-1/4*b*e^4*n/g*f^2*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*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*d^2-1/4*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))*x^2*d^2+1/4*b*e^4*n/g*f^2*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^2-3/4*b*n/g^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))+3/4*b*n/g^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))-3/2*b/g^2*f/(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^2*n/g *f^2*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^2*n/g*f^2*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+b*n/g^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))+1/2*b*e^2/g^2*f*x/(e^2*g*x^2 +e^2*f)*ln((e*x+d)^n)+b*ln((e*x+d)^n)/g^2*x+b/e/g^2*d*ln((e*x+d)^n)-1/4*b* e*n/g^2*f/(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/g^2*f^2/(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))-1/2*b*e^2/g^2*f*x/(e^2*g*x^2+e^2*f)*n*ln(e*x+d)+1/2*b*e^3*n...
\[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{4}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{4}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
1/2*a*(f*x/(g^3*x^2 + f*g^2) - 3*f*arctan(g*x/sqrt(f*g))/(sqrt(f*g)*g^2) + 2*x/g^2) + b*integrate((x^4*log((e*x + d)^n) + x^4*log(c))/(g^2*x^4 + 2*f *g*x^2 + f^2), x)
\[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{4}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int \frac {x^4\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{{\left (g\,x^2+f\right )}^2} \,d x \]