Integrand size = 20, antiderivative size = 210 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {b n x}{8 d^2 \left (d+e x^2\right )}-\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}+\frac {3 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 \left (d+e x^2\right )}+\frac {3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{5/2} \sqrt {e}}-\frac {3 i b n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{5/2} \sqrt {e}}+\frac {3 i b n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{5/2} \sqrt {e}} \]
-1/8*b*n*x/d^2/(e*x^2+d)+1/4*x*(a+b*ln(c*x^n))/d/(e*x^2+d)^2+3/8*x*(a+b*ln (c*x^n))/d^2/(e*x^2+d)-1/2*b*n*arctan(x*e^(1/2)/d^(1/2))/d^(5/2)/e^(1/2)+3 /8*arctan(x*e^(1/2)/d^(1/2))*(a+b*ln(c*x^n))/d^(5/2)/e^(1/2)-3/16*I*b*n*po lylog(2,-I*x*e^(1/2)/d^(1/2))/d^(5/2)/e^(1/2)+3/16*I*b*n*polylog(2,I*x*e^( 1/2)/d^(1/2))/d^(5/2)/e^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(544\) vs. \(2(210)=420\).
Time = 0.59 (sec) , antiderivative size = 544, normalized size of antiderivative = 2.59 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^3} \, dx=\frac {1}{16} \left (\frac {d \left (a+b \log \left (c x^n\right )\right )}{(-d)^{5/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {a+b \log \left (c x^n\right )}{(-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 \left (a+b \log \left (c x^n\right )\right )}{(-d)^{5/2} \sqrt {e}+d^2 e x}+\frac {3 \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2} d \sqrt {e}+d^2 e x}+\frac {3 b n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{5/2} \sqrt {e}}-\frac {3 b n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{(-d)^{5/2} \sqrt {e}}-\frac {b n \left (d+\left (d-\sqrt {-d} \sqrt {e} x\right ) \log (x)+\left (-d+\sqrt {-d} \sqrt {e} x\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{d^3 \left (\sqrt {-d} \sqrt {e}+e x\right )}-\frac {3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{5/2} \sqrt {e}}-\frac {b n \left (d+\left (d+\sqrt {-d} \sqrt {e} x\right ) \log (x)-\left (d+\sqrt {-d} \sqrt {e} x\right ) \log \left ((-d)^{3/2}+d \sqrt {e} x\right )\right )}{(-d)^{7/2} \sqrt {e}+d^3 e x}+\frac {3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{5/2} \sqrt {e}}+\frac {3 b n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{5/2} \sqrt {e}}-\frac {3 b n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{5/2} \sqrt {e}}\right ) \]
((d*(a + b*Log[c*x^n]))/((-d)^(5/2)*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)^2) + (a + b*Log[c*x^n])/((-d)^(3/2)*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)^2) + (3*(a + b *Log[c*x^n]))/((-d)^(5/2)*Sqrt[e] + d^2*e*x) + (3*(a + b*Log[c*x^n]))/((-d )^(3/2)*d*Sqrt[e] + d^2*e*x) + (3*b*n*(Log[x] - Log[Sqrt[-d] - Sqrt[e]*x]) )/((-d)^(5/2)*Sqrt[e]) - (3*b*n*(Log[x] - Log[Sqrt[-d] + Sqrt[e]*x]))/((-d )^(5/2)*Sqrt[e]) - (b*n*(d + (d - Sqrt[-d]*Sqrt[e]*x)*Log[x] + (-d + Sqrt[ -d]*Sqrt[e]*x)*Log[Sqrt[-d] + Sqrt[e]*x]))/(d^3*(Sqrt[-d]*Sqrt[e] + e*x)) - (3*(a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/((-d)^(5/2)*Sqrt[e] ) - (b*n*(d + (d + Sqrt[-d]*Sqrt[e]*x)*Log[x] - (d + Sqrt[-d]*Sqrt[e]*x)*L og[(-d)^(3/2) + d*Sqrt[e]*x]))/((-d)^(7/2)*Sqrt[e] + d^3*e*x) + (3*(a + b* Log[c*x^n])*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)])/((-d)^(5/2)*Sqrt[e]) + (3*b *n*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/((-d)^(5/2)*Sqrt[e]) - (3*b*n*PolyLog [2, (d*Sqrt[e]*x)/(-d)^(3/2)])/((-d)^(5/2)*Sqrt[e]))/16
Time = 0.56 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.19, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {2760, 215, 218, 2760, 218, 2761, 27, 5355, 2838}
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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 2760 |
| \(\displaystyle \frac {3 \int \frac {a+b \log \left (c x^n\right )}{\left (e x^2+d\right )^2}dx}{4 d}-\frac {b n \int \frac {1}{\left (e x^2+d\right )^2}dx}{4 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {3 \int \frac {a+b \log \left (c x^n\right )}{\left (e x^2+d\right )^2}dx}{4 d}-\frac {b n \left (\frac {\int \frac {1}{e x^2+d}dx}{2 d}+\frac {x}{2 d \left (d+e x^2\right )}\right )}{4 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {3 \int \frac {a+b \log \left (c x^n\right )}{\left (e x^2+d\right )^2}dx}{4 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {b n \left (\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}+\frac {x}{2 d \left (d+e x^2\right )}\right )}{4 d}\) |
| \(\Big \downarrow \) 2760 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{e x^2+d}dx}{2 d}-\frac {b n \int \frac {1}{e x^2+d}dx}{2 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}\right )}{4 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {b n \left (\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}+\frac {x}{2 d \left (d+e x^2\right )}\right )}{4 d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{e x^2+d}dx}{2 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}\right )}{4 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {b n \left (\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}+\frac {x}{2 d \left (d+e x^2\right )}\right )}{4 d}\) |
| \(\Big \downarrow \) 2761 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {e}}-b n \int \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x}dx}{2 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}\right )}{4 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {b n \left (\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}+\frac {x}{2 d \left (d+e x^2\right )}\right )}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {e}}-\frac {b n \int \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x}dx}{\sqrt {d} \sqrt {e}}}{2 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}\right )}{4 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {b n \left (\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}+\frac {x}{2 d \left (d+e x^2\right )}\right )}{4 d}\) |
| \(\Big \downarrow \) 5355 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {e}}-\frac {b n \left (\frac {1}{2} i \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x}dx-\frac {1}{2} i \int \frac {\log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right )}{x}dx\right )}{\sqrt {d} \sqrt {e}}}{2 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}\right )}{4 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {b n \left (\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}+\frac {x}{2 d \left (d+e x^2\right )}\right )}{4 d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {e}}-\frac {b n \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {d} \sqrt {e}}}{2 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}\right )}{4 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {b n \left (\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}+\frac {x}{2 d \left (d+e x^2\right )}\right )}{4 d}\) |
-1/4*(b*n*(x/(2*d*(d + e*x^2)) + ArcTan[(Sqrt[e]*x)/Sqrt[d]]/(2*d^(3/2)*Sq rt[e])))/d + (x*(a + b*Log[c*x^n]))/(4*d*(d + e*x^2)^2) + (3*(-1/2*(b*n*Ar cTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e]) + (x*(a + b*Log[c*x^n]))/(2*d *(d + e*x^2)) + ((ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(Sqrt[d] *Sqrt[e]) - (b*n*((I/2)*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]] - (I/2)*PolyL og[2, (I*Sqrt[e]*x)/Sqrt[d]]))/(Sqrt[d]*Sqrt[e]))/(2*d)))/(4*d)
3.3.38.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Sym bol] :> Simp[(-x)*(d + e*x^2)^(q + 1)*((a + b*Log[c*x^n])/(2*d*(q + 1))), x ] + (Simp[(2*q + 3)/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*Log[c*x^ n]), x], x] + Simp[b*(n/(2*d*(q + 1))) Int[(d + e*x^2)^(q + 1), x], x]) / ; FreeQ[{a, b, c, d, e, n}, x] && LtQ[q, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(d + e*x^2), x]}, Simp[u*(a + b*Log[c*x^n]), x] - Si mp[b*n Int[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n}, x]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Simp[I*(b/2) Int[Log[1 - I*c*x]/x, x], x] - Simp[I*(b/2) Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.63 (sec) , antiderivative size = 664, normalized size of antiderivative = 3.16
| method | result | size |
| risch | \(\frac {3 b n \ln \left (x \right ) x}{8 d \left (e \,x^{2}+d \right )^{2}}+\frac {b x \ln \left (x^{n}\right )}{4 d \left (e \,x^{2}+d \right )^{2}}-\frac {3 b x n \ln \left (x \right )}{8 d^{2} \left (e \,x^{2}+d \right )}+\frac {3 b x \ln \left (x^{n}\right )}{8 d^{2} \left (e \,x^{2}+d \right )}-\frac {3 b \arctan \left (\frac {x e}{\sqrt {d e}}\right ) n \ln \left (x \right )}{8 d^{2} \sqrt {d e}}+\frac {3 b \arctan \left (\frac {x e}{\sqrt {d e}}\right ) \ln \left (x^{n}\right )}{8 d^{2} \sqrt {d e}}-\frac {b n x}{8 d^{2} \left (e \,x^{2}+d \right )}-\frac {b n \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{2 d^{2} \sqrt {d e}}+\frac {3 b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{4} e^{2}}{16 d^{2} \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}-\frac {3 b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{4} e^{2}}{16 d^{2} \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}+\frac {3 b n \ln \left (x \right ) x^{3} e}{8 d^{2} \left (e \,x^{2}+d \right )^{2}}+\frac {3 b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2} e}{8 d \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}-\frac {3 b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2} e}{8 d \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}+\frac {3 b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{16 \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}-\frac {3 b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{16 \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}+\frac {3 b n \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{16 \sqrt {-d e}\, d^{2}}-\frac {3 b n \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{16 \sqrt {-d e}\, d^{2}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x}{4 d \left (e \,x^{2}+d \right )^{2}}+\frac {\frac {3 x}{8 d \left (e \,x^{2}+d \right )}+\frac {3 \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{8 d \sqrt {d e}}}{d}\right )\) | \(664\) |
3/8*b*n*ln(x)/d/(e*x^2+d)^2*x+1/4*b*x/d/(e*x^2+d)^2*ln(x^n)-3/8*b/d^2*x/(e *x^2+d)*n*ln(x)+3/8*b/d^2*x/(e*x^2+d)*ln(x^n)-3/8*b/d^2/(d*e)^(1/2)*arctan (x*e/(d*e)^(1/2))*n*ln(x)+3/8*b/d^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*ln (x^n)-1/8*b*n*x/d^2/(e*x^2+d)-1/2*b*n/d^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/ 2))+3/16*b*n*ln(x)/d^2/(e*x^2+d)^2/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d *e)^(1/2))*x^4*e^2-3/16*b*n*ln(x)/d^2/(e*x^2+d)^2/(-d*e)^(1/2)*ln((e*x+(-d *e)^(1/2))/(-d*e)^(1/2))*x^4*e^2+3/8*b*n*ln(x)/d^2/(e*x^2+d)^2*x^3*e+3/8*b *n*ln(x)/d/(e*x^2+d)^2/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))*x ^2*e-3/8*b*n*ln(x)/d/(e*x^2+d)^2/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e) ^(1/2))*x^2*e+3/16*b*n*ln(x)/(e*x^2+d)^2/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2 ))/(-d*e)^(1/2))-3/16*b*n*ln(x)/(e*x^2+d)^2/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1 /2))/(-d*e)^(1/2))+3/16*b*n/(-d*e)^(1/2)/d^2*dilog((-e*x+(-d*e)^(1/2))/(-d *e)^(1/2))-3/16*b*n/(-d*e)^(1/2)/d^2*dilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2) )+(-1/2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*b*Pi*csgn(I*c)*cs gn(I*c*x^n)^2+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*b*Pi*csgn(I*c*x ^n)^3+b*ln(c)+a)*(1/4*x/d/(e*x^2+d)^2+3/4/d*(1/2*x/d/(e*x^2+d)+1/2/d/(d*e) ^(1/2)*arctan(x*e/(d*e)^(1/2))))
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{\left (d + e x^{2}\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]