Integrand size = 23, antiderivative size = 224 \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^2} \, dx=-\frac {5 b n}{18 d^2 x^3}+\frac {5 b e n}{2 d^3 x}+\frac {a+b \log \left (c x^n\right )}{2 d x^3 \left (d+e x^2\right )}-\frac {5 a-b n+5 b \log \left (c x^n\right )}{6 d^2 x^3}+\frac {e \left (5 a-b n+5 b \log \left (c x^n\right )\right )}{2 d^3 x}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (5 a-b n+5 b \log \left (c x^n\right )\right )}{2 d^{7/2}}-\frac {5 i b e^{3/2} n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{7/2}}+\frac {5 i b e^{3/2} n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{7/2}} \]
-5/18*b*n/d^2/x^3+5/2*b*e*n/d^3/x+1/2*(a+b*ln(c*x^n))/d/x^3/(e*x^2+d)+1/6* (-5*a+b*n-5*b*ln(c*x^n))/d^2/x^3+1/2*e*(5*a-b*n+5*b*ln(c*x^n))/d^3/x+1/2*e ^(3/2)*arctan(x*e^(1/2)/d^(1/2))*(5*a-b*n+5*b*ln(c*x^n))/d^(7/2)-5/4*I*b*e ^(3/2)*n*polylog(2,-I*x*e^(1/2)/d^(1/2))/d^(7/2)+5/4*I*b*e^(3/2)*n*polylog (2,I*x*e^(1/2)/d^(1/2))/d^(7/2)
Time = 0.45 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.61 \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^2} \, dx=\frac {1}{36} \left (-\frac {4 b n}{d^2 x^3}+\frac {72 b e n}{d^3 x}-\frac {12 \left (a+b \log \left (c x^n\right )\right )}{d^2 x^3}+\frac {72 e \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {9 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{d^3 \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {9 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{d^3 \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {9 b e^{3/2} n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{7/2}}+\frac {9 b e^{3/2} n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{(-d)^{7/2}}+\frac {45 e^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{7/2}}-\frac {45 e^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{7/2}}-\frac {45 b e^{3/2} n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{7/2}}+\frac {45 b e^{3/2} n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{7/2}}\right ) \]
((-4*b*n)/(d^2*x^3) + (72*b*e*n)/(d^3*x) - (12*(a + b*Log[c*x^n]))/(d^2*x^ 3) + (72*e*(a + b*Log[c*x^n]))/(d^3*x) - (9*e^(3/2)*(a + b*Log[c*x^n]))/(d ^3*(Sqrt[-d] - Sqrt[e]*x)) + (9*e^(3/2)*(a + b*Log[c*x^n]))/(d^3*(Sqrt[-d] + Sqrt[e]*x)) - (9*b*e^(3/2)*n*(Log[x] - Log[Sqrt[-d] - Sqrt[e]*x]))/(-d) ^(7/2) + (9*b*e^(3/2)*n*(Log[x] - Log[Sqrt[-d] + Sqrt[e]*x]))/(-d)^(7/2) + (45*e^(3/2)*(a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(7/2) - (45*e^(3/2)*(a + b*Log[c*x^n])*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^( 7/2) - (45*b*e^(3/2)*n*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(7/2) + (45* b*e^(3/2)*n*PolyLog[2, (d*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^(7/2))/36
Time = 0.92 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2785, 25, 2780, 2741, 2780, 2741, 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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2785 |
| \(\displaystyle \frac {a+b \log \left (c x^n\right )}{2 d x^3 \left (d+e x^2\right )}-\frac {\int -\frac {5 a-b n+5 b \log \left (c x^n\right )}{x^4 \left (e x^2+d\right )}dx}{2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {5 a-b n+5 b \log \left (c x^n\right )}{x^4 \left (e x^2+d\right )}dx}{2 d}+\frac {a+b \log \left (c x^n\right )}{2 d x^3 \left (d+e x^2\right )}\) |
| \(\Big \downarrow \) 2780 |
| \(\displaystyle \frac {\frac {\int \frac {5 a-b n+5 b \log \left (c x^n\right )}{x^4}dx}{d}-\frac {e \int \frac {5 a-b n+5 b \log \left (c x^n\right )}{x^2 \left (e x^2+d\right )}dx}{d}}{2 d}+\frac {a+b \log \left (c x^n\right )}{2 d x^3 \left (d+e x^2\right )}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {\frac {-\frac {5 a+5 b \log \left (c x^n\right )-b n}{3 x^3}-\frac {5 b n}{9 x^3}}{d}-\frac {e \int \frac {5 a-b n+5 b \log \left (c x^n\right )}{x^2 \left (e x^2+d\right )}dx}{d}}{2 d}+\frac {a+b \log \left (c x^n\right )}{2 d x^3 \left (d+e x^2\right )}\) |
| \(\Big \downarrow \) 2780 |
| \(\displaystyle \frac {\frac {-\frac {5 a+5 b \log \left (c x^n\right )-b n}{3 x^3}-\frac {5 b n}{9 x^3}}{d}-\frac {e \left (\frac {\int \frac {5 a-b n+5 b \log \left (c x^n\right )}{x^2}dx}{d}-\frac {e \int \frac {5 a-b n+5 b \log \left (c x^n\right )}{e x^2+d}dx}{d}\right )}{d}}{2 d}+\frac {a+b \log \left (c x^n\right )}{2 d x^3 \left (d+e x^2\right )}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {\frac {-\frac {5 a+5 b \log \left (c x^n\right )-b n}{3 x^3}-\frac {5 b n}{9 x^3}}{d}-\frac {e \left (\frac {-\frac {5 a+5 b \log \left (c x^n\right )-b n}{x}-\frac {5 b n}{x}}{d}-\frac {e \int \frac {5 a-b n+5 b \log \left (c x^n\right )}{e x^2+d}dx}{d}\right )}{d}}{2 d}+\frac {a+b \log \left (c x^n\right )}{2 d x^3 \left (d+e x^2\right )}\) |
| \(\Big \downarrow \) 2761 |
| \(\displaystyle \frac {\frac {-\frac {5 a+5 b \log \left (c x^n\right )-b n}{3 x^3}-\frac {5 b n}{9 x^3}}{d}-\frac {e \left (\frac {-\frac {5 a+5 b \log \left (c x^n\right )-b n}{x}-\frac {5 b n}{x}}{d}-\frac {e \left (\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (5 a+5 b \log \left (c x^n\right )-b n\right )}{\sqrt {d} \sqrt {e}}-5 b n \int \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x}dx\right )}{d}\right )}{d}}{2 d}+\frac {a+b \log \left (c x^n\right )}{2 d x^3 \left (d+e x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {5 a+5 b \log \left (c x^n\right )-b n}{3 x^3}-\frac {5 b n}{9 x^3}}{d}-\frac {e \left (\frac {-\frac {5 a+5 b \log \left (c x^n\right )-b n}{x}-\frac {5 b n}{x}}{d}-\frac {e \left (\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (5 a+5 b \log \left (c x^n\right )-b n\right )}{\sqrt {d} \sqrt {e}}-\frac {5 b n \int \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x}dx}{\sqrt {d} \sqrt {e}}\right )}{d}\right )}{d}}{2 d}+\frac {a+b \log \left (c x^n\right )}{2 d x^3 \left (d+e x^2\right )}\) |
| \(\Big \downarrow \) 5355 |
| \(\displaystyle \frac {a+b \log \left (c x^n\right )}{2 d x^3 \left (d+e x^2\right )}+\frac {\frac {-\frac {5 a+5 b \log \left (c x^n\right )-b n}{3 x^3}-\frac {5 b n}{9 x^3}}{d}-\frac {e \left (\frac {-\frac {5 a+5 b \log \left (c x^n\right )-b n}{x}-\frac {5 b n}{x}}{d}-\frac {e \left (\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (5 a+5 b \log \left (c x^n\right )-b n\right )}{\sqrt {d} \sqrt {e}}-\frac {5 b n \left (\frac {1}{2} i \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x}dx-\frac {1}{2} i \int \frac {\log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right )}{x}dx\right )}{\sqrt {d} \sqrt {e}}\right )}{d}\right )}{d}}{2 d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {a+b \log \left (c x^n\right )}{2 d x^3 \left (d+e x^2\right )}+\frac {\frac {-\frac {5 a+5 b \log \left (c x^n\right )-b n}{3 x^3}-\frac {5 b n}{9 x^3}}{d}-\frac {e \left (\frac {-\frac {5 a+5 b \log \left (c x^n\right )-b n}{x}-\frac {5 b n}{x}}{d}-\frac {e \left (\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (5 a+5 b \log \left (c x^n\right )-b n\right )}{\sqrt {d} \sqrt {e}}-\frac {5 b n \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {d} \sqrt {e}}\right )}{d}\right )}{d}}{2 d}\) |
(a + b*Log[c*x^n])/(2*d*x^3*(d + e*x^2)) + (((-5*b*n)/(9*x^3) - (5*a - b*n + 5*b*Log[c*x^n])/(3*x^3))/d - (e*(((-5*b*n)/x - (5*a - b*n + 5*b*Log[c*x ^n])/x)/d - (e*((ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(5*a - b*n + 5*b*Log[c*x^n])) /(Sqrt[d]*Sqrt[e]) - (5*b*n*((I/2)*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]] - (I/2)*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]]))/(Sqrt[d]*Sqrt[e])))/d))/d)/(2*d)
3.3.30.3.1 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_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -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[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.))/((d_) + (e_.)* (x_)^(r_.)), x_Symbol] :> Simp[1/d Int[x^m*(a + b*Log[c*x^n])^p, x], x] - Simp[e/d Int[(x^(m + r)*(a + b*Log[c*x^n])^p)/(d + e*x^r), x], x] /; Fre eQ[{a, b, c, d, e, m, n, r}, x] && IGtQ[p, 0] && IGtQ[r, 0] && ILtQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^2)^(q_.), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(q + 1)*((a + b*Log[c*x^n])/(2*d*f*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(f*x)^m*(d + e*x^2)^(q + 1)*(a*(m + 2*q + 3) + b*n + b*(m + 2*q + 3)*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && ILtQ[m, 0]
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 = 0.86 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.78
| method | result | size |
| risch | \(\frac {b \ln \left (x^{n}\right ) e^{2} x}{2 d^{3} \left (e \,x^{2}+d \right )}-\frac {5 b \,e^{2} \arctan \left (\frac {x e}{\sqrt {d e}}\right ) n \ln \left (x \right )}{2 d^{3} \sqrt {d e}}+\frac {5 b \,e^{2} \arctan \left (\frac {x e}{\sqrt {d e}}\right ) \ln \left (x^{n}\right )}{2 d^{3} \sqrt {d e}}-\frac {b \ln \left (x^{n}\right )}{3 d^{2} x^{3}}+\frac {2 b \ln \left (x^{n}\right ) e}{d^{3} x}+\frac {b n \,e^{2} \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{d^{3} \sqrt {-d e}}-\frac {b n \,e^{2} \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{d^{3} \sqrt {-d e}}+\frac {5 b n \,e^{2} \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 d^{3} \sqrt {-d e}}-\frac {5 b n \,e^{2} \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 d^{3} \sqrt {-d e}}-\frac {b n}{9 d^{2} x^{3}}-\frac {b n \,e^{2} \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{2 d^{3} \sqrt {d e}}+\frac {b n \,e^{3} \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2}}{4 d^{3} \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {b n \,e^{3} \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2}}{4 d^{3} \left (e \,x^{2}+d \right ) \sqrt {-d e}}+\frac {b n \,e^{2} \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 d^{2} \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {b n \,e^{2} \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 d^{2} \left (e \,x^{2}+d \right ) \sqrt {-d e}}+\frac {2 b e n}{d^{3} x}+\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 {e^{2} \left (\frac {x}{2 e \,x^{2}+2 d}+\frac {5 \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{2 \sqrt {d e}}\right )}{d^{3}}-\frac {1}{3 d^{2} x^{3}}+\frac {2 e}{d^{3} x}\right )\) | \(622\) |
1/2*b*ln(x^n)*e^2/d^3*x/(e*x^2+d)-5/2*b*e^2/d^3/(d*e)^(1/2)*arctan(x*e/(d* e)^(1/2))*n*ln(x)+5/2*b*e^2/d^3/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*ln(x^n )-1/3*b*ln(x^n)/d^2/x^3+2*b*ln(x^n)/d^3*e/x+b*n*e^2/d^3*ln(x)/(-d*e)^(1/2) *ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-b*n*e^2/d^3*ln(x)/(-d*e)^(1/2)*ln((e *x+(-d*e)^(1/2))/(-d*e)^(1/2))+5/4*b*n*e^2/d^3/(-d*e)^(1/2)*dilog((-e*x+(- d*e)^(1/2))/(-d*e)^(1/2))-5/4*b*n*e^2/d^3/(-d*e)^(1/2)*dilog((e*x+(-d*e)^( 1/2))/(-d*e)^(1/2))-1/9*b*n/d^2/x^3-1/2*b*n*e^2/d^3/(d*e)^(1/2)*arctan(x*e /(d*e)^(1/2))+1/4*b*n*e^3/d^3*ln(x)/(e*x^2+d)/(-d*e)^(1/2)*ln((-e*x+(-d*e) ^(1/2))/(-d*e)^(1/2))*x^2-1/4*b*n*e^3/d^3*ln(x)/(e*x^2+d)/(-d*e)^(1/2)*ln( (e*x+(-d*e)^(1/2))/(-d*e)^(1/2))*x^2+1/4*b*n*e^2/d^2*ln(x)/(e*x^2+d)/(-d*e )^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/4*b*n*e^2/d^2*ln(x)/(e*x^2+ d)/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+2*b*e*n/d^3/x+(-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)*csgn(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)*(e^2/d^3*(1/2*x/(e*x^2+d)+5/2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2)))- 1/3/d^2/x^3+2/d^3*e/x)
\[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{4}} \,d x } \]
\[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^2} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{x^{4} \left (d + e x^{2}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{4}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^4\,{\left (e\,x^2+d\right )}^2} \,d x \]