Integrand size = 23, antiderivative size = 186 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {b n x}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{16 (-d)^{3/2} e^{3/2}} \] Output:
1/8*b*n*x/d/e/(e*x^2+d)-1/4*x*(a+b*ln(c*x^n))/e/(e*x^2+d)^2+1/8*x*(a+b*ln( c*x^n))/d/e/(e*x^2+d)+1/8*arctan(e^(1/2)*x/d^(1/2))*(a+b*ln(c*x^n))/d^(3/2 )/e^(3/2)+1/16*b*n*polylog(2,-e^(1/2)*x/(-d)^(1/2))/(-d)^(3/2)/e^(3/2)-1/1 6*b*n*polylog(2,e^(1/2)*x/(-d)^(1/2))/(-d)^(3/2)/e^(3/2)
Leaf count is larger than twice the leaf count of optimal. \(497\) vs. \(2(186)=372\).
Time = 1.18 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.67 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {\frac {d \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {a+b \log \left (c x^n\right )}{\sqrt {-d} \left (\sqrt {-d}+\sqrt {e} x\right )^2}-\frac {a+b \log \left (c x^n\right )}{\sqrt {-d} d-d \sqrt {e} x}+\frac {a+b \log \left (c x^n\right )}{\sqrt {-d} d+d \sqrt {e} x}+\frac {b d n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{5/2}}+\frac {b n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{(-d)^{3/2}}+\frac {b n \left (d+\left (d-\sqrt {-d} \sqrt {e} x\right ) \log (x)+\left (-d+\sqrt {-d} \sqrt {e} x\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{3/2}}-\frac {b n \left (d+\left (d+\sqrt {-d} \sqrt {e} x\right ) \log (x)-\left (d+\sqrt {-d} \sqrt {e} x\right ) \log \left ((-d)^{3/2}+d \sqrt {e} x\right )\right )}{d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{5/2}}+\frac {b d n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{5/2}}+\frac {b n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2}}}{16 e^{3/2}} \] Input:
Integrate[(x^2*(a + b*Log[c*x^n]))/(d + e*x^2)^3,x]
Output:
((d*(a + b*Log[c*x^n]))/((-d)^(3/2)*(Sqrt[-d] - Sqrt[e]*x)^2) + (a + b*Log [c*x^n])/(Sqrt[-d]*(Sqrt[-d] + Sqrt[e]*x)^2) - (a + b*Log[c*x^n])/(Sqrt[-d ]*d - d*Sqrt[e]*x) + (a + b*Log[c*x^n])/(Sqrt[-d]*d + d*Sqrt[e]*x) + (b*d* n*(Log[x] - Log[Sqrt[-d] - Sqrt[e]*x]))/(-d)^(5/2) + (b*n*(Log[x] - Log[Sq rt[-d] + Sqrt[e]*x]))/(-d)^(3/2) + (b*n*(d + (d - Sqrt[-d]*Sqrt[e]*x)*Log[ x] + (-d + Sqrt[-d]*Sqrt[e]*x)*Log[Sqrt[-d] + Sqrt[e]*x]))/(d^2*(Sqrt[-d] + Sqrt[e]*x)) + ((a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(3 /2) - (b*n*(d + (d + Sqrt[-d]*Sqrt[e]*x)*Log[x] - (d + Sqrt[-d]*Sqrt[e]*x) *Log[(-d)^(3/2) + d*Sqrt[e]*x]))/(d^2*(Sqrt[-d] - Sqrt[e]*x)) + (d*(a + b* Log[c*x^n])*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^(5/2) + (b*d*n*PolyLog [2, (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(5/2) + (b*n*PolyLog[2, (d*Sqrt[e]*x)/(-d) ^(3/2)])/(-d)^(3/2))/(16*e^(3/2))
Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2793, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 2793 |
| \(\displaystyle \int \left (\frac {a+b \log \left (c x^n\right )}{e \left (d+e x^2\right )^2}-\frac {d \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{3/2} e^{3/2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d e \left (d+e x^2\right )}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}-\frac {i b n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}+\frac {i b n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{3/2} e^{3/2}}+\frac {b n x}{8 d e \left (d+e x^2\right )}\) |
Input:
Int[(x^2*(a + b*Log[c*x^n]))/(d + e*x^2)^3,x]
Output:
(b*n*x)/(8*d*e*(d + e*x^2)) - (x*(a + b*Log[c*x^n]))/(4*e*(d + e*x^2)^2) + (x*(a + b*Log[c*x^n]))/(8*d*e*(d + e*x^2)) + (ArcTan[(Sqrt[e]*x)/Sqrt[d]] *(a + b*Log[c*x^n]))/(8*d^(3/2)*e^(3/2)) - ((I/16)*b*n*PolyLog[2, ((-I)*Sq rt[e]*x)/Sqrt[d]])/(d^(3/2)*e^(3/2)) + ((I/16)*b*n*PolyLog[2, (I*Sqrt[e]*x )/Sqrt[d]])/(d^(3/2)*e^(3/2))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && Integer Q[r]))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.49 (sec) , antiderivative size = 826, normalized size of antiderivative = 4.44
| method | result | size |
| risch | \(-\frac {b n \ln \left (x \right ) x^{3}}{2 d \left (e \,x^{2}+d \right )^{2}}+\frac {b \,x^{3} \ln \left (x^{n}\right )}{8 \left (e \,x^{2}+d \right )^{2} d}-\frac {b n \ln \left (x \right ) x}{2 e \left (e \,x^{2}+d \right )^{2}}-\frac {b x \ln \left (x^{n}\right )}{8 \left (e \,x^{2}+d \right )^{2} e}-\frac {b \arctan \left (\frac {x e}{\sqrt {d e}}\right ) n \ln \left (x \right )}{8 e d \sqrt {d e}}+\frac {b \arctan \left (\frac {x e}{\sqrt {d e}}\right ) \ln \left (x^{n}\right )}{8 e d \sqrt {d e}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2}}{4 d \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2}}{4 d \left (e \,x^{2}+d \right ) \sqrt {-d e}}+\frac {b n \ln \left (x \right ) x}{2 e d \left (e \,x^{2}+d \right )}+\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 e \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 e \left (e \,x^{2}+d \right ) \sqrt {-d e}}+\frac {b n \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{16 e d \sqrt {-d e}}-\frac {b n \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{16 e d \sqrt {-d e}}+\frac {b n x}{8 d e \left (e \,x^{2}+d \right )}-\frac {3 b n e \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{4}}{16 d \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}+\frac {3 b n e \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{4}}{16 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}}{8 \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}}{8 \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}-\frac {3 b n d \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{16 e \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}+\frac {3 b n d \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{16 e \left (e \,x^{2}+d \right )^{2} \sqrt {-d e}}+\left (\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 x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\frac {x^{3}}{8 d}-\frac {x}{8 e}}{\left (e \,x^{2}+d \right )^{2}}+\frac {\arctan \left (\frac {x e}{\sqrt {d e}}\right )}{8 e d \sqrt {d e}}\right )\) | \(826\) |
Input:
int(x^2*(a+b*ln(c*x^n))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
-1/2*b*n/d*ln(x)/(e*x^2+d)^2*x^3+1/8*b/(e*x^2+d)^2/d*x^3*ln(x^n)-1/2*b*n/e *ln(x)/(e*x^2+d)^2*x-1/8*b/(e*x^2+d)^2*x/e*ln(x^n)-1/8*b/e/d/(d*e)^(1/2)*a rctan(x*e/(d*e)^(1/2))*n*ln(x)+1/8*b/e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2 ))*ln(x^n)+1/4*b*n*ln(x)/d/(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*ln(x)/d/(e*x^2+d)/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/ 2))/(-d*e)^(1/2))*x^2+1/2*b*n/e*ln(x)/d/(e*x^2+d)*x+1/4*b*n/e*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*ln(x)/(e*x ^2+d)/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/16*b*n/e/d/(-d*e) ^(1/2)*dilog((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/16*b*n/e/d/(-d*e)^(1/2)*d ilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/8*b*n*x/d/e/(e*x^2+d)-3/16*b*n/d*e *ln(x)/(e*x^2+d)^2/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))*x^4+3 /16*b*n/d*e*ln(x)/(e*x^2+d)^2/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1 /2))*x^4-3/8*b*n*ln(x)/(e*x^2+d)^2/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d *e)^(1/2))*x^2+3/8*b*n*ln(x)/(e*x^2+d)^2/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2) )/(-d*e)^(1/2))*x^2-3/16*b*n*d/e*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*ln(x)/(e*x^2+d)^2/(-d*e)^(1/2)*ln( (e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+(1/2*I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1 /2*I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/2*I*Pi*b*csgn(I*c*x^n)^3+1 /2*I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+b*ln(c)+a)*((1/8/d*x^3-1/8*x/e)/(e*x^2 +d)^2+1/8/e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2)))
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate(x^2*(a+b*log(c*x^n))/(e*x^2+d)^3,x, algorithm="fricas")
Output:
integral((b*x^2*log(c*x^n) + a*x^2)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{2}\right )^{3}}\, dx \] Input:
integrate(x**2*(a+b*ln(c*x**n))/(e*x**2+d)**3,x)
Output:
Integral(x**2*(a + b*log(c*x**n))/(d + e*x**2)**3, x)
Exception generated. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^2*(a+b*log(c*x^n))/(e*x^2+d)^3,x, algorithm="maxima")
Output:
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
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate(x^2*(a+b*log(c*x^n))/(e*x^2+d)^3,x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)*x^2/(e*x^2 + d)^3, x)
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \] Input:
int((x^2*(a + b*log(c*x^n)))/(d + e*x^2)^3,x)
Output:
int((x^2*(a + b*log(c*x^n)))/(d + e*x^2)^3, x)
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,d^{2}+6 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d e \,x^{2}+3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,e^{2} x^{4}+4 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b \,d^{2} n +8 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b d e n \,x^{2}+4 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b \,e^{2} n \,x^{4}+8 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{5} e +16 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{4} e^{2} x^{2}+8 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{3} e^{3} x^{4}-8 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} e x -3 a \,d^{2} e x +3 a d \,e^{2} x^{3}+4 b \,d^{2} e n x +4 b d \,e^{2} n \,x^{3}}{24 d^{2} e^{2} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:
int(x^2*(a+b*log(c*x^n))/(e*x^2+d)^3,x)
Output:
(3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d**2 + 6*sqrt(e)*sqrt(d )*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d*e*x**2 + 3*sqrt(e)*sqrt(d)*atan((e*x)/ (sqrt(e)*sqrt(d)))*a*e**2*x**4 + 4*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqr t(d)))*b*d**2*n + 8*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*b*d*e*n* x**2 + 4*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*b*e**2*n*x**4 + 8*i nt(log(x**n*c)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d** 5*e + 16*int(log(x**n*c)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6 ),x)*b*d**4*e**2*x**2 + 8*int(log(x**n*c)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2 *x**4 + e**3*x**6),x)*b*d**3*e**3*x**4 - 8*log(x**n*c)*b*d**2*e*x - 3*a*d* *2*e*x + 3*a*d*e**2*x**3 + 4*b*d**2*e*n*x + 4*b*d*e**2*n*x**3)/(24*d**2*e* *2*(d**2 + 2*d*e*x**2 + e**2*x**4))