Integrand size = 23, antiderivative size = 210 \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {b n x}{8 e^2 \left (d+e x^2\right )}+\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{5/2}}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{4 e^2 \left (d+e x^2\right )^2}-\frac {5 x \left (a+b \log \left (c x^n\right )\right )}{8 e^2 \left (d+e x^2\right )}+\frac {3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 \sqrt {d} e^{5/2}}-\frac {3 b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{16 \sqrt {-d} e^{5/2}} \] Output:
-1/8*b*n*x/e^2/(e*x^2+d)+1/2*b*n*arctan(e^(1/2)*x/d^(1/2))/d^(1/2)/e^(5/2) +1/4*d*x*(a+b*ln(c*x^n))/e^2/(e*x^2+d)^2-5/8*x*(a+b*ln(c*x^n))/e^2/(e*x^2+ d)+3/8*arctan(e^(1/2)*x/d^(1/2))*(a+b*ln(c*x^n))/d^(1/2)/e^(5/2)-3/16*b*n* polylog(2,-e^(1/2)*x/(-d)^(1/2))/(-d)^(1/2)/e^(5/2)+3/16*b*n*polylog(2,e^( 1/2)*x/(-d)^(1/2))/(-d)^(1/2)/e^(5/2)
Leaf count is larger than twice the leaf count of optimal. \(495\) vs. \(2(210)=420\).
Time = 1.49 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.36 \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {-\frac {\sqrt {-d} \left (a+b \log \left (c x^n\right )\right )}{\left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {5 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-d}-\sqrt {e} x}+\frac {\sqrt {-d} \left (a+b \log \left (c x^n\right )\right )}{\left (\sqrt {-d}+\sqrt {e} x\right )^2}-\frac {5 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-d}+\sqrt {e} x}-\frac {5 b n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{\sqrt {-d}}+\frac {5 b n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{\sqrt {-d}}-\frac {b n \left (d+\left (d-\sqrt {-d} \sqrt {e} x\right ) \log (x)+\left (-d+\sqrt {-d} \sqrt {e} x\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{d \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{\sqrt {-d}}+\frac {b n \left (d+\left (d+\sqrt {-d} \sqrt {e} x\right ) \log (x)-\left (d+\sqrt {-d} \sqrt {e} x\right ) \log \left ((-d)^{3/2}+d \sqrt {e} x\right )\right )}{d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{\sqrt {-d}}+\frac {3 b n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{\sqrt {-d}}-\frac {3 b n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{\sqrt {-d}}}{16 e^{5/2}} \] Input:
Integrate[(x^4*(a + b*Log[c*x^n]))/(d + e*x^2)^3,x]
Output:
(-((Sqrt[-d]*(a + b*Log[c*x^n]))/(Sqrt[-d] - Sqrt[e]*x)^2) + (5*(a + b*Log [c*x^n]))/(Sqrt[-d] - Sqrt[e]*x) + (Sqrt[-d]*(a + b*Log[c*x^n]))/(Sqrt[-d] + Sqrt[e]*x)^2 - (5*(a + b*Log[c*x^n]))/(Sqrt[-d] + Sqrt[e]*x) - (5*b*n*( Log[x] - Log[Sqrt[-d] - Sqrt[e]*x]))/Sqrt[-d] + (5*b*n*(Log[x] - Log[Sqrt[ -d] + Sqrt[e]*x]))/Sqrt[-d] - (b*n*(d + (d - Sqrt[-d]*Sqrt[e]*x)*Log[x] + (-d + Sqrt[-d]*Sqrt[e]*x)*Log[Sqrt[-d] + Sqrt[e]*x]))/(d*(Sqrt[-d] + Sqrt[ e]*x)) - (3*(a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/Sqrt[-d] + ( 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*(Sqrt[-d] - Sqrt[e]*x)) + (3*(a + b*Log[c*x^n ])*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)])/Sqrt[-d] + (3*b*n*PolyLog[2, (Sqrt[e ]*x)/Sqrt[-d]])/Sqrt[-d] - (3*b*n*PolyLog[2, (d*Sqrt[e]*x)/(-d)^(3/2)])/Sq rt[-d])/(16*e^(5/2))
Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 2793 |
| \(\displaystyle \int \left (\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )^3}-\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )^2}+\frac {a+b \log \left (c x^n\right )}{e^2 \left (d+e x^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 \sqrt {d} e^{5/2}}-\frac {5 x \left (a+b \log \left (c x^n\right )\right )}{8 e^2 \left (d+e x^2\right )}+\frac {d x \left (a+b \log \left (c x^n\right )\right )}{4 e^2 \left (d+e x^2\right )^2}+\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{5/2}}-\frac {3 i b n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 \sqrt {d} e^{5/2}}+\frac {3 i b n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 \sqrt {d} e^{5/2}}-\frac {b n x}{8 e^2 \left (d+e x^2\right )}\) |
Input:
Int[(x^4*(a + b*Log[c*x^n]))/(d + e*x^2)^3,x]
Output:
-1/8*(b*n*x)/(e^2*(d + e*x^2)) + (b*n*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*Sqrt [d]*e^(5/2)) + (d*x*(a + b*Log[c*x^n]))/(4*e^2*(d + e*x^2)^2) - (5*x*(a + b*Log[c*x^n]))/(8*e^2*(d + e*x^2)) + (3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b *Log[c*x^n]))/(8*Sqrt[d]*e^(5/2)) - (((3*I)/16)*b*n*PolyLog[2, ((-I)*Sqrt[ e]*x)/Sqrt[d]])/(Sqrt[d]*e^(5/2)) + (((3*I)/16)*b*n*PolyLog[2, (I*Sqrt[e]* x)/Sqrt[d]])/(Sqrt[d]*e^(5/2))
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 = 1.51 (sec) , antiderivative size = 900, normalized size of antiderivative = 4.29
Input:
int(x^4*(a+b*ln(c*x^n))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
3/8*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^2-3/8*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^2+1/2*b*n/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+3/16*b* n/e^2/(-d*e)^(1/2)*dilog((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-3/16*b*n/e^2/(- d*e)^(1/2)*dilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/8*b*n*x/e^2/(e*x^2+d)- 5/8*b/(e*x^2+d)^2/e*x^3*ln(x^n)-b*n/e^2*ln(x)*x/(e*x^2+d)+1/2*b*n/e^2*ln(x )/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/2*b*n/e^2*ln(x)/(-d* e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+b*n/e*ln(x)/(e*x^2+d)^2*x^3-1 /2*b*n/e*ln(x)/(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)/(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*d/e^2*ln(x)/(e*x^2+d)/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2) )/(-d*e)^(1/2))+1/2*b*n*d/e^2*ln(x)/(e*x^2+d)/(-d*e)^(1/2)*ln((e*x+(-d*e)^ (1/2))/(-d*e)^(1/2))+3/16*b*n*d^2/e^2*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^2/e^2*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))*x^4-3/16*b*n*ln(x)/(e*x^2+d)^ 2/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))*x^4+b*n*d/e^2*ln(x)/(e* x^2+d)^2*x+3/8*b/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*ln(x^n)-3/8*b*ln( x^n)*d/e^2/(e*x^2+d)^2*x-3/8*b/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*n*l n(x)+(1/2*I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*Pi*b*csgn(I*x^n)*csg...
\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate(x^4*(a+b*log(c*x^n))/(e*x^2+d)^3,x, algorithm="fricas")
Output:
integral((b*x^4*log(c*x^n) + a*x^4)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)
\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^{4} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{2}\right )^{3}}\, dx \] Input:
integrate(x**4*(a+b*ln(c*x**n))/(e*x**2+d)**3,x)
Output:
Integral(x**4*(a + b*log(c*x**n))/(d + e*x**2)**3, x)
Exception generated. \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^4*(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
\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate(x^4*(a+b*log(c*x^n))/(e*x^2+d)^3,x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)*x^4/(e*x^2 + d)^3, x)
Timed out. \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^4\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \] Input:
int((x^4*(a + b*log(c*x^n)))/(d + e*x^2)^3,x)
Output:
int((x^4*(a + b*log(c*x^n)))/(d + e*x^2)^3, x)
\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,d^{2}+6 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d e \,x^{2}+3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,e^{2} x^{4}+8 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b \,d^{2} n +16 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b d e n \,x^{2}+8 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) b \,e^{2} n \,x^{4}+8 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{5} e +16 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{4} e^{2} x^{2}+8 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{3} e^{3} x^{4}-8 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} e x -8 \,\mathrm {log}\left (x^{n} c \right ) b d \,e^{2} x^{3}-3 a \,d^{2} e x -5 a d \,e^{2} x^{3}}{8 d \,e^{3} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:
int(x^4*(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 + 8*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqr t(d)))*b*d**2*n + 16*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*b*d*e*n *x**2 + 8*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*b*e**2*n*x**4 + 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* *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 - 8*log (x**n*c)*b*d*e**2*x**3 - 3*a*d**2*e*x - 5*a*d*e**2*x**3)/(8*d*e**3*(d**2 + 2*d*e*x**2 + e**2*x**4))