Integrand size = 23, antiderivative size = 131 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\frac {a x}{e}-\frac {b n x}{e}+\frac {b x \log \left (c x^n\right )}{e}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2}}-\frac {b \sqrt {-d} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 e^{3/2}} \] Output:
a*x/e-b*n*x/e+b*x*ln(c*x^n)/e-d^(1/2)*arctan(e^(1/2)*x/d^(1/2))*(a+b*ln(c* x^n))/e^(3/2)-1/2*b*(-d)^(1/2)*n*polylog(2,-e^(1/2)*x/(-d)^(1/2))/e^(3/2)+ 1/2*b*(-d)^(1/2)*n*polylog(2,e^(1/2)*x/(-d)^(1/2))/e^(3/2)
Time = 0.12 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.30 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\frac {2 a \sqrt {e} x-2 b \sqrt {e} n x+2 b \sqrt {e} x \log \left (c x^n\right )-\sqrt {-d} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )+\sqrt {-d} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )+b \sqrt {-d} n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )-b \sqrt {-d} n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{2 e^{3/2}} \] Input:
Integrate[(x^2*(a + b*Log[c*x^n]))/(d + e*x^2),x]
Output:
(2*a*Sqrt[e]*x - 2*b*Sqrt[e]*n*x + 2*b*Sqrt[e]*x*Log[c*x^n] - Sqrt[-d]*(a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]] + Sqrt[-d]*(a + b*Log[c*x^n] )*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)] + b*Sqrt[-d]*n*PolyLog[2, (Sqrt[e]*x)/ Sqrt[-d]] - b*Sqrt[-d]*n*PolyLog[2, (d*Sqrt[e]*x)/(-d)^(3/2)])/(2*e^(3/2))
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 132, 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.
\(\displaystyle \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx\) |
\(\Big \downarrow \) 2793 |
\(\displaystyle \int \left (\frac {a+b \log \left (c x^n\right )}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right )}{e \left (d+e x^2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{3/2}}+\frac {a x}{e}+\frac {b x \log \left (c x^n\right )}{e}+\frac {i b \sqrt {d} n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 e^{3/2}}-\frac {i b \sqrt {d} n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 e^{3/2}}-\frac {b n x}{e}\) |
Input:
Int[(x^2*(a + b*Log[c*x^n]))/(d + e*x^2),x]
Output:
(a*x)/e - (b*n*x)/e + (b*x*Log[c*x^n])/e - (Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqr t[d]]*(a + b*Log[c*x^n]))/e^(3/2) + ((I/2)*b*Sqrt[d]*n*PolyLog[2, ((-I)*Sq rt[e]*x)/Sqrt[d]])/e^(3/2) - ((I/2)*b*Sqrt[d]*n*PolyLog[2, (I*Sqrt[e]*x)/S qrt[d]])/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 = 0.50 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.42
method | result | size |
risch | \(\frac {b \ln \left (x^{n}\right ) x}{e}+\frac {b d \arctan \left (\frac {x e}{\sqrt {d e}}\right ) n \ln \left (x \right )}{e \sqrt {d e}}-\frac {b d \arctan \left (\frac {x e}{\sqrt {d e}}\right ) \ln \left (x^{n}\right )}{e \sqrt {d e}}-\frac {b n x}{e}-\frac {b n d \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e \sqrt {-d e}}+\frac {b n d \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e \sqrt {-d e}}-\frac {b n d \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e \sqrt {-d e}}+\frac {b n d \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 e \sqrt {-d e}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x}{e}-\frac {d \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{e \sqrt {d e}}\right )\) | \(317\) |
Input:
int(x^2*(a+b*ln(c*x^n))/(e*x^2+d),x,method=_RETURNVERBOSE)
Output:
b*ln(x^n)/e*x+b*d/e/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*n*ln(x)-b*d/e/(d*e )^(1/2)*arctan(x*e/(d*e)^(1/2))*ln(x^n)-b*n*x/e-1/2*b*n*d/e*ln(x)/(-d*e)^( 1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/2*b*n*d/e*ln(x)/(-d*e)^(1/2)*l n((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/2*b*n*d/e/(-d*e)^(1/2)*dilog((-e*x+(- d*e)^(1/2))/(-d*e)^(1/2))+1/2*b*n*d/e/(-d*e)^(1/2)*dilog((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)*(x/e-d/e/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2)) )
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{e x^{2} + d} \,d x } \] Input:
integrate(x^2*(a+b*log(c*x^n))/(e*x^2+d),x, algorithm="fricas")
Output:
integral((b*x^2*log(c*x^n) + a*x^2)/(e*x^2 + d), x)
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int \frac {x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )}{d + e x^{2}}\, dx \] Input:
integrate(x**2*(a+b*ln(c*x**n))/(e*x**2+d),x)
Output:
Integral(x**2*(a + b*log(c*x**n))/(d + e*x**2), x)
Exception generated. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^2*(a+b*log(c*x^n))/(e*x^2+d),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
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{e x^{2} + d} \,d x } \] Input:
integrate(x^2*(a+b*log(c*x^n))/(e*x^2+d),x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)*x^2/(e*x^2 + d), x)
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{e\,x^2+d} \,d x \] Input:
int((x^2*(a + b*log(c*x^n)))/(d + e*x^2),x)
Output:
int((x^2*(a + b*log(c*x^n)))/(d + e*x^2), x)
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx=\frac {-\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a -\left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e \,x^{2}+d}d x \right ) b d e +\mathrm {log}\left (x^{n} c \right ) b e x +a e x -b e n x}{e^{2}} \] Input:
int(x^2*(a+b*log(c*x^n))/(e*x^2+d),x)
Output:
( - sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a - int(log(x**n*c)/(d + e*x**2),x)*b*d*e + log(x**n*c)*b*e*x + a*e*x - b*e*n*x)/e**2