\(\int \frac {(a+b \log (c x^n))^3}{d+e x^2} \, dx\) [1]

Optimal result

Integrand size = 22, antiderivative size = 296 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{d+e x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {d} \sqrt {e}}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {3 b^3 n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}} \] Output:

arctan(e^(1/2)*x/d^(1/2))*(a+b*ln(c*x^n))^3/d^(1/2)/e^(1/2)-3/2*b*n*(a+b*l 
n(c*x^n))^2*polylog(2,-e^(1/2)*x/(-d)^(1/2))/(-d)^(1/2)/e^(1/2)+3/2*b*n*(a 
+b*ln(c*x^n))^2*polylog(2,e^(1/2)*x/(-d)^(1/2))/(-d)^(1/2)/e^(1/2)+3*b^2*n 
^2*(a+b*ln(c*x^n))*polylog(3,-e^(1/2)*x/(-d)^(1/2))/(-d)^(1/2)/e^(1/2)-3*b 
^2*n^2*(a+b*ln(c*x^n))*polylog(3,e^(1/2)*x/(-d)^(1/2))/(-d)^(1/2)/e^(1/2)- 
3*b^3*n^3*polylog(4,-e^(1/2)*x/(-d)^(1/2))/(-d)^(1/2)/e^(1/2)+3*b^3*n^3*po 
lylog(4,e^(1/2)*x/(-d)^(1/2))/(-d)^(1/2)/e^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.03 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.66 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{d+e x^2} \, dx=\frac {2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3+3 i b n \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left (\log (x) \left (\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )-\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )+6 i b^2 n^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\frac {1}{2} \log ^2(x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} \log ^2(x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )-\log (x) \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\log (x) \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (3,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )-\operatorname {PolyLog}\left (3,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )+i b^3 n^3 \left (\log ^3(x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )-\log ^3(x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )-3 \log ^2(x) \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+3 \log ^2(x) \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )+6 \log (x) \operatorname {PolyLog}\left (3,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )-6 \log (x) \operatorname {PolyLog}\left (3,\frac {i \sqrt {e} x}{\sqrt {d}}\right )-6 \operatorname {PolyLog}\left (4,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+6 \operatorname {PolyLog}\left (4,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{2 \sqrt {d} \sqrt {e}} \] Input:

Integrate[(a + b*Log[c*x^n])^3/(d + e*x^2),x]
 

Output:

(2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a - b*n*Log[x] + b*Log[c*x^n])^3 + (3*I)*b 
*n*(a - b*n*Log[x] + b*Log[c*x^n])^2*(Log[x]*(Log[1 - (I*Sqrt[e]*x)/Sqrt[d 
]] - Log[1 + (I*Sqrt[e]*x)/Sqrt[d]]) - PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d] 
] + PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]]) + (6*I)*b^2*n^2*(a - b*n*Log[x] + b 
*Log[c*x^n])*((Log[x]^2*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/2 - (Log[x]^2*Log[ 
1 + (I*Sqrt[e]*x)/Sqrt[d]])/2 - Log[x]*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d] 
] + Log[x]*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]] + PolyLog[3, ((-I)*Sqrt[e]*x) 
/Sqrt[d]] - PolyLog[3, (I*Sqrt[e]*x)/Sqrt[d]]) + I*b^3*n^3*(Log[x]^3*Log[1 
 - (I*Sqrt[e]*x)/Sqrt[d]] - Log[x]^3*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]] - 3*Lo 
g[x]^2*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]] + 3*Log[x]^2*PolyLog[2, (I*Sqr 
t[e]*x)/Sqrt[d]] + 6*Log[x]*PolyLog[3, ((-I)*Sqrt[e]*x)/Sqrt[d]] - 6*Log[x 
]*PolyLog[3, (I*Sqrt[e]*x)/Sqrt[d]] - 6*PolyLog[4, ((-I)*Sqrt[e]*x)/Sqrt[d 
]] + 6*PolyLog[4, (I*Sqrt[e]*x)/Sqrt[d]]))/(2*Sqrt[d]*Sqrt[e])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2767, 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.

\(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{d+e x^2} \, dx\)

\(\Big \downarrow \) 2767

\(\displaystyle \int \left (\frac {\sqrt {-d} \left (a+b \log \left (c x^n\right )\right )^3}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \left (a+b \log \left (c x^n\right )\right )^3}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-d} \sqrt {e}}-\frac {3 b^2 n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {-d} \sqrt {e}}-\frac {3 b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {-d} \sqrt {e}}+\frac {3 b n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {-d} \sqrt {e}}+\frac {\log \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt {-d} \sqrt {e}}-\frac {\log \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt {-d} \sqrt {e}}-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {3 b^3 n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}}\)

Input:

Int[(a + b*Log[c*x^n])^3/(d + e*x^2),x]
 

Output:

((a + b*Log[c*x^n])^3*Log[1 - (Sqrt[e]*x)/Sqrt[-d]])/(2*Sqrt[-d]*Sqrt[e]) 
- ((a + b*Log[c*x^n])^3*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(2*Sqrt[-d]*Sqrt[e] 
) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((Sqrt[e]*x)/Sqrt[-d])])/(2*Sq 
rt[-d]*Sqrt[e]) + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, (Sqrt[e]*x)/Sqrt[ 
-d]])/(2*Sqrt[-d]*Sqrt[e]) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((S 
qrt[e]*x)/Sqrt[-d])])/(Sqrt[-d]*Sqrt[e]) - (3*b^2*n^2*(a + b*Log[c*x^n])*P 
olyLog[3, (Sqrt[e]*x)/Sqrt[-d]])/(Sqrt[-d]*Sqrt[e]) - (3*b^3*n^3*PolyLog[4 
, -((Sqrt[e]*x)/Sqrt[-d])])/(Sqrt[-d]*Sqrt[e]) + (3*b^3*n^3*PolyLog[4, (Sq 
rt[e]*x)/Sqrt[-d]])/(Sqrt[-d]*Sqrt[e])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2767

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^( 
q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (d + e*x 
^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x] 
&& IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))
 

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{e \,x^{2}+d}d x\]

Input:

int((a+b*ln(c*x^n))^3/(e*x^2+d),x)
 

Output:

int((a+b*ln(c*x^n))^3/(e*x^2+d),x)
 

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{d+e x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{e x^{2} + d} \,d x } \] Input:

integrate((a+b*log(c*x^n))^3/(e*x^2+d),x, algorithm="fricas")
 

Output:

integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a 
^3)/(e*x^2 + d), x)
 

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{d+e x^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{3}}{d + e x^{2}}\, dx \] Input:

integrate((a+b*ln(c*x**n))**3/(e*x**2+d),x)
 

Output:

Integral((a + b*log(c*x**n))**3/(d + e*x**2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{d+e x^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*log(c*x^n))^3/(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
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{d+e x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{e x^{2} + d} \,d x } \] Input:

integrate((a+b*log(c*x^n))^3/(e*x^2+d),x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)^3/(e*x^2 + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{d+e x^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{e\,x^2+d} \,d x \] Input:

int((a + b*log(c*x^n))^3/(d + e*x^2),x)
 

Output:

int((a + b*log(c*x^n))^3/(d + e*x^2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{d+e x^2} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a^{3}+\left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{3}}{e \,x^{2}+d}d x \right ) b^{3} d e +3 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e \,x^{2}+d}d x \right ) a \,b^{2} d e +3 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e \,x^{2}+d}d x \right ) a^{2} b d e}{d e} \] Input:

int((a+b*log(c*x^n))^3/(e*x^2+d),x)
 

Output:

(sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a**3 + int(log(x**n*c)**3/( 
d + e*x**2),x)*b**3*d*e + 3*int(log(x**n*c)**2/(d + e*x**2),x)*a*b**2*d*e 
+ 3*int(log(x**n*c)/(d + e*x**2),x)*a**2*b*d*e)/(d*e)