\(\int \frac {(a+b \log (c x^n))^3 \log (d (e+f x^2)^m)}{x^2} \, dx\) [119]

Optimal result

Integrand size = 28, antiderivative size = 775 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^2} \, dx =\text {Too large to display} \] Output:

12*b^3*f^(1/2)*m*n^3*arctan(f^(1/2)*x/e^(1/2))/e^(1/2)+12*b^2*f^(1/2)*m*n^ 
2*arctan(f^(1/2)*x/e^(1/2))*(a+b*ln(c*x^n))/e^(1/2)+6*b*f^(1/2)*m*n*arctan 
(f^(1/2)*x/e^(1/2))*(a+b*ln(c*x^n))^2/e^(1/2)+2*f^(1/2)*m*arctan(f^(1/2)*x 
/e^(1/2))*(a+b*ln(c*x^n))^3/e^(1/2)-6*b^3*n^3*ln(d*(f*x^2+e)^m)/x-6*b^2*n^ 
2*(a+b*ln(c*x^n))*ln(d*(f*x^2+e)^m)/x-3*b*n*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+ 
e)^m)/x-(a+b*ln(c*x^n))^3*ln(d*(f*x^2+e)^m)/x-6*b^3*f^(1/2)*m*n^3*polylog( 
2,-f^(1/2)*x/(-e)^(1/2))/(-e)^(1/2)-6*b^2*f^(1/2)*m*n^2*(a+b*ln(c*x^n))*po 
lylog(2,-f^(1/2)*x/(-e)^(1/2))/(-e)^(1/2)-3*b*f^(1/2)*m*n*(a+b*ln(c*x^n))^ 
2*polylog(2,-f^(1/2)*x/(-e)^(1/2))/(-e)^(1/2)+6*b^3*f^(1/2)*m*n^3*polylog( 
2,f^(1/2)*x/(-e)^(1/2))/(-e)^(1/2)+6*b^2*f^(1/2)*m*n^2*(a+b*ln(c*x^n))*pol 
ylog(2,f^(1/2)*x/(-e)^(1/2))/(-e)^(1/2)+3*b*f^(1/2)*m*n*(a+b*ln(c*x^n))^2* 
polylog(2,f^(1/2)*x/(-e)^(1/2))/(-e)^(1/2)+6*b^3*f^(1/2)*m*n^3*polylog(3,- 
f^(1/2)*x/(-e)^(1/2))/(-e)^(1/2)+6*b^2*f^(1/2)*m*n^2*(a+b*ln(c*x^n))*polyl 
og(3,-f^(1/2)*x/(-e)^(1/2))/(-e)^(1/2)-6*b^3*f^(1/2)*m*n^3*polylog(3,f^(1/ 
2)*x/(-e)^(1/2))/(-e)^(1/2)-6*b^2*f^(1/2)*m*n^2*(a+b*ln(c*x^n))*polylog(3, 
f^(1/2)*x/(-e)^(1/2))/(-e)^(1/2)-6*b^3*f^(1/2)*m*n^3*polylog(4,-f^(1/2)*x/ 
(-e)^(1/2))/(-e)^(1/2)+6*b^3*f^(1/2)*m*n^3*polylog(4,f^(1/2)*x/(-e)^(1/2)) 
/(-e)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.84 (sec) , antiderivative size = 2166, normalized size of antiderivative = 2.79 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

(2*a^3*Sqrt[f]*m*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 6*a^2*b*Sqrt[f]*m*n*x*Arc 
Tan[(Sqrt[f]*x)/Sqrt[e]] + 12*a*b^2*Sqrt[f]*m*n^2*x*ArcTan[(Sqrt[f]*x)/Sqr 
t[e]] + 12*b^3*Sqrt[f]*m*n^3*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 6*a^2*b*Sqrt[ 
f]*m*n*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 12*a*b^2*Sqrt[f]*m*n^2*x*Arc 
Tan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 12*b^3*Sqrt[f]*m*n^3*x*ArcTan[(Sqrt[f]*x 
)/Sqrt[e]]*Log[x] + 6*a*b^2*Sqrt[f]*m*n^2*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Lo 
g[x]^2 + 6*b^3*Sqrt[f]*m*n^3*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 - 2*b^ 
3*Sqrt[f]*m*n^3*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^3 + 6*a^2*b*Sqrt[f]*m 
*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 12*a*b^2*Sqrt[f]*m*n*x*ArcTan[ 
(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 12*b^3*Sqrt[f]*m*n^2*x*ArcTan[(Sqrt[f]*x 
)/Sqrt[e]]*Log[c*x^n] - 12*a*b^2*Sqrt[f]*m*n*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]] 
*Log[x]*Log[c*x^n] - 12*b^3*Sqrt[f]*m*n^2*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Lo 
g[x]*Log[c*x^n] + 6*b^3*Sqrt[f]*m*n^2*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] 
^2*Log[c*x^n] + 6*a*b^2*Sqrt[f]*m*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] 
^2 + 6*b^3*Sqrt[f]*m*n*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 - 6*b^3* 
Sqrt[f]*m*n*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n]^2 + 2*b^3*Sqrt 
[f]*m*x*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^3 + (3*I)*a^2*b*Sqrt[f]*m*n 
*x*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (6*I)*a*b^2*Sqrt[f]*m*n^2*x*Log 
[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + (6*I)*b^3*Sqrt[f]*m*n^3*x*Log[x]*Log[ 
1 - (I*Sqrt[f]*x)/Sqrt[e]] - (3*I)*a*b^2*Sqrt[f]*m*n^2*x*Log[x]^2*Log[1...
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.39 (sec) , antiderivative size = 879, normalized size of antiderivative = 1.13, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2825, 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.

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

\(\Big \downarrow \) 2825

\(\displaystyle -2 f m \int \left (-\frac {6 b^3 n^3}{f x^2+e}-\frac {6 b^2 \left (a+b \log \left (c x^n\right )\right ) n^2}{f x^2+e}-\frac {3 b \left (a+b \log \left (c x^n\right )\right )^2 n}{f x^2+e}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{f x^2+e}\right )dx-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}-\frac {6 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 b^3 \log \left (d \left (f x^2+e\right )^m\right ) n^3}{x}-\frac {6 b^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (f x^2+e\right )^m\right ) n^2}{x}-\frac {3 b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (f x^2+e\right )^m\right ) n}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (f x^2+e\right )^m\right )}{x}-2 f m \left (-\frac {6 b^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) n^3}{\sqrt {e} \sqrt {f}}+\frac {3 i b^3 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) n^3}{\sqrt {e} \sqrt {f}}-\frac {3 i b^3 \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right ) n^3}{\sqrt {e} \sqrt {f}}-\frac {3 b^3 \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^3}{\sqrt {-e} \sqrt {f}}+\frac {3 b^3 \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^3}{\sqrt {-e} \sqrt {f}}+\frac {3 b^3 \operatorname {PolyLog}\left (4,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^3}{\sqrt {-e} \sqrt {f}}-\frac {3 b^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^3}{\sqrt {-e} \sqrt {f}}-\frac {6 b^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right ) n^2}{\sqrt {e} \sqrt {f}}+\frac {3 b^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^2}{\sqrt {-e} \sqrt {f}}-\frac {3 b^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^2}{\sqrt {-e} \sqrt {f}}-\frac {3 b^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^2}{\sqrt {-e} \sqrt {f}}+\frac {3 b^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^2}{\sqrt {-e} \sqrt {f}}-\frac {3 b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n}{2 \sqrt {-e} \sqrt {f}}+\frac {3 b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) n}{2 \sqrt {-e} \sqrt {f}}+\frac {3 b \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n}{2 \sqrt {-e} \sqrt {f}}-\frac {3 b \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n}{2 \sqrt {-e} \sqrt {f}}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{2 \sqrt {-e} \sqrt {f}}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right )}{2 \sqrt {-e} \sqrt {f}}\right )\)

Input:

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

Output:

(-6*b^3*n^3*Log[d*(e + f*x^2)^m])/x - (6*b^2*n^2*(a + b*Log[c*x^n])*Log[d* 
(e + f*x^2)^m])/x - (3*b*n*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/x - 
((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/x - 2*f*m*((-6*b^3*n^3*ArcTan[ 
(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*Sqrt[f]) - (6*b^2*n^2*ArcTan[(Sqrt[f]*x)/Sq 
rt[e]]*(a + b*Log[c*x^n]))/(Sqrt[e]*Sqrt[f]) - (3*b*n*(a + b*Log[c*x^n])^2 
*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(2*Sqrt[-e]*Sqrt[f]) - ((a + b*Log[c*x^n]) 
^3*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(2*Sqrt[-e]*Sqrt[f]) + (3*b*n*(a + b*Log 
[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(2*Sqrt[-e]*Sqrt[f]) + ((a + b*L 
og[c*x^n])^3*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(2*Sqrt[-e]*Sqrt[f]) + (3*b^2* 
n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(Sqrt[-e]*Sqrt 
[f]) + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(2 
*Sqrt[-e]*Sqrt[f]) - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/ 
Sqrt[-e]])/(Sqrt[-e]*Sqrt[f]) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, (Sq 
rt[f]*x)/Sqrt[-e]])/(2*Sqrt[-e]*Sqrt[f]) + ((3*I)*b^3*n^3*PolyLog[2, ((-I) 
*Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*Sqrt[f]) - ((3*I)*b^3*n^3*PolyLog[2, (I*Sqr 
t[f]*x)/Sqrt[e]])/(Sqrt[e]*Sqrt[f]) - (3*b^3*n^3*PolyLog[3, -((Sqrt[f]*x)/ 
Sqrt[-e])])/(Sqrt[-e]*Sqrt[f]) - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, 
-((Sqrt[f]*x)/Sqrt[-e])])/(Sqrt[-e]*Sqrt[f]) + (3*b^3*n^3*PolyLog[3, (Sqrt 
[f]*x)/Sqrt[-e]])/(Sqrt[-e]*Sqrt[f]) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyL 
og[3, (Sqrt[f]*x)/Sqrt[-e]])/(Sqrt[-e]*Sqrt[f]) + (3*b^3*n^3*PolyLog[4,...
 

Defintions of rubi rules used

rule 2009

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

rule 2825

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q* 
(a + b*Log[c*x^n])^p, x]}, Simp[Log[d*(e + f*x^m)^r]   u, x] - Simp[f*m*r 
 Int[x^(m - 1)/(e + f*x^m)   u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m 
, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*m*x + 6*sqrt(f)*sqrt 
(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*b*m*n*x + 12*sqrt(f)*sqrt(e)*atan(( 
f*x)/(sqrt(f)*sqrt(e)))*a*b**2*m*n**2*x + 12*sqrt(f)*sqrt(e)*atan((f*x)/(s 
qrt(f)*sqrt(e)))*b**3*m*n**3*x - 2*int(log(x**n*c)**3/(e*x**2 + f*x**4),x) 
*b**3*e**2*m*x - 6*int(log(x**n*c)**2/(e*x**2 + f*x**4),x)*a*b**2*e**2*m*x 
 - 6*int(log(x**n*c)**2/(e*x**2 + f*x**4),x)*b**3*e**2*m*n*x - 6*int(log(x 
**n*c)/(e*x**2 + f*x**4),x)*a**2*b*e**2*m*x - 12*int(log(x**n*c)/(e*x**2 + 
 f*x**4),x)*a*b**2*e**2*m*n*x - 12*int(log(x**n*c)/(e*x**2 + f*x**4),x)*b* 
*3*e**2*m*n**2*x - log((e + f*x**2)**m*d)*log(x**n*c)**3*b**3*e - 3*log((e 
 + f*x**2)**m*d)*log(x**n*c)**2*a*b**2*e - 3*log((e + f*x**2)**m*d)*log(x* 
*n*c)**2*b**3*e*n - 3*log((e + f*x**2)**m*d)*log(x**n*c)*a**2*b*e - 6*log( 
(e + f*x**2)**m*d)*log(x**n*c)*a*b**2*e*n - 6*log((e + f*x**2)**m*d)*log(x 
**n*c)*b**3*e*n**2 - log((e + f*x**2)**m*d)*a**3*e - 3*log((e + f*x**2)**m 
*d)*a**2*b*e*n - 6*log((e + f*x**2)**m*d)*a*b**2*e*n**2 - 6*log((e + f*x** 
2)**m*d)*b**3*e*n**3 - 2*log(x**n*c)**3*b**3*e*m - 6*log(x**n*c)**2*a*b**2 
*e*m - 12*log(x**n*c)**2*b**3*e*m*n - 6*log(x**n*c)*a**2*b*e*m - 24*log(x* 
*n*c)*a*b**2*e*m*n - 36*log(x**n*c)*b**3*e*m*n**2 - 6*a**2*b*e*m*n - 24*a* 
b**2*e*m*n**2 - 36*b**3*e*m*n**3)/(e*x)