\(\int \frac {(a+b \log (c x^n))^3 \log (d (e+f x^2)^m)}{x^3} \, dx\) [116]

Optimal result

Integrand size = 28, antiderivative size = 451 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=\frac {3 b^3 f m n^3 \log (x)}{4 e}-\frac {3 b^2 f m n^2 \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {3 b f m n \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac {f m \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {3 b^3 f m n^3 \log \left (e+f x^2\right )}{8 e}-\frac {3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac {3 b^3 f m n^3 \operatorname {PolyLog}\left (2,-\frac {e}{f x^2}\right )}{8 e}+\frac {3 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e}{f x^2}\right )}{4 e}+\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {e}{f x^2}\right )}{4 e}+\frac {3 b^3 f m n^3 \operatorname {PolyLog}\left (3,-\frac {e}{f x^2}\right )}{8 e}+\frac {3 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {e}{f x^2}\right )}{4 e}+\frac {3 b^3 f m n^3 \operatorname {PolyLog}\left (4,-\frac {e}{f x^2}\right )}{8 e} \] Output:

3/4*b^3*f*m*n^3*ln(x)/e-3/4*b^2*f*m*n^2*ln(1+e/f/x^2)*(a+b*ln(c*x^n))/e-3/ 
4*b*f*m*n*ln(1+e/f/x^2)*(a+b*ln(c*x^n))^2/e-1/2*f*m*ln(1+e/f/x^2)*(a+b*ln( 
c*x^n))^3/e-3/8*b^3*f*m*n^3*ln(f*x^2+e)/e-3/8*b^3*n^3*ln(d*(f*x^2+e)^m)/x^ 
2-3/4*b^2*n^2*(a+b*ln(c*x^n))*ln(d*(f*x^2+e)^m)/x^2-3/4*b*n*(a+b*ln(c*x^n) 
)^2*ln(d*(f*x^2+e)^m)/x^2-1/2*(a+b*ln(c*x^n))^3*ln(d*(f*x^2+e)^m)/x^2+3/8* 
b^3*f*m*n^3*polylog(2,-e/f/x^2)/e+3/4*b^2*f*m*n^2*(a+b*ln(c*x^n))*polylog( 
2,-e/f/x^2)/e+3/4*b*f*m*n*(a+b*ln(c*x^n))^2*polylog(2,-e/f/x^2)/e+3/8*b^3* 
f*m*n^3*polylog(3,-e/f/x^2)/e+3/4*b^2*f*m*n^2*(a+b*ln(c*x^n))*polylog(3,-e 
/f/x^2)/e+3/8*b^3*f*m*n^3*polylog(4,-e/f/x^2)/e
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.04 (sec) , antiderivative size = 2248, normalized size of antiderivative = 4.98 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

-1/8*(-8*a^3*f*m*x^2*Log[x] - 12*a^2*b*f*m*n*x^2*Log[x] - 12*a*b^2*f*m*n^2 
*x^2*Log[x] - 6*b^3*f*m*n^3*x^2*Log[x] + 12*a^2*b*f*m*n*x^2*Log[x]^2 + 12* 
a*b^2*f*m*n^2*x^2*Log[x]^2 + 6*b^3*f*m*n^3*x^2*Log[x]^2 - 8*a*b^2*f*m*n^2* 
x^2*Log[x]^3 - 4*b^3*f*m*n^3*x^2*Log[x]^3 + 2*b^3*f*m*n^3*x^2*Log[x]^4 - 2 
4*a^2*b*f*m*x^2*Log[x]*Log[c*x^n] - 24*a*b^2*f*m*n*x^2*Log[x]*Log[c*x^n] - 
 12*b^3*f*m*n^2*x^2*Log[x]*Log[c*x^n] + 24*a*b^2*f*m*n*x^2*Log[x]^2*Log[c* 
x^n] + 12*b^3*f*m*n^2*x^2*Log[x]^2*Log[c*x^n] - 8*b^3*f*m*n^2*x^2*Log[x]^3 
*Log[c*x^n] - 24*a*b^2*f*m*x^2*Log[x]*Log[c*x^n]^2 - 12*b^3*f*m*n*x^2*Log[ 
x]*Log[c*x^n]^2 + 12*b^3*f*m*n*x^2*Log[x]^2*Log[c*x^n]^2 - 8*b^3*f*m*x^2*L 
og[x]*Log[c*x^n]^3 + 12*a^2*b*f*m*n*x^2*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[ 
e]] + 12*a*b^2*f*m*n^2*x^2*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 6*b^3*f 
*m*n^3*x^2*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 12*a*b^2*f*m*n^2*x^2*Lo 
g[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 6*b^3*f*m*n^3*x^2*Log[x]^2*Log[1 - 
 (I*Sqrt[f]*x)/Sqrt[e]] + 4*b^3*f*m*n^3*x^2*Log[x]^3*Log[1 - (I*Sqrt[f]*x) 
/Sqrt[e]] + 24*a*b^2*f*m*n*x^2*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqr 
t[e]] + 12*b^3*f*m*n^2*x^2*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e] 
] - 12*b^3*f*m*n^2*x^2*Log[x]^2*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] 
+ 12*b^3*f*m*n*x^2*Log[x]*Log[c*x^n]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 12 
*a^2*b*f*m*n*x^2*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 12*a*b^2*f*m*n^2* 
x^2*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 6*b^3*f*m*n^3*x^2*Log[x]*Lo...
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.96, 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.

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

\(\Big \downarrow \) 2825

\(\displaystyle -2 f m \int \left (-\frac {3 b^3 n^3}{8 x \left (f x^2+e\right )}-\frac {3 b^2 \left (a+b \log \left (c x^n\right )\right ) n^2}{4 x \left (f x^2+e\right )}-\frac {3 b \left (a+b \log \left (c x^n\right )\right )^2 n}{4 x \left (f x^2+e\right )}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x \left (f x^2+e\right )}\right )dx-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-\frac {3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-2 f m \left (-\frac {3 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e}-\frac {3 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac {3 b^2 n^2 \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e}-\frac {3 b n \operatorname {PolyLog}\left (2,-\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{8 e}+\frac {3 b n \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{8 e}+\frac {\log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{4 e}-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (2,-\frac {e}{f x^2}\right )}{16 e}-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (3,-\frac {e}{f x^2}\right )}{16 e}-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {e}{f x^2}\right )}{16 e}+\frac {3 b^3 n^3 \log \left (e+f x^2\right )}{16 e}-\frac {3 b^3 n^3 \log (x)}{8 e}\right )-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-\frac {3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}\)

Input:

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

Output:

(-3*b^3*n^3*Log[d*(e + f*x^2)^m])/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n])* 
Log[d*(e + f*x^2)^m])/(4*x^2) - (3*b*n*(a + b*Log[c*x^n])^2*Log[d*(e + f*x 
^2)^m])/(4*x^2) - ((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/(2*x^2) - 2* 
f*m*((-3*b^3*n^3*Log[x])/(8*e) + (3*b^2*n^2*Log[1 + e/(f*x^2)]*(a + b*Log[ 
c*x^n]))/(8*e) + (3*b*n*Log[1 + e/(f*x^2)]*(a + b*Log[c*x^n])^2)/(8*e) + ( 
Log[1 + e/(f*x^2)]*(a + b*Log[c*x^n])^3)/(4*e) + (3*b^3*n^3*Log[e + f*x^2] 
)/(16*e) - (3*b^3*n^3*PolyLog[2, -(e/(f*x^2))])/(16*e) - (3*b^2*n^2*(a + b 
*Log[c*x^n])*PolyLog[2, -(e/(f*x^2))])/(8*e) - (3*b*n*(a + b*Log[c*x^n])^2 
*PolyLog[2, -(e/(f*x^2))])/(8*e) - (3*b^3*n^3*PolyLog[3, -(e/(f*x^2))])/(1 
6*e) - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(e/(f*x^2))])/(8*e) - (3* 
b^3*n^3*PolyLog[4, -(e/(f*x^2))])/(16*e))
 

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 [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 22905, normalized size of antiderivative = 50.79

\[\text {output too large to display}\]

Input:

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

Output:

result too large to display
 

Fricas [F]

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

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/8*(4*b^3*log(x^n)^3 + 6*(n^2 + 2*n*log(c) + 2*log(c)^2)*a*b^2 + (3*n^3 
+ 6*n^2*log(c) + 6*n*log(c)^2 + 4*log(c)^3)*b^3 + 6*a^2*b*(n + 2*log(c)) + 
 4*a^3 + 6*(b^3*(n + 2*log(c)) + 2*a*b^2)*log(x^n)^2 + 6*((n^2 + 2*n*log(c 
) + 2*log(c)^2)*b^3 + 2*a*b^2*(n + 2*log(c)) + 2*a^2*b)*log(x^n))*log((f*x 
^2 + e)^m)/x^2 + integrate(1/4*(4*b^3*e*log(c)^3*log(d) + 12*a*b^2*e*log(c 
)^2*log(d) + 12*a^2*b*e*log(c)*log(d) + 4*a^3*e*log(d) + 4*((f*m + f*log(d 
))*b^3*x^2 + b^3*e*log(d))*log(x^n)^3 + (4*(f*m + f*log(d))*a^3 + 6*(f*m*n 
 + 2*(f*m + f*log(d))*log(c))*a^2*b + 6*(f*m*n^2 + 2*f*m*n*log(c) + 2*(f*m 
 + f*log(d))*log(c)^2)*a*b^2 + (3*f*m*n^3 + 6*f*m*n^2*log(c) + 6*f*m*n*log 
(c)^2 + 4*(f*m + f*log(d))*log(c)^3)*b^3)*x^2 + 6*(2*b^3*e*log(c)*log(d) + 
 2*a*b^2*e*log(d) + (2*(f*m + f*log(d))*a*b^2 + (f*m*n + 2*(f*m + f*log(d) 
)*log(c))*b^3)*x^2)*log(x^n)^2 + 6*(2*b^3*e*log(c)^2*log(d) + 4*a*b^2*e*lo 
g(c)*log(d) + 2*a^2*b*e*log(d) + (2*(f*m + f*log(d))*a^2*b + 2*(f*m*n + 2* 
(f*m + f*log(d))*log(c))*a*b^2 + (f*m*n^2 + 2*f*m*n*log(c) + 2*(f*m + f*lo 
g(d))*log(c)^2)*b^3)*x^2)*log(x^n))/(f*x^5 + e*x^3), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 8*int(log(x**n*c)**3/(e*x**3 + f*x**5),x)*b**3*e**2*m*x**2 - 24*int(lo 
g(x**n*c)**2/(e*x**3 + f*x**5),x)*a*b**2*e**2*m*x**2 - 12*int(log(x**n*c)* 
*2/(e*x**3 + f*x**5),x)*b**3*e**2*m*n*x**2 - 24*int(log(x**n*c)/(e*x**3 + 
f*x**5),x)*a**2*b*e**2*m*x**2 - 24*int(log(x**n*c)/(e*x**3 + f*x**5),x)*a* 
b**2*e**2*m*n*x**2 - 12*int(log(x**n*c)/(e*x**3 + f*x**5),x)*b**3*e**2*m*n 
**2*x**2 - 4*log((e + f*x**2)**m*d)*log(x**n*c)**3*b**3*e - 12*log((e + f* 
x**2)**m*d)*log(x**n*c)**2*a*b**2*e - 6*log((e + f*x**2)**m*d)*log(x**n*c) 
**2*b**3*e*n - 12*log((e + f*x**2)**m*d)*log(x**n*c)*a**2*b*e - 12*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 - 4*log((e + f*x**2)**m*d)*a**3*e - 4*log((e + f*x**2)**m* 
d)*a**3*f*x**2 - 6*log((e + f*x**2)**m*d)*a**2*b*e*n - 6*log((e + f*x**2)* 
*m*d)*a**2*b*f*n*x**2 - 6*log((e + f*x**2)**m*d)*a*b**2*e*n**2 - 6*log((e 
+ f*x**2)**m*d)*a*b**2*f*n**2*x**2 - 3*log((e + f*x**2)**m*d)*b**3*e*n**3 
- 3*log((e + f*x**2)**m*d)*b**3*f*n**3*x**2 - 4*log(x**n*c)**3*b**3*e*m - 
12*log(x**n*c)**2*a*b**2*e*m - 12*log(x**n*c)**2*b**3*e*m*n - 12*log(x**n* 
c)*a**2*b*e*m - 24*log(x**n*c)*a*b**2*e*m*n - 18*log(x**n*c)*b**3*e*m*n**2 
 + 8*log(x)*a**3*f*m*x**2 + 12*log(x)*a**2*b*f*m*n*x**2 + 12*log(x)*a*b**2 
*f*m*n**2*x**2 + 6*log(x)*b**3*f*m*n**3*x**2 - 6*a**2*b*e*m*n - 12*a*b**2* 
e*m*n**2 - 9*b**3*e*m*n**3)/(8*e*x**2)