\(\int x (a+b \log (c x^n))^3 \log (d (e+f x^2)^m) \, dx\) [114]

Optimal result

Integrand size = 26, antiderivative size = 514 \[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\frac {3}{2} b^3 m n^3 x^2-\frac {9}{4} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 e m n^3 \log \left (e+f x^2\right )}{8 f}-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {3 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{4 f}-\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{4 f}+\frac {e m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x^2}{e}\right )}{2 f}+\frac {3 b^3 e m n^3 \operatorname {PolyLog}\left (2,-\frac {f x^2}{e}\right )}{8 f}-\frac {3 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f x^2}{e}\right )}{4 f}+\frac {3 b e m n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f x^2}{e}\right )}{4 f}+\frac {3 b^3 e m n^3 \operatorname {PolyLog}\left (3,-\frac {f x^2}{e}\right )}{8 f}-\frac {3 b^2 e m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f x^2}{e}\right )}{4 f}+\frac {3 b^3 e m n^3 \operatorname {PolyLog}\left (4,-\frac {f x^2}{e}\right )}{8 f} \] Output:

3/2*b^3*m*n^3*x^2-9/4*b^2*m*n^2*x^2*(a+b*ln(c*x^n))+3/2*b*m*n*x^2*(a+b*ln( 
c*x^n))^2-1/2*m*x^2*(a+b*ln(c*x^n))^3-3/8*b^3*e*m*n^3*ln(f*x^2+e)/f-3/8*b^ 
3*n^3*x^2*ln(d*(f*x^2+e)^m)+3/4*b^2*n^2*x^2*(a+b*ln(c*x^n))*ln(d*(f*x^2+e) 
^m)-3/4*b*n*x^2*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m)+1/2*x^2*(a+b*ln(c*x^n) 
)^3*ln(d*(f*x^2+e)^m)+3/4*b^2*e*m*n^2*(a+b*ln(c*x^n))*ln(1+f*x^2/e)/f-3/4* 
b*e*m*n*(a+b*ln(c*x^n))^2*ln(1+f*x^2/e)/f+1/2*e*m*(a+b*ln(c*x^n))^3*ln(1+f 
*x^2/e)/f+3/8*b^3*e*m*n^3*polylog(2,-f*x^2/e)/f-3/4*b^2*e*m*n^2*(a+b*ln(c* 
x^n))*polylog(2,-f*x^2/e)/f+3/4*b*e*m*n*(a+b*ln(c*x^n))^2*polylog(2,-f*x^2 
/e)/f+3/8*b^3*e*m*n^3*polylog(3,-f*x^2/e)/f-3/4*b^2*e*m*n^2*(a+b*ln(c*x^n) 
)*polylog(3,-f*x^2/e)/f+3/8*b^3*e*m*n^3*polylog(4,-f*x^2/e)/f
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.59 (sec) , antiderivative size = 1911, normalized size of antiderivative = 3.72 \[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(-4*a^3*f*m*x^2 + 12*a^2*b*f*m*n*x^2 - 18*a*b^2*f*m*n^2*x^2 + 12*b^3*f*m*n 
^3*x^2 - 12*a^2*b*f*m*x^2*Log[c*x^n] + 24*a*b^2*f*m*n*x^2*Log[c*x^n] - 18* 
b^3*f*m*n^2*x^2*Log[c*x^n] - 12*a*b^2*f*m*x^2*Log[c*x^n]^2 + 12*b^3*f*m*n* 
x^2*Log[c*x^n]^2 - 4*b^3*f*m*x^2*Log[c*x^n]^3 + 12*a^2*b*e*m*n*Log[x]*Log[ 
1 - (I*Sqrt[f]*x)/Sqrt[e]] - 12*a*b^2*e*m*n^2*Log[x]*Log[1 - (I*Sqrt[f]*x) 
/Sqrt[e]] + 6*b^3*e*m*n^3*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 12*a*b^2 
*e*m*n^2*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 6*b^3*e*m*n^3*Log[x]^2* 
Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 4*b^3*e*m*n^3*Log[x]^3*Log[1 - (I*Sqrt[f] 
*x)/Sqrt[e]] + 24*a*b^2*e*m*n*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt 
[e]] - 12*b^3*e*m*n^2*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 1 
2*b^3*e*m*n^2*Log[x]^2*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 12*b^3* 
e*m*n*Log[x]*Log[c*x^n]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 12*a^2*b*e*m*n* 
Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 12*a*b^2*e*m*n^2*Log[x]*Log[1 + (I 
*Sqrt[f]*x)/Sqrt[e]] + 6*b^3*e*m*n^3*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] 
 - 12*a*b^2*e*m*n^2*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 6*b^3*e*m*n^ 
3*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 4*b^3*e*m*n^3*Log[x]^3*Log[1 + 
 (I*Sqrt[f]*x)/Sqrt[e]] + 24*a*b^2*e*m*n*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt 
[f]*x)/Sqrt[e]] - 12*b^3*e*m*n^2*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/S 
qrt[e]] - 12*b^3*e*m*n^2*Log[x]^2*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e] 
] + 12*b^3*e*m*n*Log[x]*Log[c*x^n]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 4...
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.

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

\(\Big \downarrow \) 2825

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-2 f m \left (\frac {3 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac {3 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 f^2}-\frac {3 b^2 e n^2 \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac {9 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{8 f}-\frac {3 b e n \operatorname {PolyLog}\left (2,-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{8 f^2}+\frac {3 b e n \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{8 f^2}-\frac {e \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{4 f^2}-\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 f}-\frac {3 b^3 e n^3 \operatorname {PolyLog}\left (2,-\frac {f x^2}{e}\right )}{16 f^2}-\frac {3 b^3 e n^3 \operatorname {PolyLog}\left (3,-\frac {f x^2}{e}\right )}{16 f^2}-\frac {3 b^3 e n^3 \operatorname {PolyLog}\left (4,-\frac {f x^2}{e}\right )}{16 f^2}+\frac {3 b^3 e n^3 \log \left (e+f x^2\right )}{16 f^2}-\frac {3 b^3 n^3 x^2}{4 f}\right )-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f x^2\right )^m\right )\)

Input:

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

Output:

(-3*b^3*n^3*x^2*Log[d*(e + f*x^2)^m])/8 + (3*b^2*n^2*x^2*(a + b*Log[c*x^n] 
)*Log[d*(e + f*x^2)^m])/4 - (3*b*n*x^2*(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 - 2*f*m*((-3 
*b^3*n^3*x^2)/(4*f) + (9*b^2*n^2*x^2*(a + b*Log[c*x^n]))/(8*f) - (3*b*n*x^ 
2*(a + b*Log[c*x^n])^2)/(4*f) + (x^2*(a + b*Log[c*x^n])^3)/(4*f) + (3*b^3* 
e*n^3*Log[e + f*x^2])/(16*f^2) - (3*b^2*e*n^2*(a + b*Log[c*x^n])*Log[1 + ( 
f*x^2)/e])/(8*f^2) + (3*b*e*n*(a + b*Log[c*x^n])^2*Log[1 + (f*x^2)/e])/(8* 
f^2) - (e*(a + b*Log[c*x^n])^3*Log[1 + (f*x^2)/e])/(4*f^2) - (3*b^3*e*n^3* 
PolyLog[2, -((f*x^2)/e)])/(16*f^2) + (3*b^2*e*n^2*(a + b*Log[c*x^n])*PolyL 
og[2, -((f*x^2)/e)])/(8*f^2) - (3*b*e*n*(a + b*Log[c*x^n])^2*PolyLog[2, -( 
(f*x^2)/e)])/(8*f^2) - (3*b^3*e*n^3*PolyLog[3, -((f*x^2)/e)])/(16*f^2) + ( 
3*b^2*e*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((f*x^2)/e)])/(8*f^2) - (3*b^3* 
e*n^3*PolyLog[4, -((f*x^2)/e)])/(16*f^2))
 

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 = 0.10 (sec) , antiderivative size = 21242, normalized size of antiderivative = 41.33

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

Input:

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

Output:

result too large to display
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 8*int(log(x**n*c)**3/(e*x + f*x**3),x)*b**3*e**2*m*n - 24*int(log(x**n 
*c)**2/(e*x + f*x**3),x)*a*b**2*e**2*m*n + 12*int(log(x**n*c)**2/(e*x + f* 
x**3),x)*b**3*e**2*m*n**2 - 24*int(log(x**n*c)/(e*x + f*x**3),x)*a**2*b*e* 
*2*m*n + 24*int(log(x**n*c)/(e*x + f*x**3),x)*a*b**2*e**2*m*n**2 - 12*int( 
log(x**n*c)/(e*x + f*x**3),x)*b**3*e**2*m*n**3 + 4*log((e + f*x**2)**m*d)* 
log(x**n*c)**3*b**3*f*n*x**2 + 12*log((e + f*x**2)**m*d)*log(x**n*c)**2*a* 
b**2*f*n*x**2 - 6*log((e + f*x**2)**m*d)*log(x**n*c)**2*b**3*f*n**2*x**2 + 
 12*log((e + f*x**2)**m*d)*log(x**n*c)*a**2*b*f*n*x**2 - 12*log((e + f*x** 
2)**m*d)*log(x**n*c)*a*b**2*f*n**2*x**2 + 6*log((e + f*x**2)**m*d)*log(x** 
n*c)*b**3*f*n**3*x**2 + 4*log((e + f*x**2)**m*d)*a**3*e*n + 4*log((e + f*x 
**2)**m*d)*a**3*f*n*x**2 - 6*log((e + f*x**2)**m*d)*a**2*b*e*n**2 - 6*log( 
(e + f*x**2)**m*d)*a**2*b*f*n**2*x**2 + 6*log((e + f*x**2)**m*d)*a*b**2*e* 
n**3 + 6*log((e + f*x**2)**m*d)*a*b**2*f*n**3*x**2 - 3*log((e + f*x**2)**m 
*d)*b**3*e*n**4 - 3*log((e + f*x**2)**m*d)*b**3*f*n**4*x**2 + 2*log(x**n*c 
)**4*b**3*e*m + 8*log(x**n*c)**3*a*b**2*e*m - 4*log(x**n*c)**3*b**3*e*m*n 
- 4*log(x**n*c)**3*b**3*f*m*n*x**2 + 12*log(x**n*c)**2*a**2*b*e*m - 12*log 
(x**n*c)**2*a*b**2*e*m*n - 12*log(x**n*c)**2*a*b**2*f*m*n*x**2 + 6*log(x** 
n*c)**2*b**3*e*m*n**2 + 12*log(x**n*c)**2*b**3*f*m*n**2*x**2 - 12*log(x**n 
*c)*a**2*b*f*m*n*x**2 + 24*log(x**n*c)*a*b**2*f*m*n**2*x**2 - 18*log(x**n* 
c)*b**3*f*m*n**3*x**2 - 4*a**3*f*m*n*x**2 + 12*a**2*b*f*m*n**2*x**2 - 1...