\(\int (a+b \log (c (d (e+f x)^m)^n))^4 \, dx\) [423]

Optimal result

Integrand size = 20, antiderivative size = 160 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx=-24 a b^3 m^3 n^3 x+24 b^4 m^4 n^4 x-\frac {24 b^4 m^3 n^3 (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}+\frac {12 b^2 m^2 n^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}{f}-\frac {4 b m n (e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3}{f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4}{f} \] Output:

-24*a*b^3*m^3*n^3*x+24*b^4*m^4*n^4*x-24*b^4*m^3*n^3*(f*x+e)*ln(c*(d*(f*x+e 
)^m)^n)/f+12*b^2*m^2*n^2*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^m)^n))^2/f-4*b*m*n*( 
f*x+e)*(a+b*ln(c*(d*(f*x+e)^m)^n))^3/f+(f*x+e)*(a+b*ln(c*(d*(f*x+e)^m)^n)) 
^4/f
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.82 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4-4 b m n \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3-3 b m n \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2-2 b m n \left (f (a-b m n) x+b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )\right )\right )\right )}{f} \] Input:

Integrate[(a + b*Log[c*(d*(e + f*x)^m)^n])^4,x]
 

Output:

((e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^4 - 4*b*m*n*((e + f*x)*(a + b* 
Log[c*(d*(e + f*x)^m)^n])^3 - 3*b*m*n*((e + f*x)*(a + b*Log[c*(d*(e + f*x) 
^m)^n])^2 - 2*b*m*n*(f*(a - b*m*n)*x + b*(e + f*x)*Log[c*(d*(e + f*x)^m)^n 
]))))/f
 

Rubi [A] (warning: unable to verify)

Time = 0.85 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2895, 2836, 2733, 2733, 2733, 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.

\(\displaystyle \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx\)

\(\Big \downarrow \) 2895

\(\displaystyle \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4dx\)

\(\Big \downarrow \) 2836

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

\(\Big \downarrow \) 2733

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

\(\Big \downarrow \) 2733

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

\(\Big \downarrow \) 2733

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*Log[c*(d*(e + f*x)^m)^n])^4,x]
 

Output:

((e + f*x)*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^4 - 4*b*m*n*((e + f*x)*(a + 
b*Log[c*d^n*(e + f*x)^(m*n)])^3 - 3*b*m*n*((e + f*x)*(a + b*Log[c*d^n*(e + 
 f*x)^(m*n)])^2 - 2*b*m*n*(a*(e + f*x) - b*m*n*(e + f*x) + b*(e + f*x)*Log 
[c*d^n*(e + f*x)^(m*n)]))))/f
 

Defintions of rubi rules used

rule 2009

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

rule 2733

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b 
*Log[c*x^n])^p, x] - Simp[b*n*p   Int[(a + b*Log[c*x^n])^(p - 1), x], x] /; 
 FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
 

rule 2836

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] : 
> Simp[1/e   Subst[Int[(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{ 
a, b, c, d, e, n, p}, x]
 

rule 2895

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. 
)*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], 
 c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ 
IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(641\) vs. \(2(160)=320\).

Time = 7.27 (sec) , antiderivative size = 642, normalized size of antiderivative = 4.01

method	result	size
parallelrisch	\(\frac {24 x \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right ) a \,b^{3} e f \,m^{2} n^{2}-12 x {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} a \,b^{3} e f m n -12 x \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right ) a^{2} b^{2} e f m n +12 x {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} b^{4} e f \,m^{2} n^{2}-24 x a \,b^{3} e f \,m^{3} n^{3}-4 x {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{3} b^{4} e f m n +12 x \,a^{2} b^{2} e f \,m^{2} n^{2}-4 x \,a^{3} b e f m n -24 x \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right ) b^{4} e f \,m^{3} n^{3}-24 b^{4} e^{2} m^{4} n^{4}+{\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{4} b^{4} e^{2}-a^{4} e^{2}+4 {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{3} a \,b^{3} e^{2}+6 {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} a^{2} b^{2} e^{2}+x \,a^{4} e f +24 a \,b^{3} e^{2} m^{3} n^{3}-12 a^{2} b^{2} e^{2} m^{2} n^{2}+4 a^{3} b \,e^{2} m n +4 x {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{3} a \,b^{3} e f +6 x {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} a^{2} b^{2} e f +4 x \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right ) a^{3} b e f +24 x \,b^{4} e f \,m^{4} n^{4}-12 {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} a \,b^{3} e^{2} m n +24 \ln \left (f x +e \right ) a \,b^{3} e^{2} m^{3} n^{3}-12 \ln \left (f x +e \right ) a^{2} b^{2} e^{2} m^{2} n^{2}+4 \ln \left (f x +e \right ) a^{3} b \,e^{2} m n +x {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{4} b^{4} e f +12 {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{2} b^{4} e^{2} m^{2} n^{2}-4 {\ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}^{3} b^{4} e^{2} m n -24 \ln \left (f x +e \right ) b^{4} e^{2} m^{4} n^{4}}{e f}\)	\(642\)

Input:

int((a+b*ln(c*(d*(f*x+e)^m)^n))^4,x,method=_RETURNVERBOSE)
 

Output:

(24*x*ln(c*(d*(f*x+e)^m)^n)*a*b^3*e*f*m^2*n^2-12*x*ln(c*(d*(f*x+e)^m)^n)^2 
*a*b^3*e*f*m*n-12*x*ln(c*(d*(f*x+e)^m)^n)*a^2*b^2*e*f*m*n+12*x*ln(c*(d*(f* 
x+e)^m)^n)^2*b^4*e*f*m^2*n^2-24*x*a*b^3*e*f*m^3*n^3-4*x*ln(c*(d*(f*x+e)^m) 
^n)^3*b^4*e*f*m*n+12*x*a^2*b^2*e*f*m^2*n^2-4*x*a^3*b*e*f*m*n-24*x*ln(c*(d* 
(f*x+e)^m)^n)*b^4*e*f*m^3*n^3-24*b^4*e^2*m^4*n^4+ln(c*(d*(f*x+e)^m)^n)^4*b 
^4*e^2-a^4*e^2+4*ln(c*(d*(f*x+e)^m)^n)^3*a*b^3*e^2+6*ln(c*(d*(f*x+e)^m)^n) 
^2*a^2*b^2*e^2+x*a^4*e*f+24*a*b^3*e^2*m^3*n^3-12*a^2*b^2*e^2*m^2*n^2+4*a^3 
*b*e^2*m*n+4*x*ln(c*(d*(f*x+e)^m)^n)^3*a*b^3*e*f+6*x*ln(c*(d*(f*x+e)^m)^n) 
^2*a^2*b^2*e*f+4*x*ln(c*(d*(f*x+e)^m)^n)*a^3*b*e*f+24*x*b^4*e*f*m^4*n^4-12 
*ln(c*(d*(f*x+e)^m)^n)^2*a*b^3*e^2*m*n+24*ln(f*x+e)*a*b^3*e^2*m^3*n^3-12*l 
n(f*x+e)*a^2*b^2*e^2*m^2*n^2+4*ln(f*x+e)*a^3*b*e^2*m*n+x*ln(c*(d*(f*x+e)^m 
)^n)^4*b^4*e*f+12*ln(c*(d*(f*x+e)^m)^n)^2*b^4*e^2*m^2*n^2-4*ln(c*(d*(f*x+e 
)^m)^n)^3*b^4*e^2*m*n-24*ln(f*x+e)*b^4*e^2*m^4*n^4)/e/f
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1409 vs. \(2 (160) = 320\).

Time = 0.12 (sec) , antiderivative size = 1409, normalized size of antiderivative = 8.81 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx=\text {Too large to display} \] Input:

integrate((a+b*log(c*(d*(f*x+e)^m)^n))^4,x, algorithm="fricas")
 

Output:

(b^4*f*n^4*x*log(d)^4 + b^4*f*x*log(c)^4 + (b^4*f*m^4*n^4*x + b^4*e*m^4*n^ 
4)*log(f*x + e)^4 - 4*(b^4*f*m*n - a*b^3*f)*x*log(c)^3 - 4*(b^4*e*m^4*n^4 
- a*b^3*e*m^3*n^3 + (b^4*f*m^4*n^4 - a*b^3*f*m^3*n^3)*x - (b^4*f*m^3*n^3*x 
 + b^4*e*m^3*n^3)*log(c) - (b^4*f*m^3*n^4*x + b^4*e*m^3*n^4)*log(d))*log(f 
*x + e)^3 + 6*(2*b^4*f*m^2*n^2 - 2*a*b^3*f*m*n + a^2*b^2*f)*x*log(c)^2 + 4 
*(b^4*f*n^3*x*log(c) - (b^4*f*m*n^4 - a*b^3*f*n^3)*x)*log(d)^3 + 6*(2*b^4* 
e*m^4*n^4 - 2*a*b^3*e*m^3*n^3 + a^2*b^2*e*m^2*n^2 + (b^4*f*m^2*n^2*x + b^4 
*e*m^2*n^2)*log(c)^2 + (b^4*f*m^2*n^4*x + b^4*e*m^2*n^4)*log(d)^2 + (2*b^4 
*f*m^4*n^4 - 2*a*b^3*f*m^3*n^3 + a^2*b^2*f*m^2*n^2)*x - 2*(b^4*e*m^3*n^3 - 
 a*b^3*e*m^2*n^2 + (b^4*f*m^3*n^3 - a*b^3*f*m^2*n^2)*x)*log(c) - 2*(b^4*e* 
m^3*n^4 - a*b^3*e*m^2*n^3 + (b^4*f*m^3*n^4 - a*b^3*f*m^2*n^3)*x - (b^4*f*m 
^2*n^3*x + b^4*e*m^2*n^3)*log(c))*log(d))*log(f*x + e)^2 - 4*(6*b^4*f*m^3* 
n^3 - 6*a*b^3*f*m^2*n^2 + 3*a^2*b^2*f*m*n - a^3*b*f)*x*log(c) + 6*(b^4*f*n 
^2*x*log(c)^2 - 2*(b^4*f*m*n^3 - a*b^3*f*n^2)*x*log(c) + (2*b^4*f*m^2*n^4 
- 2*a*b^3*f*m*n^3 + a^2*b^2*f*n^2)*x)*log(d)^2 + (24*b^4*f*m^4*n^4 - 24*a* 
b^3*f*m^3*n^3 + 12*a^2*b^2*f*m^2*n^2 - 4*a^3*b*f*m*n + a^4*f)*x - 4*(6*b^4 
*e*m^4*n^4 - 6*a*b^3*e*m^3*n^3 + 3*a^2*b^2*e*m^2*n^2 - a^3*b*e*m*n - (b^4* 
f*m*n*x + b^4*e*m*n)*log(c)^3 - (b^4*f*m*n^4*x + b^4*e*m*n^4)*log(d)^3 + 3 
*(b^4*e*m^2*n^2 - a*b^3*e*m*n + (b^4*f*m^2*n^2 - a*b^3*f*m*n)*x)*log(c)^2 
+ 3*(b^4*e*m^2*n^4 - a*b^3*e*m*n^3 + (b^4*f*m^2*n^4 - a*b^3*f*m*n^3)*x ...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 609 vs. \(2 (155) = 310\).

Time = 2.76 (sec) , antiderivative size = 609, normalized size of antiderivative = 3.81 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx =\text {Too large to display} \] Input:

integrate((a+b*ln(c*(d*(f*x+e)**m)**n))**4,x)
 

Output:

Piecewise((a**4*x + 4*a**3*b*e*log(c*(d*(e + f*x)**m)**n)/f - 4*a**3*b*m*n 
*x + 4*a**3*b*x*log(c*(d*(e + f*x)**m)**n) - 12*a**2*b**2*e*m*n*log(c*(d*( 
e + f*x)**m)**n)/f + 6*a**2*b**2*e*log(c*(d*(e + f*x)**m)**n)**2/f + 12*a* 
*2*b**2*m**2*n**2*x - 12*a**2*b**2*m*n*x*log(c*(d*(e + f*x)**m)**n) + 6*a* 
*2*b**2*x*log(c*(d*(e + f*x)**m)**n)**2 + 24*a*b**3*e*m**2*n**2*log(c*(d*( 
e + f*x)**m)**n)/f - 12*a*b**3*e*m*n*log(c*(d*(e + f*x)**m)**n)**2/f + 4*a 
*b**3*e*log(c*(d*(e + f*x)**m)**n)**3/f - 24*a*b**3*m**3*n**3*x + 24*a*b** 
3*m**2*n**2*x*log(c*(d*(e + f*x)**m)**n) - 12*a*b**3*m*n*x*log(c*(d*(e + f 
*x)**m)**n)**2 + 4*a*b**3*x*log(c*(d*(e + f*x)**m)**n)**3 - 24*b**4*e*m**3 
*n**3*log(c*(d*(e + f*x)**m)**n)/f + 12*b**4*e*m**2*n**2*log(c*(d*(e + f*x 
)**m)**n)**2/f - 4*b**4*e*m*n*log(c*(d*(e + f*x)**m)**n)**3/f + b**4*e*log 
(c*(d*(e + f*x)**m)**n)**4/f + 24*b**4*m**4*n**4*x - 24*b**4*m**3*n**3*x*l 
og(c*(d*(e + f*x)**m)**n) + 12*b**4*m**2*n**2*x*log(c*(d*(e + f*x)**m)**n) 
**2 - 4*b**4*m*n*x*log(c*(d*(e + f*x)**m)**n)**3 + b**4*x*log(c*(d*(e + f* 
x)**m)**n)**4, Ne(f, 0)), (x*(a + b*log(c*(d*e**m)**n))**4, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (160) = 320\).

Time = 0.07 (sec) , antiderivative size = 559, normalized size of antiderivative = 3.49 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx =\text {Too large to display} \] Input:

integrate((a+b*log(c*(d*(f*x+e)^m)^n))^4,x, algorithm="maxima")
 

Output:

b^4*x*log(((f*x + e)^m*d)^n*c)^4 - 4*a^3*b*f*m*n*(x/f - e*log(f*x + e)/f^2 
) + 4*a*b^3*x*log(((f*x + e)^m*d)^n*c)^3 + 6*a^2*b^2*x*log(((f*x + e)^m*d) 
^n*c)^2 + 4*a^3*b*x*log(((f*x + e)^m*d)^n*c) - 6*(2*f*m*n*(x/f - e*log(f*x 
 + e)/f^2)*log(((f*x + e)^m*d)^n*c) + (e*log(f*x + e)^2 - 2*f*x + 2*e*log( 
f*x + e))*m^2*n^2/f)*a^2*b^2 - 4*(3*f*m*n*(x/f - e*log(f*x + e)/f^2)*log(( 
(f*x + e)^m*d)^n*c)^2 - ((e*log(f*x + e)^3 + 3*e*log(f*x + e)^2 - 6*f*x + 
6*e*log(f*x + e))*m^2*n^2/f^2 - 3*(e*log(f*x + e)^2 - 2*f*x + 2*e*log(f*x 
+ e))*m*n*log(((f*x + e)^m*d)^n*c)/f^2)*f*m*n)*a*b^3 - (4*f*m*n*(x/f - e*l 
og(f*x + e)/f^2)*log(((f*x + e)^m*d)^n*c)^3 + (((e*log(f*x + e)^4 + 4*e*lo 
g(f*x + e)^3 + 12*e*log(f*x + e)^2 - 24*f*x + 24*e*log(f*x + e))*m^2*n^2/f 
^3 - 4*(e*log(f*x + e)^3 + 3*e*log(f*x + e)^2 - 6*f*x + 6*e*log(f*x + e))* 
m*n*log(((f*x + e)^m*d)^n*c)/f^3)*f*m*n + 6*(e*log(f*x + e)^2 - 2*f*x + 2* 
e*log(f*x + e))*m*n*log(((f*x + e)^m*d)^n*c)^2/f^2)*f*m*n)*b^4 + a^4*x
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1697 vs. \(2 (160) = 320\).

Time = 0.15 (sec) , antiderivative size = 1697, normalized size of antiderivative = 10.61 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx=\text {Too large to display} \] Input:

integrate((a+b*log(c*(d*(f*x+e)^m)^n))^4,x, algorithm="giac")
 

Output:

(f*x + e)*b^4*m^4*n^4*log(f*x + e)^4/f - 4*(f*x + e)*b^4*m^4*n^4*log(f*x + 
 e)^3/f + 4*(f*x + e)*b^4*m^3*n^4*log(f*x + e)^3*log(d)/f + 12*(f*x + e)*b 
^4*m^4*n^4*log(f*x + e)^2/f + 4*(f*x + e)*b^4*m^3*n^3*log(f*x + e)^3*log(c 
)/f - 12*(f*x + e)*b^4*m^3*n^4*log(f*x + e)^2*log(d)/f + 6*(f*x + e)*b^4*m 
^2*n^4*log(f*x + e)^2*log(d)^2/f - 24*(f*x + e)*b^4*m^4*n^4*log(f*x + e)/f 
 + 4*(f*x + e)*a*b^3*m^3*n^3*log(f*x + e)^3/f - 12*(f*x + e)*b^4*m^3*n^3*l 
og(f*x + e)^2*log(c)/f + 24*(f*x + e)*b^4*m^3*n^4*log(f*x + e)*log(d)/f + 
12*(f*x + e)*b^4*m^2*n^3*log(f*x + e)^2*log(c)*log(d)/f - 12*(f*x + e)*b^4 
*m^2*n^4*log(f*x + e)*log(d)^2/f + 4*(f*x + e)*b^4*m*n^4*log(f*x + e)*log( 
d)^3/f + 24*(f*x + e)*b^4*m^4*n^4/f - 12*(f*x + e)*a*b^3*m^3*n^3*log(f*x + 
 e)^2/f + 24*(f*x + e)*b^4*m^3*n^3*log(f*x + e)*log(c)/f + 6*(f*x + e)*b^4 
*m^2*n^2*log(f*x + e)^2*log(c)^2/f - 24*(f*x + e)*b^4*m^3*n^4*log(d)/f + 1 
2*(f*x + e)*a*b^3*m^2*n^3*log(f*x + e)^2*log(d)/f - 24*(f*x + e)*b^4*m^2*n 
^3*log(f*x + e)*log(c)*log(d)/f + 12*(f*x + e)*b^4*m^2*n^4*log(d)^2/f + 12 
*(f*x + e)*b^4*m*n^3*log(f*x + e)*log(c)*log(d)^2/f - 4*(f*x + e)*b^4*m*n^ 
4*log(d)^3/f + (f*x + e)*b^4*n^4*log(d)^4/f + 24*(f*x + e)*a*b^3*m^3*n^3*l 
og(f*x + e)/f - 24*(f*x + e)*b^4*m^3*n^3*log(c)/f + 12*(f*x + e)*a*b^3*m^2 
*n^2*log(f*x + e)^2*log(c)/f - 12*(f*x + e)*b^4*m^2*n^2*log(f*x + e)*log(c 
)^2/f - 24*(f*x + e)*a*b^3*m^2*n^3*log(f*x + e)*log(d)/f + 24*(f*x + e)*b^ 
4*m^2*n^3*log(c)*log(d)/f + 12*(f*x + e)*b^4*m*n^2*log(f*x + e)*log(c)^...
 

Mupad [B] (verification not implemented)

Time = 25.91 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.38 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx={\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )}^3\,\left (\frac {4\,\left (a\,b^3\,e-b^4\,e\,m\,n\right )}{f}+4\,b^3\,x\,\left (a-b\,m\,n\right )\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )}^4\,\left (b^4\,x+\frac {b^4\,e}{f}\right )+x\,\left (a^4-4\,a^3\,b\,m\,n+12\,a^2\,b^2\,m^2\,n^2-24\,a\,b^3\,m^3\,n^3+24\,b^4\,m^4\,n^4\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )}^2\,\left (\frac {6\,\left (e\,a^2\,b^2-2\,e\,a\,b^3\,m\,n+2\,e\,b^4\,m^2\,n^2\right )}{f}+6\,b^2\,x\,\left (a^2-2\,a\,b\,m\,n+2\,b^2\,m^2\,n^2\right )\right )-\frac {\ln \left (e+f\,x\right )\,\left (-4\,e\,a^3\,b\,m\,n+12\,e\,a^2\,b^2\,m^2\,n^2-24\,e\,a\,b^3\,m^3\,n^3+24\,e\,b^4\,m^4\,n^4\right )}{f}+\frac {\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )\,\left (4\,b\,f\,\left (a^3-3\,a^2\,b\,m\,n+6\,a\,b^2\,m^2\,n^2-6\,b^3\,m^3\,n^3\right )\,x^2+4\,b\,e\,\left (a^3-3\,a^2\,b\,m\,n+6\,a\,b^2\,m^2\,n^2-6\,b^3\,m^3\,n^3\right )\,x\right )}{e+f\,x} \] Input:

int((a + b*log(c*(d*(e + f*x)^m)^n))^4,x)
 

Output:

log(c*(d*(e + f*x)^m)^n)^3*((4*(a*b^3*e - b^4*e*m*n))/f + 4*b^3*x*(a - b*m 
*n)) + log(c*(d*(e + f*x)^m)^n)^4*(b^4*x + (b^4*e)/f) + x*(a^4 + 24*b^4*m^ 
4*n^4 - 24*a*b^3*m^3*n^3 - 4*a^3*b*m*n + 12*a^2*b^2*m^2*n^2) + log(c*(d*(e 
 + f*x)^m)^n)^2*((6*(a^2*b^2*e + 2*b^4*e*m^2*n^2 - 2*a*b^3*e*m*n))/f + 6*b 
^2*x*(a^2 + 2*b^2*m^2*n^2 - 2*a*b*m*n)) - (log(e + f*x)*(24*b^4*e*m^4*n^4 
- 24*a*b^3*e*m^3*n^3 - 4*a^3*b*e*m*n + 12*a^2*b^2*e*m^2*n^2))/f + (log(c*( 
d*(e + f*x)^m)^n)*(4*b*f*x^2*(a^3 - 6*b^3*m^3*n^3 + 6*a*b^2*m^2*n^2 - 3*a^ 
2*b*m*n) + 4*b*e*x*(a^3 - 6*b^3*m^3*n^3 + 6*a*b^2*m^2*n^2 - 3*a^2*b*m*n))) 
/(e + f*x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 584, normalized size of antiderivative = 3.65 \[ \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^4 \, dx=\frac {-12 \mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{2} a \,b^{3} e m n +12 \mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{2} b^{4} f \,m^{2} n^{2} x -12 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) a^{2} b^{2} e m n +24 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) a \,b^{3} e \,m^{2} n^{2}-24 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) b^{4} f \,m^{3} n^{3} x -4 a^{3} b f m n x +12 a^{2} b^{2} f \,m^{2} n^{2} x -24 a \,b^{3} f \,m^{3} n^{3} x +a^{4} f x +4 \mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{3} a \,b^{3} e +6 \mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{2} a^{2} b^{2} e +4 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) a^{3} b e +4 \mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{3} a \,b^{3} f x -4 \mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{3} b^{4} e m n +6 \mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{2} a^{2} b^{2} f x +12 \mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{2} b^{4} e \,m^{2} n^{2}+4 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) a^{3} b f x -24 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) b^{4} e \,m^{3} n^{3}+24 b^{4} f \,m^{4} n^{4} x +\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{4} b^{4} e +\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{4} b^{4} f x -12 \mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{2} a \,b^{3} f m n x -12 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) a^{2} b^{2} f m n x +24 \,\mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right ) a \,b^{3} f \,m^{2} n^{2} x -4 \mathrm {log}\left (d^{n} \left (f x +e \right )^{m n} c \right )^{3} b^{4} f m n x}{f} \] Input:

int((a+b*log(c*(d*(f*x+e)^m)^n))^4,x)
 

Output:

(log(d**n*(e + f*x)**(m*n)*c)**4*b**4*e + log(d**n*(e + f*x)**(m*n)*c)**4* 
b**4*f*x + 4*log(d**n*(e + f*x)**(m*n)*c)**3*a*b**3*e + 4*log(d**n*(e + f* 
x)**(m*n)*c)**3*a*b**3*f*x - 4*log(d**n*(e + f*x)**(m*n)*c)**3*b**4*e*m*n 
- 4*log(d**n*(e + f*x)**(m*n)*c)**3*b**4*f*m*n*x + 6*log(d**n*(e + f*x)**( 
m*n)*c)**2*a**2*b**2*e + 6*log(d**n*(e + f*x)**(m*n)*c)**2*a**2*b**2*f*x - 
 12*log(d**n*(e + f*x)**(m*n)*c)**2*a*b**3*e*m*n - 12*log(d**n*(e + f*x)** 
(m*n)*c)**2*a*b**3*f*m*n*x + 12*log(d**n*(e + f*x)**(m*n)*c)**2*b**4*e*m** 
2*n**2 + 12*log(d**n*(e + f*x)**(m*n)*c)**2*b**4*f*m**2*n**2*x + 4*log(d** 
n*(e + f*x)**(m*n)*c)*a**3*b*e + 4*log(d**n*(e + f*x)**(m*n)*c)*a**3*b*f*x 
 - 12*log(d**n*(e + f*x)**(m*n)*c)*a**2*b**2*e*m*n - 12*log(d**n*(e + f*x) 
**(m*n)*c)*a**2*b**2*f*m*n*x + 24*log(d**n*(e + f*x)**(m*n)*c)*a*b**3*e*m* 
*2*n**2 + 24*log(d**n*(e + f*x)**(m*n)*c)*a*b**3*f*m**2*n**2*x - 24*log(d* 
*n*(e + f*x)**(m*n)*c)*b**4*e*m**3*n**3 - 24*log(d**n*(e + f*x)**(m*n)*c)* 
b**4*f*m**3*n**3*x + a**4*f*x - 4*a**3*b*f*m*n*x + 12*a**2*b**2*f*m**2*n** 
2*x - 24*a*b**3*f*m**3*n**3*x + 24*b**4*f*m**4*n**4*x)/f