\(\int (a+b \log (c (d (e+f x)^p)^q))^4 \, dx\) [460]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.82 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4-4 b p q \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3-3 b p q \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-2 b p q \left (f (a-b p q) x+b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )\right )}{f} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.90 (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.

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

\(\Big \downarrow \) 2895

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

\(\Big \downarrow \) 2836

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

\(\Big \downarrow \) 2733

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

\(\Big \downarrow \) 2733

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

\(\Big \downarrow \) 2733

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((e + f*x)*(a + b*Log[c*d^q*(e + f*x)^(p*q)])^4 - 4*b*p*q*((e + f*x)*(a + 
b*Log[c*d^q*(e + f*x)^(p*q)])^3 - 3*b*p*q*((e + f*x)*(a + b*Log[c*d^q*(e + 
 f*x)^(p*q)])^2 - 2*b*p*q*(a*(e + f*x) - b*p*q*(e + f*x) + b*(e + f*x)*Log 
[c*d^q*(e + f*x)^(p*q)]))))/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 = 6.70 (sec) , antiderivative size = 642, normalized size of antiderivative = 4.01

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

Input:

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

Output:

(-12*x*ln(c*(d*(f*x+e)^p)^q)*a^2*b^2*e*f*p*q+4*ln(c*(d*(f*x+e)^p)^q)^3*a*b 
^3*e^2+6*ln(c*(d*(f*x+e)^p)^q)^2*a^2*b^2*e^2+24*x*ln(c*(d*(f*x+e)^p)^q)*a* 
b^3*e*f*p^2*q^2-12*x*ln(c*(d*(f*x+e)^p)^q)^2*a*b^3*e*f*p*q+24*a*b^3*e^2*p^ 
3*q^3-12*a^2*b^2*e^2*p^2*q^2+4*a^3*b*e^2*p*q-12*ln(f*x+e)*a^2*b^2*e^2*p^2* 
q^2+4*ln(f*x+e)*a^3*b*e^2*p*q+6*x*ln(c*(d*(f*x+e)^p)^q)^2*a^2*b^2*e*f+4*x* 
ln(c*(d*(f*x+e)^p)^q)*a^3*b*e*f+12*x*ln(c*(d*(f*x+e)^p)^q)^2*b^4*e*f*p^2*q 
^2-24*x*a*b^3*e*f*p^3*q^3-4*x*ln(c*(d*(f*x+e)^p)^q)^3*b^4*e*f*p*q+12*x*a^2 
*b^2*e*f*p^2*q^2-4*x*a^3*b*e*f*p*q-24*b^4*e^2*p^4*q^4+4*x*ln(c*(d*(f*x+e)^ 
p)^q)^3*a*b^3*e*f+24*x*b^4*e*f*p^4*q^4-12*ln(c*(d*(f*x+e)^p)^q)^2*a*b^3*e^ 
2*p*q+24*ln(f*x+e)*a*b^3*e^2*p^3*q^3-24*x*ln(c*(d*(f*x+e)^p)^q)*b^4*e*f*p^ 
3*q^3+x*ln(c*(d*(f*x+e)^p)^q)^4*b^4*e*f+12*ln(c*(d*(f*x+e)^p)^q)^2*b^4*e^2 
*p^2*q^2-4*ln(c*(d*(f*x+e)^p)^q)^3*b^4*e^2*p*q-24*ln(f*x+e)*b^4*e^2*p^4*q^ 
4-a^4*e^2+x*a^4*e*f+ln(c*(d*(f*x+e)^p)^q)^4*b^4*e^2)/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 = 0.10 (sec) , antiderivative size = 1409, normalized size of antiderivative = 8.81 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx=\text {Too large to display} \] Input:

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

Output:

(b^4*f*q^4*x*log(d)^4 + b^4*f*x*log(c)^4 + (b^4*f*p^4*q^4*x + b^4*e*p^4*q^ 
4)*log(f*x + e)^4 - 4*(b^4*f*p*q - a*b^3*f)*x*log(c)^3 - 4*(b^4*e*p^4*q^4 
- a*b^3*e*p^3*q^3 + (b^4*f*p^4*q^4 - a*b^3*f*p^3*q^3)*x - (b^4*f*p^3*q^3*x 
 + b^4*e*p^3*q^3)*log(c) - (b^4*f*p^3*q^4*x + b^4*e*p^3*q^4)*log(d))*log(f 
*x + e)^3 + 6*(2*b^4*f*p^2*q^2 - 2*a*b^3*f*p*q + a^2*b^2*f)*x*log(c)^2 + 4 
*(b^4*f*q^3*x*log(c) - (b^4*f*p*q^4 - a*b^3*f*q^3)*x)*log(d)^3 + 6*(2*b^4* 
e*p^4*q^4 - 2*a*b^3*e*p^3*q^3 + a^2*b^2*e*p^2*q^2 + (b^4*f*p^2*q^2*x + b^4 
*e*p^2*q^2)*log(c)^2 + (b^4*f*p^2*q^4*x + b^4*e*p^2*q^4)*log(d)^2 + (2*b^4 
*f*p^4*q^4 - 2*a*b^3*f*p^3*q^3 + a^2*b^2*f*p^2*q^2)*x - 2*(b^4*e*p^3*q^3 - 
 a*b^3*e*p^2*q^2 + (b^4*f*p^3*q^3 - a*b^3*f*p^2*q^2)*x)*log(c) - 2*(b^4*e* 
p^3*q^4 - a*b^3*e*p^2*q^3 + (b^4*f*p^3*q^4 - a*b^3*f*p^2*q^3)*x - (b^4*f*p 
^2*q^3*x + b^4*e*p^2*q^3)*log(c))*log(d))*log(f*x + e)^2 - 4*(6*b^4*f*p^3* 
q^3 - 6*a*b^3*f*p^2*q^2 + 3*a^2*b^2*f*p*q - a^3*b*f)*x*log(c) + 6*(b^4*f*q 
^2*x*log(c)^2 - 2*(b^4*f*p*q^3 - a*b^3*f*q^2)*x*log(c) + (2*b^4*f*p^2*q^4 
- 2*a*b^3*f*p*q^3 + a^2*b^2*f*q^2)*x)*log(d)^2 + (24*b^4*f*p^4*q^4 - 24*a* 
b^3*f*p^3*q^3 + 12*a^2*b^2*f*p^2*q^2 - 4*a^3*b*f*p*q + a^4*f)*x - 4*(6*b^4 
*e*p^4*q^4 - 6*a*b^3*e*p^3*q^3 + 3*a^2*b^2*e*p^2*q^2 - a^3*b*e*p*q - (b^4* 
f*p*q*x + b^4*e*p*q)*log(c)^3 - (b^4*f*p*q^4*x + b^4*e*p*q^4)*log(d)^3 + 3 
*(b^4*e*p^2*q^2 - a*b^3*e*p*q + (b^4*f*p^2*q^2 - a*b^3*f*p*q)*x)*log(c)^2 
+ 3*(b^4*e*p^2*q^4 - a*b^3*e*p*q^3 + (b^4*f*p^2*q^4 - a*b^3*f*p*q^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 = 2.26 (sec) , antiderivative size = 609, normalized size of antiderivative = 3.81 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx =\text {Too large to display} \] Input:

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))**4,x)
 

Output:

Piecewise((a**4*x + 4*a**3*b*e*log(c*(d*(e + f*x)**p)**q)/f - 4*a**3*b*p*q 
*x + 4*a**3*b*x*log(c*(d*(e + f*x)**p)**q) - 12*a**2*b**2*e*p*q*log(c*(d*( 
e + f*x)**p)**q)/f + 6*a**2*b**2*e*log(c*(d*(e + f*x)**p)**q)**2/f + 12*a* 
*2*b**2*p**2*q**2*x - 12*a**2*b**2*p*q*x*log(c*(d*(e + f*x)**p)**q) + 6*a* 
*2*b**2*x*log(c*(d*(e + f*x)**p)**q)**2 + 24*a*b**3*e*p**2*q**2*log(c*(d*( 
e + f*x)**p)**q)/f - 12*a*b**3*e*p*q*log(c*(d*(e + f*x)**p)**q)**2/f + 4*a 
*b**3*e*log(c*(d*(e + f*x)**p)**q)**3/f - 24*a*b**3*p**3*q**3*x + 24*a*b** 
3*p**2*q**2*x*log(c*(d*(e + f*x)**p)**q) - 12*a*b**3*p*q*x*log(c*(d*(e + f 
*x)**p)**q)**2 + 4*a*b**3*x*log(c*(d*(e + f*x)**p)**q)**3 - 24*b**4*e*p**3 
*q**3*log(c*(d*(e + f*x)**p)**q)/f + 12*b**4*e*p**2*q**2*log(c*(d*(e + f*x 
)**p)**q)**2/f - 4*b**4*e*p*q*log(c*(d*(e + f*x)**p)**q)**3/f + b**4*e*log 
(c*(d*(e + f*x)**p)**q)**4/f + 24*b**4*p**4*q**4*x - 24*b**4*p**3*q**3*x*l 
og(c*(d*(e + f*x)**p)**q) + 12*b**4*p**2*q**2*x*log(c*(d*(e + f*x)**p)**q) 
**2 - 4*b**4*p*q*x*log(c*(d*(e + f*x)**p)**q)**3 + b**4*x*log(c*(d*(e + f* 
x)**p)**q)**4, Ne(f, 0)), (x*(a + b*log(c*(d*e**p)**q))**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 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx =\text {Too large to display} \] Input:

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

Output:

b^4*x*log(((f*x + e)^p*d)^q*c)^4 - 4*a^3*b*f*p*q*(x/f - e*log(f*x + e)/f^2 
) + 4*a*b^3*x*log(((f*x + e)^p*d)^q*c)^3 + 6*a^2*b^2*x*log(((f*x + e)^p*d) 
^q*c)^2 + 4*a^3*b*x*log(((f*x + e)^p*d)^q*c) - 6*(2*f*p*q*(x/f - e*log(f*x 
 + e)/f^2)*log(((f*x + e)^p*d)^q*c) + (e*log(f*x + e)^2 - 2*f*x + 2*e*log( 
f*x + e))*p^2*q^2/f)*a^2*b^2 - 4*(3*f*p*q*(x/f - e*log(f*x + e)/f^2)*log(( 
(f*x + e)^p*d)^q*c)^2 - ((e*log(f*x + e)^3 + 3*e*log(f*x + e)^2 - 6*f*x + 
6*e*log(f*x + e))*p^2*q^2/f^2 - 3*(e*log(f*x + e)^2 - 2*f*x + 2*e*log(f*x 
+ e))*p*q*log(((f*x + e)^p*d)^q*c)/f^2)*f*p*q)*a*b^3 - (4*f*p*q*(x/f - e*l 
og(f*x + e)/f^2)*log(((f*x + e)^p*d)^q*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))*p^2*q^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))* 
p*q*log(((f*x + e)^p*d)^q*c)/f^3)*f*p*q + 6*(e*log(f*x + e)^2 - 2*f*x + 2* 
e*log(f*x + e))*p*q*log(((f*x + e)^p*d)^q*c)^2/f^2)*f*p*q)*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 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx=\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 15.14 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.38 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx={\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^3\,\left (\frac {4\,\left (a\,b^3\,e-b^4\,e\,p\,q\right )}{f}+4\,b^3\,x\,\left (a-b\,p\,q\right )\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^4\,\left (b^4\,x+\frac {b^4\,e}{f}\right )+x\,\left (a^4-4\,a^3\,b\,p\,q+12\,a^2\,b^2\,p^2\,q^2-24\,a\,b^3\,p^3\,q^3+24\,b^4\,p^4\,q^4\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^2\,\left (\frac {6\,\left (e\,a^2\,b^2-2\,e\,a\,b^3\,p\,q+2\,e\,b^4\,p^2\,q^2\right )}{f}+6\,b^2\,x\,\left (a^2-2\,a\,b\,p\,q+2\,b^2\,p^2\,q^2\right )\right )+\frac {\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\,\left (4\,b\,f\,\left (a^3-3\,a^2\,b\,p\,q+6\,a\,b^2\,p^2\,q^2-6\,b^3\,p^3\,q^3\right )\,x^2+4\,b\,e\,\left (a^3-3\,a^2\,b\,p\,q+6\,a\,b^2\,p^2\,q^2-6\,b^3\,p^3\,q^3\right )\,x\right )}{e+f\,x}-\frac {\ln \left (e+f\,x\right )\,\left (-4\,e\,a^3\,b\,p\,q+12\,e\,a^2\,b^2\,p^2\,q^2-24\,e\,a\,b^3\,p^3\,q^3+24\,e\,b^4\,p^4\,q^4\right )}{f} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 584, normalized size of antiderivative = 3.65 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \, dx=\frac {-4 \mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )^{3} b^{4} f p q x -12 \mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )^{2} a \,b^{3} e p q +12 \mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )^{2} b^{4} f \,p^{2} q^{2} x -12 \,\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right ) a^{2} b^{2} e p q +24 \,\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right ) a \,b^{3} e \,p^{2} q^{2}-24 \,\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right ) b^{4} f \,p^{3} q^{3} x -4 a^{3} b f p q x +12 a^{2} b^{2} f \,p^{2} q^{2} x -24 a \,b^{3} f \,p^{3} q^{3} x -12 \,\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right ) a^{2} b^{2} f p q x +24 \,\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right ) a \,b^{3} f \,p^{2} q^{2} x +4 \mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )^{3} a \,b^{3} e +6 \mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )^{2} a^{2} b^{2} e +4 \,\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right ) a^{3} b e +\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )^{4} b^{4} e +\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )^{4} b^{4} f x +a^{4} f x -12 \mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )^{2} a \,b^{3} f p q x +6 \mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )^{2} a^{2} b^{2} f x +12 \mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )^{2} b^{4} e \,p^{2} q^{2}+4 \,\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right ) a^{3} b f x -24 \,\mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right ) b^{4} e \,p^{3} q^{3}+24 b^{4} f \,p^{4} q^{4} x +4 \mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )^{3} a \,b^{3} f x -4 \mathrm {log}\left (d^{q} \left (f x +e \right )^{p q} c \right )^{3} b^{4} e p q}{f} \] Input:

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

Output:

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