\(\int (e x)^q (a+b \log (c (d x^m)^n))^3 \, dx\) [243]

Optimal result

Integrand size = 22, antiderivative size = 135 \[ \int (e x)^q \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3 \, dx=-\frac {6 b^3 m^3 n^3 (e x)^{1+q}}{e (1+q)^4}+\frac {6 b^2 m^2 n^2 (e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{e (1+q)^3}-\frac {3 b m n (e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2}{e (1+q)^2}+\frac {(e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3}{e (1+q)} \] Output:

-6*b^3*m^3*n^3*(e*x)^(1+q)/e/(1+q)^4+6*b^2*m^2*n^2*(e*x)^(1+q)*(a+b*ln(c*( 
d*x^m)^n))/e/(1+q)^3-3*b*m*n*(e*x)^(1+q)*(a+b*ln(c*(d*x^m)^n))^2/e/(1+q)^2 
+(e*x)^(1+q)*(a+b*ln(c*(d*x^m)^n))^3/e/(1+q)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.67 \[ \int (e x)^q \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3 \, dx=\frac {x (e x)^q \left (\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3-\frac {3 b m n \left ((1+q)^2 \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2+2 b m n \left (b m n-(1+q) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )\right )\right )}{(1+q)^3}\right )}{1+q} \] Input:

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

Output:

(x*(e*x)^q*((a + b*Log[c*(d*x^m)^n])^3 - (3*b*m*n*((1 + q)^2*(a + b*Log[c* 
(d*x^m)^n])^2 + 2*b*m*n*(b*m*n - (1 + q)*(a + b*Log[c*(d*x^m)^n]))))/(1 + 
q)^3))/(1 + q)
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2895, 2742, 2742, 2741}

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.

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

\(\Big \downarrow \) 2895

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

\(\Big \downarrow \) 2742

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

\(\Big \downarrow \) 2742

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

\(\Big \downarrow \) 2741

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

Input:

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

Output:

((e*x)^(1 + q)*(a + b*Log[c*(d*x^m)^n])^3)/(e*(1 + q)) - (3*b*m*n*(((e*x)^ 
(1 + q)*(a + b*Log[c*(d*x^m)^n])^2)/(e*(1 + q)) - (2*b*m*n*(-((b*m*n*(e*x) 
^(1 + q))/(e*(1 + q)^2)) + ((e*x)^(1 + q)*(a + b*Log[c*(d*x^m)^n]))/(e*(1 
+ q))))/(1 + q)))/(1 + q)
 

Defintions of rubi rules used

rule 2741

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> 
Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( 
m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
 

rule 2742

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbo 
l] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^p/(d*(m + 1))), x] - Simp[b*n* 
(p/(m + 1))   Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b 
, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]
 

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. \(676\) vs. \(2(135)=270\).

Time = 13.19 (sec) , antiderivative size = 677, normalized size of antiderivative = 5.01

method	result	size
parallelrisch	\(-\frac {6 x \left (e x \right )^{q} \ln \left (c \left (d \,x^{m}\right )^{n}\right ) a \,b^{2} m n \,q^{2}+12 x \left (e x \right )^{q} \ln \left (c \left (d \,x^{m}\right )^{n}\right ) a \,b^{2} m n q +3 x \left (e x \right )^{q} {\ln \left (c \left (d \,x^{m}\right )^{n}\right )}^{2} b^{3} m n \,q^{2}-6 x \left (e x \right )^{q} \ln \left (c \left (d \,x^{m}\right )^{n}\right ) b^{3} m^{2} n^{2} q +6 x \left (e x \right )^{q} {\ln \left (c \left (d \,x^{m}\right )^{n}\right )}^{2} b^{3} m n q -6 x \left (e x \right )^{q} a \,b^{2} m^{2} n^{2} q +3 x \left (e x \right )^{q} a^{2} b m n \,q^{2}+6 x \left (e x \right )^{q} \ln \left (c \left (d \,x^{m}\right )^{n}\right ) a \,b^{2} m n +6 x \left (e x \right )^{q} a^{2} b m n q -x \left (e x \right )^{q} {\ln \left (c \left (d \,x^{m}\right )^{n}\right )}^{3} b^{3} q^{3}+6 x \left (e x \right )^{q} b^{3} m^{3} n^{3}-3 x \left (e x \right )^{q} {\ln \left (c \left (d \,x^{m}\right )^{n}\right )}^{3} b^{3} q^{2}-b^{3} {\ln \left (c \left (d \,x^{m}\right )^{n}\right )}^{3} \left (e x \right )^{q} x -x \left (e x \right )^{q} a^{3} q^{3}-3 x \left (e x \right )^{q} a^{3} q^{2}-3 x \left (e x \right )^{q} a^{3} q -3 x \left (e x \right )^{q} {\ln \left (c \left (d \,x^{m}\right )^{n}\right )}^{3} b^{3} q -3 x \left (e x \right )^{q} {\ln \left (c \left (d \,x^{m}\right )^{n}\right )}^{2} a \,b^{2}-3 x \left (e x \right )^{q} \ln \left (c \left (d \,x^{m}\right )^{n}\right ) a^{2} b -x \left (e x \right )^{q} a^{3}-3 x \left (e x \right )^{q} {\ln \left (c \left (d \,x^{m}\right )^{n}\right )}^{2} a \,b^{2} q^{3}-6 x \left (e x \right )^{q} \ln \left (c \left (d \,x^{m}\right )^{n}\right ) b^{3} m^{2} n^{2}-9 x \left (e x \right )^{q} {\ln \left (c \left (d \,x^{m}\right )^{n}\right )}^{2} a \,b^{2} q^{2}+3 x \left (e x \right )^{q} {\ln \left (c \left (d \,x^{m}\right )^{n}\right )}^{2} b^{3} m n -3 x \left (e x \right )^{q} \ln \left (c \left (d \,x^{m}\right )^{n}\right ) a^{2} b \,q^{3}-6 x \left (e x \right )^{q} a \,b^{2} m^{2} n^{2}-9 x \left (e x \right )^{q} {\ln \left (c \left (d \,x^{m}\right )^{n}\right )}^{2} a \,b^{2} q -9 x \left (e x \right )^{q} \ln \left (c \left (d \,x^{m}\right )^{n}\right ) a^{2} b \,q^{2}-9 x \left (e x \right )^{q} \ln \left (c \left (d \,x^{m}\right )^{n}\right ) a^{2} b q +3 x \left (e x \right )^{q} a^{2} b m n}{\left (q^{2}+2 q +1\right ) \left (1+q \right )^{2}}\)	\(677\)

Input:

int((e*x)^q*(a+b*ln(c*(d*x^m)^n))^3,x,method=_RETURNVERBOSE)
 

Output:

-(6*x*(e*x)^q*ln(c*(d*x^m)^n)*a*b^2*m*n*q^2+12*x*(e*x)^q*ln(c*(d*x^m)^n)*a 
*b^2*m*n*q+3*x*(e*x)^q*ln(c*(d*x^m)^n)^2*b^3*m*n*q^2-6*x*(e*x)^q*ln(c*(d*x 
^m)^n)*b^3*m^2*n^2*q+6*x*(e*x)^q*ln(c*(d*x^m)^n)^2*b^3*m*n*q-6*x*(e*x)^q*a 
*b^2*m^2*n^2*q+3*x*(e*x)^q*a^2*b*m*n*q^2+6*x*(e*x)^q*ln(c*(d*x^m)^n)*a*b^2 
*m*n+6*x*(e*x)^q*a^2*b*m*n*q-x*(e*x)^q*ln(c*(d*x^m)^n)^3*b^3*q^3+6*x*(e*x) 
^q*b^3*m^3*n^3-3*x*(e*x)^q*ln(c*(d*x^m)^n)^3*b^3*q^2-b^3*ln(c*(d*x^m)^n)^3 
*(e*x)^q*x-x*(e*x)^q*a^3*q^3-3*x*(e*x)^q*a^3*q^2-3*x*(e*x)^q*a^3*q-3*x*(e* 
x)^q*ln(c*(d*x^m)^n)^3*b^3*q-3*x*(e*x)^q*ln(c*(d*x^m)^n)^2*a*b^2-3*x*(e*x) 
^q*ln(c*(d*x^m)^n)*a^2*b-x*(e*x)^q*a^3-3*x*(e*x)^q*ln(c*(d*x^m)^n)^2*a*b^2 
*q^3-6*x*(e*x)^q*ln(c*(d*x^m)^n)*b^3*m^2*n^2-9*x*(e*x)^q*ln(c*(d*x^m)^n)^2 
*a*b^2*q^2+3*x*(e*x)^q*ln(c*(d*x^m)^n)^2*b^3*m*n-3*x*(e*x)^q*ln(c*(d*x^m)^ 
n)*a^2*b*q^3-6*x*(e*x)^q*a*b^2*m^2*n^2-9*x*(e*x)^q*ln(c*(d*x^m)^n)^2*a*b^2 
*q-9*x*(e*x)^q*ln(c*(d*x^m)^n)*a^2*b*q^2-9*x*(e*x)^q*ln(c*(d*x^m)^n)*a^2*b 
*q+3*x*(e*x)^q*a^2*b*m*n)/(q^2+2*q+1)/(1+q)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1358 vs. \(2 (135) = 270\).

Time = 0.13 (sec) , antiderivative size = 1358, normalized size of antiderivative = 10.06 \[ \int (e x)^q \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

((b^3*q^3 + 3*b^3*q^2 + 3*b^3*q + b^3)*x*log(c)^3 + (b^3*n^3*q^3 + 3*b^3*n 
^3*q^2 + 3*b^3*n^3*q + b^3*n^3)*x*log(d)^3 + (b^3*m^3*n^3*q^3 + 3*b^3*m^3* 
n^3*q^2 + 3*b^3*m^3*n^3*q + b^3*m^3*n^3)*x*log(x)^3 + 3*(a*b^2*q^3 - b^3*m 
*n + a*b^2 - (b^3*m*n - 3*a*b^2)*q^2 - (2*b^3*m*n - 3*a*b^2)*q)*x*log(c)^2 
 + 3*(2*b^3*m^2*n^2 + a^2*b*q^3 - 2*a*b^2*m*n + a^2*b - (2*a*b^2*m*n - 3*a 
^2*b)*q^2 + (2*b^3*m^2*n^2 - 4*a*b^2*m*n + 3*a^2*b)*q)*x*log(c) + 3*((b^3* 
n^2*q^3 + 3*b^3*n^2*q^2 + 3*b^3*n^2*q + b^3*n^2)*x*log(c) + (a*b^2*n^2*q^3 
 - b^3*m*n^3 + a*b^2*n^2 - (b^3*m*n^3 - 3*a*b^2*n^2)*q^2 - (2*b^3*m*n^3 - 
3*a*b^2*n^2)*q)*x)*log(d)^2 + 3*((b^3*m^2*n^2*q^3 + 3*b^3*m^2*n^2*q^2 + 3* 
b^3*m^2*n^2*q + b^3*m^2*n^2)*x*log(c) + (b^3*m^2*n^3*q^3 + 3*b^3*m^2*n^3*q 
^2 + 3*b^3*m^2*n^3*q + b^3*m^2*n^3)*x*log(d) + (a*b^2*m^2*n^2*q^3 - b^3*m^ 
3*n^3 + a*b^2*m^2*n^2 - (b^3*m^3*n^3 - 3*a*b^2*m^2*n^2)*q^2 - (2*b^3*m^3*n 
^3 - 3*a*b^2*m^2*n^2)*q)*x)*log(x)^2 - (6*b^3*m^3*n^3 - 6*a*b^2*m^2*n^2 - 
a^3*q^3 + 3*a^2*b*m*n - a^3 + 3*(a^2*b*m*n - a^3)*q^2 - 3*(2*a*b^2*m^2*n^2 
 - 2*a^2*b*m*n + a^3)*q)*x + 3*((b^3*n*q^3 + 3*b^3*n*q^2 + 3*b^3*n*q + b^3 
*n)*x*log(c)^2 + 2*(a*b^2*n*q^3 - b^3*m*n^2 + a*b^2*n - (b^3*m*n^2 - 3*a*b 
^2*n)*q^2 - (2*b^3*m*n^2 - 3*a*b^2*n)*q)*x*log(c) + (2*b^3*m^2*n^3 + a^2*b 
*n*q^3 - 2*a*b^2*m*n^2 + a^2*b*n - (2*a*b^2*m*n^2 - 3*a^2*b*n)*q^2 + (2*b^ 
3*m^2*n^3 - 4*a*b^2*m*n^2 + 3*a^2*b*n)*q)*x)*log(d) + 3*((b^3*m*n*q^3 + 3* 
b^3*m*n*q^2 + 3*b^3*m*n*q + b^3*m*n)*x*log(c)^2 + (b^3*m*n^3*q^3 + 3*b^...
 

Sympy [F]

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

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

Output:

Integral((e*x)**q*(a + b*log(c*(d*x**m)**n))**3, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (135) = 270\).

Time = 0.05 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.09 \[ \int (e x)^q \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3 \, dx=-\frac {3 \, a^{2} b e^{q} m n x x^{q}}{{\left (q + 1\right )}^{2}} + \frac {\left (e x\right )^{q + 1} b^{3} \log \left (\left (d x^{m}\right )^{n} c\right )^{3}}{e {\left (q + 1\right )}} + 6 \, {\left (\frac {e^{q} m^{2} n^{2} x x^{q}}{{\left (q + 1\right )}^{3}} - \frac {e^{q} m n x x^{q} \log \left (\left (d x^{m}\right )^{n} c\right )}{{\left (q + 1\right )}^{2}}\right )} a b^{2} - 3 \, {\left (\frac {e^{q} m n x x^{q} \log \left (\left (d x^{m}\right )^{n} c\right )^{2}}{{\left (q + 1\right )}^{2}} + \frac {2 \, {\left (\frac {e^{q + 1} m^{2} n^{2} x x^{q}}{{\left (q + 1\right )}^{3}} - \frac {e^{q + 1} m n x x^{q} \log \left (\left (d x^{m}\right )^{n} c\right )}{{\left (q + 1\right )}^{2}}\right )} m n}{e {\left (q + 1\right )}}\right )} b^{3} + \frac {3 \, \left (e x\right )^{q + 1} a b^{2} \log \left (\left (d x^{m}\right )^{n} c\right )^{2}}{e {\left (q + 1\right )}} + \frac {3 \, \left (e x\right )^{q + 1} a^{2} b \log \left (\left (d x^{m}\right )^{n} c\right )}{e {\left (q + 1\right )}} + \frac {\left (e x\right )^{q + 1} a^{3}}{e {\left (q + 1\right )}} \] Input:

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

Output:

-3*a^2*b*e^q*m*n*x*x^q/(q + 1)^2 + (e*x)^(q + 1)*b^3*log((d*x^m)^n*c)^3/(e 
*(q + 1)) + 6*(e^q*m^2*n^2*x*x^q/(q + 1)^3 - e^q*m*n*x*x^q*log((d*x^m)^n*c 
)/(q + 1)^2)*a*b^2 - 3*(e^q*m*n*x*x^q*log((d*x^m)^n*c)^2/(q + 1)^2 + 2*(e^ 
(q + 1)*m^2*n^2*x*x^q/(q + 1)^3 - e^(q + 1)*m*n*x*x^q*log((d*x^m)^n*c)/(q 
+ 1)^2)*m*n/(e*(q + 1)))*b^3 + 3*(e*x)^(q + 1)*a*b^2*log((d*x^m)^n*c)^2/(e 
*(q + 1)) + 3*(e*x)^(q + 1)*a^2*b*log((d*x^m)^n*c)/(e*(q + 1)) + (e*x)^(q 
+ 1)*a^3/(e*(q + 1))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1857 vs. \(2 (135) = 270\).

Time = 0.20 (sec) , antiderivative size = 1857, normalized size of antiderivative = 13.76 \[ \int (e x)^q \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

b^3*e^q*m^3*n^3*q^3*x*x^q*log(x)^3/(q^4 + 4*q^3 + 6*q^2 + 4*q + 1) + 3*b^3 
*e^q*m^3*n^3*q^2*x*x^q*log(x)^3/(q^4 + 4*q^3 + 6*q^2 + 4*q + 1) - 3*b^3*e^ 
q*m^3*n^3*q^2*x*x^q*log(x)^2/(q^4 + 4*q^3 + 6*q^2 + 4*q + 1) + 3*b^3*e^q*m 
^2*n^3*q^2*x*x^q*log(d)*log(x)^2/(q^3 + 3*q^2 + 3*q + 1) + 3*b^3*e^q*m^3*n 
^3*q*x*x^q*log(x)^3/(q^4 + 4*q^3 + 6*q^2 + 4*q + 1) - 6*b^3*e^q*m^3*n^3*q* 
x*x^q*log(x)^2/(q^4 + 4*q^3 + 6*q^2 + 4*q + 1) + 3*b^3*e^q*m^2*n^2*q^2*x*x 
^q*log(c)*log(x)^2/(q^3 + 3*q^2 + 3*q + 1) + 6*b^3*e^q*m^2*n^3*q*x*x^q*log 
(d)*log(x)^2/(q^3 + 3*q^2 + 3*q + 1) + b^3*e^q*m^3*n^3*x*x^q*log(x)^3/(q^4 
 + 4*q^3 + 6*q^2 + 4*q + 1) + 6*b^3*e^q*m^3*n^3*q*x*x^q*log(x)/(q^4 + 4*q^ 
3 + 6*q^2 + 4*q + 1) - 6*b^3*e^q*m^2*n^3*q*x*x^q*log(d)*log(x)/(q^3 + 3*q^ 
2 + 3*q + 1) + 3*b^3*e^q*m*n^3*q*x*x^q*log(d)^2*log(x)/(q^2 + 2*q + 1) - 3 
*b^3*e^q*m^3*n^3*x*x^q*log(x)^2/(q^4 + 4*q^3 + 6*q^2 + 4*q + 1) + 3*a*b^2* 
e^q*m^2*n^2*q^2*x*x^q*log(x)^2/(q^3 + 3*q^2 + 3*q + 1) + 6*b^3*e^q*m^2*n^2 
*q*x*x^q*log(c)*log(x)^2/(q^3 + 3*q^2 + 3*q + 1) + 3*b^3*e^q*m^2*n^3*x*x^q 
*log(d)*log(x)^2/(q^3 + 3*q^2 + 3*q + 1) + 6*b^3*e^q*m^3*n^3*x*x^q*log(x)/ 
(q^4 + 4*q^3 + 6*q^2 + 4*q + 1) - 6*b^3*e^q*m^2*n^2*q*x*x^q*log(c)*log(x)/ 
(q^3 + 3*q^2 + 3*q + 1) - 6*b^3*e^q*m^2*n^3*x*x^q*log(d)*log(x)/(q^3 + 3*q 
^2 + 3*q + 1) + 6*b^3*e^q*m*n^2*q*x*x^q*log(c)*log(d)*log(x)/(q^2 + 2*q + 
1) + 3*b^3*e^q*m*n^3*x*x^q*log(d)^2*log(x)/(q^2 + 2*q + 1) + 6*a*b^2*e^q*m 
^2*n^2*q*x*x^q*log(x)^2/(q^3 + 3*q^2 + 3*q + 1) + 3*b^3*e^q*m^2*n^2*x*x...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 522, normalized size of antiderivative = 3.87 \[ \int (e x)^q \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3 \, dx=\frac {x^{q} e^{q} x \left (3 \mathrm {log}\left (x^{m n} d^{n} c \right )^{2} a \,b^{2} q^{3}+\mathrm {log}\left (x^{m n} d^{n} c \right )^{3} b^{3} q^{3}+9 \mathrm {log}\left (x^{m n} d^{n} c \right )^{2} a \,b^{2} q^{2}+9 \mathrm {log}\left (x^{m n} d^{n} c \right )^{2} a \,b^{2} q -3 \mathrm {log}\left (x^{m n} d^{n} c \right )^{2} b^{3} m n +3 \,\mathrm {log}\left (x^{m n} d^{n} c \right ) a^{2} b \,q^{3}+9 \,\mathrm {log}\left (x^{m n} d^{n} c \right ) a^{2} b \,q^{2}+9 \,\mathrm {log}\left (x^{m n} d^{n} c \right ) a^{2} b q +6 \,\mathrm {log}\left (x^{m n} d^{n} c \right ) b^{3} m^{2} n^{2}-3 a^{2} b m n +6 a \,b^{2} m^{2} n^{2}+\mathrm {log}\left (x^{m n} d^{n} c \right )^{3} b^{3}+a^{3} q^{3}+3 a^{3} q^{2}+3 a^{3} q -6 \,\mathrm {log}\left (x^{m n} d^{n} c \right ) a \,b^{2} m n \,q^{2}-12 \,\mathrm {log}\left (x^{m n} d^{n} c \right ) a \,b^{2} m n q +a^{3}-3 \mathrm {log}\left (x^{m n} d^{n} c \right )^{2} b^{3} m n \,q^{2}-6 \mathrm {log}\left (x^{m n} d^{n} c \right )^{2} b^{3} m n q -6 \,\mathrm {log}\left (x^{m n} d^{n} c \right ) a \,b^{2} m n +6 \,\mathrm {log}\left (x^{m n} d^{n} c \right ) b^{3} m^{2} n^{2} q -3 a^{2} b m n \,q^{2}-6 a^{2} b m n q +6 a \,b^{2} m^{2} n^{2} q +3 \mathrm {log}\left (x^{m n} d^{n} c \right )^{3} b^{3} q^{2}+3 \mathrm {log}\left (x^{m n} d^{n} c \right )^{3} b^{3} q +3 \mathrm {log}\left (x^{m n} d^{n} c \right )^{2} a \,b^{2}+3 \,\mathrm {log}\left (x^{m n} d^{n} c \right ) a^{2} b -6 b^{3} m^{3} n^{3}\right )}{q^{4}+4 q^{3}+6 q^{2}+4 q +1} \] Input:

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

Output:

(x**q*e**q*x*(log(x**(m*n)*d**n*c)**3*b**3*q**3 + 3*log(x**(m*n)*d**n*c)** 
3*b**3*q**2 + 3*log(x**(m*n)*d**n*c)**3*b**3*q + log(x**(m*n)*d**n*c)**3*b 
**3 + 3*log(x**(m*n)*d**n*c)**2*a*b**2*q**3 + 9*log(x**(m*n)*d**n*c)**2*a* 
b**2*q**2 + 9*log(x**(m*n)*d**n*c)**2*a*b**2*q + 3*log(x**(m*n)*d**n*c)**2 
*a*b**2 - 3*log(x**(m*n)*d**n*c)**2*b**3*m*n*q**2 - 6*log(x**(m*n)*d**n*c) 
**2*b**3*m*n*q - 3*log(x**(m*n)*d**n*c)**2*b**3*m*n + 3*log(x**(m*n)*d**n* 
c)*a**2*b*q**3 + 9*log(x**(m*n)*d**n*c)*a**2*b*q**2 + 9*log(x**(m*n)*d**n* 
c)*a**2*b*q + 3*log(x**(m*n)*d**n*c)*a**2*b - 6*log(x**(m*n)*d**n*c)*a*b** 
2*m*n*q**2 - 12*log(x**(m*n)*d**n*c)*a*b**2*m*n*q - 6*log(x**(m*n)*d**n*c) 
*a*b**2*m*n + 6*log(x**(m*n)*d**n*c)*b**3*m**2*n**2*q + 6*log(x**(m*n)*d** 
n*c)*b**3*m**2*n**2 + a**3*q**3 + 3*a**3*q**2 + 3*a**3*q + a**3 - 3*a**2*b 
*m*n*q**2 - 6*a**2*b*m*n*q - 3*a**2*b*m*n + 6*a*b**2*m**2*n**2*q + 6*a*b** 
2*m**2*n**2 - 6*b**3*m**3*n**3))/(q**4 + 4*q**3 + 6*q**2 + 4*q + 1)