\(\int x (d+e x)^2 (a+b \log (c x^n))^2 \, dx\) [85]

Optimal result

Integrand size = 21, antiderivative size = 178 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{4} b^2 d^2 n^2 x^2+\frac {4}{27} b^2 d e n^2 x^3+\frac {1}{32} b^2 e^2 n^2 x^4-\frac {1}{2} b d^2 n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {4}{9} b d e n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{8} b e^2 n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} d^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2}{3} d e x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{4} e^2 x^4 \left (a+b \log \left (c x^n\right )\right )^2 \] Output:

1/4*b^2*d^2*n^2*x^2+4/27*b^2*d*e*n^2*x^3+1/32*b^2*e^2*n^2*x^4-1/2*b*d^2*n* 
x^2*(a+b*ln(c*x^n))-4/9*b*d*e*n*x^3*(a+b*ln(c*x^n))-1/8*b*e^2*n*x^4*(a+b*l 
n(c*x^n))+1/2*d^2*x^2*(a+b*ln(c*x^n))^2+2/3*d*e*x^3*(a+b*ln(c*x^n))^2+1/4* 
e^2*x^4*(a+b*ln(c*x^n))^2
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.75 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{864} x^2 \left (27 b e^2 n x^2 \left (-4 a+b n-4 b \log \left (c x^n\right )\right )+128 b d e n x \left (-3 a+b n-3 b \log \left (c x^n\right )\right )+216 b d^2 n \left (-2 a+b n-2 b \log \left (c x^n\right )\right )+432 d^2 \left (a+b \log \left (c x^n\right )\right )^2+576 d e x \left (a+b \log \left (c x^n\right )\right )^2+216 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2\right ) \] Input:

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

Output:

(x^2*(27*b*e^2*n*x^2*(-4*a + b*n - 4*b*Log[c*x^n]) + 128*b*d*e*n*x*(-3*a + 
 b*n - 3*b*Log[c*x^n]) + 216*b*d^2*n*(-2*a + b*n - 2*b*Log[c*x^n]) + 432*d 
^2*(a + b*Log[c*x^n])^2 + 576*d*e*x*(a + b*Log[c*x^n])^2 + 216*e^2*x^2*(a 
+ b*Log[c*x^n])^2))/864
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2795, 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.

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} d^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} b d^2 n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} d e x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {4}{9} b d e n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} e^2 x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} b e^2 n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} b^2 d^2 n^2 x^2+\frac {4}{27} b^2 d e n^2 x^3+\frac {1}{32} b^2 e^2 n^2 x^4\)

Input:

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

Output:

(b^2*d^2*n^2*x^2)/4 + (4*b^2*d*e*n^2*x^3)/27 + (b^2*e^2*n^2*x^4)/32 - (b*d 
^2*n*x^2*(a + b*Log[c*x^n]))/2 - (4*b*d*e*n*x^3*(a + b*Log[c*x^n]))/9 - (b 
*e^2*n*x^4*(a + b*Log[c*x^n]))/8 + (d^2*x^2*(a + b*Log[c*x^n])^2)/2 + (2*d 
*e*x^3*(a + b*Log[c*x^n])^2)/3 + (e^2*x^4*(a + b*Log[c*x^n])^2)/4
 

Defintions of rubi rules used

rule 2009

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

rule 2795

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + 
(e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ 
c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b 
, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 
] && IntegerQ[m] && IntegerQ[r]))
 

Maple [A] (verified)

Time = 104.67 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.47

method	result	size
parallelrisch	\(\frac {x^{4} b^{2} \ln \left (c \,x^{n}\right )^{2} e^{2}}{4}-\frac {\ln \left (c \,x^{n}\right ) x^{4} n \,b^{2} e^{2}}{8}+\frac {b^{2} e^{2} n^{2} x^{4}}{32}+\frac {x^{4} a b \ln \left (c \,x^{n}\right ) e^{2}}{2}-\frac {a b \,e^{2} n \,x^{4}}{8}+\frac {2 x^{3} b^{2} \ln \left (c \,x^{n}\right )^{2} e d}{3}-\frac {4 \ln \left (c \,x^{n}\right ) x^{3} n d e \,b^{2}}{9}+\frac {4 b^{2} d e \,n^{2} x^{3}}{27}+\frac {x^{4} a^{2} e^{2}}{4}+\frac {4 x^{3} a b \ln \left (c \,x^{n}\right ) e d}{3}-\frac {4 a b d e n \,x^{3}}{9}+\frac {x^{2} b^{2} \ln \left (c \,x^{n}\right )^{2} d^{2}}{2}-\frac {\ln \left (c \,x^{n}\right ) x^{2} n \,b^{2} d^{2}}{2}+\frac {b^{2} d^{2} n^{2} x^{2}}{4}+\frac {2 x^{3} a^{2} e d}{3}+x^{2} a b \ln \left (c \,x^{n}\right ) d^{2}-\frac {a b \,d^{2} n \,x^{2}}{2}+\frac {x^{2} a^{2} d^{2}}{2}\)	\(262\)
risch	\(\text {Expression too large to display}\)	\(2597\)

Input:

int(x*(e*x+d)^2*(a+b*ln(c*x^n))^2,x,method=_RETURNVERBOSE)
 

Output:

1/4*x^4*b^2*ln(c*x^n)^2*e^2-1/8*ln(c*x^n)*x^4*n*b^2*e^2+1/32*b^2*e^2*n^2*x 
^4+1/2*x^4*a*b*ln(c*x^n)*e^2-1/8*a*b*e^2*n*x^4+2/3*x^3*b^2*ln(c*x^n)^2*e*d 
-4/9*ln(c*x^n)*x^3*n*d*e*b^2+4/27*b^2*d*e*n^2*x^3+1/4*x^4*a^2*e^2+4/3*x^3* 
a*b*ln(c*x^n)*e*d-4/9*a*b*d*e*n*x^3+1/2*x^2*b^2*ln(c*x^n)^2*d^2-1/2*ln(c*x 
^n)*x^2*n*b^2*d^2+1/4*b^2*d^2*n^2*x^2+2/3*x^3*a^2*e*d+x^2*a*b*ln(c*x^n)*d^ 
2-1/2*a*b*d^2*n*x^2+1/2*x^2*a^2*d^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (160) = 320\).

Time = 0.07 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.04 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{32} \, {\left (b^{2} e^{2} n^{2} - 4 \, a b e^{2} n + 8 \, a^{2} e^{2}\right )} x^{4} + \frac {2}{27} \, {\left (2 \, b^{2} d e n^{2} - 6 \, a b d e n + 9 \, a^{2} d e\right )} x^{3} + \frac {1}{4} \, {\left (b^{2} d^{2} n^{2} - 2 \, a b d^{2} n + 2 \, a^{2} d^{2}\right )} x^{2} + \frac {1}{12} \, {\left (3 \, b^{2} e^{2} x^{4} + 8 \, b^{2} d e x^{3} + 6 \, b^{2} d^{2} x^{2}\right )} \log \left (c\right )^{2} + \frac {1}{12} \, {\left (3 \, b^{2} e^{2} n^{2} x^{4} + 8 \, b^{2} d e n^{2} x^{3} + 6 \, b^{2} d^{2} n^{2} x^{2}\right )} \log \left (x\right )^{2} - \frac {1}{72} \, {\left (9 \, {\left (b^{2} e^{2} n - 4 \, a b e^{2}\right )} x^{4} + 32 \, {\left (b^{2} d e n - 3 \, a b d e\right )} x^{3} + 36 \, {\left (b^{2} d^{2} n - 2 \, a b d^{2}\right )} x^{2}\right )} \log \left (c\right ) - \frac {1}{72} \, {\left (9 \, {\left (b^{2} e^{2} n^{2} - 4 \, a b e^{2} n\right )} x^{4} + 32 \, {\left (b^{2} d e n^{2} - 3 \, a b d e n\right )} x^{3} + 36 \, {\left (b^{2} d^{2} n^{2} - 2 \, a b d^{2} n\right )} x^{2} - 12 \, {\left (3 \, b^{2} e^{2} n x^{4} + 8 \, b^{2} d e n x^{3} + 6 \, b^{2} d^{2} n x^{2}\right )} \log \left (c\right )\right )} \log \left (x\right ) \] Input:

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

Output:

1/32*(b^2*e^2*n^2 - 4*a*b*e^2*n + 8*a^2*e^2)*x^4 + 2/27*(2*b^2*d*e*n^2 - 6 
*a*b*d*e*n + 9*a^2*d*e)*x^3 + 1/4*(b^2*d^2*n^2 - 2*a*b*d^2*n + 2*a^2*d^2)* 
x^2 + 1/12*(3*b^2*e^2*x^4 + 8*b^2*d*e*x^3 + 6*b^2*d^2*x^2)*log(c)^2 + 1/12 
*(3*b^2*e^2*n^2*x^4 + 8*b^2*d*e*n^2*x^3 + 6*b^2*d^2*n^2*x^2)*log(x)^2 - 1/ 
72*(9*(b^2*e^2*n - 4*a*b*e^2)*x^4 + 32*(b^2*d*e*n - 3*a*b*d*e)*x^3 + 36*(b 
^2*d^2*n - 2*a*b*d^2)*x^2)*log(c) - 1/72*(9*(b^2*e^2*n^2 - 4*a*b*e^2*n)*x^ 
4 + 32*(b^2*d*e*n^2 - 3*a*b*d*e*n)*x^3 + 36*(b^2*d^2*n^2 - 2*a*b*d^2*n)*x^ 
2 - 12*(3*b^2*e^2*n*x^4 + 8*b^2*d*e*n*x^3 + 6*b^2*d^2*n*x^2)*log(c))*log(x 
)
 

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.73 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {a^{2} d^{2} x^{2}}{2} + \frac {2 a^{2} d e x^{3}}{3} + \frac {a^{2} e^{2} x^{4}}{4} - \frac {a b d^{2} n x^{2}}{2} + a b d^{2} x^{2} \log {\left (c x^{n} \right )} - \frac {4 a b d e n x^{3}}{9} + \frac {4 a b d e x^{3} \log {\left (c x^{n} \right )}}{3} - \frac {a b e^{2} n x^{4}}{8} + \frac {a b e^{2} x^{4} \log {\left (c x^{n} \right )}}{2} + \frac {b^{2} d^{2} n^{2} x^{2}}{4} - \frac {b^{2} d^{2} n x^{2} \log {\left (c x^{n} \right )}}{2} + \frac {b^{2} d^{2} x^{2} \log {\left (c x^{n} \right )}^{2}}{2} + \frac {4 b^{2} d e n^{2} x^{3}}{27} - \frac {4 b^{2} d e n x^{3} \log {\left (c x^{n} \right )}}{9} + \frac {2 b^{2} d e x^{3} \log {\left (c x^{n} \right )}^{2}}{3} + \frac {b^{2} e^{2} n^{2} x^{4}}{32} - \frac {b^{2} e^{2} n x^{4} \log {\left (c x^{n} \right )}}{8} + \frac {b^{2} e^{2} x^{4} \log {\left (c x^{n} \right )}^{2}}{4} \] Input:

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

Output:

a**2*d**2*x**2/2 + 2*a**2*d*e*x**3/3 + a**2*e**2*x**4/4 - a*b*d**2*n*x**2/ 
2 + a*b*d**2*x**2*log(c*x**n) - 4*a*b*d*e*n*x**3/9 + 4*a*b*d*e*x**3*log(c* 
x**n)/3 - a*b*e**2*n*x**4/8 + a*b*e**2*x**4*log(c*x**n)/2 + b**2*d**2*n**2 
*x**2/4 - b**2*d**2*n*x**2*log(c*x**n)/2 + b**2*d**2*x**2*log(c*x**n)**2/2 
 + 4*b**2*d*e*n**2*x**3/27 - 4*b**2*d*e*n*x**3*log(c*x**n)/9 + 2*b**2*d*e* 
x**3*log(c*x**n)**2/3 + b**2*e**2*n**2*x**4/32 - b**2*e**2*n*x**4*log(c*x* 
*n)/8 + b**2*e**2*x**4*log(c*x**n)**2/4
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.40 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{4} \, b^{2} e^{2} x^{4} \log \left (c x^{n}\right )^{2} - \frac {1}{8} \, a b e^{2} n x^{4} + \frac {1}{2} \, a b e^{2} x^{4} \log \left (c x^{n}\right ) + \frac {2}{3} \, b^{2} d e x^{3} \log \left (c x^{n}\right )^{2} - \frac {4}{9} \, a b d e n x^{3} + \frac {1}{4} \, a^{2} e^{2} x^{4} + \frac {4}{3} \, a b d e x^{3} \log \left (c x^{n}\right ) + \frac {1}{2} \, b^{2} d^{2} x^{2} \log \left (c x^{n}\right )^{2} - \frac {1}{2} \, a b d^{2} n x^{2} + \frac {2}{3} \, a^{2} d e x^{3} + a b d^{2} x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a^{2} d^{2} x^{2} + \frac {1}{4} \, {\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} b^{2} d^{2} + \frac {4}{27} \, {\left (n^{2} x^{3} - 3 \, n x^{3} \log \left (c x^{n}\right )\right )} b^{2} d e + \frac {1}{32} \, {\left (n^{2} x^{4} - 4 \, n x^{4} \log \left (c x^{n}\right )\right )} b^{2} e^{2} \] Input:

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

Output:

1/4*b^2*e^2*x^4*log(c*x^n)^2 - 1/8*a*b*e^2*n*x^4 + 1/2*a*b*e^2*x^4*log(c*x 
^n) + 2/3*b^2*d*e*x^3*log(c*x^n)^2 - 4/9*a*b*d*e*n*x^3 + 1/4*a^2*e^2*x^4 + 
 4/3*a*b*d*e*x^3*log(c*x^n) + 1/2*b^2*d^2*x^2*log(c*x^n)^2 - 1/2*a*b*d^2*n 
*x^2 + 2/3*a^2*d*e*x^3 + a*b*d^2*x^2*log(c*x^n) + 1/2*a^2*d^2*x^2 + 1/4*(n 
^2*x^2 - 2*n*x^2*log(c*x^n))*b^2*d^2 + 4/27*(n^2*x^3 - 3*n*x^3*log(c*x^n)) 
*b^2*d*e + 1/32*(n^2*x^4 - 4*n*x^4*log(c*x^n))*b^2*e^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (160) = 320\).

Time = 0.12 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.29 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {1}{4} \, b^{2} e^{2} n^{2} x^{4} \log \left (x\right )^{2} - \frac {1}{8} \, b^{2} e^{2} n^{2} x^{4} \log \left (x\right ) + \frac {1}{2} \, b^{2} e^{2} n x^{4} \log \left (c\right ) \log \left (x\right ) + \frac {2}{3} \, b^{2} d e n^{2} x^{3} \log \left (x\right )^{2} + \frac {1}{32} \, b^{2} e^{2} n^{2} x^{4} - \frac {1}{8} \, b^{2} e^{2} n x^{4} \log \left (c\right ) + \frac {1}{4} \, b^{2} e^{2} x^{4} \log \left (c\right )^{2} - \frac {4}{9} \, b^{2} d e n^{2} x^{3} \log \left (x\right ) + \frac {1}{2} \, a b e^{2} n x^{4} \log \left (x\right ) + \frac {4}{3} \, b^{2} d e n x^{3} \log \left (c\right ) \log \left (x\right ) + \frac {1}{2} \, b^{2} d^{2} n^{2} x^{2} \log \left (x\right )^{2} + \frac {4}{27} \, b^{2} d e n^{2} x^{3} - \frac {1}{8} \, a b e^{2} n x^{4} - \frac {4}{9} \, b^{2} d e n x^{3} \log \left (c\right ) + \frac {1}{2} \, a b e^{2} x^{4} \log \left (c\right ) + \frac {2}{3} \, b^{2} d e x^{3} \log \left (c\right )^{2} - \frac {1}{2} \, b^{2} d^{2} n^{2} x^{2} \log \left (x\right ) + \frac {4}{3} \, a b d e n x^{3} \log \left (x\right ) + b^{2} d^{2} n x^{2} \log \left (c\right ) \log \left (x\right ) + \frac {1}{4} \, b^{2} d^{2} n^{2} x^{2} - \frac {4}{9} \, a b d e n x^{3} + \frac {1}{4} \, a^{2} e^{2} x^{4} - \frac {1}{2} \, b^{2} d^{2} n x^{2} \log \left (c\right ) + \frac {4}{3} \, a b d e x^{3} \log \left (c\right ) + \frac {1}{2} \, b^{2} d^{2} x^{2} \log \left (c\right )^{2} + a b d^{2} n x^{2} \log \left (x\right ) - \frac {1}{2} \, a b d^{2} n x^{2} + \frac {2}{3} \, a^{2} d e x^{3} + a b d^{2} x^{2} \log \left (c\right ) + \frac {1}{2} \, a^{2} d^{2} x^{2} \] Input:

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

Output:

1/4*b^2*e^2*n^2*x^4*log(x)^2 - 1/8*b^2*e^2*n^2*x^4*log(x) + 1/2*b^2*e^2*n* 
x^4*log(c)*log(x) + 2/3*b^2*d*e*n^2*x^3*log(x)^2 + 1/32*b^2*e^2*n^2*x^4 - 
1/8*b^2*e^2*n*x^4*log(c) + 1/4*b^2*e^2*x^4*log(c)^2 - 4/9*b^2*d*e*n^2*x^3* 
log(x) + 1/2*a*b*e^2*n*x^4*log(x) + 4/3*b^2*d*e*n*x^3*log(c)*log(x) + 1/2* 
b^2*d^2*n^2*x^2*log(x)^2 + 4/27*b^2*d*e*n^2*x^3 - 1/8*a*b*e^2*n*x^4 - 4/9* 
b^2*d*e*n*x^3*log(c) + 1/2*a*b*e^2*x^4*log(c) + 2/3*b^2*d*e*x^3*log(c)^2 - 
 1/2*b^2*d^2*n^2*x^2*log(x) + 4/3*a*b*d*e*n*x^3*log(x) + b^2*d^2*n*x^2*log 
(c)*log(x) + 1/4*b^2*d^2*n^2*x^2 - 4/9*a*b*d*e*n*x^3 + 1/4*a^2*e^2*x^4 - 1 
/2*b^2*d^2*n*x^2*log(c) + 4/3*a*b*d*e*x^3*log(c) + 1/2*b^2*d^2*x^2*log(c)^ 
2 + a*b*d^2*n*x^2*log(x) - 1/2*a*b*d^2*n*x^2 + 2/3*a^2*d*e*x^3 + a*b*d^2*x 
^2*log(c) + 1/2*a^2*d^2*x^2
 

Mupad [B] (verification not implemented)

Time = 26.24 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.01 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\ln \left (c\,x^n\right )\,\left (\frac {b\,\left (2\,a-b\,n\right )\,d^2\,x^2}{2}+\frac {4\,b\,\left (3\,a-b\,n\right )\,d\,e\,x^3}{9}+\frac {b\,\left (4\,a-b\,n\right )\,e^2\,x^4}{8}\right )+{\ln \left (c\,x^n\right )}^2\,\left (\frac {b^2\,d^2\,x^2}{2}+\frac {2\,b^2\,d\,e\,x^3}{3}+\frac {b^2\,e^2\,x^4}{4}\right )+\frac {d^2\,x^2\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{4}+\frac {e^2\,x^4\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{32}+\frac {2\,d\,e\,x^3\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )}{27} \] Input:

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

Output:

log(c*x^n)*((b*d^2*x^2*(2*a - b*n))/2 + (b*e^2*x^4*(4*a - b*n))/8 + (4*b*d 
*e*x^3*(3*a - b*n))/9) + log(c*x^n)^2*((b^2*d^2*x^2)/2 + (b^2*e^2*x^4)/4 + 
 (2*b^2*d*e*x^3)/3) + (d^2*x^2*(2*a^2 + b^2*n^2 - 2*a*b*n))/4 + (e^2*x^4*( 
8*a^2 + b^2*n^2 - 4*a*b*n))/32 + (2*d*e*x^3*(9*a^2 + 2*b^2*n^2 - 6*a*b*n)) 
/27
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.33 \[ \int x (d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {x^{2} \left (432 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} d^{2}+576 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} d e x +216 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} e^{2} x^{2}+864 \,\mathrm {log}\left (x^{n} c \right ) a b \,d^{2}+1152 \,\mathrm {log}\left (x^{n} c \right ) a b d e x +432 \,\mathrm {log}\left (x^{n} c \right ) a b \,e^{2} x^{2}-432 \,\mathrm {log}\left (x^{n} c \right ) b^{2} d^{2} n -384 \,\mathrm {log}\left (x^{n} c \right ) b^{2} d e n x -108 \,\mathrm {log}\left (x^{n} c \right ) b^{2} e^{2} n \,x^{2}+432 a^{2} d^{2}+576 a^{2} d e x +216 a^{2} e^{2} x^{2}-432 a b \,d^{2} n -384 a b d e n x -108 a b \,e^{2} n \,x^{2}+216 b^{2} d^{2} n^{2}+128 b^{2} d e \,n^{2} x +27 b^{2} e^{2} n^{2} x^{2}\right )}{864} \] Input:

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

Output:

(x**2*(432*log(x**n*c)**2*b**2*d**2 + 576*log(x**n*c)**2*b**2*d*e*x + 216* 
log(x**n*c)**2*b**2*e**2*x**2 + 864*log(x**n*c)*a*b*d**2 + 1152*log(x**n*c 
)*a*b*d*e*x + 432*log(x**n*c)*a*b*e**2*x**2 - 432*log(x**n*c)*b**2*d**2*n 
- 384*log(x**n*c)*b**2*d*e*n*x - 108*log(x**n*c)*b**2*e**2*n*x**2 + 432*a* 
*2*d**2 + 576*a**2*d*e*x + 216*a**2*e**2*x**2 - 432*a*b*d**2*n - 384*a*b*d 
*e*n*x - 108*a*b*e**2*n*x**2 + 216*b**2*d**2*n**2 + 128*b**2*d*e*n**2*x + 
27*b**2*e**2*n**2*x**2))/864