\(\int x (a+b \log (c x^n))^2 \log (1+e x) \, dx\) [18]

Optimal result

Integrand size = 20, antiderivative size = 327 \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=-\frac {a b n x}{e}+\frac {7 b^2 n^2 x}{4 e}-\frac {3}{8} b^2 n^2 x^2-\frac {b^2 n x \log \left (c x^n\right )}{e}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {b^2 n^2 \log (1+e x)}{4 e^2}+\frac {1}{4} b^2 n^2 x^2 \log (1+e x)+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {b^2 n^2 \operatorname {PolyLog}(2,-e x)}{2 e^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,-e x)}{e^2}+\frac {b^2 n^2 \operatorname {PolyLog}(3,-e x)}{e^2} \] Output:

-a*b*n*x/e+7/4*b^2*n^2*x/e-3/8*b^2*n^2*x^2-b^2*n*x*ln(c*x^n)/e-1/2*b*n*x*( 
a+b*ln(c*x^n))/e+1/2*b*n*x^2*(a+b*ln(c*x^n))+1/2*x*(a+b*ln(c*x^n))^2/e-1/4 
*x^2*(a+b*ln(c*x^n))^2-1/4*b^2*n^2*ln(e*x+1)/e^2+1/4*b^2*n^2*x^2*ln(e*x+1) 
+1/2*b*n*(a+b*ln(c*x^n))*ln(e*x+1)/e^2-1/2*b*n*x^2*(a+b*ln(c*x^n))*ln(e*x+ 
1)-1/2*(a+b*ln(c*x^n))^2*ln(e*x+1)/e^2+1/2*x^2*(a+b*ln(c*x^n))^2*ln(e*x+1) 
+1/2*b^2*n^2*polylog(2,-e*x)/e^2-b*n*(a+b*ln(c*x^n))*polylog(2,-e*x)/e^2+b 
^2*n^2*polylog(3,-e*x)/e^2
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.27 \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=\frac {4 a^2 e x-12 a b e n x+14 b^2 e n^2 x-2 a^2 e^2 x^2+4 a b e^2 n x^2-3 b^2 e^2 n^2 x^2+8 a b e x \log \left (c x^n\right )-12 b^2 e n x \log \left (c x^n\right )-4 a b e^2 x^2 \log \left (c x^n\right )+4 b^2 e^2 n x^2 \log \left (c x^n\right )+4 b^2 e x \log ^2\left (c x^n\right )-2 b^2 e^2 x^2 \log ^2\left (c x^n\right )-4 a^2 \log (1+e x)+4 a b n \log (1+e x)-2 b^2 n^2 \log (1+e x)+4 a^2 e^2 x^2 \log (1+e x)-4 a b e^2 n x^2 \log (1+e x)+2 b^2 e^2 n^2 x^2 \log (1+e x)-8 a b \log \left (c x^n\right ) \log (1+e x)+4 b^2 n \log \left (c x^n\right ) \log (1+e x)+8 a b e^2 x^2 \log \left (c x^n\right ) \log (1+e x)-4 b^2 e^2 n x^2 \log \left (c x^n\right ) \log (1+e x)-4 b^2 \log ^2\left (c x^n\right ) \log (1+e x)+4 b^2 e^2 x^2 \log ^2\left (c x^n\right ) \log (1+e x)+4 b n \left (-2 a+b n-2 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,-e x)+8 b^2 n^2 \operatorname {PolyLog}(3,-e x)}{8 e^2} \] Input:

Integrate[x*(a + b*Log[c*x^n])^2*Log[1 + e*x],x]
 

Output:

(4*a^2*e*x - 12*a*b*e*n*x + 14*b^2*e*n^2*x - 2*a^2*e^2*x^2 + 4*a*b*e^2*n*x 
^2 - 3*b^2*e^2*n^2*x^2 + 8*a*b*e*x*Log[c*x^n] - 12*b^2*e*n*x*Log[c*x^n] - 
4*a*b*e^2*x^2*Log[c*x^n] + 4*b^2*e^2*n*x^2*Log[c*x^n] + 4*b^2*e*x*Log[c*x^ 
n]^2 - 2*b^2*e^2*x^2*Log[c*x^n]^2 - 4*a^2*Log[1 + e*x] + 4*a*b*n*Log[1 + e 
*x] - 2*b^2*n^2*Log[1 + e*x] + 4*a^2*e^2*x^2*Log[1 + e*x] - 4*a*b*e^2*n*x^ 
2*Log[1 + e*x] + 2*b^2*e^2*n^2*x^2*Log[1 + e*x] - 8*a*b*Log[c*x^n]*Log[1 + 
 e*x] + 4*b^2*n*Log[c*x^n]*Log[1 + e*x] + 8*a*b*e^2*x^2*Log[c*x^n]*Log[1 + 
 e*x] - 4*b^2*e^2*n*x^2*Log[c*x^n]*Log[1 + e*x] - 4*b^2*Log[c*x^n]^2*Log[1 
 + e*x] + 4*b^2*e^2*x^2*Log[c*x^n]^2*Log[1 + e*x] + 4*b*n*(-2*a + b*n - 2* 
b*Log[c*x^n])*PolyLog[2, -(e*x)] + 8*b^2*n^2*PolyLog[3, -(e*x)])/(8*e^2)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2824, 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.

\(\displaystyle \int x \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2 \, dx\)

\(\Big \downarrow \) 2824

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

\(\Big \downarrow \) 2009

\(\displaystyle -2 b n \left (\frac {\operatorname {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 e}+\frac {1}{4} x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {a x}{2 e}+\frac {b x \log \left (c x^n\right )}{2 e}-\frac {b n \operatorname {PolyLog}(2,-e x)}{4 e^2}-\frac {b n \operatorname {PolyLog}(3,-e x)}{2 e^2}+\frac {b n \log (e x+1)}{8 e^2}-\frac {1}{8} b n x^2 \log (e x+1)-\frac {7 b n x}{8 e}+\frac {3}{16} b n x^2\right )-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}+\frac {1}{2} x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2\)

Input:

Int[x*(a + b*Log[c*x^n])^2*Log[1 + e*x],x]
 

Output:

(x*(a + b*Log[c*x^n])^2)/(2*e) - (x^2*(a + b*Log[c*x^n])^2)/4 - ((a + b*Lo 
g[c*x^n])^2*Log[1 + e*x])/(2*e^2) + (x^2*(a + b*Log[c*x^n])^2*Log[1 + e*x] 
)/2 - 2*b*n*((a*x)/(2*e) - (7*b*n*x)/(8*e) + (3*b*n*x^2)/16 + (b*x*Log[c*x 
^n])/(2*e) + (x*(a + b*Log[c*x^n]))/(4*e) - (x^2*(a + b*Log[c*x^n]))/4 + ( 
b*n*Log[1 + e*x])/(8*e^2) - (b*n*x^2*Log[1 + e*x])/8 - ((a + b*Log[c*x^n]) 
*Log[1 + e*x])/(4*e^2) + (x^2*(a + b*Log[c*x^n])*Log[1 + e*x])/4 - (b*n*Po 
lyLog[2, -(e*x)])/(4*e^2) + ((a + b*Log[c*x^n])*PolyLog[2, -(e*x)])/(2*e^2 
) - (b*n*PolyLog[3, -(e*x)])/(2*e^2))
 

Defintions of rubi rules used

rule 2009

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

rule 2824

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_ 
.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* 
(e + f*x^m)], x]}, Simp[(a + b*Log[c*x^n])^p   u, x] - Simp[b*n*p   Int[(a 
+ b*Log[c*x^n])^(p - 1)/x   u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, 
 q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ 
[p, 1] || (FractionQ[m] && IntegerQ[(q + 1)/m]) || (IGtQ[q, 0] && IntegerQ[ 
(q + 1)/m] && EqQ[d*e, 1]))
 

Maple [F]

\[\int x {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (e x +1\right )d x\]

Input:

int(x*(a+b*ln(c*x^n))^2*ln(e*x+1),x)
 

Output:

int(x*(a+b*ln(c*x^n))^2*ln(e*x+1),x)
 

Fricas [F]

\[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left (e x + 1\right ) \,d x } \] Input:

integrate(x*(a+b*log(c*x^n))^2*log(e*x+1),x, algorithm="fricas")
 

Output:

integral(b^2*x*log(c*x^n)^2*log(e*x + 1) + 2*a*b*x*log(c*x^n)*log(e*x + 1) 
 + a^2*x*log(e*x + 1), x)
 

Sympy [F(-1)]

Timed out. \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=\text {Timed out} \] Input:

integrate(x*(a+b*ln(c*x**n))**2*ln(e*x+1),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left (e x + 1\right ) \,d x } \] Input:

integrate(x*(a+b*log(c*x^n))^2*log(e*x+1),x, algorithm="maxima")
 

Output:

-1/4*(b^2*e^2*x^2 - 2*b^2*e*x - 2*(b^2*e^2*x^2 - b^2)*log(e*x + 1))*log(x^ 
n)^2/e^2 + 1/2*integrate((2*(b^2*e^2*log(c)^2 + 2*a*b*e^2*log(c) + a^2*e^2 
)*x^2*log(e*x + 1) + (b^2*e^2*n*x^2 - 2*b^2*e*n*x + 2*(b^2*n + (2*a*b*e^2 
- (e^2*n - 2*e^2*log(c))*b^2)*x^2)*log(e*x + 1))*log(x^n))/x, x)/e^2
 

Giac [F]

\[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left (e x + 1\right ) \,d x } \] Input:

integrate(x*(a+b*log(c*x^n))^2*log(e*x+1),x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)^2*x*log(e*x + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=\int x\,\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \] Input:

int(x*log(e*x + 1)*(a + b*log(c*x^n))^2,x)
 

Output:

int(x*log(e*x + 1)*(a + b*log(c*x^n))^2, x)
 

Reduce [F]

\[ \int x \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx=\frac {12 \,\mathrm {log}\left (e x +1\right ) \mathrm {log}\left (x^{n} c \right )^{2} b^{2} e^{2} n \,x^{2}-12 \,\mathrm {log}\left (e x +1\right ) \mathrm {log}\left (x^{n} c \right ) b^{2} e^{2} n^{2} x^{2}-12 \,\mathrm {log}\left (e x +1\right ) a b \,e^{2} n^{2} x^{2}+12 a b \,e^{2} n^{2} x^{2}-12 \,\mathrm {log}\left (x^{n} c \right ) a b \,e^{2} n \,x^{2}+24 \,\mathrm {log}\left (x^{n} c \right ) a b e n x +12 \,\mathrm {log}\left (e x +1\right ) a b \,n^{2}+12 a^{2} e n x +42 b^{2} e \,n^{3} x +12 \,\mathrm {log}\left (e x +1\right ) a^{2} e^{2} n \,x^{2}+6 \,\mathrm {log}\left (e x +1\right ) b^{2} e^{2} n^{3} x^{2}+12 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e \,x^{2}+x}d x \right ) b^{2} n -12 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e \,x^{2}+x}d x \right ) b^{2} n^{2}+24 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e \,x^{2}+x}d x \right ) a b n +24 \,\mathrm {log}\left (e x +1\right ) \mathrm {log}\left (x^{n} c \right ) a b \,e^{2} n \,x^{2}-6 a^{2} e^{2} n \,x^{2}-9 b^{2} e^{2} n^{3} x^{2}-12 \,\mathrm {log}\left (e x +1\right ) a^{2} n -6 \,\mathrm {log}\left (e x +1\right ) b^{2} n^{3}-12 \mathrm {log}\left (x^{n} c \right )^{2} a b +6 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} n -4 \mathrm {log}\left (x^{n} c \right )^{3} b^{2}+12 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} e n x -36 \,\mathrm {log}\left (x^{n} c \right ) b^{2} e \,n^{2} x -36 a b e \,n^{2} x -6 \mathrm {log}\left (x^{n} c \right )^{2} b^{2} e^{2} n \,x^{2}+12 \,\mathrm {log}\left (x^{n} c \right ) b^{2} e^{2} n^{2} x^{2}}{24 e^{2} n} \] Input:

int(x*(a+b*log(c*x^n))^2*log(e*x+1),x)
 

Output:

(12*int(log(x**n*c)**2/(e*x**2 + x),x)*b**2*n + 24*int(log(x**n*c)/(e*x**2 
 + x),x)*a*b*n - 12*int(log(x**n*c)/(e*x**2 + x),x)*b**2*n**2 + 12*log(e*x 
 + 1)*log(x**n*c)**2*b**2*e**2*n*x**2 + 24*log(e*x + 1)*log(x**n*c)*a*b*e* 
*2*n*x**2 - 12*log(e*x + 1)*log(x**n*c)*b**2*e**2*n**2*x**2 + 12*log(e*x + 
 1)*a**2*e**2*n*x**2 - 12*log(e*x + 1)*a**2*n - 12*log(e*x + 1)*a*b*e**2*n 
**2*x**2 + 12*log(e*x + 1)*a*b*n**2 + 6*log(e*x + 1)*b**2*e**2*n**3*x**2 - 
 6*log(e*x + 1)*b**2*n**3 - 4*log(x**n*c)**3*b**2 - 12*log(x**n*c)**2*a*b 
- 6*log(x**n*c)**2*b**2*e**2*n*x**2 + 12*log(x**n*c)**2*b**2*e*n*x + 6*log 
(x**n*c)**2*b**2*n - 12*log(x**n*c)*a*b*e**2*n*x**2 + 24*log(x**n*c)*a*b*e 
*n*x + 12*log(x**n*c)*b**2*e**2*n**2*x**2 - 36*log(x**n*c)*b**2*e*n**2*x - 
 6*a**2*e**2*n*x**2 + 12*a**2*e*n*x + 12*a*b*e**2*n**2*x**2 - 36*a*b*e*n** 
2*x - 9*b**2*e**2*n**3*x**2 + 42*b**2*e*n**3*x)/(24*e**2*n)