\(\int x^2 (a+b \log (c x^n))^3 \log (1+e x) \, dx\) [24]

Optimal result

Integrand size = 22, antiderivative size = 615 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x) \, dx=-\frac {8 a b^2 n^2 x}{3 e^2}+\frac {80 b^3 n^3 x}{27 e^2}-\frac {65 b^3 n^3 x^2}{216 e}+\frac {8}{81} b^3 n^3 x^3-\frac {8 b^3 n^2 x \log \left (c x^n\right )}{3 e^2}-\frac {2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac {19 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{36 e}-\frac {2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac {2 b^3 n^3 \log (1+e x)}{27 e^3}-\frac {2}{27} b^3 n^3 x^3 \log (1+e x)+\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}+\frac {2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}-\frac {1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac {2 b^3 n^3 \operatorname {PolyLog}(2,-e x)}{9 e^3}-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,-e x)}{3 e^3}+\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}(2,-e x)}{e^3}+\frac {2 b^3 n^3 \operatorname {PolyLog}(3,-e x)}{3 e^3}-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,-e x)}{e^3}+\frac {2 b^3 n^3 \operatorname {PolyLog}(4,-e x)}{e^3} \] Output:

b*n*(a+b*ln(c*x^n))^2*polylog(2,-e*x)/e^3-1/9*x^3*(a+b*ln(c*x^n))^3-8/3*a* 
b^2*n^2*x/e^2-8/3*b^3*n^2*x*ln(c*x^n)/e^2-2/9*b^2*n^2*x*(a+b*ln(c*x^n))/e^ 
2+19/36*b^2*n^2*x^2*(a+b*ln(c*x^n))/e+4/3*b*n*x*(a+b*ln(c*x^n))^2/e^2-5/12 
*b*n*x^2*(a+b*ln(c*x^n))^2/e-2*b^2*n^2*(a+b*ln(c*x^n))*polylog(3,-e*x)/e^3 
-2/3*b^2*n^2*(a+b*ln(c*x^n))*polylog(2,-e*x)/e^3+2/9*b^2*n^2*(a+b*ln(c*x^n 
))*ln(e*x+1)/e^3+2/9*b^2*n^2*x^3*(a+b*ln(c*x^n))*ln(e*x+1)-1/3*b*n*(a+b*ln 
(c*x^n))^2*ln(e*x+1)/e^3-1/3*b*n*x^3*(a+b*ln(c*x^n))^2*ln(e*x+1)-65/216*b^ 
3*n^3*x^2/e+2*b^3*n^3*polylog(4,-e*x)/e^3+2/3*b^3*n^3*polylog(3,-e*x)/e^3+ 
2/9*b^3*n^3*polylog(2,-e*x)/e^3-2/27*b^3*n^3*ln(e*x+1)/e^3-2/27*b^3*n^3*x^ 
3*ln(e*x+1)-2/9*b^2*n^2*x^3*(a+b*ln(c*x^n))+2/9*b*n*x^3*(a+b*ln(c*x^n))^2+ 
80/27*b^3*n^3*x/e^2+1/3*x^3*(a+b*ln(c*x^n))^3*ln(e*x+1)+8/81*b^3*n^3*x^3-1 
/3*x*(a+b*ln(c*x^n))^3/e^2+1/6*x^2*(a+b*ln(c*x^n))^3/e+1/3*(a+b*ln(c*x^n)) 
^3*ln(e*x+1)/e^3
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 975, normalized size of antiderivative = 1.59 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x) \, dx =\text {Too large to display} \] Input:

Integrate[x^2*(a + b*Log[c*x^n])^3*Log[1 + e*x],x]
 

Output:

(-216*a^3*e*x + 864*a^2*b*e*n*x - 1872*a*b^2*e*n^2*x + 1920*b^3*e*n^3*x + 
108*a^3*e^2*x^2 - 270*a^2*b*e^2*n*x^2 + 342*a*b^2*e^2*n^2*x^2 - 195*b^3*e^ 
2*n^3*x^2 - 72*a^3*e^3*x^3 + 144*a^2*b*e^3*n*x^3 - 144*a*b^2*e^3*n^2*x^3 + 
 64*b^3*e^3*n^3*x^3 - 648*a^2*b*e*x*Log[c*x^n] + 1728*a*b^2*e*n*x*Log[c*x^ 
n] - 1872*b^3*e*n^2*x*Log[c*x^n] + 324*a^2*b*e^2*x^2*Log[c*x^n] - 540*a*b^ 
2*e^2*n*x^2*Log[c*x^n] + 342*b^3*e^2*n^2*x^2*Log[c*x^n] - 216*a^2*b*e^3*x^ 
3*Log[c*x^n] + 288*a*b^2*e^3*n*x^3*Log[c*x^n] - 144*b^3*e^3*n^2*x^3*Log[c* 
x^n] - 648*a*b^2*e*x*Log[c*x^n]^2 + 864*b^3*e*n*x*Log[c*x^n]^2 + 324*a*b^2 
*e^2*x^2*Log[c*x^n]^2 - 270*b^3*e^2*n*x^2*Log[c*x^n]^2 - 216*a*b^2*e^3*x^3 
*Log[c*x^n]^2 + 144*b^3*e^3*n*x^3*Log[c*x^n]^2 - 216*b^3*e*x*Log[c*x^n]^3 
+ 108*b^3*e^2*x^2*Log[c*x^n]^3 - 72*b^3*e^3*x^3*Log[c*x^n]^3 + 216*a^3*Log 
[1 + e*x] - 216*a^2*b*n*Log[1 + e*x] + 144*a*b^2*n^2*Log[1 + e*x] - 48*b^3 
*n^3*Log[1 + e*x] + 216*a^3*e^3*x^3*Log[1 + e*x] - 216*a^2*b*e^3*n*x^3*Log 
[1 + e*x] + 144*a*b^2*e^3*n^2*x^3*Log[1 + e*x] - 48*b^3*e^3*n^3*x^3*Log[1 
+ e*x] + 648*a^2*b*Log[c*x^n]*Log[1 + e*x] - 432*a*b^2*n*Log[c*x^n]*Log[1 
+ e*x] + 144*b^3*n^2*Log[c*x^n]*Log[1 + e*x] + 648*a^2*b*e^3*x^3*Log[c*x^n 
]*Log[1 + e*x] - 432*a*b^2*e^3*n*x^3*Log[c*x^n]*Log[1 + e*x] + 144*b^3*e^3 
*n^2*x^3*Log[c*x^n]*Log[1 + e*x] + 648*a*b^2*Log[c*x^n]^2*Log[1 + e*x] - 2 
16*b^3*n*Log[c*x^n]^2*Log[1 + e*x] + 648*a*b^2*e^3*x^3*Log[c*x^n]^2*Log[1 
+ e*x] - 216*b^3*e^3*n*x^3*Log[c*x^n]^2*Log[1 + e*x] + 216*b^3*Log[c*x^...
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 581, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

\(\displaystyle \int x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3 \, dx\)

\(\Big \downarrow \) 2824

\(\displaystyle -3 b n \int \left (-\frac {1}{9} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{6 e}+\frac {1}{3} x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}\right )dx+\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{3 e^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac {1}{3} x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3\)

\(\Big \downarrow \) 2009

\(\displaystyle -3 b n \left (-\frac {\operatorname {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3}+\frac {2 b n \operatorname {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}+\frac {2 b n \operatorname {PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{9 e^3}-\frac {2 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{27 e^3}-\frac {4 x \left (a+b \log \left (c x^n\right )\right )^2}{9 e^2}+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{27 e^2}+\frac {1}{9} x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2-\frac {2}{27} b n x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac {5 x^2 \left (a+b \log \left (c x^n\right )\right )^2}{36 e}-\frac {19 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{108 e}-\frac {2}{27} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {8 a b n x}{9 e^2}+\frac {8 b^2 n x \log \left (c x^n\right )}{9 e^2}-\frac {2 b^2 n^2 \operatorname {PolyLog}(2,-e x)}{27 e^3}-\frac {2 b^2 n^2 \operatorname {PolyLog}(3,-e x)}{9 e^3}-\frac {2 b^2 n^2 \operatorname {PolyLog}(4,-e x)}{3 e^3}+\frac {2 b^2 n^2 \log (e x+1)}{81 e^3}-\frac {80 b^2 n^2 x}{81 e^2}+\frac {2}{81} b^2 n^2 x^3 \log (e x+1)+\frac {65 b^2 n^2 x^2}{648 e}-\frac {8}{243} b^2 n^2 x^3\right )+\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{3 e^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac {1}{3} x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3\)

Input:

Int[x^2*(a + b*Log[c*x^n])^3*Log[1 + e*x],x]
 

Output:

-1/3*(x*(a + b*Log[c*x^n])^3)/e^2 + (x^2*(a + b*Log[c*x^n])^3)/(6*e) - (x^ 
3*(a + b*Log[c*x^n])^3)/9 + ((a + b*Log[c*x^n])^3*Log[1 + e*x])/(3*e^3) + 
(x^3*(a + b*Log[c*x^n])^3*Log[1 + e*x])/3 - 3*b*n*((8*a*b*n*x)/(9*e^2) - ( 
80*b^2*n^2*x)/(81*e^2) + (65*b^2*n^2*x^2)/(648*e) - (8*b^2*n^2*x^3)/243 + 
(8*b^2*n*x*Log[c*x^n])/(9*e^2) + (2*b*n*x*(a + b*Log[c*x^n]))/(27*e^2) - ( 
19*b*n*x^2*(a + b*Log[c*x^n]))/(108*e) + (2*b*n*x^3*(a + b*Log[c*x^n]))/27 
 - (4*x*(a + b*Log[c*x^n])^2)/(9*e^2) + (5*x^2*(a + b*Log[c*x^n])^2)/(36*e 
) - (2*x^3*(a + b*Log[c*x^n])^2)/27 + (2*b^2*n^2*Log[1 + e*x])/(81*e^3) + 
(2*b^2*n^2*x^3*Log[1 + e*x])/81 - (2*b*n*(a + b*Log[c*x^n])*Log[1 + e*x])/ 
(27*e^3) - (2*b*n*x^3*(a + b*Log[c*x^n])*Log[1 + e*x])/27 + ((a + b*Log[c* 
x^n])^2*Log[1 + e*x])/(9*e^3) + (x^3*(a + b*Log[c*x^n])^2*Log[1 + e*x])/9 
- (2*b^2*n^2*PolyLog[2, -(e*x)])/(27*e^3) + (2*b*n*(a + b*Log[c*x^n])*Poly 
Log[2, -(e*x)])/(9*e^3) - ((a + b*Log[c*x^n])^2*PolyLog[2, -(e*x)])/(3*e^3 
) - (2*b^2*n^2*PolyLog[3, -(e*x)])/(9*e^3) + (2*b*n*(a + b*Log[c*x^n])*Pol 
yLog[3, -(e*x)])/(3*e^3) - (2*b^2*n^2*PolyLog[4, -(e*x)])/(3*e^3))
 

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]

\[\int x^{2} {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (e x +1\right )d x\]

Input:

int(x^2*(a+b*ln(c*x^n))^3*ln(e*x+1),x)
 

Output:

int(x^2*(a+b*ln(c*x^n))^3*ln(e*x+1),x)
 

Fricas [F]

\[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{2} \log \left (e x + 1\right ) \,d x } \] Input:

integrate(x^2*(a+b*log(c*x^n))^3*log(e*x+1),x, algorithm="fricas")
 

Output:

integral(b^3*x^2*log(c*x^n)^3*log(e*x + 1) + 3*a*b^2*x^2*log(c*x^n)^2*log( 
e*x + 1) + 3*a^2*b*x^2*log(c*x^n)*log(e*x + 1) + a^3*x^2*log(e*x + 1), x)
 

Sympy [F(-1)]

Timed out. \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x) \, dx=\text {Timed out} \] Input:

integrate(x**2*(a+b*ln(c*x**n))**3*ln(e*x+1),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{2} \log \left (e x + 1\right ) \,d x } \] Input:

integrate(x^2*(a+b*log(c*x^n))^3*log(e*x+1),x, algorithm="maxima")
 

Output:

-1/18*(2*b^3*e^3*x^3 - 3*b^3*e^2*x^2 + 6*b^3*e*x - 6*(b^3*e^3*x^3 + b^3)*l 
og(e*x + 1))*log(x^n)^3/e^3 + 1/6*integrate((18*(b^3*e^3*log(c)^2 + 2*a*b^ 
2*e^3*log(c) + a^2*b*e^3)*x^3*log(e*x + 1)*log(x^n) + 6*(b^3*e^3*log(c)^3 
+ 3*a*b^2*e^3*log(c)^2 + 3*a^2*b*e^3*log(c) + a^3*e^3)*x^3*log(e*x + 1) + 
(2*b^3*e^3*n*x^3 - 3*b^3*e^2*n*x^2 + 6*b^3*e*n*x - 6*(b^3*n - (3*a*b^2*e^3 
 - (e^3*n - 3*e^3*log(c))*b^3)*x^3)*log(e*x + 1))*log(x^n)^2)/x, x)/e^3
 

Giac [F]

\[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{2} \log \left (e x + 1\right ) \,d x } \] Input:

integrate(x^2*(a+b*log(c*x^n))^3*log(e*x+1),x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)^3*x^2*log(e*x + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x) \, dx=\int x^2\,\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \] Input:

int(x^2*log(e*x + 1)*(a + b*log(c*x^n))^3,x)
 

Output:

int(x^2*log(e*x + 1)*(a + b*log(c*x^n))^3, x)
 

Reduce [F]

\[ \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x) \, dx=\int x^{2} {\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{3} \mathrm {log}\left (e x +1\right )d x \] Input:

int(x^2*(a+b*log(c*x^n))^3*log(e*x+1),x)
 

Output:

int(x^2*(a+b*log(c*x^n))^3*log(e*x+1),x)