\(\int x^2 (a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^n)) \, dx\) [490]

Optimal result

Integrand size = 22, antiderivative size = 190 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \, dx=-\frac {b e^8 n \sqrt [3]{x}}{3 d^8}+\frac {b e^7 n x^{2/3}}{6 d^7}-\frac {b e^6 n x}{9 d^6}+\frac {b e^5 n x^{4/3}}{12 d^5}-\frac {b e^4 n x^{5/3}}{15 d^4}+\frac {b e^3 n x^2}{18 d^3}-\frac {b e^2 n x^{7/3}}{21 d^2}+\frac {b e n x^{8/3}}{24 d}+\frac {b e^9 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^9}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )+\frac {b e^9 n \log (x)}{9 d^9} \] Output:

-1/3*b*e^8*n*x^(1/3)/d^8+1/6*b*e^7*n*x^(2/3)/d^7-1/9*b*e^6*n*x/d^6+1/12*b* 
e^5*n*x^(4/3)/d^5-1/15*b*e^4*n*x^(5/3)/d^4+1/18*b*e^3*n*x^2/d^3-1/21*b*e^2 
*n*x^(7/3)/d^2+1/24*b*e*n*x^(8/3)/d+1/3*b*e^9*n*ln(d+e/x^(1/3))/d^9+1/3*x^ 
3*(a+b*ln(c*(d+e/x^(1/3))^n))+1/9*b*e^9*n*ln(x)/d^9
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.90 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \, dx=\frac {a x^3}{3}+\frac {1}{3} b x^3 \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+\frac {1}{9} b e n \left (-\frac {3 e^7 \sqrt [3]{x}}{d^8}+\frac {3 e^6 x^{2/3}}{2 d^7}-\frac {e^5 x}{d^6}+\frac {3 e^4 x^{4/3}}{4 d^5}-\frac {3 e^3 x^{5/3}}{5 d^4}+\frac {e^2 x^2}{2 d^3}-\frac {3 e x^{7/3}}{7 d^2}+\frac {3 x^{8/3}}{8 d}+\frac {3 e^8 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^9}+\frac {e^8 \log (x)}{d^9}\right ) \] Input:

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

Output:

(a*x^3)/3 + (b*x^3*Log[c*(d + e/x^(1/3))^n])/3 + (b*e*n*((-3*e^7*x^(1/3))/ 
d^8 + (3*e^6*x^(2/3))/(2*d^7) - (e^5*x)/d^6 + (3*e^4*x^(4/3))/(4*d^5) - (3 
*e^3*x^(5/3))/(5*d^4) + (e^2*x^2)/(2*d^3) - (3*e*x^(7/3))/(7*d^2) + (3*x^( 
8/3))/(8*d) + (3*e^8*Log[d + e/x^(1/3)])/d^9 + (e^8*Log[x])/d^9))/9
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2904, 2842, 54, 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.

\(\displaystyle \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \, dx\)

\(\Big \downarrow \) 2904

\(\displaystyle -3 \int x^{10/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )d\frac {1}{\sqrt [3]{x}}\)

\(\Big \downarrow \) 2842

\(\displaystyle -3 \left (\frac {1}{9} b e n \int \frac {x^3}{d+\frac {e}{\sqrt [3]{x}}}d\frac {1}{\sqrt [3]{x}}-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )\right )\)

\(\Big \downarrow \) 54

\(\displaystyle -3 \left (\frac {1}{9} b e n \int \left (-\frac {e^9}{d^9 \left (d+\frac {e}{\sqrt [3]{x}}\right )}+\frac {\sqrt [3]{x} e^8}{d^9}-\frac {x^{2/3} e^7}{d^8}+\frac {x e^6}{d^7}-\frac {x^{4/3} e^5}{d^6}+\frac {x^{5/3} e^4}{d^5}-\frac {x^2 e^3}{d^4}+\frac {x^{7/3} e^2}{d^3}-\frac {x^{8/3} e}{d^2}+\frac {x^3}{d}\right )d\frac {1}{\sqrt [3]{x}}-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -3 \left (\frac {1}{9} b e n \left (-\frac {e^8 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^9}+\frac {e^8 \log \left (\frac {1}{\sqrt [3]{x}}\right )}{d^9}+\frac {e^7 \sqrt [3]{x}}{d^8}-\frac {e^6 x^{2/3}}{2 d^7}+\frac {e^5 x}{3 d^6}-\frac {e^4 x^{4/3}}{4 d^5}+\frac {e^3 x^{5/3}}{5 d^4}-\frac {e^2 x^2}{6 d^3}+\frac {e x^{7/3}}{7 d^2}-\frac {x^{8/3}}{8 d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )\right )\)

Input:

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

Output:

-3*(-1/9*(x^3*(a + b*Log[c*(d + e/x^(1/3))^n])) + (b*e*n*((e^7*x^(1/3))/d^ 
8 - (e^6*x^(2/3))/(2*d^7) + (e^5*x)/(3*d^6) - (e^4*x^(4/3))/(4*d^5) + (e^3 
*x^(5/3))/(5*d^4) - (e^2*x^2)/(6*d^3) + (e*x^(7/3))/(7*d^2) - x^(8/3)/(8*d 
) - (e^8*Log[d + e/x^(1/3)])/d^9 + (e^8*Log[x^(-1/3)])/d^9))/9)
 

Defintions of rubi rules used

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 2009

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

rule 2842

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_ 
))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/( 
g*(q + 1))), x] - Simp[b*e*(n/(g*(q + 1)))   Int[(f + g*x)^(q + 1)/(d + e*x 
), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && 
NeQ[q, -1]
 

rule 2904

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L 
og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, 
 x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & 
&  !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.01 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \, dx=\frac {840 \, b d^{9} x^{3} \log \left (c\right ) + 140 \, b d^{6} e^{3} n x^{2} + 840 \, a d^{9} x^{3} - 280 \, b d^{3} e^{6} n x - 840 \, b d^{9} n \log \left (x^{\frac {1}{3}}\right ) + 840 \, {\left (b d^{9} + b e^{9}\right )} n \log \left (d x^{\frac {1}{3}} + e\right ) + 840 \, {\left (b d^{9} n x^{3} - b d^{9} n\right )} \log \left (\frac {d x + e x^{\frac {2}{3}}}{x}\right ) + 21 \, {\left (5 \, b d^{8} e n x^{2} - 8 \, b d^{5} e^{4} n x + 20 \, b d^{2} e^{7} n\right )} x^{\frac {2}{3}} - 30 \, {\left (4 \, b d^{7} e^{2} n x^{2} - 7 \, b d^{4} e^{5} n x + 28 \, b d e^{8} n\right )} x^{\frac {1}{3}}}{2520 \, d^{9}} \] Input:

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

Output:

1/2520*(840*b*d^9*x^3*log(c) + 140*b*d^6*e^3*n*x^2 + 840*a*d^9*x^3 - 280*b 
*d^3*e^6*n*x - 840*b*d^9*n*log(x^(1/3)) + 840*(b*d^9 + b*e^9)*n*log(d*x^(1 
/3) + e) + 840*(b*d^9*n*x^3 - b*d^9*n)*log((d*x + e*x^(2/3))/x) + 21*(5*b* 
d^8*e*n*x^2 - 8*b*d^5*e^4*n*x + 20*b*d^2*e^7*n)*x^(2/3) - 30*(4*b*d^7*e^2* 
n*x^2 - 7*b*d^4*e^5*n*x + 28*b*d*e^8*n)*x^(1/3))/d^9
 

Sympy [A] (verification not implemented)

Time = 46.21 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.85 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \, dx=\frac {a x^{3}}{3} + b \left (\frac {e n \left (\frac {3 x^{\frac {8}{3}}}{8 d} - \frac {3 e x^{\frac {7}{3}}}{7 d^{2}} + \frac {e^{2} x^{2}}{2 d^{3}} - \frac {3 e^{3} x^{\frac {5}{3}}}{5 d^{4}} + \frac {3 e^{4} x^{\frac {4}{3}}}{4 d^{5}} - \frac {e^{5} x}{d^{6}} + \frac {3 e^{6} x^{\frac {2}{3}}}{2 d^{7}} + \frac {3 e^{8} \left (\begin {cases} \frac {\sqrt [3]{x}}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d \sqrt [3]{x} + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{8}} - \frac {3 e^{7} \sqrt [3]{x}}{d^{8}}\right )}{9} + \frac {x^{3} \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}}{3}\right ) \] Input:

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

Output:

a*x**3/3 + b*(e*n*(3*x**(8/3)/(8*d) - 3*e*x**(7/3)/(7*d**2) + e**2*x**2/(2 
*d**3) - 3*e**3*x**(5/3)/(5*d**4) + 3*e**4*x**(4/3)/(4*d**5) - e**5*x/d**6 
 + 3*e**6*x**(2/3)/(2*d**7) + 3*e**8*Piecewise((x**(1/3)/e, Eq(d, 0)), (lo 
g(d*x**(1/3) + e)/d, True))/d**8 - 3*e**7*x**(1/3)/d**8)/9 + x**3*log(c*(d 
 + e/x**(1/3))**n)/3)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.67 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \, dx=\frac {1}{3} \, b x^{3} \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + \frac {1}{3} \, a x^{3} + \frac {1}{2520} \, b e n {\left (\frac {840 \, e^{8} \log \left (d x^{\frac {1}{3}} + e\right )}{d^{9}} + \frac {105 \, d^{7} x^{\frac {8}{3}} - 120 \, d^{6} e x^{\frac {7}{3}} + 140 \, d^{5} e^{2} x^{2} - 168 \, d^{4} e^{3} x^{\frac {5}{3}} + 210 \, d^{3} e^{4} x^{\frac {4}{3}} - 280 \, d^{2} e^{5} x + 420 \, d e^{6} x^{\frac {2}{3}} - 840 \, e^{7} x^{\frac {1}{3}}}{d^{8}}\right )} \] Input:

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

Output:

1/3*b*x^3*log(c*(d + e/x^(1/3))^n) + 1/3*a*x^3 + 1/2520*b*e*n*(840*e^8*log 
(d*x^(1/3) + e)/d^9 + (105*d^7*x^(8/3) - 120*d^6*e*x^(7/3) + 140*d^5*e^2*x 
^2 - 168*d^4*e^3*x^(5/3) + 210*d^3*e^4*x^(4/3) - 280*d^2*e^5*x + 420*d*e^6 
*x^(2/3) - 840*e^7*x^(1/3))/d^8)
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.71 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \, dx=\frac {1}{3} \, b x^{3} \log \left (c\right ) + \frac {1}{3} \, a x^{3} + \frac {1}{2520} \, {\left (840 \, x^{3} \log \left (d + \frac {e}{x^{\frac {1}{3}}}\right ) + e {\left (\frac {840 \, e^{8} \log \left ({\left | d x^{\frac {1}{3}} + e \right |}\right )}{d^{9}} + \frac {105 \, d^{7} x^{\frac {8}{3}} - 120 \, d^{6} e x^{\frac {7}{3}} + 140 \, d^{5} e^{2} x^{2} - 168 \, d^{4} e^{3} x^{\frac {5}{3}} + 210 \, d^{3} e^{4} x^{\frac {4}{3}} - 280 \, d^{2} e^{5} x + 420 \, d e^{6} x^{\frac {2}{3}} - 840 \, e^{7} x^{\frac {1}{3}}}{d^{8}}\right )}\right )} b n \] Input:

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

Output:

1/3*b*x^3*log(c) + 1/3*a*x^3 + 1/2520*(840*x^3*log(d + e/x^(1/3)) + e*(840 
*e^8*log(abs(d*x^(1/3) + e))/d^9 + (105*d^7*x^(8/3) - 120*d^6*e*x^(7/3) + 
140*d^5*e^2*x^2 - 168*d^4*e^3*x^(5/3) + 210*d^3*e^4*x^(4/3) - 280*d^2*e^5* 
x + 420*d*e^6*x^(2/3) - 840*e^7*x^(1/3))/d^8))*b*n
 

Mupad [B] (verification not implemented)

Time = 25.90 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.81 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \, dx=\frac {840\,a\,d^9\,x^3+1680\,b\,e^9\,n\,\mathrm {atanh}\left (\frac {2\,e}{d\,x^{1/3}}+1\right )+840\,b\,d^9\,x^3\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )-280\,b\,d^3\,e^6\,n\,x-840\,b\,d\,e^8\,n\,x^{1/3}+105\,b\,d^8\,e\,n\,x^{8/3}+140\,b\,d^6\,e^3\,n\,x^2+420\,b\,d^2\,e^7\,n\,x^{2/3}+210\,b\,d^4\,e^5\,n\,x^{4/3}-168\,b\,d^5\,e^4\,n\,x^{5/3}-120\,b\,d^7\,e^2\,n\,x^{7/3}}{2520\,d^9} \] Input:

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

Output:

(840*a*d^9*x^3 + 1680*b*e^9*n*atanh((2*e)/(d*x^(1/3)) + 1) + 840*b*d^9*x^3 
*log(c*(d + e/x^(1/3))^n) - 280*b*d^3*e^6*n*x - 840*b*d*e^8*n*x^(1/3) + 10 
5*b*d^8*e*n*x^(8/3) + 140*b*d^6*e^3*n*x^2 + 420*b*d^2*e^7*n*x^(2/3) + 210* 
b*d^4*e^5*n*x^(4/3) - 168*b*d^5*e^4*n*x^(5/3) - 120*b*d^7*e^2*n*x^(7/3))/( 
2520*d^9)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.93 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \, dx=\frac {105 x^{\frac {8}{3}} b \,d^{8} e n -168 x^{\frac {5}{3}} b \,d^{5} e^{4} n +420 x^{\frac {2}{3}} b \,d^{2} e^{7} n -120 x^{\frac {7}{3}} b \,d^{7} e^{2} n +210 x^{\frac {4}{3}} b \,d^{4} e^{5} n -840 x^{\frac {1}{3}} b d \,e^{8} n +840 \,\mathrm {log}\left (x^{\frac {1}{3}}\right ) b \,e^{9} n +840 \,\mathrm {log}\left (\frac {\left (x^{\frac {1}{3}} d +e \right )^{n} c}{x^{\frac {n}{3}}}\right ) b \,d^{9} x^{3}+840 \,\mathrm {log}\left (\frac {\left (x^{\frac {1}{3}} d +e \right )^{n} c}{x^{\frac {n}{3}}}\right ) b \,e^{9}+840 a \,d^{9} x^{3}+140 b \,d^{6} e^{3} n \,x^{2}-280 b \,d^{3} e^{6} n x}{2520 d^{9}} \] Input:

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

Output:

(105*x**(2/3)*b*d**8*e*n*x**2 - 168*x**(2/3)*b*d**5*e**4*n*x + 420*x**(2/3 
)*b*d**2*e**7*n - 120*x**(1/3)*b*d**7*e**2*n*x**2 + 210*x**(1/3)*b*d**4*e* 
*5*n*x - 840*x**(1/3)*b*d*e**8*n + 840*log(x**(1/3))*b*e**9*n + 840*log((( 
x**(1/3)*d + e)**n*c)/x**(n/3))*b*d**9*x**3 + 840*log(((x**(1/3)*d + e)**n 
*c)/x**(n/3))*b*e**9 + 840*a*d**9*x**3 + 140*b*d**6*e**3*n*x**2 - 280*b*d* 
*3*e**6*n*x)/(2520*d**9)