\(\int \frac {(a+b \log (c (d+e x^{2/3})^n))^2}{x^3} \, dx\) [474]

Optimal result
Mathematica [A] (verified)
Rubi [A] (warning: unable to verify)
Maple [F]
Fricas [F]
Sympy [F(-1)]
Maxima [F]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 24, antiderivative size = 238 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {b^2 e^2 n^2}{2 d^2 x^{2/3}}+\frac {b^2 e^3 n^2 \log \left (d+e x^{2/3}\right )}{2 d^3}-\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d x^{4/3}}+\frac {b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3 x^{2/3}}+\frac {b e^3 n \log \left (1-\frac {d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 x^2}-\frac {b^2 e^3 n^2 \log (x)}{d^3}-\frac {b^2 e^3 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e x^{2/3}}\right )}{d^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {e x^{2/3} \left (3 b d^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-6 b d e n x^{2/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+3 e^2 x^{4/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-2 b^2 e^2 n^2 x^{4/3} \left (3 \log \left (d+e x^{2/3}\right )-2 \log (x)\right )+b^2 e n^2 x^{2/3} \left (3 d-3 e x^{2/3} \log \left (d+e x^{2/3}\right )+2 e x^{2/3} \log (x)\right )-6 b e^2 n x^{4/3} \left (\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \log \left (-\frac {e x^{2/3}}{d}\right )+b n \operatorname {PolyLog}\left (2,1+\frac {e x^{2/3}}{d}\right )\right )\right )}{d^3}}{6 x^2} \] Input:

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

Output:

-1/6*(3*(a + b*Log[c*(d + e*x^(2/3))^n])^2 + (e*x^(2/3)*(3*b*d^2*n*(a + b* 
Log[c*(d + e*x^(2/3))^n]) - 6*b*d*e*n*x^(2/3)*(a + b*Log[c*(d + e*x^(2/3)) 
^n]) + 3*e^2*x^(4/3)*(a + b*Log[c*(d + e*x^(2/3))^n])^2 - 2*b^2*e^2*n^2*x^ 
(4/3)*(3*Log[d + e*x^(2/3)] - 2*Log[x]) + b^2*e*n^2*x^(2/3)*(3*d - 3*e*x^( 
2/3)*Log[d + e*x^(2/3)] + 2*e*x^(2/3)*Log[x]) - 6*b*e^2*n*x^(4/3)*((a + b* 
Log[c*(d + e*x^(2/3))^n])*Log[-((e*x^(2/3))/d)] + b*n*PolyLog[2, 1 + (e*x^ 
(2/3))/d])))/d^3)/x^2
 

Rubi [A] (warning: unable to verify)

Time = 1.64 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {2904, 2845, 2858, 25, 27, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}

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.

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

\(\Big \downarrow \) 2904

\(\displaystyle \frac {3}{2} \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^{8/3}}dx^{2/3}\)

\(\Big \downarrow \) 2845

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

\(\Big \downarrow \) 2858

\(\displaystyle \frac {3}{2} \left (\frac {2}{3} b n \int \frac {a+b \log \left (c x^{2 n/3}\right )}{x^{8/3}}d\left (d+e x^{2/3}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3}{2} \left (-\frac {2}{3} b n \int -\frac {a+b \log \left (c x^{2 n/3}\right )}{x^{8/3}}d\left (d+e x^{2/3}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{2} \left (-\frac {2}{3} b e^3 n \int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e^3 x^{8/3}}d\left (d+e x^{2/3}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^2}\right )\)

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 16
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25
Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27
Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 2751
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x 
_Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* 
(n/d)   Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, 
x] && EqQ[r*(q + 1) + 1, 0]
 

rule 2756
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), 
x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] 
- Simp[b*n*(p/(e*(q + 1)))   Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 
 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, 
 -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] & 
& NeQ[q, 1]))
 

rule 2779
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

rule 2789
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

rule 2838
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2845
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. 
)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ 
n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1)))   Int[(f + g*x)^(q + 1) 
*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, 
d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In 
tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
 

rule 2858
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/2*b^2*log((e*x^(2/3) + d)^n)^2/x^2 + integrate(1/3*(3*(b^2*e*log(c)^2 + 
 2*a*b*e*log(c) + a^2*e)*x + 2*(b^2*e*n*x + 3*(b^2*e*log(c) + a*b*e)*x + 3 
*(b^2*d*log(c) + a*b*d)*x^(1/3))*log((e*x^(2/3) + d)^n) + 3*(b^2*d*log(c)^ 
2 + 2*a*b*d*log(c) + a^2*d)*x^(1/3))/(e*x^4 + d*x^(10/3)), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^3} \, dx=\frac {-3 x^{\frac {2}{3}} \mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) b^{2} d^{2} e n -3 x^{\frac {2}{3}} a b \,d^{2} e n +6 x^{\frac {4}{3}} \mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) b^{2} d \,e^{2} n +6 x^{\frac {4}{3}} a b d \,e^{2} n -3 x^{\frac {4}{3}} b^{2} d \,e^{2} n^{2}+4 \left (\int \frac {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}{x^{\frac {5}{3}} e +d x}d x \right ) b^{2} d \,e^{3} n \,x^{2}+12 \,\mathrm {log}\left (x^{\frac {1}{3}}\right ) a b \,e^{3} n \,x^{2}-18 \,\mathrm {log}\left (x^{\frac {1}{3}}\right ) b^{2} e^{3} n^{2} x^{2}-3 {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}^{2} b^{2} d^{3}-6 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) a b \,d^{3}-6 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) a b \,e^{3} x^{2}+9 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) b^{2} e^{3} n \,x^{2}-3 a^{2} d^{3}}{6 d^{3} x^{2}} \] Input:

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

Output:

( - 3*x**(2/3)*log((x**(2/3)*e + d)**n*c)*b**2*d**2*e*n - 3*x**(2/3)*a*b*d 
**2*e*n + 6*x**(1/3)*log((x**(2/3)*e + d)**n*c)*b**2*d*e**2*n*x + 6*x**(1/ 
3)*a*b*d*e**2*n*x - 3*x**(1/3)*b**2*d*e**2*n**2*x + 4*int(log((x**(2/3)*e 
+ d)**n*c)/(x**(2/3)*e*x + d*x),x)*b**2*d*e**3*n*x**2 + 12*log(x**(1/3))*a 
*b*e**3*n*x**2 - 18*log(x**(1/3))*b**2*e**3*n**2*x**2 - 3*log((x**(2/3)*e 
+ d)**n*c)**2*b**2*d**3 - 6*log((x**(2/3)*e + d)**n*c)*a*b*d**3 - 6*log((x 
**(2/3)*e + d)**n*c)*a*b*e**3*x**2 + 9*log((x**(2/3)*e + d)**n*c)*b**2*e** 
3*n*x**2 - 3*a**2*d**3)/(6*d**3*x**2)