\(\int \frac {(a+b \log (c (d+\frac {e}{x^{2/3}})^n))^2}{x} \, dx\) [518]

Optimal result

Integrand size = 24, antiderivative size = 95 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x} \, dx=-\frac {3}{2} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \log \left (-\frac {e}{d x^{2/3}}\right )-3 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e}{d x^{2/3}}\right )+3 b^2 n^2 \operatorname {PolyLog}\left (3,1+\frac {e}{d x^{2/3}}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 1701, normalized size of antiderivative = 17.91 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x} \, dx =\text {Too large to display} \] Input:

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

Output:

(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2*Log[x] + 2*b*n 
*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])*((Log[d + e/x^( 
2/3)] - Log[1 + e/(d*x^(2/3))])*Log[x] + (3*PolyLog[2, -(e/(d*x^(2/3)))])/ 
2) + 3*b^2*n^2*(Log[((-I)*Sqrt[e])/Sqrt[d] + x^(1/3)]^2*Log[((-I)*Sqrt[d]* 
x^(1/3))/Sqrt[e]] + 2*Log[((-I)*Sqrt[e])/Sqrt[d] + x^(1/3)]*Log[(I*Sqrt[e] 
)/Sqrt[d] + x^(1/3)]*Log[((-I)*Sqrt[d]*x^(1/3))/Sqrt[e]] + Log[1 - (I*Sqrt 
[d]*x^(1/3))/Sqrt[e]]*(-2*Log[((-I)*Sqrt[e])/Sqrt[d] + x^(1/3)] + Log[1 - 
(I*Sqrt[d]*x^(1/3))/Sqrt[e]])*(Log[((-I)*Sqrt[d]*x^(1/3))/Sqrt[e]] - Log[( 
I*Sqrt[d]*x^(1/3))/Sqrt[e]]) + Log[(I*Sqrt[e])/Sqrt[d] + x^(1/3)]^2*Log[(I 
*Sqrt[d]*x^(1/3))/Sqrt[e]] + 2*Log[(Sqrt[e] - I*Sqrt[d]*x^(1/3))/(Sqrt[e] 
+ I*Sqrt[d]*x^(1/3))]*Log[1 - (I*Sqrt[d]*x^(1/3))/Sqrt[e]]*(-Log[((-I)*Sqr 
t[d]*x^(1/3))/Sqrt[e]] + Log[(I*Sqrt[d]*x^(1/3))/Sqrt[e]]) + Log[(Sqrt[e] 
- I*Sqrt[d]*x^(1/3))/(Sqrt[e] + I*Sqrt[d]*x^(1/3))]^2*(Log[(2*Sqrt[e])/(Sq 
rt[e] + I*Sqrt[d]*x^(1/3))] + Log[((-I)*Sqrt[d]*x^(1/3))/Sqrt[e]] - Log[(2 
*x^(1/3))/(((-I)*Sqrt[e])/Sqrt[d] + x^(1/3))]) + ((-Log[d + e/x^(2/3)] + L 
og[((-I)*Sqrt[e])/Sqrt[d] + x^(1/3)] + Log[(I*Sqrt[e])/Sqrt[d] + x^(1/3)] 
- (2*Log[x])/3)^2*Log[x])/3 + (4*Log[x]^3)/81 + 2*Log[(Sqrt[e] - I*Sqrt[d] 
*x^(1/3))/(Sqrt[e] + I*Sqrt[d]*x^(1/3))]*(-PolyLog[2, (I*Sqrt[e] + Sqrt[d] 
*x^(1/3))/(I*Sqrt[e] - Sqrt[d]*x^(1/3))] + PolyLog[2, (I*Sqrt[e] + Sqrt[d] 
*x^(1/3))/((-I)*Sqrt[e] + Sqrt[d]*x^(1/3))]) + 2*Log[(I*Sqrt[e])/Sqrt[d...
 

Rubi [A] (warning: unable to verify)

Time = 0.75 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2904, 2843, 2881, 2821, 7143}

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.

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

\(\Big \downarrow \) 2904

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

\(\Big \downarrow \) 2843

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

\(\Big \downarrow \) 2881

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c 
*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 
0] && EqQ[d*e, 1]
 

rule 2843

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

rule 2881

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log 
[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym 
bol] :> Simp[1/e   Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[h* 
((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, 
 f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 0]
 

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])
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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