\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt {x}})^n))^3}{x} \, dx\) [438]

Optimal result

Integrand size = 24, antiderivative size = 135 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x} \, dx=-2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \log \left (-\frac {e}{d \sqrt {x}}\right )-6 b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \operatorname {PolyLog}\left (2,1+\frac {e}{d \sqrt {x}}\right )+12 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \operatorname {PolyLog}\left (3,1+\frac {e}{d \sqrt {x}}\right )-12 b^3 n^3 \operatorname {PolyLog}\left (4,1+\frac {e}{d \sqrt {x}}\right ) \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.94 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2904, 2843, 2881, 2821, 2830, 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.

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

\(\Big \downarrow \) 2904

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

\(\Big \downarrow \) 2843

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

\(\Big \downarrow \) 2881

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

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 2830

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ 
.)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) 
, x] - Simp[b*n*(p/q)   Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(p - 
1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((a + b*log(c*(d + e/sqrt(x))**n))**3/x, x)
 

Maxima [F]

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

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

Output:

b^3*log((d*sqrt(x) + e)^n)^3*log(x) - integrate(1/2*(2*(b^3*d*x + b^3*e*sq 
rt(x))*log(x^(1/2*n))^3 + 3*(b^3*d*n*x*log(x) - 2*(b^3*d*log(c) + a*b^2*d) 
*x + 2*(b^3*d*x + b^3*e*sqrt(x))*log(x^(1/2*n)) - 2*(b^3*e*log(c) + a*b^2* 
e)*sqrt(x))*log((d*sqrt(x) + e)^n)^2 - 6*((b^3*d*log(c) + a*b^2*d)*x + (b^ 
3*e*log(c) + a*b^2*e)*sqrt(x))*log(x^(1/2*n))^2 - 2*(b^3*d*log(c)^3 + 3*a* 
b^2*d*log(c)^2 + 3*a^2*b*d*log(c) + a^3*d)*x - 6*((b^3*d*x + b^3*e*sqrt(x) 
)*log(x^(1/2*n))^2 + (b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x - 2*( 
(b^3*d*log(c) + a*b^2*d)*x + (b^3*e*log(c) + a*b^2*e)*sqrt(x))*log(x^(1/2* 
n)) + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*sqrt(x))*log((d*sqrt(x 
) + e)^n) + 6*((b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x + (b^3*e*lo 
g(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*sqrt(x))*log(x^(1/2*n)) - 2*(b^3*e*lo 
g(c)^3 + 3*a*b^2*e*log(c)^2 + 3*a^2*b*e*log(c) + a^3*e)*sqrt(x))/(d*x^2 + 
e*x^(3/2)), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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