\(\int \frac {(a+b \log (c (d+e \sqrt {x})^n))^3}{x} \, dx\) [418]

Optimal result

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(333\) vs. \(2(135)=270\).

Time = 0.18 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.47 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x} \, dx=\left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \log (x)+3 b n \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \left (\left (\log \left (d+e \sqrt {x}\right )-\log \left (1+\frac {e \sqrt {x}}{d}\right )\right ) \log (x)-2 \operatorname {PolyLog}\left (2,-\frac {e \sqrt {x}}{d}\right )\right )+6 b^2 n^2 \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (\log ^2\left (d+e \sqrt {x}\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+2 \log \left (d+e \sqrt {x}\right ) \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {e \sqrt {x}}{d}\right )\right )+2 b^3 n^3 \left (\log ^3\left (d+e \sqrt {x}\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+3 \log ^2\left (d+e \sqrt {x}\right ) \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )-6 \log \left (d+e \sqrt {x}\right ) \operatorname {PolyLog}\left (3,1+\frac {e \sqrt {x}}{d}\right )+6 \operatorname {PolyLog}\left (4,1+\frac {e \sqrt {x}}{d}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.97 (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.

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

\(\Big \downarrow \) 2904

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

\(\Big \downarrow \) 2843

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

\(\Big \downarrow \) 2881

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 7143

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

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*Sqrt[x])/d)] - 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]

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

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]

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

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

Output:

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

Sympy [F]

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

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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