\(\int (a+b \log (c (d+\frac {e}{f+g x})^p))^3 \, dx\) [724]

Optimal result

Integrand size = 22, antiderivative size = 168 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3 \, dx=-\frac {3 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}-\frac {6 b^2 e p^2 \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e}{d (f+g x)}\right )}{d g}+\frac {6 b^3 e p^3 \operatorname {PolyLog}\left (3,1+\frac {e}{d (f+g x)}\right )}{d g} \] Output:

-3*b*e*p*ln(-e/d/(g*x+f))*(a+b*ln(c*(d+e/(g*x+f))^p))^2/d/g+(e+d*(g*x+f))* 
(a+b*ln(c*(d+e/(g*x+f))^p))^3/d/g-6*b^2*e*p^2*(a+b*ln(c*(d+e/(g*x+f))^p))* 
polylog(2,1+e/d/(g*x+f))/d/g+6*b^3*e*p^3*polylog(3,1+e/d/(g*x+f))/d/g
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(441\) vs. \(2(168)=336\).

Time = 0.75 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.62 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3 \, dx=\frac {3 b d p (f+g x) \log \left (d+\frac {e}{f+g x}\right ) \left (a-b p \log \left (d+\frac {e}{f+g x}\right )+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2+d (f+g x) \left (a-b p \log \left (d+\frac {e}{f+g x}\right )+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3+3 b e p \left (a-b p \log \left (d+\frac {e}{f+g x}\right )+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2 \log (e+d (f+g x))+3 b^2 p^2 \left (a-b p \log \left (d+\frac {e}{f+g x}\right )+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right ) \left (d (f+g x) \log ^2\left (d+\frac {e}{f+g x}\right )+e \left (\log ^2\left (\frac {e}{d}+f+g x\right )+2 \left (\log (f+g x)-\log \left (\frac {e}{d}+f+g x\right )+\log \left (d+\frac {e}{f+g x}\right )\right ) \log (e+d (f+g x))-2 \left (\log (f+g x) \log \left (1+\frac {d (f+g x)}{e}\right )+\operatorname {PolyLog}\left (2,-\frac {d (f+g x)}{e}\right )\right )\right )\right )+b^3 p^3 \left (\log ^2\left (d+\frac {e}{f+g x}\right ) \left (-3 e \log \left (-\frac {e}{d f+d g x}\right )+(e+d f+d g x) \log \left (d+\frac {e}{f+g x}\right )\right )-6 e \log \left (d+\frac {e}{f+g x}\right ) \operatorname {PolyLog}\left (2,1+\frac {e}{d f+d g x}\right )+6 e \operatorname {PolyLog}\left (3,1+\frac {e}{d f+d g x}\right )\right )}{d g} \] Input:

Integrate[(a + b*Log[c*(d + e/(f + g*x))^p])^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.89, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2933, 2899, 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.

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

\(\Big \downarrow \) 2933

\(\displaystyle \frac {\int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3d(f+g x)}{g}\)

\(\Big \downarrow \) 2899

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

\(\Big \downarrow \) 2904

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

\(\Big \downarrow \) 2843

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

\(\Big \downarrow \) 2881

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 7143

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

Input:

Int[(a + b*Log[c*(d + e/(f + g*x))^p])^3,x]
 

Output:

(((e + d*(f + g*x))*(a + b*Log[c*(d + e/(f + g*x))^p])^3)/d - (3*b*e*p*(Lo 
g[-(e/(d*(f + g*x)))]*(a + b*Log[c*(d + e/(f + g*x))^p])^2 - 2*b*p*(-((a + 
 b*Log[c*(d + e/(f + g*x))^p])*PolyLog[2, (d + e/(f + g*x))/d]) + b*p*Poly 
Log[3, (d + e/(f + g*x))/d])))/d)/g
 

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 2899

Int[((a_.) + Log[(c_.)*((d_) + (e_.)/(x_))^(p_.)]*(b_.))^(q_), x_Symbol] :> 
 Simp[(e + d*x)*((a + b*Log[c*(d + e/x)^p])^q/d), x] + Simp[b*e*p*(q/d)   I 
nt[(a + b*Log[c*(d + e/x)^p])^(q - 1)/x, x], x] /; FreeQ[{a, b, c, d, e, p} 
, x] && IGtQ[q, 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 2933

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

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]

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

Input:

int((a+b*ln(c*(d+e/(g*x+f))^p))^3,x)
 

Output:

int((a+b*ln(c*(d+e/(g*x+f))^p))^3,x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

integrate((a+b*ln(c*(d+e/(g*x+f))**p))**3,x)
 

Output:

Integral((a + b*log(c*(d + e/(f + g*x))**p))**3, x)
 

Maxima [F]

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

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

Output:

-3*a^2*b*e*g*p*(f*log(g*x + f)/(e*g^2) - (d*f + e)*log(d*g*x + d*f + e)/(d 
*e*g^2)) + 3*a^2*b*x*log(c*(d + e/(g*x + f))^p) + a^3*x + (b^3*d*g*x*log(( 
d*g*x + d*f + e)^p)^3 - 3*(b^3*d*f*p*log(g*x + f) + b^3*d*g*x*log((g*x + f 
)^p) - (d*f*p + e*p)*b^3*log(d*g*x + d*f + e) - (b^3*d*g*log(c) + a*b^2*d* 
g)*x)*log((d*g*x + d*f + e)^p)^2)/(d*g) + integrate(((d*f + e)*b^3*log(c)^ 
3 + 3*(d*f + e)*a*b^2*log(c)^2 - (b^3*d*g*x + (d*f + e)*b^3)*log((g*x + f) 
^p)^3 + 3*((d*f + e)*b^3*log(c) + (d*f + e)*a*b^2 + (b^3*d*g*log(c) + a*b^ 
2*d*g)*x)*log((g*x + f)^p)^2 + (b^3*d*g*log(c)^3 + 3*a*b^2*d*g*log(c)^2)*x 
 + 3*(2*b^3*d*f*p^2*log(g*x + f) + (d*f + e)*b^3*log(c)^2 - 2*(d*f*p^2 + e 
*p^2)*b^3*log(d*g*x + d*f + e) + 2*(d*f + e)*a*b^2*log(c) + (b^3*d*g*x + ( 
d*f + e)*b^3)*log((g*x + f)^p)^2 - (2*(d*g*p - d*g*log(c))*a*b^2 + (2*d*g* 
p*log(c) - d*g*log(c)^2)*b^3)*x - 2*((d*f + e)*b^3*log(c) + (d*f + e)*a*b^ 
2 + (a*b^2*d*g - (d*g*p - d*g*log(c))*b^3)*x)*log((g*x + f)^p))*log((d*g*x 
 + d*f + e)^p) - 3*((d*f + e)*b^3*log(c)^2 + 2*(d*f + e)*a*b^2*log(c) + (b 
^3*d*g*log(c)^2 + 2*a*b^2*d*g*log(c))*x)*log((g*x + f)^p))/(d*g*x + d*f + 
e), x)
 

Giac [F]

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

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

Output:

integrate((b*log(c*(d + e/(g*x + f))^p) + a)^3, x)
 

Mupad [F(-1)]

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

int((a + b*log(c*(d + e/(f + g*x))^p))^3,x)
 

Output:

int((a + b*log(c*(d + e/(f + g*x))^p))^3, x)
 

Reduce [F]

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

int((a+b*log(c*(d+e/(g*x+f))^p))^3,x)
 

Output:

(3*int((log(((d*f + d*g*x + e)**p*c)/(f + g*x)**p)**2*x)/(d*f**2 + 2*d*f*g 
*x + d*g**2*x**2 + e*f + e*g*x),x)*b**3*d*e*g**2*p + 6*int((log(((d*f + d* 
g*x + e)**p*c)/(f + g*x)**p)*x)/(d*f**2 + 2*d*f*g*x + d*g**2*x**2 + e*f + 
e*g*x),x)*a*b**2*d*e*g**2*p + 3*log(f + g*x)*a**2*b*e*p + log(((d*f + d*g* 
x + e)**p*c)/(f + g*x)**p)**3*b**3*d*g*x + 3*log(((d*f + d*g*x + e)**p*c)/ 
(f + g*x)**p)**2*a*b**2*d*g*x + 3*log(((d*f + d*g*x + e)**p*c)/(f + g*x)** 
p)*a**2*b*d*f + 3*log(((d*f + d*g*x + e)**p*c)/(f + g*x)**p)*a**2*b*d*g*x 
+ 3*log(((d*f + d*g*x + e)**p*c)/(f + g*x)**p)*a**2*b*e + a**3*d*g*x)/(d*g 
)