\(\int \frac {(a+b \log (c (d+e x)^n))^3}{(f+g x)^2} \, dx\) [57]

Optimal result

Integrand size = 24, antiderivative size = 190 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(e f-d g) (f+g x)}-\frac {3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g (e f-d g)}-\frac {6 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)}+\frac {6 b^3 e n^3 \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(410\) vs. \(2(190)=380\).

Time = 0.42 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.16 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\frac {-3 b (e f-d g) n \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+3 b e n (f+g x) \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-(e f-d g) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3-3 b e n (f+g x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log (f+g x)+3 b^2 n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\log (d+e x) \left (g (d+e x) \log (d+e x)-2 e (f+g x) \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )-2 e (f+g x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+b^3 n^3 \left (\log ^2(d+e x) \left (g (d+e x) \log (d+e x)-3 e (f+g x) \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )-6 e (f+g x) \log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )+6 e (f+g x) \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )}{g (e f-d g) (f+g x)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.85, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2844, 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.

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

\(\Big \downarrow \) 2844

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

\(\Big \downarrow \) 2843

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

\(\Big \downarrow \) 2881

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

((d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/((e*f - d*g)*(f + g*x)) - (3*b*e* 
n*(((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(f + g*x))/(e*f - d*g)])/g - (2*b* 
n*(-((a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))]) 
+ b*n*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))]))/g))/(e*f - 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 2844

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

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 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 [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 4.70 (sec) , antiderivative size = 1268, normalized size of antiderivative = 6.67

method	result	size
risch	\(\text {Expression too large to display}\)	\(1268\)

Input:

int((a+b*ln(c*(e*x+d)^n))^3/(g*x+f)^2,x,method=_RETURNVERBOSE)
 

Output:

-b^3*ln((e*x+d)^n)^3/(g*x+f)/g+3*b^3/g*n^2*e/(d*g-e*f)*ln(e*x+d)^2*ln((e*x 
+d)^n)-3*b^3/g*n*e/(d*g-e*f)*ln(e*x+d)*ln((e*x+d)^n)^2+3*b^3/g*n^3*e/(d*g- 
e*f)*ln(g*(e*x+d)-d*g+e*f)*ln(e*x+d)^2-6*b^3/g*n^2*e/(d*g-e*f)*ln(g*(e*x+d 
)-d*g+e*f)*ln((e*x+d)^n)*ln(e*x+d)+3*b^3/g*n*e/(d*g-e*f)*ln(g*(e*x+d)-d*g+ 
e*f)*ln((e*x+d)^n)^2+b^3/g*n^3*e/(-d*g+e*f)*ln(e*x+d)^3+3*b^3/g*n^3*e/(d*g 
-e*f)*ln(e*x+d)^2*ln(1+g*(e*x+d)/(-d*g+e*f))+6*b^3/g*n^3*e/(d*g-e*f)*ln(e* 
x+d)*polylog(2,-g*(e*x+d)/(-d*g+e*f))-6*b^3/g*n^3*e/(d*g-e*f)*polylog(3,-g 
*(e*x+d)/(-d*g+e*f))-6*b^3/g*n^3*e/(d*g-e*f)*dilog((g*(e*x+d)-d*g+e*f)/(-d 
*g+e*f))*ln(e*x+d)+6*b^3/g*n^2*e/(d*g-e*f)*dilog((g*(e*x+d)-d*g+e*f)/(-d*g 
+e*f))*ln((e*x+d)^n)-6*b^3/g*n^3*e/(d*g-e*f)*ln(e*x+d)^2*ln((g*(e*x+d)-d*g 
+e*f)/(-d*g+e*f))+6*b^3/g*n^2*e/(d*g-e*f)*ln(e*x+d)*ln((g*(e*x+d)-d*g+e*f) 
/(-d*g+e*f))*ln((e*x+d)^n)-1/8*(I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^ 
n)^2-I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-I*b*Pi*csgn(I* 
c*(e*x+d)^n)^3+I*b*Pi*csgn(I*c*(e*x+d)^n)^2*csgn(I*c)+2*b*ln(c)+2*a)^3/(g* 
x+f)/g+3/2*(I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*( 
e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+I*b*P 
i*csgn(I*c*(e*x+d)^n)^2*csgn(I*c)+2*b*ln(c)+2*a)*b^2*(-ln((e*x+d)^n)^2/(g* 
x+f)/g+2/g*n*e*(ln((e*x+d)^n)/(d*g-e*f)*ln(g*x+f)-ln((e*x+d)^n)/(d*g-e*f)* 
ln(e*x+d)-e*n*(1/(d*g-e*f)*(dilog(((g*x+f)*e+d*g-e*f)/(d*g-e*f))/e+ln(g*x+ 
f)*ln(((g*x+f)*e+d*g-e*f)/(d*g-e*f))/e)-1/2/(d*g-e*f)/e*ln(e*x+d)^2)))+...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\text {too large to display} \] Input:

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

Output:

( - 6*int(log((d + e*x)**n*c)/(d*f**2 + 2*d*f*g*x + d*g**2*x**2 + e*f**2*x 
 + 2*e*f*g*x**2 + e*g**2*x**3),x)*a*b**2*d**3*f**2*g**3*n - 6*int(log((d + 
 e*x)**n*c)/(d*f**2 + 2*d*f*g*x + d*g**2*x**2 + e*f**2*x + 2*e*f*g*x**2 + 
e*g**2*x**3),x)*a*b**2*d**3*f*g**4*n*x + 12*int(log((d + e*x)**n*c)/(d*f** 
2 + 2*d*f*g*x + d*g**2*x**2 + e*f**2*x + 2*e*f*g*x**2 + e*g**2*x**3),x)*a* 
b**2*d**2*e*f**3*g**2*n + 12*int(log((d + e*x)**n*c)/(d*f**2 + 2*d*f*g*x + 
 d*g**2*x**2 + e*f**2*x + 2*e*f*g*x**2 + e*g**2*x**3),x)*a*b**2*d**2*e*f** 
2*g**3*n*x - 6*int(log((d + e*x)**n*c)/(d*f**2 + 2*d*f*g*x + d*g**2*x**2 + 
 e*f**2*x + 2*e*f*g*x**2 + e*g**2*x**3),x)*a*b**2*d*e**2*f**4*g*n - 6*int( 
log((d + e*x)**n*c)/(d*f**2 + 2*d*f*g*x + d*g**2*x**2 + e*f**2*x + 2*e*f*g 
*x**2 + e*g**2*x**3),x)*a*b**2*d*e**2*f**3*g**2*n*x - 6*int(log((d + e*x)* 
*n*c)/(d*f**2 + 2*d*f*g*x + d*g**2*x**2 + e*f**2*x + 2*e*f*g*x**2 + e*g**2 
*x**3),x)*b**3*d**2*e*f**3*g**2*n**2 - 6*int(log((d + e*x)**n*c)/(d*f**2 + 
 2*d*f*g*x + d*g**2*x**2 + e*f**2*x + 2*e*f*g*x**2 + e*g**2*x**3),x)*b**3* 
d**2*e*f**2*g**3*n**2*x + 12*int(log((d + e*x)**n*c)/(d*f**2 + 2*d*f*g*x + 
 d*g**2*x**2 + e*f**2*x + 2*e*f*g*x**2 + e*g**2*x**3),x)*b**3*d*e**2*f**4* 
g*n**2 + 12*int(log((d + e*x)**n*c)/(d*f**2 + 2*d*f*g*x + d*g**2*x**2 + e* 
f**2*x + 2*e*f*g*x**2 + e*g**2*x**3),x)*b**3*d*e**2*f**3*g**2*n**2*x - 6*i 
nt(log((d + e*x)**n*c)/(d*f**2 + 2*d*f*g*x + d*g**2*x**2 + e*f**2*x + 2*e* 
f*g*x**2 + e*g**2*x**3),x)*b**3*e**3*f**5*n**2 - 6*int(log((d + e*x)**n...