\(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^3}{(g+h x)^2} \, dx\) [458]

Optimal result

Integrand size = 28, antiderivative size = 209 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x)^2} \, dx=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(f g-e h) (g+h x)}-\frac {3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)}-\frac {6 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}+\frac {6 b^3 f p^3 q^3 \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(444\) vs. \(2(209)=418\).

Time = 0.55 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.12 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x)^2} \, dx=\frac {-3 b (f g-e h) p q \log (e+f x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+3 b f p q (g+h x) \log (e+f x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-(f g-e h) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3-3 b f p q (g+h x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log (g+h x)+3 b^2 p^2 q^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left (\log (e+f x) \left (h (e+f x) \log (e+f x)-2 f (g+h x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )-2 f (g+h x) \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )\right )+b^3 p^3 q^3 \left (\log ^2(e+f x) \left (h (e+f x) \log (e+f x)-3 f (g+h x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )-6 f (g+h x) \log (e+f x) \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )+6 f (g+h x) \operatorname {PolyLog}\left (3,\frac {h (e+f x)}{-f g+e h}\right )\right )}{h (f g-e h) (g+h x)} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 1.61 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.85, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2895, 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.

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

\(\Big \downarrow \) 2895

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

\(\Big \downarrow \) 2844

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

\(\Big \downarrow \) 2843

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

\(\Big \downarrow \) 2881

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

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 2895

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. 
)*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], 
 c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ 
IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
 

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]

\[\int \frac {{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{3}}{\left (h x +g \right )^{2}}d x\]

Input:

int((a+b*ln(c*(d*(f*x+e)^p)^q))^3/(h*x+g)^2,x)
 

Output:

int((a+b*ln(c*(d*(f*x+e)^p)^q))^3/(h*x+g)^2,x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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