3.5.43 \(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^4}{(g+h x)^2} \, dx\) [443]

3.5.43.1 Optimal result

Integrand size = 28, antiderivative size = 274 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x)^2} \, dx=\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(f g-e h) (g+h x)}-\frac {4 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)}-\frac {12 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}+\frac {24 b^3 f p^3 q^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)}-\frac {24 b^4 f p^4 q^4 \operatorname {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)} \]

output

(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^4/(-e*h+f*g)/(h*x+g)-4*b*f*p*q*(a+b*ln 
(c*(d*(f*x+e)^p)^q))^3*ln(f*(h*x+g)/(-e*h+f*g))/h/(-e*h+f*g)-12*b^2*f*p^2* 
q^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^2*polylog(2,-h*(f*x+e)/(-e*h+f*g))/h/(-e*h 
+f*g)+24*b^3*f*p^3*q^3*(a+b*ln(c*(d*(f*x+e)^p)^q))*polylog(3,-h*(f*x+e)/(- 
e*h+f*g))/h/(-e*h+f*g)-24*b^4*f*p^4*q^4*polylog(4,-h*(f*x+e)/(-e*h+f*g))/h 
/(-e*h+f*g)
 

3.5.43.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1301\) vs. \(2(274)=548\).

Time = 0.38 (sec) , antiderivative size = 1301, normalized size of antiderivative = 4.75 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x)^2} \, dx =\text {Too large to display} \]

input

Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])^4/(g + h*x)^2,x]
 

output

(a^4*f*g - a^4*e*h - 4*a^3*b*f*g*p*q*Log[e + f*x] - 4*a^3*b*f*h*p*q*x*Log[ 
e + f*x] + 6*a^2*b^2*f*g*p^2*q^2*Log[e + f*x]^2 + 6*a^2*b^2*f*h*p^2*q^2*x* 
Log[e + f*x]^2 - 4*a*b^3*f*g*p^3*q^3*Log[e + f*x]^3 - 4*a*b^3*f*h*p^3*q^3* 
x*Log[e + f*x]^3 + b^4*f*g*p^4*q^4*Log[e + f*x]^4 + b^4*f*h*p^4*q^4*x*Log[ 
e + f*x]^4 + 4*a^3*b*f*g*Log[c*(d*(e + f*x)^p)^q] - 4*a^3*b*e*h*Log[c*(d*( 
e + f*x)^p)^q] - 12*a^2*b^2*f*g*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q] 
- 12*a^2*b^2*f*h*p*q*x*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q] + 12*a*b^3*f* 
g*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q] + 12*a*b^3*f*h*p^2*q^2*x 
*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q] - 4*b^4*f*g*p^3*q^3*Log[e + f*x]^ 
3*Log[c*(d*(e + f*x)^p)^q] - 4*b^4*f*h*p^3*q^3*x*Log[e + f*x]^3*Log[c*(d*( 
e + f*x)^p)^q] + 6*a^2*b^2*f*g*Log[c*(d*(e + f*x)^p)^q]^2 - 6*a^2*b^2*e*h* 
Log[c*(d*(e + f*x)^p)^q]^2 - 12*a*b^3*f*g*p*q*Log[e + f*x]*Log[c*(d*(e + f 
*x)^p)^q]^2 - 12*a*b^3*f*h*p*q*x*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^2 + 
 6*b^4*f*g*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q]^2 + 6*b^4*f*h*p 
^2*q^2*x*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q]^2 + 4*a*b^3*f*g*Log[c*(d* 
(e + f*x)^p)^q]^3 - 4*a*b^3*e*h*Log[c*(d*(e + f*x)^p)^q]^3 - 4*b^4*f*g*p*q 
*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^3 - 4*b^4*f*h*p*q*x*Log[e + f*x]*Lo 
g[c*(d*(e + f*x)^p)^q]^3 + b^4*f*g*Log[c*(d*(e + f*x)^p)^q]^4 - b^4*e*h*Lo 
g[c*(d*(e + f*x)^p)^q]^4 + 4*a^3*b*f*g*p*q*Log[(f*(g + h*x))/(f*g - e*h)] 
+ 4*a^3*b*f*h*p*q*x*Log[(f*(g + h*x))/(f*g - e*h)] + 12*a^2*b^2*f*g*p*q...
 

3.5.43.3 Rubi [A] (warning: unable to verify)

Time = 1.13 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.82, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2895, 2844, 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.

\(\displaystyle \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x)^2} \, dx\)

\(\Big \downarrow \) 2895

\(\displaystyle \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x)^2}dx\)

\(\Big \downarrow \) 2844

\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x) (f g-e h)}-\frac {4 b f p q \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x}dx}{f g-e h}\)

\(\Big \downarrow \) 2843

\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x) (f g-e h)}-\frac {4 b f p q \left (\frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h}-\frac {3 b f p q \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x}dx}{h}\right )}{f g-e h}\)

\(\Big \downarrow \) 2881

\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x) (f g-e h)}-\frac {4 b f p q \left (\frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h}-\frac {3 b p q \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \log \left (\frac {f \left (g-\frac {e h}{f}\right )+h (e+f x)}{f g-e h}\right )}{e+f x}d(e+f x)}{h}\right )}{f g-e h}\)

\(\Big \downarrow \) 2821

\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x) (f g-e h)}-\frac {4 b f p q \left (\frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h}-\frac {3 b p q \left (2 b p q \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{e+f x}d(e+f x)-\operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2\right )}{h}\right )}{f g-e h}\)

\(\Big \downarrow \) 2830

\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x) (f g-e h)}-\frac {4 b f p q \left (\frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h}-\frac {3 b p q \left (2 b p q \left (\operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )-b p q \int \frac {\operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{e+f x}d(e+f x)\right )-\operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2\right )}{h}\right )}{f g-e h}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x) (f g-e h)}-\frac {4 b f p q \left (\frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h}-\frac {3 b p q \left (2 b p q \left (\operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )-b p q \operatorname {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2\right )}{h}\right )}{f g-e h}\)

input

Int[(a + b*Log[c*(d*(e + f*x)^p)^q])^4/(g + h*x)^2,x]
 

output

((e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^4)/((f*g - e*h)*(g + h*x)) - ( 
4*b*f*p*q*(((a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(g + h*x))/(f*g - e* 
h)])/h - (3*b*p*q*(-((a + b*Log[c*d^q*(e + f*x)^(p*q)])^2*PolyLog[2, -((h* 
(e + f*x))/(f*g - e*h))]) + 2*b*p*q*((a + b*Log[c*d^q*(e + f*x)^(p*q)])*Po 
lyLog[3, -((h*(e + f*x))/(f*g - e*h))] - b*p*q*PolyLog[4, -((h*(e + f*x))/ 
(f*g - e*h))])))/h))/(f*g - e*h)
 

3.5.43.3.1 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 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]
 

3.5.43.4 Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{4}}{\left (h x +g \right )^{2}}d x\]

input

int((a+b*ln(c*(d*(f*x+e)^p)^q))^4/(h*x+g)^2,x)
 

output

int((a+b*ln(c*(d*(f*x+e)^p)^q))^4/(h*x+g)^2,x)
 

3.5.43.5 Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x)^2} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{4}}{{\left (h x + g\right )}^{2}} \,d x } \]

input

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^4/(h*x+g)^2,x, algorithm="fricas")
 

output

integral((b^4*log(((f*x + e)^p*d)^q*c)^4 + 4*a*b^3*log(((f*x + e)^p*d)^q*c 
)^3 + 6*a^2*b^2*log(((f*x + e)^p*d)^q*c)^2 + 4*a^3*b*log(((f*x + e)^p*d)^q 
*c) + a^4)/(h^2*x^2 + 2*g*h*x + g^2), x)
 

3.5.43.6 Sympy [F]

\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x)^2} \, dx=\int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{4}}{\left (g + h x\right )^{2}}\, dx \]

input

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))**4/(h*x+g)**2,x)
 

output

Integral((a + b*log(c*(d*(e + f*x)**p)**q))**4/(g + h*x)**2, x)
 

3.5.43.7 Maxima [F]

\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x)^2} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{4}}{{\left (h x + g\right )}^{2}} \,d x } \]

input

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^4/(h*x+g)^2,x, algorithm="maxima")
 

output

4*a^3*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^4*log(((f*x + e)^p)^q)^4/(h^2*x + g*h) - 4*a^3*b*log(((f*x + e)^p*d) 
^q*c)/(h^2*x + g*h) - a^4/(h^2*x + g*h) + integrate((6*(e*h*q^2*log(d)^2 + 
 2*e*h*q*log(c)*log(d) + e*h*log(c)^2)*a^2*b^2 + 4*(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)*a*b^3 + ( 
e*h*q^4*log(d)^4 + 4*e*h*q^3*log(c)*log(d)^3 + 6*e*h*q^2*log(c)^2*log(d)^2 
 + 4*e*h*q*log(c)^3*log(d) + e*h*log(c)^4)*b^4 + 4*(a*b^3*e*h + (f*g*p*q + 
 e*h*q*log(d) + e*h*log(c))*b^4 + (a*b^3*f*h + (f*h*p*q + f*h*q*log(d) + f 
*h*log(c))*b^4)*x)*log(((f*x + e)^p)^q)^3 + 6*(a^2*b^2*e*h + 2*(e*h*q*log( 
d) + e*h*log(c))*a*b^3 + (e*h*q^2*log(d)^2 + 2*e*h*q*log(c)*log(d) + e*h*l 
og(c)^2)*b^4 + (a^2*b^2*f*h + 2*(f*h*q*log(d) + f*h*log(c))*a*b^3 + (f*h*q 
^2*log(d)^2 + 2*f*h*q*log(c)*log(d) + f*h*log(c)^2)*b^4)*x)*log(((f*x + e) 
^p)^q)^2 + (6*(f*h*q^2*log(d)^2 + 2*f*h*q*log(c)*log(d) + f*h*log(c)^2)*a^ 
2*b^2 + 4*(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)*a*b^3 + (f*h*q^4*log(d)^4 + 4*f*h*q^3*log(c)*log(d 
)^3 + 6*f*h*q^2*log(c)^2*log(d)^2 + 4*f*h*q*log(c)^3*log(d) + f*h*log(c)^4 
)*b^4)*x + 4*(3*(e*h*q*log(d) + e*h*log(c))*a^2*b^2 + 3*(e*h*q^2*log(d)^2 
+ 2*e*h*q*log(c)*log(d) + e*h*log(c)^2)*a*b^3 + (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^4 + (3*(f* 
h*q*log(d) + f*h*log(c))*a^2*b^2 + 3*(f*h*q^2*log(d)^2 + 2*f*h*q*log(c)...
 

3.5.43.8 Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x)^2} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{4}}{{\left (h x + g\right )}^{2}} \,d x } \]

input

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^4/(h*x+g)^2,x, algorithm="giac")
 

output

integrate((b*log(((f*x + e)^p*d)^q*c) + a)^4/(h*x + g)^2, x)
 

3.5.43.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^4}{{\left (g+h\,x\right )}^2} \,d x \]

input

int((a + b*log(c*(d*(e + f*x)^p)^q))^4/(g + h*x)^2,x)
 

output

int((a + b*log(c*(d*(e + f*x)^p)^q))^4/(g + h*x)^2, x)