3.1.51 \(\int \frac {(a+b \log (c (d+e x)^n))^2}{(f+g x)^4} \, dx\) [51]

3.1.51.1 Optimal result

Integrand size = 24, antiderivative size = 317 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^4} \, dx=-\frac {b^2 e^2 n^2}{3 g (e f-d g)^2 (f+g x)}-\frac {b^2 e^3 n^2 \log (d+e x)}{3 g (e f-d g)^3}+\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) (f+g x)^2}-\frac {2 b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g)^3 (f+g x)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^3}+\frac {b^2 e^3 n^2 \log (f+g x)}{g (e f-d g)^3}-\frac {2 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1+\frac {e f-d g}{g (d+e x)}\right )}{3 g (e f-d g)^3}+\frac {2 b^2 e^3 n^2 \operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right )}{3 g (e f-d g)^3} \]

output

-1/3*b^2*e^2*n^2/g/(-d*g+e*f)^2/(g*x+f)-1/3*b^2*e^3*n^2*ln(e*x+d)/g/(-d*g+ 
e*f)^3+1/3*b*e*n*(a+b*ln(c*(e*x+d)^n))/g/(-d*g+e*f)/(g*x+f)^2-2/3*b*e^2*n* 
(e*x+d)*(a+b*ln(c*(e*x+d)^n))/(-d*g+e*f)^3/(g*x+f)-1/3*(a+b*ln(c*(e*x+d)^n 
))^2/g/(g*x+f)^3+b^2*e^3*n^2*ln(g*x+f)/g/(-d*g+e*f)^3-2/3*b*e^3*n*(a+b*ln( 
c*(e*x+d)^n))*ln(1+(-d*g+e*f)/g/(e*x+d))/g/(-d*g+e*f)^3+2/3*b^2*e^3*n^2*po 
lylog(2,(d*g-e*f)/g/(e*x+d))/g/(-d*g+e*f)^3
 

3.1.51.2 Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^4} \, dx=\frac {-\left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac {e (f+g x) \left (b (e f-d g)^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )+2 b e (e f-d g) n (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )+e^2 (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b^2 e^2 n^2 (f+g x)^2 (\log (d+e x)-\log (f+g x))-b^2 e n^2 (f+g x) (e f-d g+e (f+g x) \log (d+e x)-e (f+g x) \log (f+g x))-2 b e^2 n (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )-2 b^2 e^2 n^2 (f+g x)^2 \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )}{(e f-d g)^3}}{3 g (f+g x)^3} \]

input

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

output

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

3.1.51.3 Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.20, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2845, 2858, 27, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}

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.

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

\(\Big \downarrow \) 2845

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

\(\Big \downarrow \) 2858

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

input

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

output

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

3.1.51.3.1 Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2751

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x 
_Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* 
(n/d)   Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, 
x] && EqQ[r*(q + 1) + 1, 0]
 

rule 2756

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), 
x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] 
- Simp[b*n*(p/(e*(q + 1)))   Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 
 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, 
 -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] & 
& NeQ[q, 1]))
 

rule 2779

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

rule 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2845

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

rule 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

3.1.51.4 Maple [C] (warning: unable to verify)

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

Time = 2.19 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.38

method	result	size
risch	\(-\frac {b^{2} \ln \left (\left (e x +d \right )^{n}\right )^{2}}{3 \left (g x +f \right )^{3} g}-\frac {2 b^{2} n \,e^{3} \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (e x +d \right )}{3 g \left (d g -e f \right )^{3}}-\frac {b^{2} n e \ln \left (\left (e x +d \right )^{n}\right )}{3 g \left (d g -e f \right ) \left (g x +f \right )^{2}}+\frac {2 b^{2} n \,e^{3} \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{3 g \left (d g -e f \right )^{3}}+\frac {2 b^{2} n \,e^{2} \ln \left (\left (e x +d \right )^{n}\right )}{3 g \left (d g -e f \right )^{2} \left (g x +f \right )}+\frac {b^{2} n^{2} e^{3} \ln \left (e x +d \right )}{g \left (d g -e f \right )^{3}}-\frac {b^{2} n^{2} e^{2}}{3 g \left (d g -e f \right )^{2} \left (g x +f \right )}-\frac {b^{2} n^{2} e^{3} \ln \left (g x +f \right )}{g \left (d g -e f \right )^{3}}+\frac {b^{2} n^{2} e^{3} \ln \left (e x +d \right )^{2}}{3 g \left (d g -e f \right )^{3}}-\frac {2 b^{2} n^{2} e^{3} \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{3 g \left (d g -e f \right )^{3}}-\frac {2 b^{2} n^{2} e^{3} \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{3 g \left (d g -e f \right )^{3}}+\left (-i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )+i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b +i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b -i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b +2 b \ln \left (c \right )+2 a \right ) b \left (-\frac {\ln \left (\left (e x +d \right )^{n}\right )}{3 \left (g x +f \right )^{3} g}+\frac {n e \left (-\frac {e^{2} \ln \left (e x +d \right )}{\left (d g -e f \right )^{3}}-\frac {1}{2 \left (d g -e f \right ) \left (g x +f \right )^{2}}+\frac {e^{2} \ln \left (g x +f \right )}{\left (d g -e f \right )^{3}}+\frac {e}{\left (d g -e f \right )^{2} \left (g x +f \right )}\right )}{3 g}\right )-\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )+i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b +i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b -i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b +2 b \ln \left (c \right )+2 a \right )}^{2}}{12 \left (g x +f \right )^{3} g}\)	\(755\)

input

int((a+b*ln(c*(e*x+d)^n))^2/(g*x+f)^4,x,method=_RETURNVERBOSE)
 

output

-1/3*b^2*ln((e*x+d)^n)^2/(g*x+f)^3/g-2/3*b^2/g*n*e^3*ln((e*x+d)^n)/(d*g-e* 
f)^3*ln(e*x+d)-1/3*b^2/g*n*e*ln((e*x+d)^n)/(d*g-e*f)/(g*x+f)^2+2/3*b^2/g*n 
*e^3*ln((e*x+d)^n)/(d*g-e*f)^3*ln(g*x+f)+2/3*b^2/g*n*e^2*ln((e*x+d)^n)/(d* 
g-e*f)^2/(g*x+f)+b^2/g*n^2*e^3/(d*g-e*f)^3*ln(e*x+d)-1/3*b^2/g*n^2*e^2/(d* 
g-e*f)^2/(g*x+f)-b^2/g*n^2*e^3/(d*g-e*f)^3*ln(g*x+f)+1/3*b^2/g*n^2*e^3/(d* 
g-e*f)^3*ln(e*x+d)^2-2/3*b^2/g*n^2*e^3/(d*g-e*f)^3*dilog(((g*x+f)*e+d*g-e* 
f)/(d*g-e*f))-2/3*b^2/g*n^2*e^3/(d*g-e*f)^3*ln(g*x+f)*ln(((g*x+f)*e+d*g-e* 
f)/(d*g-e*f))+(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*P 
i*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d 
)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*b*ln(c)+2*a)*b*(-1/3*ln((e*x+d)^n) 
/(g*x+f)^3/g+1/3/g*n*e*(-e^2/(d*g-e*f)^3*ln(e*x+d)-1/2/(d*g-e*f)/(g*x+f)^2 
+e^2/(d*g-e*f)^3*ln(g*x+f)+e/(d*g-e*f)^2/(g*x+f)))-1/12*(-I*b*Pi*csgn(I*c* 
(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^ 
2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n 
)^3*b+2*b*ln(c)+2*a)^2/(g*x+f)^3/g
 

3.1.51.5 Fricas [F]

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

input

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

output

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

3.1.51.6 Sympy [F]

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

input

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

output

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

3.1.51.7 Maxima [F]

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

input

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

output

1/3*(2*e^2*log(e*x + d)/(e^3*f^3*g - 3*d*e^2*f^2*g^2 + 3*d^2*e*f*g^3 - d^3 
*g^4) - 2*e^2*log(g*x + f)/(e^3*f^3*g - 3*d*e^2*f^2*g^2 + 3*d^2*e*f*g^3 - 
d^3*g^4) + (2*e*g*x + 3*e*f - d*g)/(e^2*f^4*g - 2*d*e*f^3*g^2 + d^2*f^2*g^ 
3 + (e^2*f^2*g^3 - 2*d*e*f*g^4 + d^2*g^5)*x^2 + 2*(e^2*f^3*g^2 - 2*d*e*f^2 
*g^3 + d^2*f*g^4)*x))*a*b*e*n - 1/3*b^2*(log((e*x + d)^n)^2/(g^4*x^3 + 3*f 
*g^3*x^2 + 3*f^2*g^2*x + f^3*g) - 3*integrate(1/3*(3*e*g*x*log(c)^2 + 3*d* 
g*log(c)^2 + 2*(e*f*n + 3*d*g*log(c) + (e*g*n + 3*e*g*log(c))*x)*log((e*x 
+ d)^n))/(e*g^5*x^5 + d*f^4*g + (4*e*f*g^4 + d*g^5)*x^4 + 2*(3*e*f^2*g^3 + 
 2*d*f*g^4)*x^3 + 2*(2*e*f^3*g^2 + 3*d*f^2*g^3)*x^2 + (e*f^4*g + 4*d*f^3*g 
^2)*x), x)) - 2/3*a*b*log((e*x + d)^n*c)/(g^4*x^3 + 3*f*g^3*x^2 + 3*f^2*g^ 
2*x + f^3*g) - 1/3*a^2/(g^4*x^3 + 3*f*g^3*x^2 + 3*f^2*g^2*x + f^3*g)
 

3.1.51.8 Giac [F]

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

input

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

output

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

3.1.51.9 Mupad [F(-1)]

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

input

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

output

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