3.2.54 \(\int \frac {A+B \log (e (a+b x)^n (c+d x)^{-n})}{(a+b x)^4} \, dx\) [154]

3.2.54.1 Optimal result

Integrand size = 31, antiderivative size = 166 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx=-\frac {B n}{9 b (a+b x)^3}+\frac {B d n}{6 b (b c-a d) (a+b x)^2}-\frac {B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac {B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac {B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3} \]

output

-1/9*B*n/b/(b*x+a)^3+1/6*B*d*n/b/(-a*d+b*c)/(b*x+a)^2-1/3*B*d^2*n/b/(-a*d+ 
b*c)^2/(b*x+a)-1/3*B*d^3*n*ln(b*x+a)/b/(-a*d+b*c)^3+1/3*B*d^3*n*ln(d*x+c)/ 
b/(-a*d+b*c)^3+1/3*(-A-B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b/(b*x+a)^3
 

3.2.54.2 Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.86 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx=-\frac {\frac {6 A}{(a+b x)^3}+B n \left (\frac {2+\frac {3 d (a+b x)}{-b c+a d}+\frac {6 d^2 (a+b x)^2}{(b c-a d)^2}}{(a+b x)^3}+\frac {6 d^3 \log (a+b x)}{(b c-a d)^3}-\frac {6 d^3 \log (c+d x)}{(b c-a d)^3}\right )+\frac {6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3}}{18 b} \]

input

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(a + b*x)^4,x]
 

output

-1/18*((6*A)/(a + b*x)^3 + B*n*((2 + (3*d*(a + b*x))/(-(b*c) + a*d) + (6*d 
^2*(a + b*x)^2)/(b*c - a*d)^2)/(a + b*x)^3 + (6*d^3*Log[a + b*x])/(b*c - a 
*d)^3 - (6*d^3*Log[c + d*x])/(b*c - a*d)^3) + (6*B*Log[(e*(a + b*x)^n)/(c 
+ d*x)^n])/(a + b*x)^3)/b
 

3.2.54.3 Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2948, 54, 2009}

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.

\(\displaystyle \int \frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A}{(a+b x)^4} \, dx\)

\(\Big \downarrow \) 2948

\(\displaystyle \frac {B n (b c-a d) \int \frac {1}{(a+b x)^4 (c+d x)}dx}{3 b}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A}{3 b (a+b x)^3}\)

\(\Big \downarrow \) 54

\(\displaystyle \frac {B n (b c-a d) \int \left (\frac {d^4}{(b c-a d)^4 (c+d x)}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b}{(b c-a d) (a+b x)^4}\right )dx}{3 b}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A}{3 b (a+b x)^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {B n (b c-a d) \left (-\frac {d^3 \log (a+b x)}{(b c-a d)^4}+\frac {d^3 \log (c+d x)}{(b c-a d)^4}-\frac {d^2}{(a+b x) (b c-a d)^3}+\frac {d}{2 (a+b x)^2 (b c-a d)^2}-\frac {1}{3 (a+b x)^3 (b c-a d)}\right )}{3 b}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A}{3 b (a+b x)^3}\)

input

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(a + b*x)^4,x]
 

output

(B*(b*c - a*d)*n*(-1/3*1/((b*c - a*d)*(a + b*x)^3) + d/(2*(b*c - a*d)^2*(a 
 + b*x)^2) - d^2/((b*c - a*d)^3*(a + b*x)) - (d^3*Log[a + b*x])/(b*c - a*d 
)^4 + (d^3*Log[c + d*x])/(b*c - a*d)^4))/(3*b) - (A + B*Log[(e*(a + b*x)^n 
)/(c + d*x)^n])/(3*b*(a + b*x)^3)
 

3.2.54.3.1 Defintions of rubi rules used

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 2948

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*( 
(A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Simp[B*n*((b*c 
- a*d)/(g*(m + 1)))   Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] / 
; FreeQ[{a, b, c, d, e, f, g, A, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c 
- a*d, 0] && NeQ[m, -1] &&  !(EqQ[m, -2] && IntegerQ[n])
 

3.2.54.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(503\) vs. \(2(157)=314\).

Time = 54.43 (sec) , antiderivative size = 504, normalized size of antiderivative = 3.04

method	result	size
parallelrisch	\(-\frac {-18 A \,a^{2} b^{5} c \,d^{3}+18 A a \,b^{6} c^{2} d^{2}-6 B \ln \left (b x +a \right ) x^{3} b^{7} d^{4} n +6 B \,x^{2} a \,b^{6} d^{4} n -6 B \,x^{2} b^{7} c \,d^{3} n +15 B x \,a^{2} b^{5} d^{4} n +3 B x \,b^{7} c^{2} d^{2} n +6 B \ln \left (d x +c \right ) x^{3} b^{7} d^{4} n -6 B \ln \left (b x +a \right ) a^{3} b^{4} d^{4} n +6 B \ln \left (d x +c \right ) a^{3} b^{4} d^{4} n -18 B x a \,b^{6} c \,d^{3} n -18 B \ln \left (b x +a \right ) x^{2} a \,b^{6} d^{4} n +18 B \ln \left (d x +c \right ) x^{2} a \,b^{6} d^{4} n -18 B \ln \left (b x +a \right ) x \,a^{2} b^{5} d^{4} n +18 B \ln \left (d x +c \right ) x \,a^{2} b^{5} d^{4} n -18 B \,a^{2} b^{5} c \,d^{3} n +9 B a \,b^{6} c^{2} d^{2} n -18 B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) a^{2} b^{5} c \,d^{3}+18 B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) a \,b^{6} c^{2} d^{2}+6 A \,a^{3} b^{4} d^{4}-6 A \,b^{7} c^{3} d +6 B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) a^{3} b^{4} d^{4}-6 B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) b^{7} c^{3} d +11 B \,a^{3} b^{4} d^{4} n -2 B \,b^{7} c^{3} d n}{18 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (b x +a \right )^{3} b^{5} d}\)	\(504\)
risch	\(\text {Expression too large to display}\)	\(1976\)

input

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

output

-1/18*(-18*A*a^2*b^5*c*d^3+18*A*a*b^6*c^2*d^2+6*B*ln(e*(b*x+a)^n/((d*x+c)^ 
n))*a^3*b^4*d^4-6*B*ln(e*(b*x+a)^n/((d*x+c)^n))*b^7*c^3*d-6*B*ln(b*x+a)*x^ 
3*b^7*d^4*n+6*B*x^2*a*b^6*d^4*n-6*B*x^2*b^7*c*d^3*n+15*B*x*a^2*b^5*d^4*n+3 
*B*x*b^7*c^2*d^2*n-18*B*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2*b^5*c*d^3+18*B*ln( 
e*(b*x+a)^n/((d*x+c)^n))*a*b^6*c^2*d^2+6*B*ln(d*x+c)*x^3*b^7*d^4*n-6*B*ln( 
b*x+a)*a^3*b^4*d^4*n+6*B*ln(d*x+c)*a^3*b^4*d^4*n-18*B*x*a*b^6*c*d^3*n-18*B 
*ln(b*x+a)*x^2*a*b^6*d^4*n+18*B*ln(d*x+c)*x^2*a*b^6*d^4*n-18*B*ln(b*x+a)*x 
*a^2*b^5*d^4*n+18*B*ln(d*x+c)*x*a^2*b^5*d^4*n-18*B*a^2*b^5*c*d^3*n+9*B*a*b 
^6*c^2*d^2*n+6*A*a^3*b^4*d^4-6*A*b^7*c^3*d+11*B*a^3*b^4*d^4*n-2*B*b^7*c^3* 
d*n)/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(b*x+a)^3/b^5/d
 

3.2.54.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (154) = 308\).

Time = 0.29 (sec) , antiderivative size = 540, normalized size of antiderivative = 3.25 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx=-\frac {6 \, A b^{3} c^{3} - 18 \, A a b^{2} c^{2} d + 18 \, A a^{2} b c d^{2} - 6 \, A a^{3} d^{3} + 6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n x^{2} - 3 \, {\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} n x + {\left (2 \, B b^{3} c^{3} - 9 \, B a b^{2} c^{2} d + 18 \, B a^{2} b c d^{2} - 11 \, B a^{3} d^{3}\right )} n + 6 \, {\left (B b^{3} d^{3} n x^{3} + 3 \, B a b^{2} d^{3} n x^{2} + 3 \, B a^{2} b d^{3} n x + {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} n\right )} \log \left (b x + a\right ) - 6 \, {\left (B b^{3} d^{3} n x^{3} + 3 \, B a b^{2} d^{3} n x^{2} + 3 \, B a^{2} b d^{3} n x + {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} n\right )} \log \left (d x + c\right ) + 6 \, {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2} - B a^{3} d^{3}\right )} \log \left (e\right )}{18 \, {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3} + {\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} x^{3} + 3 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} x\right )}} \]

input

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))/(b*x+a)^4,x, algorithm="frica 
s")
 

output

-1/18*(6*A*b^3*c^3 - 18*A*a*b^2*c^2*d + 18*A*a^2*b*c*d^2 - 6*A*a^3*d^3 + 6 
*(B*b^3*c*d^2 - B*a*b^2*d^3)*n*x^2 - 3*(B*b^3*c^2*d - 6*B*a*b^2*c*d^2 + 5* 
B*a^2*b*d^3)*n*x + (2*B*b^3*c^3 - 9*B*a*b^2*c^2*d + 18*B*a^2*b*c*d^2 - 11* 
B*a^3*d^3)*n + 6*(B*b^3*d^3*n*x^3 + 3*B*a*b^2*d^3*n*x^2 + 3*B*a^2*b*d^3*n* 
x + (B*b^3*c^3 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2)*n)*log(b*x + a) - 6*(B 
*b^3*d^3*n*x^3 + 3*B*a*b^2*d^3*n*x^2 + 3*B*a^2*b*d^3*n*x + (B*b^3*c^3 - 3* 
B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2)*n)*log(d*x + c) + 6*(B*b^3*c^3 - 3*B*a*b^ 
2*c^2*d + 3*B*a^2*b*c*d^2 - B*a^3*d^3)*log(e))/(a^3*b^4*c^3 - 3*a^4*b^3*c^ 
2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3 + (b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c 
*d^2 - a^3*b^4*d^3)*x^3 + 3*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 
 - a^4*b^3*d^3)*x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - 
 a^5*b^2*d^3)*x)
 

3.2.54.6 Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx=\text {Timed out} \]

input

integrate((A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))/(b*x+a)**4,x)
 

output

Timed out
 

3.2.54.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (154) = 308\).

Time = 0.20 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.41 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx=-\frac {{\left (\frac {6 \, d^{3} e n \log \left (b x + a\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac {6 \, d^{3} e n \log \left (d x + c\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} + \frac {6 \, b^{2} d^{2} e n x^{2} + 2 \, b^{2} c^{2} e n - 7 \, a b c d e n + 11 \, a^{2} d^{2} e n - 3 \, {\left (b^{2} c d e n - 5 \, a b d^{2} e n\right )} x}{a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{3} + 3 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{2} + 3 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x}\right )} B}{18 \, e} - \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac {A}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} \]

input

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))/(b*x+a)^4,x, algorithm="maxim 
a")
 

output

-1/18*(6*d^3*e*n*log(b*x + a)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - 
 a^3*b*d^3) - 6*d^3*e*n*log(d*x + c)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2* 
c*d^2 - a^3*b*d^3) + (6*b^2*d^2*e*n*x^2 + 2*b^2*c^2*e*n - 7*a*b*c*d*e*n + 
11*a^2*d^2*e*n - 3*(b^2*c*d*e*n - 5*a*b*d^2*e*n)*x)/(a^3*b^3*c^2 - 2*a^4*b 
^2*c*d + a^5*b*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^3 + 3*(a*b^5* 
c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + 
a^4*b^2*d^2)*x))*B/e - 1/3*B*log((b*x + a)^n*e/(d*x + c)^n)/(b^4*x^3 + 3*a 
*b^3*x^2 + 3*a^2*b^2*x + a^3*b) - 1/3*A/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2 
*x + a^3*b)
 

3.2.54.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (154) = 308\).

Time = 0.29 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.73 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx=-\frac {B d^{3} n \log \left (b x + a\right )}{3 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} + \frac {B d^{3} n \log \left (d x + c\right )}{3 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} - \frac {B n \log \left (b x + a\right )}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} + \frac {B n \log \left (d x + c\right )}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac {6 \, B b^{2} d^{2} n x^{2} - 3 \, B b^{2} c d n x + 15 \, B a b d^{2} n x + 2 \, B b^{2} c^{2} n - 7 \, B a b c d n + 11 \, B a^{2} d^{2} n + 6 \, B b^{2} c^{2} \log \left (e\right ) - 12 \, B a b c d \log \left (e\right ) + 6 \, B a^{2} d^{2} \log \left (e\right ) + 6 \, A b^{2} c^{2} - 12 \, A a b c d + 6 \, A a^{2} d^{2}}{18 \, {\left (b^{6} c^{2} x^{3} - 2 \, a b^{5} c d x^{3} + a^{2} b^{4} d^{2} x^{3} + 3 \, a b^{5} c^{2} x^{2} - 6 \, a^{2} b^{4} c d x^{2} + 3 \, a^{3} b^{3} d^{2} x^{2} + 3 \, a^{2} b^{4} c^{2} x - 6 \, a^{3} b^{3} c d x + 3 \, a^{4} b^{2} d^{2} x + a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )}} \]

input

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

output

-1/3*B*d^3*n*log(b*x + a)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3 
*b*d^3) + 1/3*B*d^3*n*log(d*x + c)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c* 
d^2 - a^3*b*d^3) - 1/3*B*n*log(b*x + a)/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2 
*x + a^3*b) + 1/3*B*n*log(d*x + c)/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + 
a^3*b) - 1/18*(6*B*b^2*d^2*n*x^2 - 3*B*b^2*c*d*n*x + 15*B*a*b*d^2*n*x + 2* 
B*b^2*c^2*n - 7*B*a*b*c*d*n + 11*B*a^2*d^2*n + 6*B*b^2*c^2*log(e) - 12*B*a 
*b*c*d*log(e) + 6*B*a^2*d^2*log(e) + 6*A*b^2*c^2 - 12*A*a*b*c*d + 6*A*a^2* 
d^2)/(b^6*c^2*x^3 - 2*a*b^5*c*d*x^3 + a^2*b^4*d^2*x^3 + 3*a*b^5*c^2*x^2 - 
6*a^2*b^4*c*d*x^2 + 3*a^3*b^3*d^2*x^2 + 3*a^2*b^4*c^2*x - 6*a^3*b^3*c*d*x 
+ 3*a^4*b^2*d^2*x + a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)
 

3.2.54.9 Mupad [B] (verification not implemented)

Time = 1.58 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.91 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx=\frac {2\,A\,a\,c\,d}{3\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {A\,b\,c^2}{3\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}{3\,b\,{\left (a+b\,x\right )}^3}-\frac {A\,a^2\,d^2}{3\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,b\,c^2\,n}{9\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {5\,B\,a\,d^2\,n\,x}{6\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,b\,d^2\,n\,x^2}{3\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {7\,B\,a\,c\,d\,n}{18\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {11\,B\,a^2\,d^2\,n}{18\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {B\,b\,c\,d\,n\,x}{6\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,d^3\,n\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{3\,b\,{\left (a\,d-b\,c\right )}^3} \]

input

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))/(a + b*x)^4,x)
 

output

(2*A*a*c*d)/(3*(a*d - b*c)^2*(a + b*x)^3) - (A*b*c^2)/(3*(a*d - b*c)^2*(a 
+ b*x)^3) - (B*log((e*(a + b*x)^n)/(c + d*x)^n))/(3*b*(a + b*x)^3) - (A*a^ 
2*d^2)/(3*b*(a*d - b*c)^2*(a + b*x)^3) - (B*b*c^2*n)/(9*(a*d - b*c)^2*(a + 
 b*x)^3) - (B*d^3*n*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*2i)/(3* 
b*(a*d - b*c)^3) - (5*B*a*d^2*n*x)/(6*(a*d - b*c)^2*(a + b*x)^3) - (B*b*d^ 
2*n*x^2)/(3*(a*d - b*c)^2*(a + b*x)^3) + (7*B*a*c*d*n)/(18*(a*d - b*c)^2*( 
a + b*x)^3) - (11*B*a^2*d^2*n)/(18*b*(a*d - b*c)^2*(a + b*x)^3) + (B*b*c*d 
*n*x)/(6*(a*d - b*c)^2*(a + b*x)^3)