3.3.28 \(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^3}{(a+b x) (c+d x)} \, dx\) [228]

3.3.28.1 Optimal result

Integrand size = 40, antiderivative size = 45 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x) (c+d x)} \, dx=\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^4}{4 B (b c-a d) n} \]

output

1/4*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^4/B/(-a*d+b*c)/n
 

3.3.28.2 Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x) (c+d x)} \, dx=\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^4}{4 (b B c n-a B d n)} \]

input

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/((a + b*x)*(c + d*x)) 
,x]
 

output

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

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

Time = 0.45 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2973, 2961, 2739, 15}

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.

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

\(\Big \downarrow \) 2973

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2739

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {(a+b x)^4}{4 B n (c+d x)^4 (b c-a d)}\)

input

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/((a + b*x)*(c + d*x)),x]
 

output

(a + b*x)^4/(4*B*(b*c - a*d)*n*(c + d*x)^4)
 

3.3.28.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 2739

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[1/( 
b*n)   Subst[Int[x^p, x], x, a + b*Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p} 
, x]
 

rule 2961

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + B*L 
og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ 
b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
 

rule 2973

Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] 
 :> Subst[Int[w*(A + B*Log[e*(u/v)^n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; Fr 
eeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !Intege 
rQ[n]
 

3.3.28.4 Maple [A] (verified)

Time = 33.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(-\frac {{\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{4}}{4 n \left (a d -c b \right ) B}\)	\(44\)
default	\(-\frac {{\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{4}}{4 n \left (a d -c b \right ) B}\)	\(44\)
parallelrisch	\(-\frac {B^{3} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{4} b^{2} d^{2}+4 A \,B^{2} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3} b^{2} d^{2}+6 A^{2} B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2} b^{2} d^{2}+4 A^{3} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) b^{2} d^{2}}{4 n \,b^{2} d^{2} \left (a d -c b \right )}\)	\(150\)
parts	\(\frac {A^{3} \ln \left (d x +c \right )}{a d -c b}-\frac {A^{3} \ln \left (b x +a \right )}{a d -c b}-\frac {B^{3} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{4}}{4 n \left (a d -c b \right )}-\frac {A \,B^{2} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}}{n \left (a d -c b \right )}-\frac {3 A^{2} B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}}{2 n \left (a d -c b \right )}\)	\(162\)
risch	\(\text {Expression too large to display}\)	\(64288\)

input

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)/(d*x+c),x,method=_RETURNVE 
RBOSE)
 

output

-1/4/n/(a*d-b*c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^4/B
 

3.3.28.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (43) = 86\).

Time = 0.32 (sec) , antiderivative size = 375, normalized size of antiderivative = 8.33 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x) (c+d x)} \, dx=\frac {B^{3} n^{3} \log \left (b x + a\right )^{4} + B^{3} n^{3} \log \left (d x + c\right )^{4} + 4 \, {\left (B^{3} n^{2} \log \left (e\right ) + A B^{2} n^{2}\right )} \log \left (b x + a\right )^{3} - 4 \, {\left (B^{3} n^{3} \log \left (b x + a\right ) + B^{3} n^{2} \log \left (e\right ) + A B^{2} n^{2}\right )} \log \left (d x + c\right )^{3} + 6 \, {\left (B^{3} n \log \left (e\right )^{2} + 2 \, A B^{2} n \log \left (e\right ) + A^{2} B n\right )} \log \left (b x + a\right )^{2} + 6 \, {\left (B^{3} n^{3} \log \left (b x + a\right )^{2} + B^{3} n \log \left (e\right )^{2} + 2 \, A B^{2} n \log \left (e\right ) + A^{2} B n + 2 \, {\left (B^{3} n^{2} \log \left (e\right ) + A B^{2} n^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )^{2} + 4 \, {\left (B^{3} \log \left (e\right )^{3} + 3 \, A B^{2} \log \left (e\right )^{2} + 3 \, A^{2} B \log \left (e\right ) + A^{3}\right )} \log \left (b x + a\right ) - 4 \, {\left (B^{3} n^{3} \log \left (b x + a\right )^{3} + B^{3} \log \left (e\right )^{3} + 3 \, A B^{2} \log \left (e\right )^{2} + 3 \, A^{2} B \log \left (e\right ) + A^{3} + 3 \, {\left (B^{3} n^{2} \log \left (e\right ) + A B^{2} n^{2}\right )} \log \left (b x + a\right )^{2} + 3 \, {\left (B^{3} n \log \left (e\right )^{2} + 2 \, A B^{2} n \log \left (e\right ) + A^{2} B n\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{4 \, {\left (b c - a d\right )}} \]

input

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)/(d*x+c),x, algorith 
m="fricas")
 

output

1/4*(B^3*n^3*log(b*x + a)^4 + B^3*n^3*log(d*x + c)^4 + 4*(B^3*n^2*log(e) + 
 A*B^2*n^2)*log(b*x + a)^3 - 4*(B^3*n^3*log(b*x + a) + B^3*n^2*log(e) + A* 
B^2*n^2)*log(d*x + c)^3 + 6*(B^3*n*log(e)^2 + 2*A*B^2*n*log(e) + A^2*B*n)* 
log(b*x + a)^2 + 6*(B^3*n^3*log(b*x + a)^2 + B^3*n*log(e)^2 + 2*A*B^2*n*lo 
g(e) + A^2*B*n + 2*(B^3*n^2*log(e) + A*B^2*n^2)*log(b*x + a))*log(d*x + c) 
^2 + 4*(B^3*log(e)^3 + 3*A*B^2*log(e)^2 + 3*A^2*B*log(e) + A^3)*log(b*x + 
a) - 4*(B^3*n^3*log(b*x + a)^3 + B^3*log(e)^3 + 3*A*B^2*log(e)^2 + 3*A^2*B 
*log(e) + A^3 + 3*(B^3*n^2*log(e) + A*B^2*n^2)*log(b*x + a)^2 + 3*(B^3*n*l 
og(e)^2 + 2*A*B^2*n*log(e) + A^2*B*n)*log(b*x + a))*log(d*x + c))/(b*c - a 
*d)
 

3.3.28.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x) (c+d x)} \, dx=\text {Timed out} \]

input

integrate((A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**3/(b*x+a)/(d*x+c),x)
 

output

Timed out
 

3.3.28.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 766 vs. \(2 (43) = 86\).

Time = 0.25 (sec) , antiderivative size = 766, normalized size of antiderivative = 17.02 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x) (c+d x)} \, dx=B^{3} {\left (\frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{3} + 3 \, A B^{2} {\left (\frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 3 \, A^{2} B {\left (\frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) - \frac {1}{4} \, B^{3} {\left (\frac {6 \, {\left (e n \log \left (b x + a\right )^{2} - 2 \, e n \log \left (b x + a\right ) \log \left (d x + c\right ) + e n \log \left (d x + c\right )^{2}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2}}{{\left (b c - a d\right )} e} - \frac {\frac {4 \, {\left (e^{2} n^{2} \log \left (b x + a\right )^{3} - 3 \, e^{2} n^{2} \log \left (b x + a\right )^{2} \log \left (d x + c\right ) + 3 \, e^{2} n^{2} \log \left (b x + a\right ) \log \left (d x + c\right )^{2} - e^{2} n^{2} \log \left (d x + c\right )^{3}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{{\left (b c - a d\right )} e} - \frac {e^{3} n^{3} \log \left (b x + a\right )^{4} - 4 \, e^{3} n^{3} \log \left (b x + a\right )^{3} \log \left (d x + c\right ) + 6 \, e^{3} n^{3} \log \left (b x + a\right )^{2} \log \left (d x + c\right )^{2} - 4 \, e^{3} n^{3} \log \left (b x + a\right ) \log \left (d x + c\right )^{3} + e^{3} n^{3} \log \left (d x + c\right )^{4}}{{\left (b c - a d\right )} e^{2}}}{e}\right )} + A^{3} {\left (\frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d}\right )} - A B^{2} {\left (\frac {3 \, {\left (e n \log \left (b x + a\right )^{2} - 2 \, e n \log \left (b x + a\right ) \log \left (d x + c\right ) + e n \log \left (d x + c\right )^{2}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{{\left (b c - a d\right )} e} - \frac {e^{2} n^{2} \log \left (b x + a\right )^{3} - 3 \, e^{2} n^{2} \log \left (b x + a\right )^{2} \log \left (d x + c\right ) + 3 \, e^{2} n^{2} \log \left (b x + a\right ) \log \left (d x + c\right )^{2} - e^{2} n^{2} \log \left (d x + c\right )^{3}}{{\left (b c - a d\right )} e^{2}}\right )} - \frac {3 \, {\left (e n \log \left (b x + a\right )^{2} - 2 \, e n \log \left (b x + a\right ) \log \left (d x + c\right ) + e n \log \left (d x + c\right )^{2}\right )} A^{2} B}{2 \, {\left (b c - a d\right )} e} \]

input

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)/(d*x+c),x, algorith 
m="maxima")
 

output

B^3*(log(b*x + a)/(b*c - a*d) - log(d*x + c)/(b*c - a*d))*log((b*x + a)^n* 
e/(d*x + c)^n)^3 + 3*A*B^2*(log(b*x + a)/(b*c - a*d) - log(d*x + c)/(b*c - 
 a*d))*log((b*x + a)^n*e/(d*x + c)^n)^2 + 3*A^2*B*(log(b*x + a)/(b*c - a*d 
) - log(d*x + c)/(b*c - a*d))*log((b*x + a)^n*e/(d*x + c)^n) - 1/4*B^3*(6* 
(e*n*log(b*x + a)^2 - 2*e*n*log(b*x + a)*log(d*x + c) + e*n*log(d*x + c)^2 
)*log((b*x + a)^n*e/(d*x + c)^n)^2/((b*c - a*d)*e) - (4*(e^2*n^2*log(b*x + 
 a)^3 - 3*e^2*n^2*log(b*x + a)^2*log(d*x + c) + 3*e^2*n^2*log(b*x + a)*log 
(d*x + c)^2 - e^2*n^2*log(d*x + c)^3)*log((b*x + a)^n*e/(d*x + c)^n)/((b*c 
 - a*d)*e) - (e^3*n^3*log(b*x + a)^4 - 4*e^3*n^3*log(b*x + a)^3*log(d*x + 
c) + 6*e^3*n^3*log(b*x + a)^2*log(d*x + c)^2 - 4*e^3*n^3*log(b*x + a)*log( 
d*x + c)^3 + e^3*n^3*log(d*x + c)^4)/((b*c - a*d)*e^2))/e) + A^3*(log(b*x 
+ a)/(b*c - a*d) - log(d*x + c)/(b*c - a*d)) - A*B^2*(3*(e*n*log(b*x + a)^ 
2 - 2*e*n*log(b*x + a)*log(d*x + c) + e*n*log(d*x + c)^2)*log((b*x + a)^n* 
e/(d*x + c)^n)/((b*c - a*d)*e) - (e^2*n^2*log(b*x + a)^3 - 3*e^2*n^2*log(b 
*x + a)^2*log(d*x + c) + 3*e^2*n^2*log(b*x + a)*log(d*x + c)^2 - e^2*n^2*l 
og(d*x + c)^3)/((b*c - a*d)*e^2)) - 3/2*(e*n*log(b*x + a)^2 - 2*e*n*log(b* 
x + a)*log(d*x + c) + e*n*log(d*x + c)^2)*A^2*B/((b*c - a*d)*e)
 

3.3.28.8 Giac [F]

\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x) (c+d x)} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]

input

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)/(d*x+c),x, algorith 
m="giac")
 

output

integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)^3/((b*x + a)*(d*x + c)), 
x)
 

3.3.28.9 Mupad [B] (verification not implemented)

Time = 2.61 (sec) , antiderivative size = 141, normalized size of antiderivative = 3.13 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x) (c+d x)} \, dx=-\frac {\frac {3\,A^2\,B\,{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^2}{2}+A\,B^2\,{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^3+\frac {B^3\,{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^4}{4}}{n\,\left (a\,d-b\,c\right )}+\frac {A^3\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{a\,d-b\,c} \]

input

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^3/((a + b*x)*(c + d*x)),x)
 

output

(A^3*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*2i)/(a*d - b*c) - ((B^ 
3*log((e*(a + b*x)^n)/(c + d*x)^n)^4)/4 + (3*A^2*B*log((e*(a + b*x)^n)/(c 
+ d*x)^n)^2)/2 + A*B^2*log((e*(a + b*x)^n)/(c + d*x)^n)^3)/(n*(a*d - b*c))