\(\int (a g+b g x) (c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\) [22]

Optimal result

Integrand size = 38, antiderivative size = 271 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {B (b c-a d)^4 g i^3 x}{20 b^3 d}+\frac {B (b c-a d)^3 g i^3 (c+d x)^2}{40 b^2 d^2}+\frac {B (b c-a d)^2 g i^3 (c+d x)^3}{60 b d^2}-\frac {B (b c-a d) g i^3 (c+d x)^4}{20 d^2}+\frac {B (b c-a d)^5 g i^3 \log \left (\frac {a+b x}{c+d x}\right )}{20 b^4 d^2}-\frac {(b c-a d) g i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d^2}+\frac {b g i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}+\frac {B (b c-a d)^5 g i^3 \log (c+d x)}{20 b^4 d^2} \] Output:

1/20*B*(-a*d+b*c)^4*g*i^3*x/b^3/d+1/40*B*(-a*d+b*c)^3*g*i^3*(d*x+c)^2/b^2/ 
d^2+1/60*B*(-a*d+b*c)^2*g*i^3*(d*x+c)^3/b/d^2-1/20*B*(-a*d+b*c)*g*i^3*(d*x 
+c)^4/d^2+1/20*B*(-a*d+b*c)^5*g*i^3*ln((b*x+a)/(d*x+c))/b^4/d^2-1/4*(-a*d+ 
b*c)*g*i^3*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^2+1/5*b*g*i^3*(d*x+c)^5 
*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^2+1/20*B*(-a*d+b*c)^5*g*i^3*ln(d*x+c)/b^4/d 
^2
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.96 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g i^3 \left (\frac {5 B (b c-a d)^2 \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )}{b^4}-\frac {2 B (b c-a d) \left (12 b d (b c-a d)^3 x+6 b^2 (b c-a d)^2 (c+d x)^2+4 b^3 (b c-a d) (c+d x)^3+3 b^4 (c+d x)^4+12 (b c-a d)^4 \log (a+b x)\right )}{b^4}-30 (b c-a d) (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+24 b (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right )}{120 d^2} \] Input:

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

Output:

(g*i^3*((5*B*(b*c - a*d)^2*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c - a*d)*(c + 
 d*x)^2 + 2*b^3*(c + d*x)^3 + 6*(b*c - a*d)^3*Log[a + b*x]))/b^4 - (2*B*(b 
*c - a*d)*(12*b*d*(b*c - a*d)^3*x + 6*b^2*(b*c - a*d)^2*(c + d*x)^2 + 4*b^ 
3*(b*c - a*d)*(c + d*x)^3 + 3*b^4*(c + d*x)^4 + 12*(b*c - a*d)^4*Log[a + b 
*x]))/b^4 - 30*(b*c - a*d)*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)] 
) + 24*b*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])))/(120*d^2)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2962, 2782, 27, 86, 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.

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

\(\Big \downarrow \) 2962

\(\displaystyle g i^3 (b c-a d)^5 \int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}\)

\(\Big \downarrow \) 2782

\(\displaystyle g i^3 (b c-a d)^5 \left (-B \int -\frac {(c+d x) \left (b-\frac {5 d (a+b x)}{c+d x}\right )}{20 d^2 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle g i^3 (b c-a d)^5 \left (\frac {B \int \frac {(c+d x) \left (b-\frac {5 d (a+b x)}{c+d x}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{20 d^2}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 86

\(\displaystyle g i^3 (b c-a d)^5 \left (\frac {B \int \left (\frac {d}{b^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {4 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {c+d x}{b^4 (a+b x)}\right )d\frac {a+b x}{c+d x}}{20 d^2}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle g i^3 (b c-a d)^5 \left (-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^4}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^4}+\frac {1}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{3 b \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{20 d^2}\right )\)

Input:

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

Output:

(b*c - a*d)^5*g*i^3*((b*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*d^2*(b - 
(d*(a + b*x))/(c + d*x))^5) - (A + B*Log[(e*(a + b*x))/(c + d*x)])/(4*d^2* 
(b - (d*(a + b*x))/(c + d*x))^4) + (B*(-(b - (d*(a + b*x))/(c + d*x))^(-4) 
 + 1/(3*b*(b - (d*(a + b*x))/(c + d*x))^3) + 1/(2*b^2*(b - (d*(a + b*x))/( 
c + d*x))^2) + 1/(b^3*(b - (d*(a + b*x))/(c + d*x))) + Log[(a + b*x)/(c + 
d*x)]/b^4 - Log[b - (d*(a + b*x))/(c + d*x)]/b^4))/(20*d^2))
 

Defintions of rubi rules used

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 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 2009

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

rule 2782

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q 
_), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x)^q, x]}, Simp[(a + b*Log[c* 
x^n])   u, x] - Simp[b*n   Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[ 
{a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]
 

rule 2962

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy 
mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + 
 B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; 
 FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt 
Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I 
ntegersQ[m, q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(567\) vs. \(2(255)=510\).

Time = 1.44 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.10

method	result	size
risch	\(-\frac {i^{3} g B a \,c^{3} x}{4}+\frac {i^{3} g \,d^{2} B \ln \left (b x +a \right ) a^{4} c}{4 b^{3}}-\frac {i^{3} g d B \ln \left (b x +a \right ) a^{3} c^{2}}{2 b^{2}}+i^{3} g A a \,c^{3} x +\frac {i^{3} g \,d^{2} B a c \,x^{3}}{6}+\frac {3 i^{3} g d A a \,c^{2} x^{2}}{2}+\frac {i^{3} g \,d^{2} B \,a^{2} c \,x^{2}}{8 b}+\frac {i^{3} g d B a \,c^{2} x^{2}}{8}+\frac {i^{3} g \,d^{3} A a \,x^{4}}{4}+i^{3} g \,d^{2} A a c \,x^{3}-\frac {i^{3} g \,d^{2} B \,a^{3} c x}{4 b^{2}}+\frac {i^{3} g d B \,a^{2} c^{2} x}{2 b}-\frac {i^{3} g b B \,c^{4} x}{20 d}-\frac {i^{3} g B \ln \left (-d x -c \right ) a \,c^{4}}{4 d}+\frac {i^{3} g B \ln \left (b x +a \right ) a^{2} c^{3}}{2 b}+\frac {i^{3} g b B \ln \left (-d x -c \right ) c^{5}}{20 d^{2}}+\frac {3 i^{3} g b \,d^{2} A c \,x^{4}}{4}+\frac {i^{3} g \,d^{3} B a \,x^{4}}{20}-\frac {i^{3} g b \,d^{2} B c \,x^{4}}{20}+i^{3} g b d A \,c^{2} x^{3}+\frac {i^{3} g \,d^{3} B \,a^{2} x^{3}}{60 b}-\frac {11 i^{3} g b d B \,c^{2} x^{3}}{60}+\frac {i^{3} g b A \,c^{3} x^{2}}{2}-\frac {i^{3} g \,d^{3} B \,a^{3} x^{2}}{40 b^{2}}-\frac {9 i^{3} g b B \,c^{3} x^{2}}{40}+\frac {i^{3} g \,d^{3} B \,a^{4} x}{20 b^{3}}-\frac {i^{3} g \,d^{3} B \ln \left (b x +a \right ) a^{5}}{20 b^{4}}+\frac {g \,i^{3} B x \left (4 d^{3} b \,x^{4}+5 x^{3} a \,d^{3}+15 b c \,d^{2} x^{3}+20 a c \,d^{2} x^{2}+20 b \,c^{2} d \,x^{2}+30 a \,c^{2} d x +10 b \,c^{3} x +20 c^{3} a \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{20}+\frac {i^{3} g b \,d^{3} A \,x^{5}}{5}\)	\(568\)
parallelrisch	\(\frac {27 B \,a^{4} b c \,d^{4} g \,i^{3}-45 B \,a^{3} b^{2} c^{2} d^{3} g \,i^{3}-45 B \,a^{2} b^{3} c^{3} d^{2} g \,i^{3}+63 B a \,b^{4} c^{4} d g \,i^{3}+120 A x a \,b^{4} c^{3} d^{2} g \,i^{3}-300 A \,a^{2} b^{3} c^{3} d^{2} g \,i^{3}-180 A a \,b^{4} c^{4} d g \,i^{3}-6 B \,a^{5} d^{5} g \,i^{3}+6 B \,b^{5} c^{5} g \,i^{3}+6 B \,x^{4} a \,b^{4} d^{5} g \,i^{3}-6 B \,x^{4} b^{5} c \,d^{4} g \,i^{3}+120 A \,x^{3} b^{5} c^{2} d^{3} g \,i^{3}+2 B \,x^{3} a^{2} b^{3} d^{5} g \,i^{3}-60 B \ln \left (b x +a \right ) a^{3} b^{2} c^{2} d^{3} g \,i^{3}-30 B x \,a^{3} b^{2} c \,d^{4} g \,i^{3}+60 B x \,a^{2} b^{3} c^{2} d^{3} g \,i^{3}+120 A \,x^{3} a \,b^{4} c \,d^{4} g \,i^{3}+20 B \,x^{3} a \,b^{4} c \,d^{4} g \,i^{3}+60 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c^{3} d^{2} g \,i^{3}+180 A \,x^{2} a \,b^{4} c^{2} d^{3} g \,i^{3}-3 B \,x^{2} a^{3} b^{2} d^{5} g \,i^{3}-27 B \,x^{2} b^{5} c^{3} d^{2} g \,i^{3}+6 B x \,a^{4} b \,d^{5} g \,i^{3}-6 B x \,b^{5} c^{4} d g \,i^{3}+15 B \,x^{2} a^{2} b^{3} c \,d^{4} g \,i^{3}+15 B \,x^{2} a \,b^{4} c^{2} d^{3} g \,i^{3}-30 B x a \,b^{4} c^{3} d^{2} g \,i^{3}+30 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{4} c^{4} d g \,i^{3}+30 B \ln \left (b x +a \right ) a^{4} b c \,d^{4} g \,i^{3}-22 B \,x^{3} b^{5} c^{2} d^{3} g \,i^{3}+60 A \,x^{2} b^{5} c^{3} d^{2} g \,i^{3}+24 B \,x^{5} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} d^{5} g \,i^{3}+30 A \,x^{4} a \,b^{4} d^{5} g \,i^{3}+90 A \,x^{4} b^{5} c \,d^{4} g \,i^{3}+60 B \ln \left (b x +a \right ) a^{2} b^{3} c^{3} d^{2} g \,i^{3}-30 B \ln \left (b x +a \right ) a \,b^{4} c^{4} d g \,i^{3}+30 B \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{4} d^{5} g \,i^{3}+90 B \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c \,d^{4} g \,i^{3}+120 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c^{2} d^{3} g \,i^{3}+24 A \,x^{5} b^{5} d^{5} g \,i^{3}-6 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c^{5} g \,i^{3}-6 B \ln \left (b x +a \right ) a^{5} d^{5} g \,i^{3}+6 B \ln \left (b x +a \right ) b^{5} c^{5} g \,i^{3}+120 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{4} c \,d^{4} g \,i^{3}+180 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{4} c^{2} d^{3} g \,i^{3}+120 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{4} c^{3} d^{2} g \,i^{3}}{120 b^{4} d^{2}}\)	\(988\)
parts	\(\text {Expression too large to display}\)	\(1019\)
derivativedivides	\(\text {Expression too large to display}\)	\(1124\)
default	\(\text {Expression too large to display}\)	\(1124\)

Input:

int((b*g*x+a*g)*(d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c))),x,method=_RETURN 
VERBOSE)
 

Output:

-1/4*i^3*g*B*a*c^3*x+1/4*i^3*g/b^3*d^2*B*ln(b*x+a)*a^4*c-1/2*i^3*g/b^2*d*B 
*ln(b*x+a)*a^3*c^2+i^3*g*A*a*c^3*x+1/6*i^3*g*d^2*B*a*c*x^3+3/2*i^3*g*d*A*a 
*c^2*x^2+1/8*i^3*g/b*d^2*B*a^2*c*x^2+1/8*i^3*g*d*B*a*c^2*x^2+1/4*i^3*g*d^3 
*A*a*x^4+i^3*g*d^2*A*a*c*x^3-1/4*i^3*g/b^2*d^2*B*a^3*c*x+1/2*i^3*g/b*d*B*a 
^2*c^2*x-1/20*i^3*g*b/d*B*c^4*x-1/4*i^3*g/d*B*ln(-d*x-c)*a*c^4+1/2*i^3*g/b 
*B*ln(b*x+a)*a^2*c^3+1/20*i^3*g*b/d^2*B*ln(-d*x-c)*c^5+3/4*i^3*g*b*d^2*A*c 
*x^4+1/20*i^3*g*d^3*B*a*x^4-1/20*i^3*g*b*d^2*B*c*x^4+i^3*g*b*d*A*c^2*x^3+1 
/60*i^3*g/b*d^3*B*a^2*x^3-11/60*i^3*g*b*d*B*c^2*x^3+1/2*i^3*g*b*A*c^3*x^2- 
1/40*i^3*g/b^2*d^3*B*a^3*x^2-9/40*i^3*g*b*B*c^3*x^2+1/20*i^3*g/b^3*d^3*B*a 
^4*x-1/20*i^3*g/b^4*d^3*B*ln(b*x+a)*a^5+1/20*g*i^3*B*x*(4*b*d^3*x^4+5*a*d^ 
3*x^3+15*b*c*d^2*x^3+20*a*c*d^2*x^2+20*b*c^2*d*x^2+30*a*c^2*d*x+10*b*c^3*x 
+20*a*c^3)*ln(e*(b*x+a)/(d*x+c))+1/5*i^3*g*b*d^3*A*x^5
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.85 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {24 \, A b^{5} d^{5} g i^{3} x^{5} + 6 \, {\left ({\left (15 \, A - B\right )} b^{5} c d^{4} + {\left (5 \, A + B\right )} a b^{4} d^{5}\right )} g i^{3} x^{4} + 2 \, {\left ({\left (60 \, A - 11 \, B\right )} b^{5} c^{2} d^{3} + 10 \, {\left (6 \, A + B\right )} a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g i^{3} x^{3} + 3 \, {\left ({\left (20 \, A - 9 \, B\right )} b^{5} c^{3} d^{2} + 5 \, {\left (12 \, A + B\right )} a b^{4} c^{2} d^{3} + 5 \, B a^{2} b^{3} c d^{4} - B a^{3} b^{2} d^{5}\right )} g i^{3} x^{2} - 6 \, {\left (B b^{5} c^{4} d - 5 \, {\left (4 \, A - B\right )} a b^{4} c^{3} d^{2} - 10 \, B a^{2} b^{3} c^{2} d^{3} + 5 \, B a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g i^{3} x + 6 \, {\left (10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4} - B a^{5} d^{5}\right )} g i^{3} \log \left (b x + a\right ) + 6 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d\right )} g i^{3} \log \left (d x + c\right ) + 6 \, {\left (4 \, B b^{5} d^{5} g i^{3} x^{5} + 20 \, B a b^{4} c^{3} d^{2} g i^{3} x + 5 \, {\left (3 \, B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g i^{3} x^{4} + 20 \, {\left (B b^{5} c^{2} d^{3} + B a b^{4} c d^{4}\right )} g i^{3} x^{3} + 10 \, {\left (B b^{5} c^{3} d^{2} + 3 \, B a b^{4} c^{2} d^{3}\right )} g i^{3} x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{120 \, b^{4} d^{2}} \] Input:

integrate((b*g*x+a*g)*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algori 
thm="fricas")
 

Output:

1/120*(24*A*b^5*d^5*g*i^3*x^5 + 6*((15*A - B)*b^5*c*d^4 + (5*A + B)*a*b^4* 
d^5)*g*i^3*x^4 + 2*((60*A - 11*B)*b^5*c^2*d^3 + 10*(6*A + B)*a*b^4*c*d^4 + 
 B*a^2*b^3*d^5)*g*i^3*x^3 + 3*((20*A - 9*B)*b^5*c^3*d^2 + 5*(12*A + B)*a*b 
^4*c^2*d^3 + 5*B*a^2*b^3*c*d^4 - B*a^3*b^2*d^5)*g*i^3*x^2 - 6*(B*b^5*c^4*d 
 - 5*(4*A - B)*a*b^4*c^3*d^2 - 10*B*a^2*b^3*c^2*d^3 + 5*B*a^3*b^2*c*d^4 - 
B*a^4*b*d^5)*g*i^3*x + 6*(10*B*a^2*b^3*c^3*d^2 - 10*B*a^3*b^2*c^2*d^3 + 5* 
B*a^4*b*c*d^4 - B*a^5*d^5)*g*i^3*log(b*x + a) + 6*(B*b^5*c^5 - 5*B*a*b^4*c 
^4*d)*g*i^3*log(d*x + c) + 6*(4*B*b^5*d^5*g*i^3*x^5 + 20*B*a*b^4*c^3*d^2*g 
*i^3*x + 5*(3*B*b^5*c*d^4 + B*a*b^4*d^5)*g*i^3*x^4 + 20*(B*b^5*c^2*d^3 + B 
*a*b^4*c*d^4)*g*i^3*x^3 + 10*(B*b^5*c^3*d^2 + 3*B*a*b^4*c^2*d^3)*g*i^3*x^2 
)*log((b*e*x + a*e)/(d*x + c)))/(b^4*d^2)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1158 vs. \(2 (252) = 504\).

Time = 3.53 (sec) , antiderivative size = 1158, normalized size of antiderivative = 4.27 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

A*b*d**3*g*i**3*x**5/5 - B*a**2*g*i**3*(a**3*d**3 - 5*a**2*b*c*d**2 + 10*a 
*b**2*c**2*d - 10*b**3*c**3)*log(x + (B*a**5*c*d**4*g*i**3 - 5*B*a**4*b*c* 
*2*d**3*g*i**3 + 10*B*a**3*b**2*c**3*d**2*g*i**3 + B*a**3*d**2*g*i**3*(a** 
3*d**3 - 5*a**2*b*c*d**2 + 10*a*b**2*c**2*d - 10*b**3*c**3)/b - 15*B*a**2* 
b**3*c**4*d*g*i**3 - B*a**2*c*d*g*i**3*(a**3*d**3 - 5*a**2*b*c*d**2 + 10*a 
*b**2*c**2*d - 10*b**3*c**3) + B*a*b**4*c**5*g*i**3)/(B*a**5*d**5*g*i**3 - 
 5*B*a**4*b*c*d**4*g*i**3 + 10*B*a**3*b**2*c**2*d**3*g*i**3 - 10*B*a**2*b* 
*3*c**3*d**2*g*i**3 - 5*B*a*b**4*c**4*d*g*i**3 + B*b**5*c**5*g*i**3))/(20* 
b**4) - B*c**4*g*i**3*(5*a*d - b*c)*log(x + (B*a**5*c*d**4*g*i**3 - 5*B*a* 
*4*b*c**2*d**3*g*i**3 + 10*B*a**3*b**2*c**3*d**2*g*i**3 - 15*B*a**2*b**3*c 
**4*d*g*i**3 + B*a*b**4*c**5*g*i**3 + B*a*b**3*c**4*g*i**3*(5*a*d - b*c) - 
 B*b**4*c**5*g*i**3*(5*a*d - b*c)/d)/(B*a**5*d**5*g*i**3 - 5*B*a**4*b*c*d* 
*4*g*i**3 + 10*B*a**3*b**2*c**2*d**3*g*i**3 - 10*B*a**2*b**3*c**3*d**2*g*i 
**3 - 5*B*a*b**4*c**4*d*g*i**3 + B*b**5*c**5*g*i**3))/(20*d**2) + x**4*(A* 
a*d**3*g*i**3/4 + 3*A*b*c*d**2*g*i**3/4 + B*a*d**3*g*i**3/20 - B*b*c*d**2* 
g*i**3/20) + x**3*(A*a*c*d**2*g*i**3 + A*b*c**2*d*g*i**3 + B*a**2*d**3*g*i 
**3/(60*b) + B*a*c*d**2*g*i**3/6 - 11*B*b*c**2*d*g*i**3/60) + x**2*(3*A*a* 
c**2*d*g*i**3/2 + A*b*c**3*g*i**3/2 - B*a**3*d**3*g*i**3/(40*b**2) + B*a** 
2*c*d**2*g*i**3/(8*b) + B*a*c**2*d*g*i**3/8 - 9*B*b*c**3*g*i**3/40) + x*(A 
*a*c**3*g*i**3 + B*a**4*d**3*g*i**3/(20*b**3) - B*a**3*c*d**2*g*i**3/(4...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1022 vs. \(2 (255) = 510\).

Time = 0.06 (sec) , antiderivative size = 1022, normalized size of antiderivative = 3.77 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx =\text {Too large to display} \] Input:

integrate((b*g*x+a*g)*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algori 
thm="maxima")
 

Output:

1/5*A*b*d^3*g*i^3*x^5 + 3/4*A*b*c*d^2*g*i^3*x^4 + 1/4*A*a*d^3*g*i^3*x^4 + 
A*b*c^2*d*g*i^3*x^3 + A*a*c*d^2*g*i^3*x^3 + 1/2*A*b*c^3*g*i^3*x^2 + 3/2*A* 
a*c^2*d*g*i^3*x^2 + (x*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + a*log(b*x + 
a)/b - c*log(d*x + c)/d)*B*a*c^3*g*i^3 + 1/2*(x^2*log(b*e*x/(d*x + c) + a* 
e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x 
/(b*d))*B*b*c^3*g*i^3 + 3/2*(x^2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - a^ 
2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*B*a*c^2*d 
*g*i^3 + 1/2*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + 
 a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - 
 a^2*d^2)*x)/(b^2*d^2))*B*b*c^2*d*g*i^3 + 1/2*(2*x^3*log(b*e*x/(d*x + c) + 
 a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2* 
c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*a*c*d^2*g*i^3 + 
 1/8*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 
+ 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - 
 a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*b*c*d^2*g*i^3 + 1/ 
24*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 
6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a 
^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*a*d^3*g*i^3 + 1/60*( 
12*x^5*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 12*a^5*log(b*x + a)/b^5 - 12 
*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2589 vs. \(2 (255) = 510\).

Time = 0.30 (sec) , antiderivative size = 2589, normalized size of antiderivative = 9.55 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

integrate((b*g*x+a*g)*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algori 
thm="giac")
 

Output:

-1/120*(6*(B*b^7*c^6*e^6*g*i^3 - 6*B*a*b^6*c^5*d*e^6*g*i^3 + 15*B*a^2*b^5* 
c^4*d^2*e^6*g*i^3 - 20*B*a^3*b^4*c^3*d^3*e^6*g*i^3 + 15*B*a^4*b^3*c^2*d^4* 
e^6*g*i^3 - 6*B*a^5*b^2*c*d^5*e^6*g*i^3 + B*a^6*b*d^6*e^6*g*i^3 - 5*(b*e*x 
 + a*e)*B*b^6*c^6*d*e^5*g*i^3/(d*x + c) + 30*(b*e*x + a*e)*B*a*b^5*c^5*d^2 
*e^5*g*i^3/(d*x + c) - 75*(b*e*x + a*e)*B*a^2*b^4*c^4*d^3*e^5*g*i^3/(d*x + 
 c) + 100*(b*e*x + a*e)*B*a^3*b^3*c^3*d^4*e^5*g*i^3/(d*x + c) - 75*(b*e*x 
+ a*e)*B*a^4*b^2*c^2*d^5*e^5*g*i^3/(d*x + c) + 30*(b*e*x + a*e)*B*a^5*b*c* 
d^6*e^5*g*i^3/(d*x + c) - 5*(b*e*x + a*e)*B*a^6*d^7*e^5*g*i^3/(d*x + c))*l 
og((b*e*x + a*e)/(d*x + c))/(b^5*d^2*e^5 - 5*(b*e*x + a*e)*b^4*d^3*e^4/(d* 
x + c) + 10*(b*e*x + a*e)^2*b^3*d^4*e^3/(d*x + c)^2 - 10*(b*e*x + a*e)^3*b 
^2*d^5*e^2/(d*x + c)^3 + 5*(b*e*x + a*e)^4*b*d^6*e/(d*x + c)^4 - (b*e*x + 
a*e)^5*d^7/(d*x + c)^5) + (6*A*b^10*c^6*e^6*g*i^3 - 5*B*b^10*c^6*e^6*g*i^3 
 - 36*A*a*b^9*c^5*d*e^6*g*i^3 + 30*B*a*b^9*c^5*d*e^6*g*i^3 + 90*A*a^2*b^8* 
c^4*d^2*e^6*g*i^3 - 75*B*a^2*b^8*c^4*d^2*e^6*g*i^3 - 120*A*a^3*b^7*c^3*d^3 
*e^6*g*i^3 + 100*B*a^3*b^7*c^3*d^3*e^6*g*i^3 + 90*A*a^4*b^6*c^2*d^4*e^6*g* 
i^3 - 75*B*a^4*b^6*c^2*d^4*e^6*g*i^3 - 36*A*a^5*b^5*c*d^5*e^6*g*i^3 + 30*B 
*a^5*b^5*c*d^5*e^6*g*i^3 + 6*A*a^6*b^4*d^6*e^6*g*i^3 - 5*B*a^6*b^4*d^6*e^6 
*g*i^3 - 30*(b*e*x + a*e)*A*b^9*c^6*d*e^5*g*i^3/(d*x + c) + 31*(b*e*x + a* 
e)*B*b^9*c^6*d*e^5*g*i^3/(d*x + c) + 180*(b*e*x + a*e)*A*a*b^8*c^5*d^2*e^5 
*g*i^3/(d*x + c) - 186*(b*e*x + a*e)*B*a*b^8*c^5*d^2*e^5*g*i^3/(d*x + c...
 

Mupad [B] (verification not implemented)

Time = 27.98 (sec) , antiderivative size = 1192, normalized size of antiderivative = 4.40 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

x^4*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d - B*b*c))/20 - (A*d^2*g*i^3*( 
20*a*d + 20*b*c))/80) + x*((a*c*(((20*a*d + 20*b*c)*((d^2*g*i^3*(10*A*a*d 
+ 20*A*b*c + B*a*d - B*b*c))/5 - (A*d^2*g*i^3*(20*a*d + 20*b*c))/20))/(20* 
b*d) - (d*g*i^3*(4*A*a^2*d^2 + 24*A*b^2*c^2 + B*a^2*d^2 - 3*B*b^2*c^2 + 32 
*A*a*b*c*d + 2*B*a*b*c*d))/(4*b) + A*a*c*d^2*g*i^3))/(b*d) - ((20*a*d + 20 
*b*c)*(((20*a*d + 20*b*c)*(((20*a*d + 20*b*c)*((d^2*g*i^3*(10*A*a*d + 20*A 
*b*c + B*a*d - B*b*c))/5 - (A*d^2*g*i^3*(20*a*d + 20*b*c))/20))/(20*b*d) - 
 (d*g*i^3*(4*A*a^2*d^2 + 24*A*b^2*c^2 + B*a^2*d^2 - 3*B*b^2*c^2 + 32*A*a*b 
*c*d + 2*B*a*b*c*d))/(4*b) + A*a*c*d^2*g*i^3))/(20*b*d) - (a*c*((d^2*g*i^3 
*(10*A*a*d + 20*A*b*c + B*a*d - B*b*c))/5 - (A*d^2*g*i^3*(20*a*d + 20*b*c) 
)/20))/(b*d) + (c*g*i^3*(4*A*a^2*d^2 + 4*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 
 + 12*A*a*b*c*d))/b))/(20*b*d) + (c^2*g*i^3*(12*A*a^2*d^2 + 2*A*b^2*c^2 + 
3*B*a^2*d^2 - B*b^2*c^2 + 16*A*a*b*c*d - 2*B*a*b*c*d))/(2*b*d)) - x^3*(((2 
0*a*d + 20*b*c)*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d - B*b*c))/5 - (A* 
d^2*g*i^3*(20*a*d + 20*b*c))/20))/(60*b*d) - (d*g*i^3*(4*A*a^2*d^2 + 24*A* 
b^2*c^2 + B*a^2*d^2 - 3*B*b^2*c^2 + 32*A*a*b*c*d + 2*B*a*b*c*d))/(12*b) + 
(A*a*c*d^2*g*i^3)/3) + x^2*(((20*a*d + 20*b*c)*(((20*a*d + 20*b*c)*((d^2*g 
*i^3*(10*A*a*d + 20*A*b*c + B*a*d - B*b*c))/5 - (A*d^2*g*i^3*(20*a*d + 20* 
b*c))/20))/(20*b*d) - (d*g*i^3*(4*A*a^2*d^2 + 24*A*b^2*c^2 + B*a^2*d^2 - 3 
*B*b^2*c^2 + 32*A*a*b*c*d + 2*B*a*b*c*d))/(4*b) + A*a*c*d^2*g*i^3))/(40...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 703, normalized size of antiderivative = 2.59 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g i \left (-24 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{5} d^{5} x^{5}-6 a^{4} b \,d^{5} x +3 a^{3} b^{2} d^{5} x^{2}-2 a^{2} b^{3} d^{5} x^{3}-6 a \,b^{4} d^{5} x^{4}+6 b^{5} c^{4} d x +27 b^{5} c^{3} d^{2} x^{2}+22 b^{5} c^{2} d^{3} x^{3}+6 b^{5} c \,d^{4} x^{4}-30 \,\mathrm {log}\left (b x +a \right ) a^{4} b c \,d^{4}+60 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{2} c^{2} d^{3}-60 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} c^{3} d^{2}+30 \,\mathrm {log}\left (b x +a \right ) a \,b^{4} c^{4} d -30 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{4} c^{4} d -90 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{5} c \,d^{4} x^{4}-120 a^{2} b^{3} c \,d^{4} x^{3}-90 a \,b^{4} c \,d^{4} x^{4}-120 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{5} c^{2} d^{3} x^{3}-180 a^{2} b^{3} c^{2} d^{3} x^{2}-120 a \,b^{4} c^{2} d^{3} x^{3}-6 \,\mathrm {log}\left (b x +a \right ) b^{5} c^{5}+6 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{5} c^{5}+6 \,\mathrm {log}\left (b x +a \right ) a^{5} d^{5}-15 a \,b^{4} c^{2} d^{3} x^{2}-20 a \,b^{4} c \,d^{4} x^{3}-120 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{4} c^{3} d^{2} x -30 a^{2} b^{3} d^{5} x^{4}-24 a \,b^{4} d^{5} x^{5}-180 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{4} c^{2} d^{3} x^{2}-30 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{4} d^{5} x^{4}+30 a^{3} b^{2} c \,d^{4} x -60 a^{2} b^{3} c^{2} d^{3} x -15 a^{2} b^{3} c \,d^{4} x^{2}+30 a \,b^{4} c^{3} d^{2} x -120 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{4} c \,d^{4} x^{3}-60 a \,b^{4} c^{3} d^{2} x^{2}-60 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{5} c^{3} d^{2} x^{2}-120 a^{2} b^{3} c^{3} d^{2} x \right )}{120 b^{3} d^{2}} \] Input:

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

Output:

(g*i*(6*log(a + b*x)*a**5*d**5 - 30*log(a + b*x)*a**4*b*c*d**4 + 60*log(a 
+ b*x)*a**3*b**2*c**2*d**3 - 60*log(a + b*x)*a**2*b**3*c**3*d**2 + 30*log( 
a + b*x)*a*b**4*c**4*d - 6*log(a + b*x)*b**5*c**5 - 30*log((a*e + b*e*x)/( 
c + d*x))*a*b**4*c**4*d - 120*log((a*e + b*e*x)/(c + d*x))*a*b**4*c**3*d** 
2*x - 180*log((a*e + b*e*x)/(c + d*x))*a*b**4*c**2*d**3*x**2 - 120*log((a* 
e + b*e*x)/(c + d*x))*a*b**4*c*d**4*x**3 - 30*log((a*e + b*e*x)/(c + d*x)) 
*a*b**4*d**5*x**4 + 6*log((a*e + b*e*x)/(c + d*x))*b**5*c**5 - 60*log((a*e 
 + b*e*x)/(c + d*x))*b**5*c**3*d**2*x**2 - 120*log((a*e + b*e*x)/(c + d*x) 
)*b**5*c**2*d**3*x**3 - 90*log((a*e + b*e*x)/(c + d*x))*b**5*c*d**4*x**4 - 
 24*log((a*e + b*e*x)/(c + d*x))*b**5*d**5*x**5 - 6*a**4*b*d**5*x + 30*a** 
3*b**2*c*d**4*x + 3*a**3*b**2*d**5*x**2 - 120*a**2*b**3*c**3*d**2*x - 180* 
a**2*b**3*c**2*d**3*x**2 - 60*a**2*b**3*c**2*d**3*x - 120*a**2*b**3*c*d**4 
*x**3 - 15*a**2*b**3*c*d**4*x**2 - 30*a**2*b**3*d**5*x**4 - 2*a**2*b**3*d* 
*5*x**3 - 60*a*b**4*c**3*d**2*x**2 + 30*a*b**4*c**3*d**2*x - 120*a*b**4*c* 
*2*d**3*x**3 - 15*a*b**4*c**2*d**3*x**2 - 90*a*b**4*c*d**4*x**4 - 20*a*b** 
4*c*d**4*x**3 - 24*a*b**4*d**5*x**5 - 6*a*b**4*d**5*x**4 + 6*b**5*c**4*d*x 
 + 27*b**5*c**3*d**2*x**2 + 22*b**5*c**2*d**3*x**3 + 6*b**5*c*d**4*x**4))/ 
(120*b**3*d**2)