\(\int (a g+b g x)^3 (c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\) [1]

Optimal result

Integrand size = 38, antiderivative size = 212 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=-\frac {B (b c-a d)^4 g^3 i x}{20 b d^3}+\frac {B (b c-a d)^3 g^3 i (a+b x)^2}{40 b^2 d^2}-\frac {B (b c-a d)^2 g^3 i (a+b x)^3}{60 b^2 d}+\frac {g^3 i (a+b x)^4 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b}+\frac {(b c-a d) g^3 i (a+b x)^4 \left (A-B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^2}+\frac {B (b c-a d)^5 g^3 i \log (c+d x)}{20 b^2 d^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.23 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g^3 i \left (30 (b c-a d) (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+24 d (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-\frac {5 B (b c-a d)^2 \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )}{d^4}+\frac {2 B (b c-a d) \left (12 b d (b c-a d)^3 x-6 d^2 (b c-a d)^2 (a+b x)^2+4 d^3 (b c-a d) (a+b x)^3-3 d^4 (a+b x)^4-12 (b c-a d)^4 \log (c+d x)\right )}{d^4}\right )}{120 b^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2960, 27, 2948, 49, 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.

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

\(\Big \downarrow \) 2960

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2948

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 49

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

rule 2960

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(573\) vs. \(2(200)=400\).

Time = 1.48 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.71

method	result	size
risch	\(\frac {i \,g^{3} b B \ln \left (-d x -c \right ) a^{2} c^{3}}{2 d^{2}}-\frac {i \,g^{3} b^{2} B \ln \left (-d x -c \right ) a \,c^{4}}{4 d^{3}}+\frac {i \,g^{3} B \,a^{3} c x}{4}-\frac {i \,g^{3} b B \,a^{2} c^{2} x}{2 d}+\frac {i \,g^{3} b^{2} B a \,c^{3} x}{4 d^{2}}+i \,g^{3} b^{2} A a c \,x^{3}-\frac {i \,g^{3} b^{2} B a c \,x^{3}}{6}+\frac {3 i \,g^{3} b A \,a^{2} c \,x^{2}}{2}-\frac {i \,g^{3} b B \,a^{2} c \,x^{2}}{8}-\frac {i \,g^{3} b^{2} B a \,c^{2} x^{2}}{8 d}+i \,g^{3} A \,a^{3} c x +\frac {i \,g^{3} b^{3} B \,c^{3} x^{2}}{40 d^{2}}+\frac {i \,g^{3} d B \,a^{4} x}{20 b}-\frac {i \,g^{3} b^{3} B \,c^{4} x}{20 d^{3}}+\frac {i \,g^{3} B \ln \left (b x +a \right ) a^{4} c}{4 b}-\frac {i \,g^{3} B \ln \left (-d x -c \right ) a^{3} c^{2}}{2 d}-\frac {i \,g^{3} d B \ln \left (b x +a \right ) a^{5}}{20 b^{2}}+\frac {i \,g^{3} b^{3} B \ln \left (-d x -c \right ) c^{5}}{20 d^{4}}+\frac {3 i \,g^{3} b^{2} d A a \,x^{4}}{4}+\frac {i \,g^{3} b^{3} A c \,x^{4}}{4}+\frac {i \,g^{3} b^{2} d B a \,x^{4}}{20}-\frac {i \,g^{3} b^{3} B c \,x^{4}}{20}+i \,g^{3} b d A \,a^{2} x^{3}+\frac {11 i \,g^{3} b d B \,a^{2} x^{3}}{60}-\frac {i \,g^{3} b^{3} B \,c^{2} x^{3}}{60 d}+\frac {i \,g^{3} d A \,a^{3} x^{2}}{2}+\frac {9 i \,g^{3} d B \,a^{3} x^{2}}{40}+\frac {i \,g^{3} b^{3} d A \,x^{5}}{5}+\frac {i \,g^{3} B x \left (4 d \,b^{3} x^{4}+15 a \,b^{2} d \,x^{3}+5 b^{3} c \,x^{3}+20 a^{2} b d \,x^{2}+20 a \,b^{2} c \,x^{2}+10 a^{3} d x +30 a^{2} b c x +20 c \,a^{3}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{20}\)	\(574\)
parallelrisch	\(\text {Expression too large to display}\)	\(1056\)
parts	\(\text {Expression too large to display}\)	\(1698\)
derivativedivides	\(\text {Expression too large to display}\)	\(1777\)
default	\(\text {Expression too large to display}\)	\(1777\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (200) = 400\).

Time = 0.13 (sec) , antiderivative size = 504, normalized size of antiderivative = 2.38 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {24 \, A b^{5} d^{5} g^{3} i x^{5} + 6 \, {\left ({\left (5 \, A - B\right )} b^{5} c d^{4} + {\left (15 \, A + B\right )} a b^{4} d^{5}\right )} g^{3} i x^{4} - 2 \, {\left (B b^{5} c^{2} d^{3} - 10 \, {\left (6 \, A - B\right )} a b^{4} c d^{4} - {\left (60 \, A + 11 \, B\right )} a^{2} b^{3} d^{5}\right )} g^{3} i x^{3} + 3 \, {\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} + 5 \, {\left (12 \, A - B\right )} a^{2} b^{3} c d^{4} + {\left (20 \, A + 9 \, B\right )} a^{3} b^{2} d^{5}\right )} g^{3} i x^{2} - 6 \, {\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 10 \, B a^{2} b^{3} c^{2} d^{3} - 5 \, {\left (4 \, A + B\right )} a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g^{3} i x + 6 \, {\left (5 \, B a^{4} b c d^{4} - B a^{5} d^{5}\right )} g^{3} i \log \left (b x + a\right ) + 6 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3}\right )} g^{3} i \log \left (d x + c\right ) + 6 \, {\left (4 \, B b^{5} d^{5} g^{3} i x^{5} + 20 \, B a^{3} b^{2} c d^{4} g^{3} i x + 5 \, {\left (B b^{5} c d^{4} + 3 \, B a b^{4} d^{5}\right )} g^{3} i x^{4} + 20 \, {\left (B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{3} i x^{3} + 10 \, {\left (3 \, B a^{2} b^{3} c d^{4} + B a^{3} b^{2} d^{5}\right )} g^{3} i x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{120 \, b^{2} d^{4}} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1158 vs. \(2 (194) = 388\).

Time = 3.79 (sec) , antiderivative size = 1158, normalized size of antiderivative = 5.46 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1022 vs. \(2 (200) = 400\).

Time = 0.07 (sec) , antiderivative size = 1022, normalized size of antiderivative = 4.82 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

1/5*A*b^3*d*g^3*i*x^5 + 1/4*A*b^3*c*g^3*i*x^4 + 3/4*A*a*b^2*d*g^3*i*x^4 + 
A*a*b^2*c*g^3*i*x^3 + A*a^2*b*d*g^3*i*x^3 + 3/2*A*a^2*b*c*g^3*i*x^2 + 1/2* 
A*a^3*d*g^3*i*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^3*c*g^3*i + 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^2*b*c*g^3*i + 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*b^2*c*g^3*i + 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*b^3*c*g^3*i + 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*a^3*d*g^3*i + 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^2*b*d*g^3*i + 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*a*b^2*d*g^3*i + 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. 3637 vs. \(2 (200) = 400\).

Time = 0.31 (sec) , antiderivative size = 3637, normalized size of antiderivative = 17.16 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/120*(6*(B*b^9*c^6*e^6*g^3*i - 6*B*a*b^8*c^5*d*e^6*g^3*i + 15*B*a^2*b^7* 
c^4*d^2*e^6*g^3*i - 20*B*a^3*b^6*c^3*d^3*e^6*g^3*i + 15*B*a^4*b^5*c^2*d^4* 
e^6*g^3*i - 6*B*a^5*b^4*c*d^5*e^6*g^3*i + B*a^6*b^3*d^6*e^6*g^3*i - 5*(b*e 
*x + a*e)*B*b^8*c^6*d*e^5*g^3*i/(d*x + c) + 30*(b*e*x + a*e)*B*a*b^7*c^5*d 
^2*e^5*g^3*i/(d*x + c) - 75*(b*e*x + a*e)*B*a^2*b^6*c^4*d^3*e^5*g^3*i/(d*x 
 + c) + 100*(b*e*x + a*e)*B*a^3*b^5*c^3*d^4*e^5*g^3*i/(d*x + c) - 75*(b*e* 
x + a*e)*B*a^4*b^4*c^2*d^5*e^5*g^3*i/(d*x + c) + 30*(b*e*x + a*e)*B*a^5*b^ 
3*c*d^6*e^5*g^3*i/(d*x + c) - 5*(b*e*x + a*e)*B*a^6*b^2*d^7*e^5*g^3*i/(d*x 
 + c) + 10*(b*e*x + a*e)^2*B*b^7*c^6*d^2*e^4*g^3*i/(d*x + c)^2 - 60*(b*e*x 
 + a*e)^2*B*a*b^6*c^5*d^3*e^4*g^3*i/(d*x + c)^2 + 150*(b*e*x + a*e)^2*B*a^ 
2*b^5*c^4*d^4*e^4*g^3*i/(d*x + c)^2 - 200*(b*e*x + a*e)^2*B*a^3*b^4*c^3*d^ 
5*e^4*g^3*i/(d*x + c)^2 + 150*(b*e*x + a*e)^2*B*a^4*b^3*c^2*d^6*e^4*g^3*i/ 
(d*x + c)^2 - 60*(b*e*x + a*e)^2*B*a^5*b^2*c*d^7*e^4*g^3*i/(d*x + c)^2 + 1 
0*(b*e*x + a*e)^2*B*a^6*b*d^8*e^4*g^3*i/(d*x + c)^2 - 10*(b*e*x + a*e)^3*B 
*b^6*c^6*d^3*e^3*g^3*i/(d*x + c)^3 + 60*(b*e*x + a*e)^3*B*a*b^5*c^5*d^4*e^ 
3*g^3*i/(d*x + c)^3 - 150*(b*e*x + a*e)^3*B*a^2*b^4*c^4*d^5*e^3*g^3*i/(d*x 
 + c)^3 + 200*(b*e*x + a*e)^3*B*a^3*b^3*c^3*d^6*e^3*g^3*i/(d*x + c)^3 - 15 
0*(b*e*x + a*e)^3*B*a^4*b^2*c^2*d^7*e^3*g^3*i/(d*x + c)^3 + 60*(b*e*x + a* 
e)^3*B*a^5*b*c*d^8*e^3*g^3*i/(d*x + c)^3 - 10*(b*e*x + a*e)^3*B*a^6*d^9*e^ 
3*g^3*i/(d*x + c)^3)*log((b*e*x + a*e)/(d*x + c))/(b^5*d^4*e^5 - 5*(b*e...
 

Mupad [B] (verification not implemented)

Time = 27.20 (sec) , antiderivative size = 1195, normalized size of antiderivative = 5.64 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 757, normalized size of antiderivative = 3.57 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g^{3} 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 +27 a^{3} b^{2} d^{5} x^{2}+22 a^{2} b^{3} d^{5} x^{3}+6 a \,b^{4} d^{5} x^{4}-6 b^{5} c^{4} d x +3 b^{5} c^{3} d^{2} x^{2}-2 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 +60 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{3} b^{2} c^{2} d^{3}-60 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b^{3} c^{3} d^{2}+30 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{4} c^{4} d +30 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{5} c \,d^{4} x^{4}+120 a^{4} b c \,d^{4} x +180 a^{3} b^{2} c \,d^{4} x^{2}+120 a^{2} b^{3} c \,d^{4} x^{3}+30 a \,b^{4} c \,d^{4} x^{4}+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}+60 a^{4} b \,d^{5} x^{2}+120 a^{3} b^{2} d^{5} x^{3}+90 a^{2} b^{3} d^{5} x^{4}+24 a \,b^{4} d^{5} x^{5}+60 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{3} b^{2} d^{5} x^{2}+120 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b^{3} d^{5} x^{3}+90 \,\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^{3} b^{2} c \,d^{4} x +180 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b^{3} c \,d^{4} x^{2}+120 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{4} c \,d^{4} x^{3}\right )}{120 b \,d^{4}} \] Input:

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

Output:

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