\(\int (a g+b g x)^3 (c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\) [20]

Optimal result

Integrand size = 40, antiderivative size = 457 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {B (b c-a d)^6 g^3 i^3 x}{140 b^3 d^3}+\frac {B (b c-a d)^5 g^3 i^3 (c+d x)^2}{280 b^2 d^4}+\frac {B (b c-a d)^4 g^3 i^3 (c+d x)^3}{420 b d^4}-\frac {17 B (b c-a d)^3 g^3 i^3 (c+d x)^4}{280 d^4}+\frac {b B (b c-a d)^2 g^3 i^3 (c+d x)^5}{14 d^4}-\frac {b^2 B (b c-a d) g^3 i^3 (c+d x)^6}{42 d^4}+\frac {B (b c-a d)^7 g^3 i^3 \log \left (\frac {a+b x}{c+d x}\right )}{140 b^4 d^4}-\frac {(b c-a d)^3 g^3 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d^4}+\frac {3 b (b c-a d)^2 g^3 i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^4}-\frac {b^2 (b c-a d) g^3 i^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^4}+\frac {b^3 g^3 i^3 (c+d x)^7 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{7 d^4}+\frac {B (b c-a d)^7 g^3 i^3 \log (c+d x)}{140 b^4 d^4} \] Output:

1/140*B*(-a*d+b*c)^6*g^3*i^3*x/b^3/d^3+1/280*B*(-a*d+b*c)^5*g^3*i^3*(d*x+c 
)^2/b^2/d^4+1/420*B*(-a*d+b*c)^4*g^3*i^3*(d*x+c)^3/b/d^4-17/280*B*(-a*d+b* 
c)^3*g^3*i^3*(d*x+c)^4/d^4+1/14*b*B*(-a*d+b*c)^2*g^3*i^3*(d*x+c)^5/d^4-1/4 
2*b^2*B*(-a*d+b*c)*g^3*i^3*(d*x+c)^6/d^4+1/140*B*(-a*d+b*c)^7*g^3*i^3*ln(( 
b*x+a)/(d*x+c))/b^4/d^4-1/4*(-a*d+b*c)^3*g^3*i^3*(d*x+c)^4*(A+B*ln(e*(b*x+ 
a)/(d*x+c)))/d^4+3/5*b*(-a*d+b*c)^2*g^3*i^3*(d*x+c)^5*(A+B*ln(e*(b*x+a)/(d 
*x+c)))/d^4-1/2*b^2*(-a*d+b*c)*g^3*i^3*(d*x+c)^6*(A+B*ln(e*(b*x+a)/(d*x+c) 
))/d^4+1/7*b^3*g^3*i^3*(d*x+c)^7*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^4+1/140*B*( 
-a*d+b*c)^7*g^3*i^3*ln(d*x+c)/b^4/d^4
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.28 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g^3 i^3 \left (\frac {120 b^2 B c (b c-a d)^5 x}{d^3}-\frac {126 b B (b c-a d)^6 x}{d^3}+\frac {120 a b B (-b c+a d)^5 x}{d^2}-\frac {60 b B c (b c-a d)^4 (a+b x)^2}{d^2}+\frac {60 a B (b c-a d)^4 (a+b x)^2}{d}+\frac {63 B (b c-a d)^5 (a+b x)^2}{d^2}+\frac {40 b B c (b c-a d)^3 (a+b x)^3}{d}-\frac {42 B (b c-a d)^4 (a+b x)^3}{d}+40 a B (-b c+a d)^3 (a+b x)^3-30 b B c (b c-a d)^2 (a+b x)^4+30 a B d (b c-a d)^2 (a+b x)^4+21 B (-b c+a d)^3 (a+b x)^4+24 b B c d (b c-a d) (a+b x)^5-84 B d (b c-a d)^2 (a+b x)^5+24 a B d^2 (-b c+a d) (a+b x)^5-20 b B c d^2 (a+b x)^6+20 a B d^3 (a+b x)^6+210 (b c-a d)^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+504 d (b c-a d)^2 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+420 d^2 (b c-a d) (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+120 d^3 (a+b x)^7 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-\frac {120 b B c (b c-a d)^6 \log (c+d x)}{d^4}+\frac {120 a B (b c-a d)^6 \log (c+d x)}{d^3}+\frac {126 B (b c-a d)^7 \log (c+d x)}{d^4}\right )}{840 b^4} \] Input:

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

Output:

(g^3*i^3*((120*b^2*B*c*(b*c - a*d)^5*x)/d^3 - (126*b*B*(b*c - a*d)^6*x)/d^ 
3 + (120*a*b*B*(-(b*c) + a*d)^5*x)/d^2 - (60*b*B*c*(b*c - a*d)^4*(a + b*x) 
^2)/d^2 + (60*a*B*(b*c - a*d)^4*(a + b*x)^2)/d + (63*B*(b*c - a*d)^5*(a + 
b*x)^2)/d^2 + (40*b*B*c*(b*c - a*d)^3*(a + b*x)^3)/d - (42*B*(b*c - a*d)^4 
*(a + b*x)^3)/d + 40*a*B*(-(b*c) + a*d)^3*(a + b*x)^3 - 30*b*B*c*(b*c - a* 
d)^2*(a + b*x)^4 + 30*a*B*d*(b*c - a*d)^2*(a + b*x)^4 + 21*B*(-(b*c) + a*d 
)^3*(a + b*x)^4 + 24*b*B*c*d*(b*c - a*d)*(a + b*x)^5 - 84*B*d*(b*c - a*d)^ 
2*(a + b*x)^5 + 24*a*B*d^2*(-(b*c) + a*d)*(a + b*x)^5 - 20*b*B*c*d^2*(a + 
b*x)^6 + 20*a*B*d^3*(a + b*x)^6 + 210*(b*c - a*d)^3*(a + b*x)^4*(A + B*Log 
[(e*(a + b*x))/(c + d*x)]) + 504*d*(b*c - a*d)^2*(a + b*x)^5*(A + B*Log[(e 
*(a + b*x))/(c + d*x)]) + 420*d^2*(b*c - a*d)*(a + b*x)^6*(A + B*Log[(e*(a 
 + b*x))/(c + d*x)]) + 120*d^3*(a + b*x)^7*(A + B*Log[(e*(a + b*x))/(c + d 
*x)]) - (120*b*B*c*(b*c - a*d)^6*Log[c + d*x])/d^4 + (120*a*B*(b*c - a*d)^ 
6*Log[c + d*x])/d^3 + (126*B*(b*c - a*d)^7*Log[c + d*x])/d^4))/(840*b^4)
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.88, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2962, 2782, 27, 2123, 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.

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

\(\Big \downarrow \) 2962

\(\displaystyle g^3 i^3 (b c-a d)^7 \int \frac {(a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^8}d\frac {a+b x}{c+d x}\)

\(\Big \downarrow \) 2782

\(\displaystyle g^3 i^3 (b c-a d)^7 \left (-B \int -\frac {(c+d x) \left (b^3-\frac {7 d (a+b x) b^2}{c+d x}+\frac {21 d^2 (a+b x)^2 b}{(c+d x)^2}-\frac {35 d^3 (a+b x)^3}{(c+d x)^3}\right )}{140 d^4 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^7}d\frac {a+b x}{c+d x}+\frac {b^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{7 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {3 b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle g^3 i^3 (b c-a d)^7 \left (\frac {B \int \frac {(c+d x) \left (b^3-\frac {7 d (a+b x) b^2}{c+d x}+\frac {21 d^2 (a+b x)^2 b}{(c+d x)^2}-\frac {35 d^3 (a+b x)^3}{(c+d x)^3}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^7}d\frac {a+b x}{c+d x}}{140 d^4}+\frac {b^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{7 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {3 b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\)

\(\Big \downarrow \) 2123

\(\displaystyle g^3 i^3 (b c-a d)^7 \left (\frac {B \int \left (-\frac {20 d b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^7}+\frac {50 d b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {34 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4 b}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3 b^2}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2 b^3}+\frac {c+d x}{(a+b x) b^4}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^4}\right )d\frac {a+b x}{c+d x}}{140 d^4}+\frac {b^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{7 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {3 b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle g^3 i^3 (b c-a d)^7 \left (\frac {b^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{7 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {3 b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^4}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^4}+\frac {1}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {10 b^2}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {1}{2 b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {10 b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {17}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {1}{3 b \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{140 d^4}\right )\)

Input:

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

Output:

(b*c - a*d)^7*g^3*i^3*((b^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(7*d^4*( 
b - (d*(a + b*x))/(c + d*x))^7) - (b^2*(A + B*Log[(e*(a + b*x))/(c + d*x)] 
))/(2*d^4*(b - (d*(a + b*x))/(c + d*x))^6) + (3*b*(A + B*Log[(e*(a + b*x)) 
/(c + d*x)]))/(5*d^4*(b - (d*(a + b*x))/(c + d*x))^5) - (A + B*Log[(e*(a + 
 b*x))/(c + d*x)])/(4*d^4*(b - (d*(a + b*x))/(c + d*x))^4) + (B*((-10*b^2) 
/(3*(b - (d*(a + b*x))/(c + d*x))^6) + (10*b)/(b - (d*(a + b*x))/(c + d*x) 
)^5 - 17/(2*(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))/(140*d^4))
 

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 2009

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

rule 2123

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c 
, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
 

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. \(1193\) vs. \(2(433)=866\).

Time = 2.13 (sec) , antiderivative size = 1194, normalized size of antiderivative = 2.61

method	result	size
risch	\(\text {Expression too large to display}\)	\(1194\)
parallelrisch	\(\text {Expression too large to display}\)	\(1958\)
parts	\(\text {Expression too large to display}\)	\(2513\)
derivativedivides	\(\text {Expression too large to display}\)	\(2536\)
default	\(\text {Expression too large to display}\)	\(2536\)

Input:

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

Output:

9/5*i^3*g^3*d^2*b^2*A*a*c*x^5+i^3*g^3*b^2*A*a*c^3*x^3+7/30*i^3*g^3*d^2*B*a 
^3*c*x^3-7/30*i^3*g^3*b^2*B*a*c^3*x^3+3/2*i^3*g^3*d*A*a^3*c^2*x^2+3/2*i^3* 
g^3*b*A*a^2*c^3*x^2+1/40*i^3*g^3*d^2/b*B*a^4*c*x^2+3/10*i^3*g^3*d*B*a^3*c^ 
2*x^2-3/10*i^3*g^3*b*B*a^2*c^3*x^2-1/40*i^3*g^3/d*b^2*B*a*c^4*x^2+i^3*g^3* 
A*a^3*c^3*x+9/4*i^3*g^3*d*b^2*A*a*c^2*x^4+7/40*i^3*g^3*d^2*b*B*a^2*c*x^4-7 
/40*i^3*g^3*d*b^2*B*a*c^2*x^4+1/4*i^3*g^3/b*B*ln(b*x+a)*a^4*c^3+1/140*i^3* 
g^3/d^4*b^3*B*ln(-d*x-c)*c^7-1/140*i^3*g^3*d^3/b^4*B*ln(b*x+a)*a^7+17/280* 
i^3*g^3*d^3*B*a^3*x^4+i^3*g^3*d^2*A*a^3*c*x^3+3*i^3*g^3*d*b*A*a^2*c^2*x^3+ 
9/4*i^3*g^3*d^2*b*A*a^2*c*x^4-1/20*i^3*g^3*d^2/b^2*B*a^5*c*x+3/20*i^3*g^3* 
d/b*B*a^4*c^2*x-3/20*i^3*g^3/d*b*B*a^2*c^4*x+1/20*i^3*g^3/d^2*b^2*B*a*c^5* 
x+3/20*i^3*g^3/d^2*b*B*ln(-d*x-c)*a^2*c^5-1/20*i^3*g^3/d^3*b^2*B*ln(-d*x-c 
)*a*c^6+1/20*i^3*g^3*d^2/b^3*B*ln(b*x+a)*a^6*c-3/20*i^3*g^3*d/b^2*B*ln(b*x 
+a)*a^5*c^2+3/5*i^3*g^3*d^3*b*A*a^2*x^5+1/2*i^3*g^3*d^3*b^2*A*a*x^6+1/2*i^ 
3*g^3*d^2*b^3*A*c*x^6+3/5*i^3*g^3*d*b^3*A*c^2*x^5+1/14*i^3*g^3*d^3*b*B*a^2 
*x^5-1/14*i^3*g^3*d*b^3*B*c^2*x^5+1/4*i^3*g^3*d^3*A*a^3*x^4+1/4*i^3*g^3*b^ 
3*A*c^3*x^4-17/280*i^3*g^3*b^3*B*c^3*x^4+1/420*i^3*g^3*d^3/b*B*a^4*x^3-1/4 
20*i^3*g^3/d*b^3*B*c^4*x^3-1/280*i^3*g^3*d^3/b^2*B*a^5*x^2+1/280*i^3*g^3/d 
^2*b^3*B*c^5*x^2+1/140*i^3*g^3*d^3/b^3*B*a^6*x-1/140*i^3*g^3/d^3*b^3*B*c^6 
*x-1/4*i^3*g^3/d*B*ln(-d*x-c)*a^3*c^4+1/42*i^3*g^3*d^3*b^2*B*a*x^6-1/42*i^ 
3*g^3*d^2*b^3*B*c*x^6+1/140*i^3*g^3*B*x*(20*b^3*d^3*x^6+70*a*b^2*d^3*x^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (433) = 866\).

Time = 0.31 (sec) , antiderivative size = 912, normalized size of antiderivative = 2.00 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

1/840*(120*A*b^7*d^7*g^3*i^3*x^7 + 20*((21*A - B)*b^7*c*d^6 + (21*A + B)*a 
*b^6*d^7)*g^3*i^3*x^6 + 12*((42*A - 5*B)*b^7*c^2*d^5 + 126*A*a*b^6*c*d^6 + 
 (42*A + 5*B)*a^2*b^5*d^7)*g^3*i^3*x^5 + 3*((70*A - 17*B)*b^7*c^3*d^4 + 7* 
(90*A - 7*B)*a*b^6*c^2*d^5 + 7*(90*A + 7*B)*a^2*b^5*c*d^6 + (70*A + 17*B)* 
a^3*b^4*d^7)*g^3*i^3*x^4 - 2*(B*b^7*c^4*d^3 - 14*(30*A - 7*B)*a*b^6*c^3*d^ 
4 - 1260*A*a^2*b^5*c^2*d^5 - 14*(30*A + 7*B)*a^3*b^4*c*d^6 - B*a^4*b^3*d^7 
)*g^3*i^3*x^3 + 3*(B*b^7*c^5*d^2 - 7*B*a*b^6*c^4*d^3 + 84*(5*A - B)*a^2*b^ 
5*c^3*d^4 + 84*(5*A + B)*a^3*b^4*c^2*d^5 + 7*B*a^4*b^3*c*d^6 - B*a^5*b^2*d 
^7)*g^3*i^3*x^2 - 6*(B*b^7*c^6*d - 7*B*a*b^6*c^5*d^2 + 21*B*a^2*b^5*c^4*d^ 
3 - 140*A*a^3*b^4*c^3*d^4 - 21*B*a^4*b^3*c^2*d^5 + 7*B*a^5*b^2*c*d^6 - B*a 
^6*b*d^7)*g^3*i^3*x + 6*(35*B*a^4*b^3*c^3*d^4 - 21*B*a^5*b^2*c^2*d^5 + 7*B 
*a^6*b*c*d^6 - B*a^7*d^7)*g^3*i^3*log(b*x + a) + 6*(B*b^7*c^7 - 7*B*a*b^6* 
c^6*d + 21*B*a^2*b^5*c^5*d^2 - 35*B*a^3*b^4*c^4*d^3)*g^3*i^3*log(d*x + c) 
+ 6*(20*B*b^7*d^7*g^3*i^3*x^7 + 140*B*a^3*b^4*c^3*d^4*g^3*i^3*x + 70*(B*b^ 
7*c*d^6 + B*a*b^6*d^7)*g^3*i^3*x^6 + 84*(B*b^7*c^2*d^5 + 3*B*a*b^6*c*d^6 + 
 B*a^2*b^5*d^7)*g^3*i^3*x^5 + 35*(B*b^7*c^3*d^4 + 9*B*a*b^6*c^2*d^5 + 9*B* 
a^2*b^5*c*d^6 + B*a^3*b^4*d^7)*g^3*i^3*x^4 + 140*(B*a*b^6*c^3*d^4 + 3*B*a^ 
2*b^5*c^2*d^5 + B*a^3*b^4*c*d^6)*g^3*i^3*x^3 + 210*(B*a^2*b^5*c^3*d^4 + B* 
a^3*b^4*c^2*d^5)*g^3*i^3*x^2)*log((b*e*x + a*e)/(d*x + c)))/(b^4*d^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2161 vs. \(2 (427) = 854\).

Time = 9.20 (sec) , antiderivative size = 2161, normalized size of antiderivative = 4.73 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

A*b**3*d**3*g**3*i**3*x**7/7 - B*a**4*g**3*i**3*(a**3*d**3 - 7*a**2*b*c*d* 
*2 + 21*a*b**2*c**2*d - 35*b**3*c**3)*log(x + (B*a**7*c*d**6*g**3*i**3 - 7 
*B*a**6*b*c**2*d**5*g**3*i**3 + 21*B*a**5*b**2*c**3*d**4*g**3*i**3 + B*a** 
5*d**4*g**3*i**3*(a**3*d**3 - 7*a**2*b*c*d**2 + 21*a*b**2*c**2*d - 35*b**3 
*c**3)/b - 70*B*a**4*b**3*c**4*d**3*g**3*i**3 - B*a**4*c*d**3*g**3*i**3*(a 
**3*d**3 - 7*a**2*b*c*d**2 + 21*a*b**2*c**2*d - 35*b**3*c**3) + 21*B*a**3* 
b**4*c**5*d**2*g**3*i**3 - 7*B*a**2*b**5*c**6*d*g**3*i**3 + B*a*b**6*c**7* 
g**3*i**3)/(B*a**7*d**7*g**3*i**3 - 7*B*a**6*b*c*d**6*g**3*i**3 + 21*B*a** 
5*b**2*c**2*d**5*g**3*i**3 - 35*B*a**4*b**3*c**3*d**4*g**3*i**3 - 35*B*a** 
3*b**4*c**4*d**3*g**3*i**3 + 21*B*a**2*b**5*c**5*d**2*g**3*i**3 - 7*B*a*b* 
*6*c**6*d*g**3*i**3 + B*b**7*c**7*g**3*i**3))/(140*b**4) - B*c**4*g**3*i** 
3*(35*a**3*d**3 - 21*a**2*b*c*d**2 + 7*a*b**2*c**2*d - b**3*c**3)*log(x + 
(B*a**7*c*d**6*g**3*i**3 - 7*B*a**6*b*c**2*d**5*g**3*i**3 + 21*B*a**5*b**2 
*c**3*d**4*g**3*i**3 - 70*B*a**4*b**3*c**4*d**3*g**3*i**3 + 21*B*a**3*b**4 
*c**5*d**2*g**3*i**3 - 7*B*a**2*b**5*c**6*d*g**3*i**3 + B*a*b**6*c**7*g**3 
*i**3 + B*a*b**3*c**4*g**3*i**3*(35*a**3*d**3 - 21*a**2*b*c*d**2 + 7*a*b** 
2*c**2*d - b**3*c**3) - B*b**4*c**5*g**3*i**3*(35*a**3*d**3 - 21*a**2*b*c* 
d**2 + 7*a*b**2*c**2*d - b**3*c**3)/d)/(B*a**7*d**7*g**3*i**3 - 7*B*a**6*b 
*c*d**6*g**3*i**3 + 21*B*a**5*b**2*c**2*d**5*g**3*i**3 - 35*B*a**4*b**3*c* 
*3*d**4*g**3*i**3 - 35*B*a**3*b**4*c**4*d**3*g**3*i**3 + 21*B*a**2*b**5...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2637 vs. \(2 (433) = 866\).

Time = 0.10 (sec) , antiderivative size = 2637, normalized size of antiderivative = 5.77 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/7*A*b^3*d^3*g^3*i^3*x^7 + 1/2*A*b^3*c*d^2*g^3*i^3*x^6 + 1/2*A*a*b^2*d^3* 
g^3*i^3*x^6 + 3/5*A*b^3*c^2*d*g^3*i^3*x^5 + 9/5*A*a*b^2*c*d^2*g^3*i^3*x^5 
+ 3/5*A*a^2*b*d^3*g^3*i^3*x^5 + 1/4*A*b^3*c^3*g^3*i^3*x^4 + 9/4*A*a*b^2*c^ 
2*d*g^3*i^3*x^4 + 9/4*A*a^2*b*c*d^2*g^3*i^3*x^4 + 1/4*A*a^3*d^3*g^3*i^3*x^ 
4 + A*a*b^2*c^3*g^3*i^3*x^3 + 3*A*a^2*b*c^2*d*g^3*i^3*x^3 + A*a^3*c*d^2*g^ 
3*i^3*x^3 + 3/2*A*a^2*b*c^3*g^3*i^3*x^2 + 3/2*A*a^3*c^2*d*g^3*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^3*c^3*g^3*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^2*b*c^ 
3*g^3*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*b^2*c^3*g^3*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*b^3*c^3*g^3*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^3*c^2*d*g^3*i^3 + 3/2*(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*c^2*d*g^3*i^3 
 + 3/8*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5520 vs. \(2 (433) = 866\).

Time = 0.43 (sec) , antiderivative size = 5520, normalized size of antiderivative = 12.08 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/840*(6*(B*b^11*c^8*e^8*g^3*i^3 - 8*B*a*b^10*c^7*d*e^8*g^3*i^3 + 28*B*a^ 
2*b^9*c^6*d^2*e^8*g^3*i^3 - 56*B*a^3*b^8*c^5*d^3*e^8*g^3*i^3 + 70*B*a^4*b^ 
7*c^4*d^4*e^8*g^3*i^3 - 56*B*a^5*b^6*c^3*d^5*e^8*g^3*i^3 + 28*B*a^6*b^5*c^ 
2*d^6*e^8*g^3*i^3 - 8*B*a^7*b^4*c*d^7*e^8*g^3*i^3 + B*a^8*b^3*d^8*e^8*g^3* 
i^3 - 7*(b*e*x + a*e)*B*b^10*c^8*d*e^7*g^3*i^3/(d*x + c) + 56*(b*e*x + a*e 
)*B*a*b^9*c^7*d^2*e^7*g^3*i^3/(d*x + c) - 196*(b*e*x + a*e)*B*a^2*b^8*c^6* 
d^3*e^7*g^3*i^3/(d*x + c) + 392*(b*e*x + a*e)*B*a^3*b^7*c^5*d^4*e^7*g^3*i^ 
3/(d*x + c) - 490*(b*e*x + a*e)*B*a^4*b^6*c^4*d^5*e^7*g^3*i^3/(d*x + c) + 
392*(b*e*x + a*e)*B*a^5*b^5*c^3*d^6*e^7*g^3*i^3/(d*x + c) - 196*(b*e*x + a 
*e)*B*a^6*b^4*c^2*d^7*e^7*g^3*i^3/(d*x + c) + 56*(b*e*x + a*e)*B*a^7*b^3*c 
*d^8*e^7*g^3*i^3/(d*x + c) - 7*(b*e*x + a*e)*B*a^8*b^2*d^9*e^7*g^3*i^3/(d* 
x + c) + 21*(b*e*x + a*e)^2*B*b^9*c^8*d^2*e^6*g^3*i^3/(d*x + c)^2 - 168*(b 
*e*x + a*e)^2*B*a*b^8*c^7*d^3*e^6*g^3*i^3/(d*x + c)^2 + 588*(b*e*x + a*e)^ 
2*B*a^2*b^7*c^6*d^4*e^6*g^3*i^3/(d*x + c)^2 - 1176*(b*e*x + a*e)^2*B*a^3*b 
^6*c^5*d^5*e^6*g^3*i^3/(d*x + c)^2 + 1470*(b*e*x + a*e)^2*B*a^4*b^5*c^4*d^ 
6*e^6*g^3*i^3/(d*x + c)^2 - 1176*(b*e*x + a*e)^2*B*a^5*b^4*c^3*d^7*e^6*g^3 
*i^3/(d*x + c)^2 + 588*(b*e*x + a*e)^2*B*a^6*b^3*c^2*d^8*e^6*g^3*i^3/(d*x 
+ c)^2 - 168*(b*e*x + a*e)^2*B*a^7*b^2*c*d^9*e^6*g^3*i^3/(d*x + c)^2 + 21* 
(b*e*x + a*e)^2*B*a^8*b*d^10*e^6*g^3*i^3/(d*x + c)^2 - 35*(b*e*x + a*e)^3* 
B*b^8*c^8*d^3*e^5*g^3*i^3/(d*x + c)^3 + 280*(b*e*x + a*e)^3*B*a*b^7*c^7...
 

Mupad [B] (verification not implemented)

Time = 29.20 (sec) , antiderivative size = 4347, normalized size of antiderivative = 9.51 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

x*(((140*a*d + 140*b*c)*(((140*a*d + 140*b*c)*((a*c*((((b^2*d^2*g^3*i^3*(2 
8*A*a*d + 28*A*b*c + B*a*d - B*b*c))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140 
*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12*A*a^2*d^2 + 
12*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^ 
3*i^3))/(b*d) - ((140*a*d + 140*b*c)*((g^3*i^3*(20*A*a^3*d^3 + 20*A*b^3*c^ 
3 + 3*B*a^3*d^3 - 3*B*b^3*c^3 + 120*A*a*b^2*c^2*d + 120*A*a^2*b*c*d^2 - 6* 
B*a*b^2*c^2*d + 6*B*a^2*b*c*d^2))/5 + ((140*a*d + 140*b*c)*((((b^2*d^2*g^3 
*i^3*(28*A*a*d + 28*A*b*c + B*a*d - B*b*c))/7 - (A*b^2*d^2*g^3*i^3*(140*a* 
d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12*A*a^2 
*d^2 + 12*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*c*d))/2 + A*a*b^2*c 
*d^2*g^3*i^3))/(140*b*d) - (a*c*((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B 
*a*d - B*b*c))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140))/(b*d)))/( 
140*b*d) + (g^3*i^3*(4*A*a^4*d^4 + 4*A*b^4*c^4 + B*a^4*d^4 - B*b^4*c^4 + 1 
44*A*a^2*b^2*c^2*d^2 + 64*A*a*b^3*c^3*d + 64*A*a^3*b*c*d^3 - 8*B*a*b^3*c^3 
*d + 8*B*a^3*b*c*d^3))/(4*b*d)))/(140*b*d) + (a*c*((g^3*i^3*(20*A*a^3*d^3 
+ 20*A*b^3*c^3 + 3*B*a^3*d^3 - 3*B*b^3*c^3 + 120*A*a*b^2*c^2*d + 120*A*a^2 
*b*c*d^2 - 6*B*a*b^2*c^2*d + 6*B*a^2*b*c*d^2))/5 + ((140*a*d + 140*b*c)*(( 
((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d - B*b*c))/7 - (A*b^2*d^2*g^ 
3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3* 
i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*c*d...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 1347, normalized size of antiderivative = 2.95 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(g**3*i*(6*log(a + b*x)*a**7*d**7 - 42*log(a + b*x)*a**6*b*c*d**6 + 126*lo 
g(a + b*x)*a**5*b**2*c**2*d**5 - 210*log(a + b*x)*a**4*b**3*c**3*d**4 + 21 
0*log(a + b*x)*a**3*b**4*c**4*d**3 - 126*log(a + b*x)*a**2*b**5*c**5*d**2 
+ 42*log(a + b*x)*a*b**6*c**6*d - 6*log(a + b*x)*b**7*c**7 - 210*log((a*e 
+ b*e*x)/(c + d*x))*a**3*b**4*c**4*d**3 - 840*log((a*e + b*e*x)/(c + d*x)) 
*a**3*b**4*c**3*d**4*x - 1260*log((a*e + b*e*x)/(c + d*x))*a**3*b**4*c**2* 
d**5*x**2 - 840*log((a*e + b*e*x)/(c + d*x))*a**3*b**4*c*d**6*x**3 - 210*l 
og((a*e + b*e*x)/(c + d*x))*a**3*b**4*d**7*x**4 + 126*log((a*e + b*e*x)/(c 
 + d*x))*a**2*b**5*c**5*d**2 - 1260*log((a*e + b*e*x)/(c + d*x))*a**2*b**5 
*c**3*d**4*x**2 - 2520*log((a*e + b*e*x)/(c + d*x))*a**2*b**5*c**2*d**5*x* 
*3 - 1890*log((a*e + b*e*x)/(c + d*x))*a**2*b**5*c*d**6*x**4 - 504*log((a* 
e + b*e*x)/(c + d*x))*a**2*b**5*d**7*x**5 - 42*log((a*e + b*e*x)/(c + d*x) 
)*a*b**6*c**6*d - 840*log((a*e + b*e*x)/(c + d*x))*a*b**6*c**3*d**4*x**3 - 
 1890*log((a*e + b*e*x)/(c + d*x))*a*b**6*c**2*d**5*x**4 - 1512*log((a*e + 
 b*e*x)/(c + d*x))*a*b**6*c*d**6*x**5 - 420*log((a*e + b*e*x)/(c + d*x))*a 
*b**6*d**7*x**6 + 6*log((a*e + b*e*x)/(c + d*x))*b**7*c**7 - 210*log((a*e 
+ b*e*x)/(c + d*x))*b**7*c**3*d**4*x**4 - 504*log((a*e + b*e*x)/(c + d*x)) 
*b**7*c**2*d**5*x**5 - 420*log((a*e + b*e*x)/(c + d*x))*b**7*c*d**6*x**6 - 
 120*log((a*e + b*e*x)/(c + d*x))*b**7*d**7*x**7 - 6*a**6*b*d**7*x + 42*a* 
*5*b**2*c*d**6*x + 3*a**5*b**2*d**7*x**2 - 840*a**4*b**3*c**3*d**4*x - ...