\(\int (a g+b g x)^2 (c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\) [21]

Optimal result

Integrand size = 40, antiderivative size = 371 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=-\frac {B (b c-a d)^5 g^2 i^3 x}{60 b^3 d^2}-\frac {B (b c-a d)^4 g^2 i^3 (c+d x)^2}{120 b^2 d^3}-\frac {B (b c-a d)^3 g^2 i^3 (c+d x)^3}{180 b d^3}+\frac {7 B (b c-a d)^2 g^2 i^3 (c+d x)^4}{120 d^3}-\frac {b B (b c-a d) g^2 i^3 (c+d x)^5}{30 d^3}-\frac {B (b c-a d)^6 g^2 i^3 \log \left (\frac {a+b x}{c+d x}\right )}{60 b^4 d^3}+\frac {(b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d^3}-\frac {2 b (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^3}+\frac {b^2 g^2 i^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^3}-\frac {B (b c-a d)^6 g^2 i^3 \log (c+d x)}{60 b^4 d^3} \] Output:

-1/60*B*(-a*d+b*c)^5*g^2*i^3*x/b^3/d^2-1/120*B*(-a*d+b*c)^4*g^2*i^3*(d*x+c 
)^2/b^2/d^3-1/180*B*(-a*d+b*c)^3*g^2*i^3*(d*x+c)^3/b/d^3+7/120*B*(-a*d+b*c 
)^2*g^2*i^3*(d*x+c)^4/d^3-1/30*b*B*(-a*d+b*c)*g^2*i^3*(d*x+c)^5/d^3-1/60*B 
*(-a*d+b*c)^6*g^2*i^3*ln((b*x+a)/(d*x+c))/b^4/d^3+1/4*(-a*d+b*c)^2*g^2*i^3 
*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^3-2/5*b*(-a*d+b*c)*g^2*i^3*(d*x+c 
)^5*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^3+1/6*b^2*g^2*i^3*(d*x+c)^6*(A+B*ln(e*(b 
*x+a)/(d*x+c)))/d^3-1/60*B*(-a*d+b*c)^6*g^2*i^3*ln(d*x+c)/b^4/d^3
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.16 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g^2 i^3 \left (-15 B (b c-a d)^3 \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )+12 B (b c-a d)^2 \left (12 b d (b c-a d)^3 x+6 b^2 (b c-a d)^2 (c+d x)^2+4 b^3 (b c-a d) (c+d x)^3+3 b^4 (c+d x)^4+12 (b c-a d)^4 \log (a+b x)\right )-B (b c-a d) \left (60 b d (b c-a d)^4 x+30 b^2 (b c-a d)^3 (c+d x)^2+20 b^3 (b c-a d)^2 (c+d x)^3+15 b^4 (b c-a d) (c+d x)^4+12 b^5 (c+d x)^5+60 (b c-a d)^5 \log (a+b x)\right )+90 b^4 (b c-a d)^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-144 b^5 (b c-a d) (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+60 b^6 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right )}{360 b^4 d^3} \] Input:

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

Output:

(g^2*i^3*(-15*B*(b*c - a*d)^3*(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]) + 12*B*(b*c 
 - a*d)^2*(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*(b*c - a*d)*(60*b*d*(b*c - a*d)^4*x + 30*b^2*(b*c - a*d)^3*(c + d 
*x)^2 + 20*b^3*(b*c - a*d)^2*(c + d*x)^3 + 15*b^4*(b*c - a*d)*(c + d*x)^4 
+ 12*b^5*(c + d*x)^5 + 60*(b*c - a*d)^5*Log[a + b*x]) + 90*b^4*(b*c - a*d) 
^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 144*b^5*(b*c - a*d)* 
(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 60*b^6*(c + d*x)^6*(A + 
 B*Log[(e*(a + b*x))/(c + d*x)])))/(360*b^4*d^3)
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 327, 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, 1195, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2782

\(\displaystyle g^2 i^3 (b c-a d)^6 \left (-B \int \frac {(c+d x) \left (b^2-\frac {6 d (a+b x) b}{c+d x}+\frac {15 d^2 (a+b x)^2}{(c+d x)^2}\right )}{60 d^3 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}+\frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {2 b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle g^2 i^3 (b c-a d)^6 \left (-\frac {B \int \frac {(c+d x) \left (b^2-\frac {6 d (a+b x) b}{c+d x}+\frac {15 d^2 (a+b x)^2}{(c+d x)^2}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{60 d^3}+\frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {2 b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\)

\(\Big \downarrow \) 1195

\(\displaystyle g^2 i^3 (b c-a d)^6 \left (-\frac {B \int \left (\frac {d}{b^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {14 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {10 b d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {c+d x}{b^4 (a+b x)}\right )d\frac {a+b x}{c+d x}}{60 d^3}+\frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {2 b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle g^2 i^3 (b c-a d)^6 \left (\frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {2 b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^4}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^4}+\frac {1}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {2 b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {7}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {1}{3 b \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{60 d^3}\right )\)

Input:

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

Output:

(b*c - a*d)^6*g^2*i^3*((b^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*d^3*( 
b - (d*(a + b*x))/(c + d*x))^6) - (2*b*(A + B*Log[(e*(a + b*x))/(c + d*x)] 
))/(5*d^3*(b - (d*(a + b*x))/(c + d*x))^5) + (A + B*Log[(e*(a + b*x))/(c + 
 d*x)])/(4*d^3*(b - (d*(a + b*x))/(c + d*x))^4) - (B*((2*b)/(b - (d*(a + b 
*x))/(c + d*x))^5 - 7/(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))/(60*d^3))
 

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 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

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. \(924\) vs. \(2(351)=702\).

Time = 1.68 (sec) , antiderivative size = 925, normalized size of antiderivative = 2.49

method	result	size
risch	\(\frac {3 i^{3} g^{2} b^{2} d^{2} A c \,x^{5}}{5}+\frac {i^{3} g^{2} b \,d^{3} B a \,x^{5}}{30}-\frac {i^{3} g^{2} b^{2} d^{2} B c \,x^{5}}{30}+\frac {i^{3} g^{2} d^{3} A \,a^{2} x^{4}}{4}+\frac {3 i^{3} g^{2} b^{2} d A \,c^{2} x^{4}}{4}+\frac {i^{3} g^{2} b B \ln \left (-d x -c \right ) a \,c^{5}}{10 d^{2}}+\frac {3 i^{3} g^{2} b \,d^{2} A a c \,x^{4}}{2}+\frac {i^{3} g^{2} b \,d^{2} B a c \,x^{4}}{20}+i^{3} g^{2} d^{2} A \,a^{2} c \,x^{3}+2 i^{3} g^{2} b d A a \,c^{2} x^{3}+\frac {13 i^{3} g^{2} d^{2} B \,a^{2} c \,x^{3}}{60}-\frac {7 i^{3} g^{2} b d B a \,c^{2} x^{3}}{60}+\frac {3 i^{3} g^{2} d A \,a^{2} c^{2} x^{2}}{2}+i^{3} g^{2} b A a \,c^{3} x^{2}+\frac {2 i^{3} g^{2} b \,d^{3} A a \,x^{5}}{5}+\frac {i^{3} g^{2} d B \,a^{3} c^{2} x}{4 b}-\frac {i^{3} g^{2} B \,a^{2} c^{3} x}{12}-\frac {i^{3} g^{2} b B a \,c^{4} x}{10 d}+\frac {i^{3} g^{2} d^{2} B \ln \left (b x +a \right ) a^{5} c}{10 b^{3}}-\frac {i^{3} g^{2} d B \ln \left (b x +a \right ) a^{4} c^{2}}{4 b^{2}}+\frac {i^{3} g^{2} d^{2} B \,a^{3} c \,x^{2}}{20 b}+\frac {i^{3} g^{2} d B \,a^{2} c^{2} x^{2}}{4}-\frac {17 i^{3} g^{2} b B a \,c^{3} x^{2}}{60}+i^{3} g^{2} A \,a^{2} c^{3} x -\frac {i^{3} g^{2} d^{2} B \,a^{4} c x}{10 b^{2}}+\frac {7 i^{3} g^{2} d^{3} B \,a^{2} x^{4}}{120}-\frac {13 i^{3} g^{2} b^{2} d B \,c^{2} x^{4}}{120}+\frac {i^{3} g^{2} b^{2} A \,c^{3} x^{3}}{3}+\frac {i^{3} g^{2} d^{3} B \,a^{3} x^{3}}{180 b}-\frac {19 i^{3} g^{2} b^{2} B \,c^{3} x^{3}}{180}-\frac {i^{3} g^{2} d^{3} B \,a^{4} x^{2}}{120 b^{2}}-\frac {i^{3} g^{2} b^{2} B \,c^{4} x^{2}}{120 d}+\frac {i^{3} g^{2} d^{3} B \,a^{5} x}{60 b^{3}}+\frac {i^{3} g^{2} b^{2} B \,c^{5} x}{60 d^{2}}+\frac {i^{3} g^{2} B \ln \left (b x +a \right ) a^{3} c^{3}}{3 b}-\frac {i^{3} g^{2} B \ln \left (-d x -c \right ) a^{2} c^{4}}{4 d}-\frac {i^{3} g^{2} d^{3} B \ln \left (b x +a \right ) a^{6}}{60 b^{4}}-\frac {i^{3} g^{2} b^{2} B \ln \left (-d x -c \right ) c^{6}}{60 d^{3}}+\frac {i^{3} g^{2} B x \left (10 d^{3} b^{2} x^{5}+24 a b \,d^{3} x^{4}+36 b^{2} c \,d^{2} x^{4}+15 a^{2} d^{3} x^{3}+90 a b c \,d^{2} x^{3}+45 b^{2} c^{2} d \,x^{3}+60 a^{2} c \,d^{2} x^{2}+120 a b \,c^{2} d \,x^{2}+20 b^{2} c^{3} x^{2}+90 a^{2} c^{2} d x +60 a b \,c^{3} x +60 c^{3} a^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{60}+\frac {i^{3} g^{2} b^{2} d^{3} A \,x^{6}}{6}\)	\(925\)
parallelrisch	\(\text {Expression too large to display}\)	\(1542\)
parts	\(\text {Expression too large to display}\)	\(1708\)
derivativedivides	\(\text {Expression too large to display}\)	\(1796\)
default	\(\text {Expression too large to display}\)	\(1796\)

Input:

int((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c))),x,method=_RETU 
RNVERBOSE)
 

Output:

3/5*i^3*g^2*b^2*d^2*A*c*x^5+1/30*i^3*g^2*b*d^3*B*a*x^5-1/30*i^3*g^2*b^2*d^ 
2*B*c*x^5+1/4*i^3*g^2*d^3*A*a^2*x^4+3/4*i^3*g^2*b^2*d*A*c^2*x^4+1/10*i^3*g 
^2*b/d^2*B*ln(-d*x-c)*a*c^5+3/2*i^3*g^2*b*d^2*A*a*c*x^4+1/20*i^3*g^2*b*d^2 
*B*a*c*x^4+i^3*g^2*d^2*A*a^2*c*x^3+2*i^3*g^2*b*d*A*a*c^2*x^3+13/60*i^3*g^2 
*d^2*B*a^2*c*x^3-7/60*i^3*g^2*b*d*B*a*c^2*x^3+3/2*i^3*g^2*d*A*a^2*c^2*x^2+ 
i^3*g^2*b*A*a*c^3*x^2+2/5*i^3*g^2*b*d^3*A*a*x^5+1/4*i^3*g^2/b*d*B*a^3*c^2* 
x-1/12*i^3*g^2*B*a^2*c^3*x-1/10*i^3*g^2*b/d*B*a*c^4*x+1/10*i^3*g^2/b^3*d^2 
*B*ln(b*x+a)*a^5*c-1/4*i^3*g^2/b^2*d*B*ln(b*x+a)*a^4*c^2+1/20*i^3*g^2/b*d^ 
2*B*a^3*c*x^2+1/4*i^3*g^2*d*B*a^2*c^2*x^2-17/60*i^3*g^2*b*B*a*c^3*x^2+i^3* 
g^2*A*a^2*c^3*x-1/10*i^3*g^2/b^2*d^2*B*a^4*c*x+7/120*i^3*g^2*d^3*B*a^2*x^4 
-13/120*i^3*g^2*b^2*d*B*c^2*x^4+1/3*i^3*g^2*b^2*A*c^3*x^3+1/180*i^3*g^2/b* 
d^3*B*a^3*x^3-19/180*i^3*g^2*b^2*B*c^3*x^3-1/120*i^3*g^2/b^2*d^3*B*a^4*x^2 
-1/120*i^3*g^2*b^2/d*B*c^4*x^2+1/60*i^3*g^2/b^3*d^3*B*a^5*x+1/60*i^3*g^2*b 
^2/d^2*B*c^5*x+1/3*i^3*g^2/b*B*ln(b*x+a)*a^3*c^3-1/4*i^3*g^2/d*B*ln(-d*x-c 
)*a^2*c^4-1/60*i^3*g^2/b^4*d^3*B*ln(b*x+a)*a^6-1/60*i^3*g^2*b^2/d^3*B*ln(- 
d*x-c)*c^6+1/60*i^3*g^2*B*x*(10*b^2*d^3*x^5+24*a*b*d^3*x^4+36*b^2*c*d^2*x^ 
4+15*a^2*d^3*x^3+90*a*b*c*d^2*x^3+45*b^2*c^2*d*x^3+60*a^2*c*d^2*x^2+120*a* 
b*c^2*d*x^2+20*b^2*c^3*x^2+90*a^2*c^2*d*x+60*a*b*c^3*x+60*a^2*c^3)*ln(e*(b 
*x+a)/(d*x+c))+1/6*i^3*g^2*b^2*d^3*A*x^6
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 722 vs. \(2 (351) = 702\).

Time = 0.19 (sec) , antiderivative size = 722, normalized size of antiderivative = 1.95 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {60 \, A b^{6} d^{6} g^{2} i^{3} x^{6} + 12 \, {\left ({\left (18 \, A - B\right )} b^{6} c d^{5} + {\left (12 \, A + B\right )} a b^{5} d^{6}\right )} g^{2} i^{3} x^{5} + 3 \, {\left ({\left (90 \, A - 13 \, B\right )} b^{6} c^{2} d^{4} + 6 \, {\left (30 \, A + B\right )} a b^{5} c d^{5} + {\left (30 \, A + 7 \, B\right )} a^{2} b^{4} d^{6}\right )} g^{2} i^{3} x^{4} + 2 \, {\left ({\left (60 \, A - 19 \, B\right )} b^{6} c^{3} d^{3} + 3 \, {\left (120 \, A - 7 \, B\right )} a b^{5} c^{2} d^{4} + 3 \, {\left (60 \, A + 13 \, B\right )} a^{2} b^{4} c d^{5} + B a^{3} b^{3} d^{6}\right )} g^{2} i^{3} x^{3} - 3 \, {\left (B b^{6} c^{4} d^{2} - 2 \, {\left (60 \, A - 17 \, B\right )} a b^{5} c^{3} d^{3} - 30 \, {\left (6 \, A + B\right )} a^{2} b^{4} c^{2} d^{4} - 6 \, B a^{3} b^{3} c d^{5} + B a^{4} b^{2} d^{6}\right )} g^{2} i^{3} x^{2} + 6 \, {\left (B b^{6} c^{5} d - 6 \, B a b^{5} c^{4} d^{2} + 5 \, {\left (12 \, A - B\right )} a^{2} b^{4} c^{3} d^{3} + 15 \, B a^{3} b^{3} c^{2} d^{4} - 6 \, B a^{4} b^{2} c d^{5} + B a^{5} b d^{6}\right )} g^{2} i^{3} x + 6 \, {\left (20 \, B a^{3} b^{3} c^{3} d^{3} - 15 \, B a^{4} b^{2} c^{2} d^{4} + 6 \, B a^{5} b c d^{5} - B a^{6} d^{6}\right )} g^{2} i^{3} \log \left (b x + a\right ) - 6 \, {\left (B b^{6} c^{6} - 6 \, B a b^{5} c^{5} d + 15 \, B a^{2} b^{4} c^{4} d^{2}\right )} g^{2} i^{3} \log \left (d x + c\right ) + 6 \, {\left (10 \, B b^{6} d^{6} g^{2} i^{3} x^{6} + 60 \, B a^{2} b^{4} c^{3} d^{3} g^{2} i^{3} x + 12 \, {\left (3 \, B b^{6} c d^{5} + 2 \, B a b^{5} d^{6}\right )} g^{2} i^{3} x^{5} + 15 \, {\left (3 \, B b^{6} c^{2} d^{4} + 6 \, B a b^{5} c d^{5} + B a^{2} b^{4} d^{6}\right )} g^{2} i^{3} x^{4} + 20 \, {\left (B b^{6} c^{3} d^{3} + 6 \, B a b^{5} c^{2} d^{4} + 3 \, B a^{2} b^{4} c d^{5}\right )} g^{2} i^{3} x^{3} + 30 \, {\left (2 \, B a b^{5} c^{3} d^{3} + 3 \, B a^{2} b^{4} c^{2} d^{4}\right )} g^{2} i^{3} x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{360 \, b^{4} d^{3}} \] Input:

integrate((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algo 
rithm="fricas")
 

Output:

1/360*(60*A*b^6*d^6*g^2*i^3*x^6 + 12*((18*A - B)*b^6*c*d^5 + (12*A + B)*a* 
b^5*d^6)*g^2*i^3*x^5 + 3*((90*A - 13*B)*b^6*c^2*d^4 + 6*(30*A + B)*a*b^5*c 
*d^5 + (30*A + 7*B)*a^2*b^4*d^6)*g^2*i^3*x^4 + 2*((60*A - 19*B)*b^6*c^3*d^ 
3 + 3*(120*A - 7*B)*a*b^5*c^2*d^4 + 3*(60*A + 13*B)*a^2*b^4*c*d^5 + B*a^3* 
b^3*d^6)*g^2*i^3*x^3 - 3*(B*b^6*c^4*d^2 - 2*(60*A - 17*B)*a*b^5*c^3*d^3 - 
30*(6*A + B)*a^2*b^4*c^2*d^4 - 6*B*a^3*b^3*c*d^5 + B*a^4*b^2*d^6)*g^2*i^3* 
x^2 + 6*(B*b^6*c^5*d - 6*B*a*b^5*c^4*d^2 + 5*(12*A - B)*a^2*b^4*c^3*d^3 + 
15*B*a^3*b^3*c^2*d^4 - 6*B*a^4*b^2*c*d^5 + B*a^5*b*d^6)*g^2*i^3*x + 6*(20* 
B*a^3*b^3*c^3*d^3 - 15*B*a^4*b^2*c^2*d^4 + 6*B*a^5*b*c*d^5 - B*a^6*d^6)*g^ 
2*i^3*log(b*x + a) - 6*(B*b^6*c^6 - 6*B*a*b^5*c^5*d + 15*B*a^2*b^4*c^4*d^2 
)*g^2*i^3*log(d*x + c) + 6*(10*B*b^6*d^6*g^2*i^3*x^6 + 60*B*a^2*b^4*c^3*d^ 
3*g^2*i^3*x + 12*(3*B*b^6*c*d^5 + 2*B*a*b^5*d^6)*g^2*i^3*x^5 + 15*(3*B*b^6 
*c^2*d^4 + 6*B*a*b^5*c*d^5 + B*a^2*b^4*d^6)*g^2*i^3*x^4 + 20*(B*b^6*c^3*d^ 
3 + 6*B*a*b^5*c^2*d^4 + 3*B*a^2*b^4*c*d^5)*g^2*i^3*x^3 + 30*(2*B*a*b^5*c^3 
*d^3 + 3*B*a^2*b^4*c^2*d^4)*g^2*i^3*x^2)*log((b*e*x + a*e)/(d*x + c)))/(b^ 
4*d^3)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1727 vs. \(2 (347) = 694\).

Time = 5.84 (sec) , antiderivative size = 1727, normalized size of antiderivative = 4.65 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

integrate((b*g*x+a*g)**2*(d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c))),x)
 

Output:

A*b**2*d**3*g**2*i**3*x**6/6 - B*a**3*g**2*i**3*(a**3*d**3 - 6*a**2*b*c*d* 
*2 + 15*a*b**2*c**2*d - 20*b**3*c**3)*log(x + (B*a**6*c*d**5*g**2*i**3 - 6 
*B*a**5*b*c**2*d**4*g**2*i**3 + 15*B*a**4*b**2*c**3*d**3*g**2*i**3 + B*a** 
4*d**3*g**2*i**3*(a**3*d**3 - 6*a**2*b*c*d**2 + 15*a*b**2*c**2*d - 20*b**3 
*c**3)/b - 35*B*a**3*b**3*c**4*d**2*g**2*i**3 - B*a**3*c*d**2*g**2*i**3*(a 
**3*d**3 - 6*a**2*b*c*d**2 + 15*a*b**2*c**2*d - 20*b**3*c**3) + 6*B*a**2*b 
**4*c**5*d*g**2*i**3 - B*a*b**5*c**6*g**2*i**3)/(B*a**6*d**6*g**2*i**3 - 6 
*B*a**5*b*c*d**5*g**2*i**3 + 15*B*a**4*b**2*c**2*d**4*g**2*i**3 - 20*B*a** 
3*b**3*c**3*d**3*g**2*i**3 - 15*B*a**2*b**4*c**4*d**2*g**2*i**3 + 6*B*a*b* 
*5*c**5*d*g**2*i**3 - B*b**6*c**6*g**2*i**3))/(60*b**4) - B*c**4*g**2*i**3 
*(15*a**2*d**2 - 6*a*b*c*d + b**2*c**2)*log(x + (B*a**6*c*d**5*g**2*i**3 - 
 6*B*a**5*b*c**2*d**4*g**2*i**3 + 15*B*a**4*b**2*c**3*d**3*g**2*i**3 - 35* 
B*a**3*b**3*c**4*d**2*g**2*i**3 + 6*B*a**2*b**4*c**5*d*g**2*i**3 - B*a*b** 
5*c**6*g**2*i**3 + B*a*b**3*c**4*g**2*i**3*(15*a**2*d**2 - 6*a*b*c*d + b** 
2*c**2) - B*b**4*c**5*g**2*i**3*(15*a**2*d**2 - 6*a*b*c*d + b**2*c**2)/d)/ 
(B*a**6*d**6*g**2*i**3 - 6*B*a**5*b*c*d**5*g**2*i**3 + 15*B*a**4*b**2*c**2 
*d**4*g**2*i**3 - 20*B*a**3*b**3*c**3*d**3*g**2*i**3 - 15*B*a**2*b**4*c**4 
*d**2*g**2*i**3 + 6*B*a*b**5*c**5*d*g**2*i**3 - B*b**6*c**6*g**2*i**3))/(6 
0*d**3) + x**5*(2*A*a*b*d**3*g**2*i**3/5 + 3*A*b**2*c*d**2*g**2*i**3/5 + B 
*a*b*d**3*g**2*i**3/30 - B*b**2*c*d**2*g**2*i**3/30) + x**4*(A*a**2*d**...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1789 vs. \(2 (351) = 702\).

Time = 0.08 (sec) , antiderivative size = 1789, normalized size of antiderivative = 4.82 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

integrate((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algo 
rithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4110 vs. \(2 (351) = 702\).

Time = 0.36 (sec) , antiderivative size = 4110, normalized size of antiderivative = 11.08 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

integrate((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algo 
rithm="giac")
 

Output:

1/360*(6*(B*b^9*c^7*e^7*g^2*i^3 - 7*B*a*b^8*c^6*d*e^7*g^2*i^3 + 21*B*a^2*b 
^7*c^5*d^2*e^7*g^2*i^3 - 35*B*a^3*b^6*c^4*d^3*e^7*g^2*i^3 + 35*B*a^4*b^5*c 
^3*d^4*e^7*g^2*i^3 - 21*B*a^5*b^4*c^2*d^5*e^7*g^2*i^3 + 7*B*a^6*b^3*c*d^6* 
e^7*g^2*i^3 - B*a^7*b^2*d^7*e^7*g^2*i^3 - 6*(b*e*x + a*e)*B*b^8*c^7*d*e^6* 
g^2*i^3/(d*x + c) + 42*(b*e*x + a*e)*B*a*b^7*c^6*d^2*e^6*g^2*i^3/(d*x + c) 
 - 126*(b*e*x + a*e)*B*a^2*b^6*c^5*d^3*e^6*g^2*i^3/(d*x + c) + 210*(b*e*x 
+ a*e)*B*a^3*b^5*c^4*d^4*e^6*g^2*i^3/(d*x + c) - 210*(b*e*x + a*e)*B*a^4*b 
^4*c^3*d^5*e^6*g^2*i^3/(d*x + c) + 126*(b*e*x + a*e)*B*a^5*b^3*c^2*d^6*e^6 
*g^2*i^3/(d*x + c) - 42*(b*e*x + a*e)*B*a^6*b^2*c*d^7*e^6*g^2*i^3/(d*x + c 
) + 6*(b*e*x + a*e)*B*a^7*b*d^8*e^6*g^2*i^3/(d*x + c) + 15*(b*e*x + a*e)^2 
*B*b^7*c^7*d^2*e^5*g^2*i^3/(d*x + c)^2 - 105*(b*e*x + a*e)^2*B*a*b^6*c^6*d 
^3*e^5*g^2*i^3/(d*x + c)^2 + 315*(b*e*x + a*e)^2*B*a^2*b^5*c^5*d^4*e^5*g^2 
*i^3/(d*x + c)^2 - 525*(b*e*x + a*e)^2*B*a^3*b^4*c^4*d^5*e^5*g^2*i^3/(d*x 
+ c)^2 + 525*(b*e*x + a*e)^2*B*a^4*b^3*c^3*d^6*e^5*g^2*i^3/(d*x + c)^2 - 3 
15*(b*e*x + a*e)^2*B*a^5*b^2*c^2*d^7*e^5*g^2*i^3/(d*x + c)^2 + 105*(b*e*x 
+ a*e)^2*B*a^6*b*c*d^8*e^5*g^2*i^3/(d*x + c)^2 - 15*(b*e*x + a*e)^2*B*a^7* 
d^9*e^5*g^2*i^3/(d*x + c)^2)*log((b*e*x + a*e)/(d*x + c))/(b^6*d^3*e^6 - 6 
*(b*e*x + a*e)*b^5*d^4*e^5/(d*x + c) + 15*(b*e*x + a*e)^2*b^4*d^5*e^4/(d*x 
 + c)^2 - 20*(b*e*x + a*e)^3*b^3*d^6*e^3/(d*x + c)^3 + 15*(b*e*x + a*e)^4* 
b^2*d^7*e^2/(d*x + c)^4 - 6*(b*e*x + a*e)^5*b*d^8*e/(d*x + c)^5 + (b*e*...
 

Mupad [B] (verification not implemented)

Time = 28.41 (sec) , antiderivative size = 2465, normalized size of antiderivative = 6.64 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

int((a*g + b*g*x)^2*(c*i + d*i*x)^3*(A + B*log((e*(a + b*x))/(c + d*x))),x 
)
 

Output:

x^3*((g^2*i^3*(4*A*a^3*d^3 + 16*A*b^3*c^3 + B*a^3*d^3 - 3*B*b^3*c^3 + 72*A 
*a*b^2*c^2*d + 48*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d + 5*B*a^2*b*c*d^2))/(12* 
b) + ((60*a*d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d - B 
*b*c))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60* 
b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2 - 3*B*b^2*c^2 
 + 60*A*a*b*c*d + B*a*b*c*d))/5 + A*a*b*c*d^2*g^2*i^3))/(180*b*d) - (a*c*( 
(b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d - B*b*c))/6 - (A*b*d^2*g^2*i^3 
*(60*a*d + 60*b*c))/60))/(3*b*d)) - x^4*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A 
*b*c + B*a*d - B*b*c))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d 
 + 60*b*c))/(240*b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2* 
d^2 - 3*B*b^2*c^2 + 60*A*a*b*c*d + B*a*b*c*d))/20 + (A*a*b*c*d^2*g^2*i^3)/ 
4) + x^2*((a*c*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d - B*b*c))/6 
- (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d 
*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2 - 3*B*b^2*c^2 + 60*A*a 
*b*c*d + B*a*b*c*d))/5 + A*a*b*c*d^2*g^2*i^3))/(2*b*d) - ((60*a*d + 60*b*c 
)*((g^2*i^3*(4*A*a^3*d^3 + 16*A*b^3*c^3 + B*a^3*d^3 - 3*B*b^3*c^3 + 72*A*a 
*b^2*c^2*d + 48*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d + 5*B*a^2*b*c*d^2))/(4*b) 
+ ((60*a*d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d - B*b* 
c))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d 
) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2 - 3*B*b^2*c^2...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 1040, normalized size of antiderivative = 2.80 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx =\text {Too large to display} \] Input:

int((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x)
 

Output:

(g**2*i*(6*log(a + b*x)*a**6*d**6 - 36*log(a + b*x)*a**5*b*c*d**5 + 90*log 
(a + b*x)*a**4*b**2*c**2*d**4 - 120*log(a + b*x)*a**3*b**3*c**3*d**3 + 90* 
log(a + b*x)*a**2*b**4*c**4*d**2 - 36*log(a + b*x)*a*b**5*c**5*d + 6*log(a 
 + b*x)*b**6*c**6 - 90*log((a*e + b*e*x)/(c + d*x))*a**2*b**4*c**4*d**2 - 
360*log((a*e + b*e*x)/(c + d*x))*a**2*b**4*c**3*d**3*x - 540*log((a*e + b* 
e*x)/(c + d*x))*a**2*b**4*c**2*d**4*x**2 - 360*log((a*e + b*e*x)/(c + d*x) 
)*a**2*b**4*c*d**5*x**3 - 90*log((a*e + b*e*x)/(c + d*x))*a**2*b**4*d**6*x 
**4 + 36*log((a*e + b*e*x)/(c + d*x))*a*b**5*c**5*d - 360*log((a*e + b*e*x 
)/(c + d*x))*a*b**5*c**3*d**3*x**2 - 720*log((a*e + b*e*x)/(c + d*x))*a*b* 
*5*c**2*d**4*x**3 - 540*log((a*e + b*e*x)/(c + d*x))*a*b**5*c*d**5*x**4 - 
144*log((a*e + b*e*x)/(c + d*x))*a*b**5*d**6*x**5 - 6*log((a*e + b*e*x)/(c 
 + d*x))*b**6*c**6 - 120*log((a*e + b*e*x)/(c + d*x))*b**6*c**3*d**3*x**3 
- 270*log((a*e + b*e*x)/(c + d*x))*b**6*c**2*d**4*x**4 - 216*log((a*e + b* 
e*x)/(c + d*x))*b**6*c*d**5*x**5 - 60*log((a*e + b*e*x)/(c + d*x))*b**6*d* 
*6*x**6 - 6*a**5*b*d**6*x + 36*a**4*b**2*c*d**5*x + 3*a**4*b**2*d**6*x**2 
- 360*a**3*b**3*c**3*d**3*x - 540*a**3*b**3*c**2*d**4*x**2 - 90*a**3*b**3* 
c**2*d**4*x - 360*a**3*b**3*c*d**5*x**3 - 18*a**3*b**3*c*d**5*x**2 - 90*a* 
*3*b**3*d**6*x**4 - 2*a**3*b**3*d**6*x**3 - 360*a**2*b**4*c**3*d**3*x**2 + 
 30*a**2*b**4*c**3*d**3*x - 720*a**2*b**4*c**2*d**4*x**3 - 90*a**2*b**4*c* 
*2*d**4*x**2 - 540*a**2*b**4*c*d**5*x**4 - 78*a**2*b**4*c*d**5*x**3 - 1...