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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2562, 45, 2382, 12, 78} \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=-\frac {g i^3 (c+d x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d^2}+\frac {b g i^3 (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^2}+\frac {B g i^3 (b c-a d)^5 \log \left (\frac {a+b x}{c+d x}\right )}{20 b^4 d^2}+\frac {B g i^3 (b c-a d)^5 \log (c+d x)}{20 b^4 d^2}+\frac {B g i^3 x (b c-a d)^4}{20 b^3 d}+\frac {B g i^3 (c+d x)^2 (b c-a d)^3}{40 b^2 d^2}+\frac {B g i^3 (c+d x)^3 (b c-a d)^2}{60 b d^2}-\frac {B g i^3 (c+d x)^4 (b c-a d)}{20 d^2} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((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 2382

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]}, Dist[a + b*Log[c*x^n], u, x] - Dist[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 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(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] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i
, 0] && IntegersQ[m, q]

Rubi steps \begin{align*} \text {integral}& = \left ((b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {x (A+B \log (e x))}{(b-d x)^6} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = -\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}-\left (B (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {-b+5 d x}{20 d^2 x (b-d x)^5} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = -\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 {\left (B (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {-b+5 d x}{x (b-d x)^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{20 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 {\left (B (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \left (-\frac {1}{b^4 x}+\frac {4 d}{(b-d x)^5}-\frac {d}{b (b-d x)^4}-\frac {d}{b^2 (b-d x)^3}-\frac {d}{b^3 (b-d x)^2}-\frac {d}{b^4 (b-d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{20 d^2} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.96 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g i^3 \left (\frac {5 B (b c-a d)^2 \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )}{b^4}-\frac {2 B (b c-a d) \left (12 b d (b c-a d)^3 x+6 b^2 (b c-a d)^2 (c+d x)^2+4 b^3 (b c-a d) (c+d x)^3+3 b^4 (c+d x)^4+12 (b c-a d)^4 \log (a+b x)\right )}{b^4}-30 (b c-a d) (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+24 b (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right )}{120 d^2} \]

[In]

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

[Out]

(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)

Maple [B] (verified)

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

Time = 0.89 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.10

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.85 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {24 \, A b^{5} d^{5} g i^{3} x^{5} + 6 \, {\left ({\left (15 \, A - B\right )} b^{5} c d^{4} + {\left (5 \, A + B\right )} a b^{4} d^{5}\right )} g i^{3} x^{4} + 2 \, {\left ({\left (60 \, A - 11 \, B\right )} b^{5} c^{2} d^{3} + 10 \, {\left (6 \, A + B\right )} a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g i^{3} x^{3} + 3 \, {\left ({\left (20 \, A - 9 \, B\right )} b^{5} c^{3} d^{2} + 5 \, {\left (12 \, A + B\right )} a b^{4} c^{2} d^{3} + 5 \, B a^{2} b^{3} c d^{4} - B a^{3} b^{2} d^{5}\right )} g i^{3} x^{2} - 6 \, {\left (B b^{5} c^{4} d - 5 \, {\left (4 \, A - B\right )} a b^{4} c^{3} d^{2} - 10 \, B a^{2} b^{3} c^{2} d^{3} + 5 \, B a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g i^{3} x + 6 \, {\left (10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4} - B a^{5} d^{5}\right )} g i^{3} \log \left (b x + a\right ) + 6 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d\right )} g i^{3} \log \left (d x + c\right ) + 6 \, {\left (4 \, B b^{5} d^{5} g i^{3} x^{5} + 20 \, B a b^{4} c^{3} d^{2} g i^{3} x + 5 \, {\left (3 \, B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g i^{3} x^{4} + 20 \, {\left (B b^{5} c^{2} d^{3} + B a b^{4} c d^{4}\right )} g i^{3} x^{3} + 10 \, {\left (B b^{5} c^{3} d^{2} + 3 \, B a b^{4} c^{2} d^{3}\right )} g i^{3} x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{120 \, b^{4} d^{2}} \]

[In]

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

[Out]

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 = 3.84 (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=\frac {A b d^{3} g i^{3} x^{5}}{5} - \frac {B a^{2} g i^{3} \left (a^{3} d^{3} - 5 a^{2} b c d^{2} + 10 a b^{2} c^{2} d - 10 b^{3} c^{3}\right ) \log {\left (x + \frac {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} + \frac {B a^{3} d^{2} g i^{3} \left (a^{3} d^{3} - 5 a^{2} b c d^{2} + 10 a b^{2} c^{2} d - 10 b^{3} c^{3}\right )}{b} - 15 B a^{2} b^{3} c^{4} d g i^{3} - B a^{2} c d g i^{3} \left (a^{3} d^{3} - 5 a^{2} b c d^{2} + 10 a b^{2} c^{2} d - 10 b^{3} c^{3}\right ) + 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}} \right )}}{20 b^{4}} - \frac {B c^{4} g i^{3} \cdot \left (5 a d - b c\right ) \log {\left (x + \frac {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} \cdot \left (5 a d - b c\right ) - \frac {B b^{4} c^{5} g i^{3} \cdot \left (5 a d - b c\right )}{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}} \right )}}{20 d^{2}} + x^{4} \left (\frac {A a d^{3} g i^{3}}{4} + \frac {3 A b c d^{2} g i^{3}}{4} + \frac {B a d^{3} g i^{3}}{20} - \frac {B b c d^{2} g i^{3}}{20}\right ) + x^{3} \left (A a c d^{2} g i^{3} + A b c^{2} d g i^{3} + \frac {B a^{2} d^{3} g i^{3}}{60 b} + \frac {B a c d^{2} g i^{3}}{6} - \frac {11 B b c^{2} d g i^{3}}{60}\right ) + x^{2} \cdot \left (\frac {3 A a c^{2} d g i^{3}}{2} + \frac {A b c^{3} g i^{3}}{2} - \frac {B a^{3} d^{3} g i^{3}}{40 b^{2}} + \frac {B a^{2} c d^{2} g i^{3}}{8 b} + \frac {B a c^{2} d g i^{3}}{8} - \frac {9 B b c^{3} g i^{3}}{40}\right ) + x \left (A a c^{3} g i^{3} + \frac {B a^{4} d^{3} g i^{3}}{20 b^{3}} - \frac {B a^{3} c d^{2} g i^{3}}{4 b^{2}} + \frac {B a^{2} c^{2} d g i^{3}}{2 b} - \frac {B a c^{3} g i^{3}}{4} - \frac {B b c^{4} g i^{3}}{20 d}\right ) + \left (B a c^{3} g i^{3} x + \frac {3 B a c^{2} d g i^{3} x^{2}}{2} + B a c d^{2} g i^{3} x^{3} + \frac {B a d^{3} g i^{3} x^{4}}{4} + \frac {B b c^{3} g i^{3} x^{2}}{2} + B b c^{2} d g i^{3} x^{3} + \frac {3 B b c d^{2} g i^{3} x^{4}}{4} + \frac {B b d^{3} g i^{3} x^{5}}{5}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \]

[In]

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

[Out]

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*b**2) + B*a**2*c**2*d*g*i**3/(2*b) - B*a*c**3*g*i**3/4 - B*b*c
**4*g*i**3/(20*d)) + (B*a*c**3*g*i**3*x + 3*B*a*c**2*d*g*i**3*x**2/2 + B*a*c*d**2*g*i**3*x**3 + B*a*d**3*g*i**
3*x**4/4 + B*b*c**3*g*i**3*x**2/2 + B*b*c**2*d*g*i**3*x**3 + 3*B*b*c*d**2*g*i**3*x**4/4 + B*b*d**3*g*i**3*x**5
/5)*log(e*(a + b*x)/(c + d*x))

Maxima [B] (verification not implemented)

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

Time = 0.31 (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=\frac {1}{5} \, A b d^{3} g i^{3} x^{5} + \frac {3}{4} \, A b c d^{2} g i^{3} x^{4} + \frac {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} + \frac {1}{2} \, A b c^{3} g i^{3} x^{2} + \frac {3}{2} \, A a c^{2} d g i^{3} x^{2} + {\left (x \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B a c^{3} g i^{3} + \frac {1}{2} \, {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B b c^{3} g i^{3} + \frac {3}{2} \, {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B a c^{2} d g i^{3} + \frac {1}{2} \, {\left (2 \, x^{3} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B b c^{2} d g i^{3} + \frac {1}{2} \, {\left (2 \, x^{3} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B a c d^{2} g i^{3} + \frac {1}{8} \, {\left (6 \, x^{4} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B b c d^{2} g i^{3} + \frac {1}{24} \, {\left (6 \, x^{4} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B a d^{3} g i^{3} + \frac {1}{60} \, {\left (12 \, x^{5} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {12 \, a^{5} \log \left (b x + a\right )}{b^{5}} - \frac {12 \, c^{5} \log \left (d x + c\right )}{d^{5}} - \frac {3 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{4} - 4 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x^{2} - 12 \, {\left (b^{4} c^{4} - a^{4} d^{4}\right )} x}{b^{4} d^{4}}\right )} B b d^{3} g i^{3} + A a c^{3} g i^{3} x \]

[In]

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

[Out]

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 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c
^4 - a^4*d^4)*x)/(b^4*d^4))*B*b*d^3*g*i^3 + A*a*c^3*g*i^3*x

Giac [B] (verification not implemented)

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

Time = 0.48 (sec) , antiderivative size = 2589, normalized size of antiderivative = 9.55 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-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))*log((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) - 450*(b*e*x + a*e)*A*a^2*b^7*c^4*d^3*e^5*g*i^3/(d*x + c) + 4
65*(b*e*x + a*e)*B*a^2*b^7*c^4*d^3*e^5*g*i^3/(d*x + c) + 600*(b*e*x + a*e)*A*a^3*b^6*c^3*d^4*e^5*g*i^3/(d*x +
c) - 620*(b*e*x + a*e)*B*a^3*b^6*c^3*d^4*e^5*g*i^3/(d*x + c) - 450*(b*e*x + a*e)*A*a^4*b^5*c^2*d^5*e^5*g*i^3/(
d*x + c) + 465*(b*e*x + a*e)*B*a^4*b^5*c^2*d^5*e^5*g*i^3/(d*x + c) + 180*(b*e*x + a*e)*A*a^5*b^4*c*d^6*e^5*g*i
^3/(d*x + c) - 186*(b*e*x + a*e)*B*a^5*b^4*c*d^6*e^5*g*i^3/(d*x + c) - 30*(b*e*x + a*e)*A*a^6*b^3*d^7*e^5*g*i^
3/(d*x + c) + 31*(b*e*x + a*e)*B*a^6*b^3*d^7*e^5*g*i^3/(d*x + c) - 47*(b*e*x + a*e)^2*B*b^8*c^6*d^2*e^4*g*i^3/
(d*x + c)^2 + 282*(b*e*x + a*e)^2*B*a*b^7*c^5*d^3*e^4*g*i^3/(d*x + c)^2 - 705*(b*e*x + a*e)^2*B*a^2*b^6*c^4*d^
4*e^4*g*i^3/(d*x + c)^2 + 940*(b*e*x + a*e)^2*B*a^3*b^5*c^3*d^5*e^4*g*i^3/(d*x + c)^2 - 705*(b*e*x + a*e)^2*B*
a^4*b^4*c^2*d^6*e^4*g*i^3/(d*x + c)^2 + 282*(b*e*x + a*e)^2*B*a^5*b^3*c*d^7*e^4*g*i^3/(d*x + c)^2 - 47*(b*e*x
+ a*e)^2*B*a^6*b^2*d^8*e^4*g*i^3/(d*x + c)^2 + 27*(b*e*x + a*e)^3*B*b^7*c^6*d^3*e^3*g*i^3/(d*x + c)^3 - 162*(b
*e*x + a*e)^3*B*a*b^6*c^5*d^4*e^3*g*i^3/(d*x + c)^3 + 405*(b*e*x + a*e)^3*B*a^2*b^5*c^4*d^5*e^3*g*i^3/(d*x + c
)^3 - 540*(b*e*x + a*e)^3*B*a^3*b^4*c^3*d^6*e^3*g*i^3/(d*x + c)^3 + 405*(b*e*x + a*e)^3*B*a^4*b^3*c^2*d^7*e^3*
g*i^3/(d*x + c)^3 - 162*(b*e*x + a*e)^3*B*a^5*b^2*c*d^8*e^3*g*i^3/(d*x + c)^3 + 27*(b*e*x + a*e)^3*B*a^6*b*d^9
*e^3*g*i^3/(d*x + c)^3 - 6*(b*e*x + a*e)^4*B*b^6*c^6*d^4*e^2*g*i^3/(d*x + c)^4 + 36*(b*e*x + a*e)^4*B*a*b^5*c^
5*d^5*e^2*g*i^3/(d*x + c)^4 - 90*(b*e*x + a*e)^4*B*a^2*b^4*c^4*d^6*e^2*g*i^3/(d*x + c)^4 + 120*(b*e*x + a*e)^4
*B*a^3*b^3*c^3*d^7*e^2*g*i^3/(d*x + c)^4 - 90*(b*e*x + a*e)^4*B*a^4*b^2*c^2*d^8*e^2*g*i^3/(d*x + c)^4 + 36*(b*
e*x + a*e)^4*B*a^5*b*c*d^9*e^2*g*i^3/(d*x + c)^4 - 6*(b*e*x + a*e)^4*B*a^6*d^10*e^2*g*i^3/(d*x + c)^4)/(b^8*d^
2*e^5 - 5*(b*e*x + a*e)*b^7*d^3*e^4/(d*x + c) + 10*(b*e*x + a*e)^2*b^6*d^4*e^3/(d*x + c)^2 - 10*(b*e*x + a*e)^
3*b^5*d^5*e^2/(d*x + c)^3 + 5*(b*e*x + a*e)^4*b^4*d^6*e/(d*x + c)^4 - (b*e*x + a*e)^5*b^3*d^7/(d*x + c)^5) + 6
*(B*b^6*c^6*e*g*i^3 - 6*B*a*b^5*c^5*d*e*g*i^3 + 15*B*a^2*b^4*c^4*d^2*e*g*i^3 - 20*B*a^3*b^3*c^3*d^3*e*g*i^3 +
15*B*a^4*b^2*c^2*d^4*e*g*i^3 - 6*B*a^5*b*c*d^5*e*g*i^3 + B*a^6*d^6*e*g*i^3)*log(-b*e + (b*e*x + a*e)*d/(d*x +
c))/(b^4*d^2) - 6*(B*b^6*c^6*e*g*i^3 - 6*B*a*b^5*c^5*d*e*g*i^3 + 15*B*a^2*b^4*c^4*d^2*e*g*i^3 - 20*B*a^3*b^3*c
^3*d^3*e*g*i^3 + 15*B*a^4*b^2*c^2*d^4*e*g*i^3 - 6*B*a^5*b*c*d^5*e*g*i^3 + B*a^6*d^6*e*g*i^3)*log((b*e*x + a*e)
/(d*x + c))/(b^4*d^2))*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))

Mupad [B] (verification not implemented)

Time = 2.06 (sec) , antiderivative size = 1192, normalized size of antiderivative = 4.40 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

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*(((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))/(2
0*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*(((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))/(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*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))/(2*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))/(2*b)) + log((e*(a + b*x))/(c + d*x))*((B*c^2*g*i^3*x^2*(3*a*d + b*c))/2 + (B*d^2*g*i^3*x^4
*(a*d + 3*b*c))/4 + B*a*c^3*g*i^3*x + (B*b*d^3*g*i^3*x^5)/5 + B*c*d*g*i^3*x^3*(a*d + b*c)) + (log(c + d*x)*(B*
b*c^5*g*i^3 - 5*B*a*c^4*d*g*i^3))/(20*d^2) - (log(a + b*x)*(B*a^5*d^3*g*i^3 - 10*B*a^2*b^3*c^3*g*i^3 - 5*B*a^4
*b*c*d^2*g*i^3 + 10*B*a^3*b^2*c^2*d*g*i^3))/(20*b^4) + (A*b*d^3*g*i^3*x^5)/5