\(\int (a g+b g x)^3 (c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\) [10]

Optimal result

Integrand size = 40, antiderivative size = 423 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {B (b c-a d)^5 g^3 i^2 x}{60 b^2 d^3}+\frac {B (b c-a d)^4 g^3 i^2 (c+d x)^2}{120 b d^4}-\frac {19 B (b c-a d)^3 g^3 i^2 (c+d x)^3}{180 d^4}+\frac {13 b B (b c-a d)^2 g^3 i^2 (c+d x)^4}{120 d^4}-\frac {b^2 B (b c-a d) g^3 i^2 (c+d x)^5}{30 d^4}+\frac {B (b c-a d)^6 g^3 i^2 \log \left (\frac {a+b x}{c+d x}\right )}{60 b^3 d^4}-\frac {(b c-a d)^3 g^3 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d^4}+\frac {3 b (b c-a d)^2 g^3 i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d^4}-\frac {3 b^2 (b c-a d) g^3 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^4}+\frac {b^3 g^3 i^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^4}+\frac {B (b c-a d)^6 g^3 i^2 \log (c+d x)}{60 b^3 d^4} \]

[Out]

1/60*B*(-a*d+b*c)^5*g^3*i^2*x/b^2/d^3+1/120*B*(-a*d+b*c)^4*g^3*i^2*(d*x+c)^2/b/d^4-19/180*B*(-a*d+b*c)^3*g^3*i
^2*(d*x+c)^3/d^4+13/120*b*B*(-a*d+b*c)^2*g^3*i^2*(d*x+c)^4/d^4-1/30*b^2*B*(-a*d+b*c)*g^3*i^2*(d*x+c)^5/d^4+1/6
0*B*(-a*d+b*c)^6*g^3*i^2*ln((b*x+a)/(d*x+c))/b^3/d^4-1/3*(-a*d+b*c)^3*g^3*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x
+c)))/d^4+3/4*b*(-a*d+b*c)^2*g^3*i^2*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^4-3/5*b^2*(-a*d+b*c)*g^3*i^2*(d*x
+c)^5*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^4+1/6*b^3*g^3*i^2*(d*x+c)^6*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^4+1/60*B*(-a*d+b
*c)^6*g^3*i^2*ln(d*x+c)/b^3/d^4

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2562, 45, 2382, 12, 1634} \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {b^3 g^3 i^2 (c+d x)^6 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 d^4}-\frac {3 b^2 g^3 i^2 (c+d x)^5 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^4}-\frac {g^3 i^2 (c+d x)^3 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d^4}+\frac {3 b g^3 i^2 (c+d x)^4 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d^4}+\frac {B g^3 i^2 (b c-a d)^6 \log \left (\frac {a+b x}{c+d x}\right )}{60 b^3 d^4}+\frac {B g^3 i^2 (b c-a d)^6 \log (c+d x)}{60 b^3 d^4}-\frac {b^2 B g^3 i^2 (c+d x)^5 (b c-a d)}{30 d^4}+\frac {B g^3 i^2 x (b c-a d)^5}{60 b^2 d^3}+\frac {B g^3 i^2 (c+d x)^2 (b c-a d)^4}{120 b d^4}-\frac {19 B g^3 i^2 (c+d x)^3 (b c-a d)^3}{180 d^4}+\frac {13 b B g^3 i^2 (c+d x)^4 (b c-a d)^2}{120 d^4} \]

[In]

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

[Out]

(B*(b*c - a*d)^5*g^3*i^2*x)/(60*b^2*d^3) + (B*(b*c - a*d)^4*g^3*i^2*(c + d*x)^2)/(120*b*d^4) - (19*B*(b*c - a*
d)^3*g^3*i^2*(c + d*x)^3)/(180*d^4) + (13*b*B*(b*c - a*d)^2*g^3*i^2*(c + d*x)^4)/(120*d^4) - (b^2*B*(b*c - a*d
)*g^3*i^2*(c + d*x)^5)/(30*d^4) + (B*(b*c - a*d)^6*g^3*i^2*Log[(a + b*x)/(c + d*x)])/(60*b^3*d^4) - ((b*c - a*
d)^3*g^3*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*d^4) + (3*b*(b*c - a*d)^2*g^3*i^2*(c + d*x)^
4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*d^4) - (3*b^2*(b*c - a*d)*g^3*i^2*(c + d*x)^5*(A + B*Log[(e*(a + b*
x))/(c + d*x)]))/(5*d^4) + (b^3*g^3*i^2*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*d^4) + (B*(b*c -
a*d)^6*g^3*i^2*Log[c + d*x])/(60*b^3*d^4)

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 1634

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]) &&
GtQ[Expon[Px, x], 2]

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)^6 g^3 i^2\right ) \text {Subst}\left (\int \frac {x^3 (A+B \log (e x))}{(b-d x)^7} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = -\frac {(b c-a d)^3 g^3 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d^4}+\frac {3 b (b c-a d)^2 g^3 i^2 (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^2 (b c-a d) g^3 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^4}+\frac {b^3 g^3 i^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^4}-\left (B (b c-a d)^6 g^3 i^2\right ) \text {Subst}\left (\int \frac {-b^3+6 b^2 d x-15 b d^2 x^2+20 d^3 x^3}{60 d^4 x (b-d x)^6} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = -\frac {(b c-a d)^3 g^3 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d^4}+\frac {3 b (b c-a d)^2 g^3 i^2 (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^2 (b c-a d) g^3 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^4}+\frac {b^3 g^3 i^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^4}-\frac {\left (B (b c-a d)^6 g^3 i^2\right ) \text {Subst}\left (\int \frac {-b^3+6 b^2 d x-15 b d^2 x^2+20 d^3 x^3}{x (b-d x)^6} \, dx,x,\frac {a+b x}{c+d x}\right )}{60 d^4} \\ & = -\frac {(b c-a d)^3 g^3 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d^4}+\frac {3 b (b c-a d)^2 g^3 i^2 (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^2 (b c-a d) g^3 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^4}+\frac {b^3 g^3 i^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^4}-\frac {\left (B (b c-a d)^6 g^3 i^2\right ) \text {Subst}\left (\int \left (-\frac {1}{b^3 x}+\frac {10 b^2 d}{(b-d x)^6}-\frac {26 b d}{(b-d x)^5}+\frac {19 d}{(b-d x)^4}-\frac {d}{b (b-d x)^3}-\frac {d}{b^2 (b-d x)^2}-\frac {d}{b^3 (b-d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{60 d^4} \\ & = \frac {B (b c-a d)^5 g^3 i^2 x}{60 b^2 d^3}+\frac {B (b c-a d)^4 g^3 i^2 (c+d x)^2}{120 b d^4}-\frac {19 B (b c-a d)^3 g^3 i^2 (c+d x)^3}{180 d^4}+\frac {13 b B (b c-a d)^2 g^3 i^2 (c+d x)^4}{120 d^4}-\frac {b^2 B (b c-a d) g^3 i^2 (c+d x)^5}{30 d^4}+\frac {B (b c-a d)^6 g^3 i^2 \log \left (\frac {a+b x}{c+d x}\right )}{60 b^3 d^4}-\frac {(b c-a d)^3 g^3 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d^4}+\frac {3 b (b c-a d)^2 g^3 i^2 (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^2 (b c-a d) g^3 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^4}+\frac {b^3 g^3 i^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^4}+\frac {B (b c-a d)^6 g^3 i^2 \log (c+d x)}{60 b^3 d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.01 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g^3 i^2 \left (90 d^4 (b c-a d)^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+144 d^5 (b c-a d) (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+60 d^6 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-15 B (b c-a d)^3 \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )+12 B (b c-a d)^2 \left (12 b d (b c-a d)^3 x-6 d^2 (b c-a d)^2 (a+b x)^2+4 d^3 (b c-a d) (a+b x)^3-3 d^4 (a+b x)^4-12 (b c-a d)^4 \log (c+d x)\right )-B (b c-a d) \left (60 b d (b c-a d)^4 x+30 d^2 (-b c+a d)^3 (a+b x)^2+20 d^3 (b c-a d)^2 (a+b x)^3+15 d^4 (-b c+a d) (a+b x)^4+12 d^5 (a+b x)^5-60 (b c-a d)^5 \log (c+d x)\right )\right )}{360 b^3 d^4} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(924\) vs. \(2(401)=802\).

Time = 1.18 (sec) , antiderivative size = 925, normalized size of antiderivative = 2.19

method	result	size
risch	\(\frac {i^{2} g^{3} B \,a^{3} c^{2} x}{12}-\frac {i^{2} g^{3} b B \,a^{2} c^{3} x}{4 d}+\frac {i^{2} g^{3} b^{2} B a \,c^{4} x}{10 d^{2}}+\frac {i^{2} g^{3} b B \ln \left (d x +c \right ) a^{2} c^{4}}{4 d^{2}}+\frac {7 i^{2} g^{3} b d B \,a^{2} c \,x^{3}}{60}-\frac {13 i^{2} g^{3} b^{2} B a \,c^{2} x^{3}}{60}+i^{2} g^{3} d A \,a^{3} c \,x^{2}+\frac {3 i^{2} g^{3} b A \,a^{2} c^{2} x^{2}}{2}+\frac {17 i^{2} g^{3} d B \,a^{3} c \,x^{2}}{60}-\frac {i^{2} g^{3} b B \,a^{2} c^{2} x^{2}}{4}-\frac {i^{2} g^{3} b^{2} B a \,c^{3} x^{2}}{20 d}+i^{2} g^{3} A \,a^{3} c^{2} x +\frac {i^{2} g^{3} d B \,a^{4} c x}{10 b}-\frac {i^{2} g^{3} b^{3} B \,c^{5} x}{60 d^{3}}-\frac {i^{2} g^{3} B \ln \left (d x +c \right ) a^{3} c^{3}}{3 d}+\frac {i^{2} g^{3} B \ln \left (-b x -a \right ) a^{4} c^{2}}{4 b}+\frac {i^{2} g^{3} b^{3} B \ln \left (d x +c \right ) c^{6}}{60 d^{4}}+\frac {i^{2} g^{3} d^{2} B \ln \left (-b x -a \right ) a^{6}}{60 b^{3}}+\frac {3 i^{2} g^{3} b^{2} d^{2} A a \,x^{5}}{5}+\frac {2 i^{2} g^{3} b^{3} d A c \,x^{5}}{5}+\frac {i^{2} g^{3} b^{2} d^{2} B a \,x^{5}}{30}-\frac {i^{2} g^{3} b^{3} d B c \,x^{5}}{30}+\frac {3 i^{2} g^{3} b \,d^{2} A \,a^{2} x^{4}}{4}+\frac {i^{2} g^{3} b^{3} A \,c^{2} x^{4}}{4}+\frac {13 i^{2} g^{3} b \,d^{2} B \,a^{2} x^{4}}{120}-\frac {7 i^{2} g^{3} b^{3} B \,c^{2} x^{4}}{120}+\frac {i^{2} g^{3} d^{2} A \,a^{3} x^{3}}{3}+\frac {19 i^{2} g^{3} d^{2} B \,a^{3} x^{3}}{180}-\frac {i^{2} g^{3} b^{3} B \,c^{3} x^{3}}{180 d}+\frac {i^{2} g^{3} d^{2} B \,a^{4} x^{2}}{120 b}+\frac {i^{2} g^{3} b^{3} B \,c^{4} x^{2}}{120 d^{2}}-\frac {i^{2} g^{3} d^{2} B \,a^{5} x}{60 b^{2}}-\frac {i^{2} g^{3} b^{2} B \ln \left (d x +c \right ) a \,c^{5}}{10 d^{3}}-\frac {i^{2} g^{3} d B \ln \left (-b x -a \right ) a^{5} c}{10 b^{2}}+\frac {i^{2} g^{3} b^{3} d^{2} A \,x^{6}}{6}+\frac {3 i^{2} g^{3} b^{2} d A a c \,x^{4}}{2}-\frac {i^{2} g^{3} b^{2} d B a c \,x^{4}}{20}+2 i^{2} g^{3} b d A \,a^{2} c \,x^{3}+i^{2} g^{3} b^{2} A a \,c^{2} x^{3}+\frac {i^{2} g^{3} B x \left (10 d^{2} b^{3} x^{5}+36 a \,b^{2} d^{2} x^{4}+24 b^{3} c d \,x^{4}+45 a^{2} b \,d^{2} x^{3}+90 a \,b^{2} c d \,x^{3}+15 b^{3} c^{2} x^{3}+20 a^{3} d^{2} x^{2}+120 a^{2} b c d \,x^{2}+60 a \,b^{2} c^{2} x^{2}+60 a^{3} c d x +90 a^{2} b \,c^{2} x +60 c^{2} a^{3}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{60}\)	\(925\)
parallelrisch	\(\text {Expression too large to display}\)	\(1578\)
parts	\(\text {Expression too large to display}\)	\(2106\)
derivativedivides	\(\text {Expression too large to display}\)	\(2168\)
default	\(\text {Expression too large to display}\)	\(2168\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.51 (sec) , antiderivative size = 724, normalized size of antiderivative = 1.71 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {60 \, A b^{6} d^{6} g^{3} i^{2} x^{6} + 12 \, {\left ({\left (12 \, A - B\right )} b^{6} c d^{5} + {\left (18 \, A + B\right )} a b^{5} d^{6}\right )} g^{3} i^{2} x^{5} + 3 \, {\left ({\left (30 \, A - 7 \, B\right )} b^{6} c^{2} d^{4} + 6 \, {\left (30 \, A - B\right )} a b^{5} c d^{5} + {\left (90 \, A + 13 \, B\right )} a^{2} b^{4} d^{6}\right )} g^{3} i^{2} x^{4} - 2 \, {\left (B b^{6} c^{3} d^{3} - 3 \, {\left (60 \, A - 13 \, B\right )} a b^{5} c^{2} d^{4} - 3 \, {\left (120 \, A + 7 \, B\right )} a^{2} b^{4} c d^{5} - {\left (60 \, A + 19 \, B\right )} a^{3} b^{3} d^{6}\right )} g^{3} i^{2} x^{3} + 3 \, {\left (B b^{6} c^{4} d^{2} - 6 \, B a b^{5} c^{3} d^{3} + 30 \, {\left (6 \, A - B\right )} a^{2} b^{4} c^{2} d^{4} + 2 \, {\left (60 \, A + 17 \, B\right )} a^{3} b^{3} c d^{5} + B a^{4} b^{2} d^{6}\right )} g^{3} i^{2} x^{2} - 6 \, {\left (B b^{6} c^{5} d - 6 \, B a b^{5} c^{4} d^{2} + 15 \, B a^{2} b^{4} c^{3} d^{3} - 5 \, {\left (12 \, A + B\right )} 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^{3} i^{2} x + 6 \, {\left (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^{3} i^{2} \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} - 20 \, B a^{3} b^{3} c^{3} d^{3}\right )} g^{3} i^{2} \log \left (d x + c\right ) + 6 \, {\left (10 \, B b^{6} d^{6} g^{3} i^{2} x^{6} + 60 \, B a^{3} b^{3} c^{2} d^{4} g^{3} i^{2} x + 12 \, {\left (2 \, B b^{6} c d^{5} + 3 \, B a b^{5} d^{6}\right )} g^{3} i^{2} x^{5} + 15 \, {\left (B b^{6} c^{2} d^{4} + 6 \, B a b^{5} c d^{5} + 3 \, B a^{2} b^{4} d^{6}\right )} g^{3} i^{2} x^{4} + 20 \, {\left (3 \, B a b^{5} c^{2} d^{4} + 6 \, B a^{2} b^{4} c d^{5} + B a^{3} b^{3} d^{6}\right )} g^{3} i^{2} x^{3} + 30 \, {\left (3 \, B a^{2} b^{4} c^{2} d^{4} + 2 \, B a^{3} b^{3} c d^{5}\right )} g^{3} i^{2} x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{360 \, b^{3} d^{4}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1727 vs. \(2 (398) = 796\).

Time = 7.85 (sec) , antiderivative size = 1727, normalized size of antiderivative = 4.08 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

A*b**3*d**2*g**3*i**2*x**6/6 + B*a**4*g**3*i**2*(a**2*d**2 - 6*a*b*c*d + 15*b**2*c**2)*log(x + (B*a**6*c*d**5*
g**3*i**2 - 6*B*a**5*b*c**2*d**4*g**3*i**2 + B*a**5*d**4*g**3*i**2*(a**2*d**2 - 6*a*b*c*d + 15*b**2*c**2)/b +
35*B*a**4*b**2*c**3*d**3*g**3*i**2 - B*a**4*c*d**3*g**3*i**2*(a**2*d**2 - 6*a*b*c*d + 15*b**2*c**2) - 15*B*a**
3*b**3*c**4*d**2*g**3*i**2 + 6*B*a**2*b**4*c**5*d*g**3*i**2 - B*a*b**5*c**6*g**3*i**2)/(B*a**6*d**6*g**3*i**2
- 6*B*a**5*b*c*d**5*g**3*i**2 + 15*B*a**4*b**2*c**2*d**4*g**3*i**2 + 20*B*a**3*b**3*c**3*d**3*g**3*i**2 - 15*B
*a**2*b**4*c**4*d**2*g**3*i**2 + 6*B*a*b**5*c**5*d*g**3*i**2 - B*b**6*c**6*g**3*i**2))/(60*b**3) - B*c**3*g**3
*i**2*(20*a**3*d**3 - 15*a**2*b*c*d**2 + 6*a*b**2*c**2*d - b**3*c**3)*log(x + (B*a**6*c*d**5*g**3*i**2 - 6*B*a
**5*b*c**2*d**4*g**3*i**2 + 35*B*a**4*b**2*c**3*d**3*g**3*i**2 - 15*B*a**3*b**3*c**4*d**2*g**3*i**2 + 6*B*a**2
*b**4*c**5*d*g**3*i**2 - B*a*b**5*c**6*g**3*i**2 - B*a*b**2*c**3*g**3*i**2*(20*a**3*d**3 - 15*a**2*b*c*d**2 +
6*a*b**2*c**2*d - b**3*c**3) + B*b**3*c**4*g**3*i**2*(20*a**3*d**3 - 15*a**2*b*c*d**2 + 6*a*b**2*c**2*d - b**3
*c**3)/d)/(B*a**6*d**6*g**3*i**2 - 6*B*a**5*b*c*d**5*g**3*i**2 + 15*B*a**4*b**2*c**2*d**4*g**3*i**2 + 20*B*a**
3*b**3*c**3*d**3*g**3*i**2 - 15*B*a**2*b**4*c**4*d**2*g**3*i**2 + 6*B*a*b**5*c**5*d*g**3*i**2 - B*b**6*c**6*g*
*3*i**2))/(60*d**4) + x**5*(3*A*a*b**2*d**2*g**3*i**2/5 + 2*A*b**3*c*d*g**3*i**2/5 + B*a*b**2*d**2*g**3*i**2/3
0 - B*b**3*c*d*g**3*i**2/30) + x**4*(3*A*a**2*b*d**2*g**3*i**2/4 + 3*A*a*b**2*c*d*g**3*i**2/2 + A*b**3*c**2*g*
*3*i**2/4 + 13*B*a**2*b*d**2*g**3*i**2/120 - B*a*b**2*c*d*g**3*i**2/20 - 7*B*b**3*c**2*g**3*i**2/120) + x**3*(
A*a**3*d**2*g**3*i**2/3 + 2*A*a**2*b*c*d*g**3*i**2 + A*a*b**2*c**2*g**3*i**2 + 19*B*a**3*d**2*g**3*i**2/180 +
7*B*a**2*b*c*d*g**3*i**2/60 - 13*B*a*b**2*c**2*g**3*i**2/60 - B*b**3*c**3*g**3*i**2/(180*d)) + x**2*(A*a**3*c*
d*g**3*i**2 + 3*A*a**2*b*c**2*g**3*i**2/2 + B*a**4*d**2*g**3*i**2/(120*b) + 17*B*a**3*c*d*g**3*i**2/60 - B*a**
2*b*c**2*g**3*i**2/4 - B*a*b**2*c**3*g**3*i**2/(20*d) + B*b**3*c**4*g**3*i**2/(120*d**2)) + x*(A*a**3*c**2*g**
3*i**2 - B*a**5*d**2*g**3*i**2/(60*b**2) + B*a**4*c*d*g**3*i**2/(10*b) + B*a**3*c**2*g**3*i**2/12 - B*a**2*b*c
**3*g**3*i**2/(4*d) + B*a*b**2*c**4*g**3*i**2/(10*d**2) - B*b**3*c**5*g**3*i**2/(60*d**3)) + (B*a**3*c**2*g**3
*i**2*x + B*a**3*c*d*g**3*i**2*x**2 + B*a**3*d**2*g**3*i**2*x**3/3 + 3*B*a**2*b*c**2*g**3*i**2*x**2/2 + 2*B*a*
*2*b*c*d*g**3*i**2*x**3 + 3*B*a**2*b*d**2*g**3*i**2*x**4/4 + B*a*b**2*c**2*g**3*i**2*x**3 + 3*B*a*b**2*c*d*g**
3*i**2*x**4/2 + 3*B*a*b**2*d**2*g**3*i**2*x**5/5 + B*b**3*c**2*g**3*i**2*x**4/4 + 2*B*b**3*c*d*g**3*i**2*x**5/
5 + B*b**3*d**2*g**3*i**2*x**6/6)*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. 1789 vs. \(2 (401) = 802\).

Time = 0.24 (sec) , antiderivative size = 1789, normalized size of antiderivative = 4.23 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/6*A*b^3*d^2*g^3*i^2*x^6 + 2/5*A*b^3*c*d*g^3*i^2*x^5 + 3/5*A*a*b^2*d^2*g^3*i^2*x^5 + 1/4*A*b^3*c^2*g^3*i^2*x^
4 + 3/2*A*a*b^2*c*d*g^3*i^2*x^4 + 3/4*A*a^2*b*d^2*g^3*i^2*x^4 + A*a*b^2*c^2*g^3*i^2*x^3 + 2*A*a^2*b*c*d*g^3*i^
2*x^3 + 1/3*A*a^3*d^2*g^3*i^2*x^3 + 3/2*A*a^2*b*c^2*g^3*i^2*x^2 + A*a^3*c*d*g^3*i^2*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^2*g^3*i^2 + 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^2*g^3*i^2 + 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^2*g^3*i^2 + 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^2*g^3*i^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*d*g^3*i^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*d*g^3*i^2 + 1/4*(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*c*d*g^3*i^2 + 1/30*(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^3*c*d*g^3*i^2 + 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*a^3*d^2*g^3*i^2 + 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^2*b*d^2*g^3*i^2 + 1/20*(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*a*b^2*d^2*g^3*i^2 +
1/360*(60*x^6*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 60*a^6*log(b*x + a)/b^6 + 60*c^6*log(d*x + c)/d^6 - (12*(
b^5*c*d^4 - a*b^4*d^5)*x^5 - 15*(b^5*c^2*d^3 - a^2*b^3*d^5)*x^4 + 20*(b^5*c^3*d^2 - a^3*b^2*d^5)*x^3 - 30*(b^5
*c^4*d - a^4*b*d^5)*x^2 + 60*(b^5*c^5 - a^5*d^5)*x)/(b^5*d^5))*B*b^3*d^2*g^3*i^2 + A*a^3*c^2*g^3*i^2*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4746 vs. \(2 (401) = 802\).

Time = 0.60 (sec) , antiderivative size = 4746, normalized size of antiderivative = 11.22 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/360*(6*(B*b^10*c^7*e^7*g^3*i^2 - 7*B*a*b^9*c^6*d*e^7*g^3*i^2 + 21*B*a^2*b^8*c^5*d^2*e^7*g^3*i^2 - 35*B*a^3*
b^7*c^4*d^3*e^7*g^3*i^2 + 35*B*a^4*b^6*c^3*d^4*e^7*g^3*i^2 - 21*B*a^5*b^5*c^2*d^5*e^7*g^3*i^2 + 7*B*a^6*b^4*c*
d^6*e^7*g^3*i^2 - B*a^7*b^3*d^7*e^7*g^3*i^2 - 6*(b*e*x + a*e)*B*b^9*c^7*d*e^6*g^3*i^2/(d*x + c) + 42*(b*e*x +
a*e)*B*a*b^8*c^6*d^2*e^6*g^3*i^2/(d*x + c) - 126*(b*e*x + a*e)*B*a^2*b^7*c^5*d^3*e^6*g^3*i^2/(d*x + c) + 210*(
b*e*x + a*e)*B*a^3*b^6*c^4*d^4*e^6*g^3*i^2/(d*x + c) - 210*(b*e*x + a*e)*B*a^4*b^5*c^3*d^5*e^6*g^3*i^2/(d*x +
c) + 126*(b*e*x + a*e)*B*a^5*b^4*c^2*d^6*e^6*g^3*i^2/(d*x + c) - 42*(b*e*x + a*e)*B*a^6*b^3*c*d^7*e^6*g^3*i^2/
(d*x + c) + 6*(b*e*x + a*e)*B*a^7*b^2*d^8*e^6*g^3*i^2/(d*x + c) + 15*(b*e*x + a*e)^2*B*b^8*c^7*d^2*e^5*g^3*i^2
/(d*x + c)^2 - 105*(b*e*x + a*e)^2*B*a*b^7*c^6*d^3*e^5*g^3*i^2/(d*x + c)^2 + 315*(b*e*x + a*e)^2*B*a^2*b^6*c^5
*d^4*e^5*g^3*i^2/(d*x + c)^2 - 525*(b*e*x + a*e)^2*B*a^3*b^5*c^4*d^5*e^5*g^3*i^2/(d*x + c)^2 + 525*(b*e*x + a*
e)^2*B*a^4*b^4*c^3*d^6*e^5*g^3*i^2/(d*x + c)^2 - 315*(b*e*x + a*e)^2*B*a^5*b^3*c^2*d^7*e^5*g^3*i^2/(d*x + c)^2
+ 105*(b*e*x + a*e)^2*B*a^6*b^2*c*d^8*e^5*g^3*i^2/(d*x + c)^2 - 15*(b*e*x + a*e)^2*B*a^7*b*d^9*e^5*g^3*i^2/(d
*x + c)^2 - 20*(b*e*x + a*e)^3*B*b^7*c^7*d^3*e^4*g^3*i^2/(d*x + c)^3 + 140*(b*e*x + a*e)^3*B*a*b^6*c^6*d^4*e^4
*g^3*i^2/(d*x + c)^3 - 420*(b*e*x + a*e)^3*B*a^2*b^5*c^5*d^5*e^4*g^3*i^2/(d*x + c)^3 + 700*(b*e*x + a*e)^3*B*a
^3*b^4*c^4*d^6*e^4*g^3*i^2/(d*x + c)^3 - 700*(b*e*x + a*e)^3*B*a^4*b^3*c^3*d^7*e^4*g^3*i^2/(d*x + c)^3 + 420*(
b*e*x + a*e)^3*B*a^5*b^2*c^2*d^8*e^4*g^3*i^2/(d*x + c)^3 - 140*(b*e*x + a*e)^3*B*a^6*b*c*d^9*e^4*g^3*i^2/(d*x
+ c)^3 + 20*(b*e*x + a*e)^3*B*a^7*d^10*e^4*g^3*i^2/(d*x + c)^3)*log((b*e*x + a*e)/(d*x + c))/(b^6*d^4*e^6 - 6*
(b*e*x + a*e)*b^5*d^5*e^5/(d*x + c) + 15*(b*e*x + a*e)^2*b^4*d^6*e^4/(d*x + c)^2 - 20*(b*e*x + a*e)^3*b^3*d^7*
e^3/(d*x + c)^3 + 15*(b*e*x + a*e)^4*b^2*d^8*e^2/(d*x + c)^4 - 6*(b*e*x + a*e)^5*b*d^9*e/(d*x + c)^5 + (b*e*x
+ a*e)^6*d^10/(d*x + c)^6) + (6*A*b^12*c^7*e^7*g^3*i^2 + 2*B*b^12*c^7*e^7*g^3*i^2 - 42*A*a*b^11*c^6*d*e^7*g^3*
i^2 - 14*B*a*b^11*c^6*d*e^7*g^3*i^2 + 126*A*a^2*b^10*c^5*d^2*e^7*g^3*i^2 + 42*B*a^2*b^10*c^5*d^2*e^7*g^3*i^2 -
210*A*a^3*b^9*c^4*d^3*e^7*g^3*i^2 - 70*B*a^3*b^9*c^4*d^3*e^7*g^3*i^2 + 210*A*a^4*b^8*c^3*d^4*e^7*g^3*i^2 + 70
*B*a^4*b^8*c^3*d^4*e^7*g^3*i^2 - 126*A*a^5*b^7*c^2*d^5*e^7*g^3*i^2 - 42*B*a^5*b^7*c^2*d^5*e^7*g^3*i^2 + 42*A*a
^6*b^6*c*d^6*e^7*g^3*i^2 + 14*B*a^6*b^6*c*d^6*e^7*g^3*i^2 - 6*A*a^7*b^5*d^7*e^7*g^3*i^2 - 2*B*a^7*b^5*d^7*e^7*
g^3*i^2 - 36*(b*e*x + a*e)*A*b^11*c^7*d*e^6*g^3*i^2/(d*x + c) - 6*(b*e*x + a*e)*B*b^11*c^7*d*e^6*g^3*i^2/(d*x
+ c) + 252*(b*e*x + a*e)*A*a*b^10*c^6*d^2*e^6*g^3*i^2/(d*x + c) + 42*(b*e*x + a*e)*B*a*b^10*c^6*d^2*e^6*g^3*i^
2/(d*x + c) - 756*(b*e*x + a*e)*A*a^2*b^9*c^5*d^3*e^6*g^3*i^2/(d*x + c) - 126*(b*e*x + a*e)*B*a^2*b^9*c^5*d^3*
e^6*g^3*i^2/(d*x + c) + 1260*(b*e*x + a*e)*A*a^3*b^8*c^4*d^4*e^6*g^3*i^2/(d*x + c) + 210*(b*e*x + a*e)*B*a^3*b
^8*c^4*d^4*e^6*g^3*i^2/(d*x + c) - 1260*(b*e*x + a*e)*A*a^4*b^7*c^3*d^5*e^6*g^3*i^2/(d*x + c) - 210*(b*e*x + a
*e)*B*a^4*b^7*c^3*d^5*e^6*g^3*i^2/(d*x + c) + 756*(b*e*x + a*e)*A*a^5*b^6*c^2*d^6*e^6*g^3*i^2/(d*x + c) + 126*
(b*e*x + a*e)*B*a^5*b^6*c^2*d^6*e^6*g^3*i^2/(d*x + c) - 252*(b*e*x + a*e)*A*a^6*b^5*c*d^7*e^6*g^3*i^2/(d*x + c
) - 42*(b*e*x + a*e)*B*a^6*b^5*c*d^7*e^6*g^3*i^2/(d*x + c) + 36*(b*e*x + a*e)*A*a^7*b^4*d^8*e^6*g^3*i^2/(d*x +
c) + 6*(b*e*x + a*e)*B*a^7*b^4*d^8*e^6*g^3*i^2/(d*x + c) + 90*(b*e*x + a*e)^2*A*b^10*c^7*d^2*e^5*g^3*i^2/(d*x
+ c)^2 - 3*(b*e*x + a*e)^2*B*b^10*c^7*d^2*e^5*g^3*i^2/(d*x + c)^2 - 630*(b*e*x + a*e)^2*A*a*b^9*c^6*d^3*e^5*g
^3*i^2/(d*x + c)^2 + 21*(b*e*x + a*e)^2*B*a*b^9*c^6*d^3*e^5*g^3*i^2/(d*x + c)^2 + 1890*(b*e*x + a*e)^2*A*a^2*b
^8*c^5*d^4*e^5*g^3*i^2/(d*x + c)^2 - 63*(b*e*x + a*e)^2*B*a^2*b^8*c^5*d^4*e^5*g^3*i^2/(d*x + c)^2 - 3150*(b*e*
x + a*e)^2*A*a^3*b^7*c^4*d^5*e^5*g^3*i^2/(d*x + c)^2 + 105*(b*e*x + a*e)^2*B*a^3*b^7*c^4*d^5*e^5*g^3*i^2/(d*x
+ c)^2 + 3150*(b*e*x + a*e)^2*A*a^4*b^6*c^3*d^6*e^5*g^3*i^2/(d*x + c)^2 - 105*(b*e*x + a*e)^2*B*a^4*b^6*c^3*d^
6*e^5*g^3*i^2/(d*x + c)^2 - 1890*(b*e*x + a*e)^2*A*a^5*b^5*c^2*d^7*e^5*g^3*i^2/(d*x + c)^2 + 63*(b*e*x + a*e)^
2*B*a^5*b^5*c^2*d^7*e^5*g^3*i^2/(d*x + c)^2 + 630*(b*e*x + a*e)^2*A*a^6*b^4*c*d^8*e^5*g^3*i^2/(d*x + c)^2 - 21
*(b*e*x + a*e)^2*B*a^6*b^4*c*d^8*e^5*g^3*i^2/(d*x + c)^2 - 90*(b*e*x + a*e)^2*A*a^7*b^3*d^9*e^5*g^3*i^2/(d*x +
c)^2 + 3*(b*e*x + a*e)^2*B*a^7*b^3*d^9*e^5*g^3*i^2/(d*x + c)^2 - 120*(b*e*x + a*e)^3*A*b^9*c^7*d^3*e^4*g^3*i^
2/(d*x + c)^3 + 34*(b*e*x + a*e)^3*B*b^9*c^7*d^3*e^4*g^3*i^2/(d*x + c)^3 + 840*(b*e*x + a*e)^3*A*a*b^8*c^6*d^4
*e^4*g^3*i^2/(d*x + c)^3 - 238*(b*e*x + a*e)^3*B*a*b^8*c^6*d^4*e^4*g^3*i^2/(d*x + c)^3 - 2520*(b*e*x + a*e)^3*
A*a^2*b^7*c^5*d^5*e^4*g^3*i^2/(d*x + c)^3 + 714*(b*e*x + a*e)^3*B*a^2*b^7*c^5*d^5*e^4*g^3*i^2/(d*x + c)^3 + 42
00*(b*e*x + a*e)^3*A*a^3*b^6*c^4*d^6*e^4*g^3*i^2/(d*x + c)^3 - 1190*(b*e*x + a*e)^3*B*a^3*b^6*c^4*d^6*e^4*g^3*
i^2/(d*x + c)^3 - 4200*(b*e*x + a*e)^3*A*a^4*b^5*c^3*d^7*e^4*g^3*i^2/(d*x + c)^3 + 1190*(b*e*x + a*e)^3*B*a^4*
b^5*c^3*d^7*e^4*g^3*i^2/(d*x + c)^3 + 2520*(b*e*x + a*e)^3*A*a^5*b^4*c^2*d^8*e^4*g^3*i^2/(d*x + c)^3 - 714*(b*
e*x + a*e)^3*B*a^5*b^4*c^2*d^8*e^4*g^3*i^2/(d*x + c)^3 - 840*(b*e*x + a*e)^3*A*a^6*b^3*c*d^9*e^4*g^3*i^2/(d*x
+ c)^3 + 238*(b*e*x + a*e)^3*B*a^6*b^3*c*d^9*e^4*g^3*i^2/(d*x + c)^3 + 120*(b*e*x + a*e)^3*A*a^7*b^2*d^10*e^4*
g^3*i^2/(d*x + c)^3 - 34*(b*e*x + a*e)^3*B*a^7*b^2*d^10*e^4*g^3*i^2/(d*x + c)^3 - 33*(b*e*x + a*e)^4*B*b^8*c^7
*d^4*e^3*g^3*i^2/(d*x + c)^4 + 231*(b*e*x + a*e)^4*B*a*b^7*c^6*d^5*e^3*g^3*i^2/(d*x + c)^4 - 693*(b*e*x + a*e)
^4*B*a^2*b^6*c^5*d^6*e^3*g^3*i^2/(d*x + c)^4 + 1155*(b*e*x + a*e)^4*B*a^3*b^5*c^4*d^7*e^3*g^3*i^2/(d*x + c)^4
- 1155*(b*e*x + a*e)^4*B*a^4*b^4*c^3*d^8*e^3*g^3*i^2/(d*x + c)^4 + 693*(b*e*x + a*e)^4*B*a^5*b^3*c^2*d^9*e^3*g
^3*i^2/(d*x + c)^4 - 231*(b*e*x + a*e)^4*B*a^6*b^2*c*d^10*e^3*g^3*i^2/(d*x + c)^4 + 33*(b*e*x + a*e)^4*B*a^7*b
*d^11*e^3*g^3*i^2/(d*x + c)^4 + 6*(b*e*x + a*e)^5*B*b^7*c^7*d^5*e^2*g^3*i^2/(d*x + c)^5 - 42*(b*e*x + a*e)^5*B
*a*b^6*c^6*d^6*e^2*g^3*i^2/(d*x + c)^5 + 126*(b*e*x + a*e)^5*B*a^2*b^5*c^5*d^7*e^2*g^3*i^2/(d*x + c)^5 - 210*(
b*e*x + a*e)^5*B*a^3*b^4*c^4*d^8*e^2*g^3*i^2/(d*x + c)^5 + 210*(b*e*x + a*e)^5*B*a^4*b^3*c^3*d^9*e^2*g^3*i^2/(
d*x + c)^5 - 126*(b*e*x + a*e)^5*B*a^5*b^2*c^2*d^10*e^2*g^3*i^2/(d*x + c)^5 + 42*(b*e*x + a*e)^5*B*a^6*b*c*d^1
1*e^2*g^3*i^2/(d*x + c)^5 - 6*(b*e*x + a*e)^5*B*a^7*d^12*e^2*g^3*i^2/(d*x + c)^5)/(b^8*d^4*e^6 - 6*(b*e*x + a*
e)*b^7*d^5*e^5/(d*x + c) + 15*(b*e*x + a*e)^2*b^6*d^6*e^4/(d*x + c)^2 - 20*(b*e*x + a*e)^3*b^5*d^7*e^3/(d*x +
c)^3 + 15*(b*e*x + a*e)^4*b^4*d^8*e^2/(d*x + c)^4 - 6*(b*e*x + a*e)^5*b^3*d^9*e/(d*x + c)^5 + (b*e*x + a*e)^6*
b^2*d^10/(d*x + c)^6) + 6*(B*b^7*c^7*e*g^3*i^2 - 7*B*a*b^6*c^6*d*e*g^3*i^2 + 21*B*a^2*b^5*c^5*d^2*e*g^3*i^2 -
35*B*a^3*b^4*c^4*d^3*e*g^3*i^2 + 35*B*a^4*b^3*c^3*d^4*e*g^3*i^2 - 21*B*a^5*b^2*c^2*d^5*e*g^3*i^2 + 7*B*a^6*b*c
*d^6*e*g^3*i^2 - B*a^7*d^7*e*g^3*i^2)*log(-b*e + (b*e*x + a*e)*d/(d*x + c))/(b^3*d^4) - 6*(B*b^7*c^7*e*g^3*i^2
- 7*B*a*b^6*c^6*d*e*g^3*i^2 + 21*B*a^2*b^5*c^5*d^2*e*g^3*i^2 - 35*B*a^3*b^4*c^4*d^3*e*g^3*i^2 + 35*B*a^4*b^3*
c^3*d^4*e*g^3*i^2 - 21*B*a^5*b^2*c^2*d^5*e*g^3*i^2 + 7*B*a^6*b*c*d^6*e*g^3*i^2 - B*a^7*d^7*e*g^3*i^2)*log((b*e
*x + a*e)/(d*x + c))/(b^3*d^4))*(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 = 2.52 (sec) , antiderivative size = 2473, normalized size of antiderivative = 5.85 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

x^3*((g^3*i^2*(16*A*a^3*d^3 + 4*A*b^3*c^3 + 3*B*a^3*d^3 - B*b^3*c^3 + 48*A*a*b^2*c^2*d + 72*A*a^2*b*c*d^2 - 5*
B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2))/(12*d) + ((60*a*d + 60*b*c)*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d -
B*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (b*g^3*i^2*(30*A*a^2*d^2 +
15*A*b^2*c^2 + 3*B*a^2*d^2 - 2*B*b^2*c^2 + 60*A*a*b*c*d - B*a*b*c*d))/5 + A*a*b^2*c*d*g^3*i^2))/(180*b*d) - (a
*c*((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60))/(3*b*d)
) - x^4*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(
60*a*d + 60*b*c))/(240*b*d) - (b*g^3*i^2*(30*A*a^2*d^2 + 15*A*b^2*c^2 + 3*B*a^2*d^2 - 2*B*b^2*c^2 + 60*A*a*b*c
*d - B*a*b*c*d))/20 + (A*a*b^2*c*d*g^3*i^2)/4) + x^2*((a*c*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*
b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (b*g^3*i^2*(30*A*a^2*d^2 + 15*
A*b^2*c^2 + 3*B*a^2*d^2 - 2*B*b^2*c^2 + 60*A*a*b*c*d - B*a*b*c*d))/5 + A*a*b^2*c*d*g^3*i^2))/(2*b*d) - ((60*a*
d + 60*b*c)*((g^3*i^2*(16*A*a^3*d^3 + 4*A*b^3*c^3 + 3*B*a^3*d^3 - B*b^3*c^3 + 48*A*a*b^2*c^2*d + 72*A*a^2*b*c*
d^2 - 5*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2))/(4*d) + ((60*a*d + 60*b*c)*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c +
B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (b*g^3*i^2*(30*A*a^2
*d^2 + 15*A*b^2*c^2 + 3*B*a^2*d^2 - 2*B*b^2*c^2 + 60*A*a*b*c*d - B*a*b*c*d))/5 + A*a*b^2*c*d*g^3*i^2))/(60*b*d
) - (a*c*((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60))/(
b*d)))/(120*b*d) + (a*g^3*i^2*(3*A*a^3*d^3 + 12*A*b^3*c^3 + B*a^3*d^3 - 3*B*b^3*c^3 + 54*A*a*b^2*c^2*d + 36*A*
a^2*b*c*d^2 - 3*B*a*b^2*c^2*d + 5*B*a^2*b*c*d^2))/(6*b*d)) + log((e*(a + b*x))/(c + d*x))*(B*a^3*c^2*g^3*i^2*x
+ (B*a*g^3*i^2*x^3*(a^2*d^2 + 3*b^2*c^2 + 6*a*b*c*d))/3 + (B*b*g^3*i^2*x^4*(3*a^2*d^2 + b^2*c^2 + 6*a*b*c*d))
/4 + (B*b^3*d^2*g^3*i^2*x^6)/6 + (B*a^2*c*g^3*i^2*x^2*(2*a*d + 3*b*c))/2 + (B*b^2*d*g^3*i^2*x^5*(3*a*d + 2*b*c
))/5) + x^5*((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/30 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/30
0) - x*(((60*a*d + 60*b*c)*((a*c*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2
*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (b*g^3*i^2*(30*A*a^2*d^2 + 15*A*b^2*c^2 + 3*B*a^2*d^2 -
2*B*b^2*c^2 + 60*A*a*b*c*d - B*a*b*c*d))/5 + A*a*b^2*c*d*g^3*i^2))/(b*d) - ((60*a*d + 60*b*c)*((g^3*i^2*(16*A*
a^3*d^3 + 4*A*b^3*c^3 + 3*B*a^3*d^3 - B*b^3*c^3 + 48*A*a*b^2*c^2*d + 72*A*a^2*b*c*d^2 - 5*B*a*b^2*c^2*d + 3*B*
a^2*b*c*d^2))/(4*d) + ((60*a*d + 60*b*c)*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*d
*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (b*g^3*i^2*(30*A*a^2*d^2 + 15*A*b^2*c^2 + 3*B*a^
2*d^2 - 2*B*b^2*c^2 + 60*A*a*b*c*d - B*a*b*c*d))/5 + A*a*b^2*c*d*g^3*i^2))/(60*b*d) - (a*c*((b^2*d*g^3*i^2*(24
*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60))/(b*d)))/(60*b*d) + (a*g^3*i^2
*(3*A*a^3*d^3 + 12*A*b^3*c^3 + B*a^3*d^3 - 3*B*b^3*c^3 + 54*A*a*b^2*c^2*d + 36*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d
+ 5*B*a^2*b*c*d^2))/(3*b*d)))/(60*b*d) + (a*c*((g^3*i^2*(16*A*a^3*d^3 + 4*A*b^3*c^3 + 3*B*a^3*d^3 - B*b^3*c^3
+ 48*A*a*b^2*c^2*d + 72*A*a^2*b*c*d^2 - 5*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2))/(4*d) + ((60*a*d + 60*b*c)*((((b^
2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*
c))/(60*b*d) - (b*g^3*i^2*(30*A*a^2*d^2 + 15*A*b^2*c^2 + 3*B*a^2*d^2 - 2*B*b^2*c^2 + 60*A*a*b*c*d - B*a*b*c*d)
)/5 + A*a*b^2*c*d*g^3*i^2))/(60*b*d) - (a*c*((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*
d*g^3*i^2*(60*a*d + 60*b*c))/60))/(b*d)))/(b*d) - (a^2*c*g^3*i^2*(6*A*a^2*d^2 + 12*A*b^2*c^2 + 2*B*a^2*d^2 - 3
*B*b^2*c^2 + 24*A*a*b*c*d + B*a*b*c*d))/(2*b*d)) + (log(a + b*x)*(B*a^6*d^2*g^3*i^2 + 15*B*a^4*b^2*c^2*g^3*i^2
- 6*B*a^5*b*c*d*g^3*i^2))/(60*b^3) + (log(c + d*x)*(B*b^3*c^6*g^3*i^2 - 20*B*a^3*c^3*d^3*g^3*i^2 - 6*B*a*b^2*
c^5*d*g^3*i^2 + 15*B*a^2*b*c^4*d^2*g^3*i^2))/(60*d^4) + (A*b^3*d^2*g^3*i^2*x^6)/6