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

Optimal. Leaf size=457 \[ \frac {B (b c-a d)^6 g^3 i^3 x}{140 b^3 d^3}+\frac {B (b c-a d)^5 g^3 i^3 (c+d x)^2}{280 b^2 d^4}+\frac {B (b c-a d)^4 g^3 i^3 (c+d x)^3}{420 b d^4}-\frac {17 B (b c-a d)^3 g^3 i^3 (c+d x)^4}{280 d^4}+\frac {b B (b c-a d)^2 g^3 i^3 (c+d x)^5}{14 d^4}-\frac {b^2 B (b c-a d) g^3 i^3 (c+d x)^6}{42 d^4}+\frac {B (b c-a d)^7 g^3 i^3 \log \left (\frac {a+b x}{c+d x}\right )}{140 b^4 d^4}-\frac {(b c-a d)^3 g^3 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d^4}+\frac {3 b (b c-a d)^2 g^3 i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^4}-\frac {b^2 (b c-a d) g^3 i^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^4}+\frac {b^3 g^3 i^3 (c+d x)^7 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{7 d^4}+\frac {B (b c-a d)^7 g^3 i^3 \log (c+d x)}{140 b^4 d^4} \]

[Out]

1/140*B*(-a*d+b*c)^6*g^3*i^3*x/b^3/d^3+1/280*B*(-a*d+b*c)^5*g^3*i^3*(d*x+c)^2/b^2/d^4+1/420*B*(-a*d+b*c)^4*g^3
*i^3*(d*x+c)^3/b/d^4-17/280*B*(-a*d+b*c)^3*g^3*i^3*(d*x+c)^4/d^4+1/14*b*B*(-a*d+b*c)^2*g^3*i^3*(d*x+c)^5/d^4-1
/42*b^2*B*(-a*d+b*c)*g^3*i^3*(d*x+c)^6/d^4+1/140*B*(-a*d+b*c)^7*g^3*i^3*ln((b*x+a)/(d*x+c))/b^4/d^4-1/4*(-a*d+
b*c)^3*g^3*i^3*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^4+3/5*b*(-a*d+b*c)^2*g^3*i^3*(d*x+c)^5*(A+B*ln(e*(b*x+a
)/(d*x+c)))/d^4-1/2*b^2*(-a*d+b*c)*g^3*i^3*(d*x+c)^6*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^4+1/7*b^3*g^3*i^3*(d*x+c)^7
*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^4+1/140*B*(-a*d+b*c)^7*g^3*i^3*ln(d*x+c)/b^4/d^4

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Rubi [A]
time = 0.31, antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2562, 45, 2382, 12, 1634} \begin {gather*} \frac {b^3 g^3 i^3 (c+d x)^7 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{7 d^4}-\frac {b^2 g^3 i^3 (c+d x)^6 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^4}-\frac {g^3 i^3 (c+d x)^4 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d^4}+\frac {3 b g^3 i^3 (c+d x)^5 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^4}+\frac {B g^3 i^3 (b c-a d)^7 \log \left (\frac {a+b x}{c+d x}\right )}{140 b^4 d^4}+\frac {B g^3 i^3 (b c-a d)^7 \log (c+d x)}{140 b^4 d^4}+\frac {B g^3 i^3 x (b c-a d)^6}{140 b^3 d^3}+\frac {B g^3 i^3 (c+d x)^2 (b c-a d)^5}{280 b^2 d^4}-\frac {b^2 B g^3 i^3 (c+d x)^6 (b c-a d)}{42 d^4}+\frac {B g^3 i^3 (c+d x)^3 (b c-a d)^4}{420 b d^4}-\frac {17 B g^3 i^3 (c+d x)^4 (b c-a d)^3}{280 d^4}+\frac {b B g^3 i^3 (c+d x)^5 (b c-a d)^2}{14 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B*(b*c - a*d)^6*g^3*i^3*x)/(140*b^3*d^3) + (B*(b*c - a*d)^5*g^3*i^3*(c + d*x)^2)/(280*b^2*d^4) + (B*(b*c - a*
d)^4*g^3*i^3*(c + d*x)^3)/(420*b*d^4) - (17*B*(b*c - a*d)^3*g^3*i^3*(c + d*x)^4)/(280*d^4) + (b*B*(b*c - a*d)^
2*g^3*i^3*(c + d*x)^5)/(14*d^4) - (b^2*B*(b*c - a*d)*g^3*i^3*(c + d*x)^6)/(42*d^4) + (B*(b*c - a*d)^7*g^3*i^3*
Log[(a + b*x)/(c + d*x)])/(140*b^4*d^4) - ((b*c - a*d)^3*g^3*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x
)]))/(4*d^4) + (3*b*(b*c - a*d)^2*g^3*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*d^4) - (b^2*(b*
c - a*d)*g^3*i^3*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d^4) + (b^3*g^3*i^3*(c + d*x)^7*(A + B*L
og[(e*(a + b*x))/(c + d*x)]))/(7*d^4) + (B*(b*c - a*d)^7*g^3*i^3*Log[c + d*x])/(140*b^4*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*} \int (20 c+20 d x)^3 (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx &=\int \left (\frac {(-b c+a d)^3 g^3 (20 c+20 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}+\frac {3 b (b c-a d)^2 g^3 (20 c+20 d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 d^3}-\frac {3 b^2 (b c-a d) g^3 (20 c+20 d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{400 d^3}+\frac {b^3 g^3 (20 c+20 d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8000 d^3}\right ) \, dx\\ &=\frac {\left (b^3 g^3\right ) \int (20 c+20 d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{8000 d^3}-\frac {\left (3 b^2 (b c-a d) g^3\right ) \int (20 c+20 d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{400 d^3}+\frac {\left (3 b (b c-a d)^2 g^3\right ) \int (20 c+20 d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{20 d^3}-\frac {\left ((b c-a d)^3 g^3\right ) \int (20 c+20 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{d^3}\\ &=-\frac {2000 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac {4800 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4}-\frac {4000 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac {8000 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{7 d^4}-\frac {\left (b^3 B g^3\right ) \int \frac {1280000000 (b c-a d) (c+d x)^6}{a+b x} \, dx}{1120000 d^4}+\frac {\left (b^2 B (b c-a d) g^3\right ) \int \frac {64000000 (b c-a d) (c+d x)^5}{a+b x} \, dx}{16000 d^4}-\frac {\left (3 b B (b c-a d)^2 g^3\right ) \int \frac {3200000 (b c-a d) (c+d x)^4}{a+b x} \, dx}{2000 d^4}+\frac {\left (B (b c-a d)^3 g^3\right ) \int \frac {160000 (b c-a d) (c+d x)^3}{a+b x} \, dx}{80 d^4}\\ &=-\frac {2000 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac {4800 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4}-\frac {4000 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac {8000 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{7 d^4}-\frac {\left (8000 b^3 B (b c-a d) g^3\right ) \int \frac {(c+d x)^6}{a+b x} \, dx}{7 d^4}+\frac {\left (4000 b^2 B (b c-a d)^2 g^3\right ) \int \frac {(c+d x)^5}{a+b x} \, dx}{d^4}-\frac {\left (4800 b B (b c-a d)^3 g^3\right ) \int \frac {(c+d x)^4}{a+b x} \, dx}{d^4}+\frac {\left (2000 B (b c-a d)^4 g^3\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{d^4}\\ &=-\frac {2000 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac {4800 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4}-\frac {4000 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac {8000 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{7 d^4}-\frac {\left (8000 b^3 B (b c-a d) g^3\right ) \int \left (\frac {d (b c-a d)^5}{b^6}+\frac {(b c-a d)^6}{b^6 (a+b x)}+\frac {d (b c-a d)^4 (c+d x)}{b^5}+\frac {d (b c-a d)^3 (c+d x)^2}{b^4}+\frac {d (b c-a d)^2 (c+d x)^3}{b^3}+\frac {d (b c-a d) (c+d x)^4}{b^2}+\frac {d (c+d x)^5}{b}\right ) \, dx}{7 d^4}+\frac {\left (4000 b^2 B (b c-a d)^2 g^3\right ) \int \left (\frac {d (b c-a d)^4}{b^5}+\frac {(b c-a d)^5}{b^5 (a+b x)}+\frac {d (b c-a d)^3 (c+d x)}{b^4}+\frac {d (b c-a d)^2 (c+d x)^2}{b^3}+\frac {d (b c-a d) (c+d x)^3}{b^2}+\frac {d (c+d x)^4}{b}\right ) \, dx}{d^4}-\frac {\left (4800 b B (b c-a d)^3 g^3\right ) \int \left (\frac {d (b c-a d)^3}{b^4}+\frac {(b c-a d)^4}{b^4 (a+b x)}+\frac {d (b c-a d)^2 (c+d x)}{b^3}+\frac {d (b c-a d) (c+d x)^2}{b^2}+\frac {d (c+d x)^3}{b}\right ) \, dx}{d^4}+\frac {\left (2000 B (b c-a d)^4 g^3\right ) \int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx}{d^4}\\ &=\frac {400 B (b c-a d)^6 g^3 x}{7 b^3 d^3}+\frac {200 B (b c-a d)^5 g^3 (c+d x)^2}{7 b^2 d^4}+\frac {400 B (b c-a d)^4 g^3 (c+d x)^3}{21 b d^4}-\frac {3400 B (b c-a d)^3 g^3 (c+d x)^4}{7 d^4}+\frac {4000 b B (b c-a d)^2 g^3 (c+d x)^5}{7 d^4}-\frac {4000 b^2 B (b c-a d) g^3 (c+d x)^6}{21 d^4}+\frac {400 B (b c-a d)^7 g^3 \log (a+b x)}{7 b^4 d^4}-\frac {2000 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac {4800 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4}-\frac {4000 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac {8000 b^3 g^3 (c+d x)^7 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{7 d^4}\\ \end {align*}

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Mathematica [A]
time = 0.41, size = 586, normalized size = 1.28 \begin {gather*} \frac {g^3 i^3 \left (\frac {120 b^2 B c (b c-a d)^5 x}{d^3}-\frac {126 b B (b c-a d)^6 x}{d^3}+\frac {120 a b B (-b c+a d)^5 x}{d^2}-\frac {60 b B c (b c-a d)^4 (a+b x)^2}{d^2}+\frac {60 a B (b c-a d)^4 (a+b x)^2}{d}+\frac {63 B (b c-a d)^5 (a+b x)^2}{d^2}+\frac {40 b B c (b c-a d)^3 (a+b x)^3}{d}-\frac {42 B (b c-a d)^4 (a+b x)^3}{d}+40 a B (-b c+a d)^3 (a+b x)^3-30 b B c (b c-a d)^2 (a+b x)^4+30 a B d (b c-a d)^2 (a+b x)^4+21 B (-b c+a d)^3 (a+b x)^4+24 b B c d (b c-a d) (a+b x)^5-84 B d (b c-a d)^2 (a+b x)^5+24 a B d^2 (-b c+a d) (a+b x)^5-20 b B c d^2 (a+b x)^6+20 a B d^3 (a+b x)^6+210 (b c-a d)^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+504 d (b c-a d)^2 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+420 d^2 (b c-a d) (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+120 d^3 (a+b x)^7 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-\frac {120 b B c (b c-a d)^6 \log (c+d x)}{d^4}+\frac {120 a B (b c-a d)^6 \log (c+d x)}{d^3}+\frac {126 B (b c-a d)^7 \log (c+d x)}{d^4}\right )}{840 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(20504\) vs. \(2(433)=866\).
time = 0.71, size = 20505, normalized size = 44.87

method	result	size
risch	\(\frac {9 i^{3} g^{3} d^{2} b^{2} A a c \,x^{5}}{5}+\frac {9 i^{3} g^{3} d^{2} b A \,a^{2} c \,x^{4}}{4}-\frac {3 i^{3} g^{3} b B \,a^{2} c^{3} x^{2}}{10}-\frac {i^{3} g^{3} b^{2} B a \,c^{4} x^{2}}{40 d}+i^{3} g^{3} A \,a^{3} c^{3} x +\frac {9 i^{3} g^{3} d \,b^{2} A a \,c^{2} x^{4}}{4}+\frac {7 i^{3} g^{3} d^{2} b B \,a^{2} c \,x^{4}}{40}-\frac {7 i^{3} g^{3} d \,b^{2} B a \,c^{2} x^{4}}{40}+i^{3} g^{3} d^{2} A \,a^{3} c \,x^{3}+3 i^{3} g^{3} d b A \,a^{2} c^{2} x^{3}+i^{3} g^{3} b^{2} A a \,c^{3} x^{3}+\frac {7 i^{3} g^{3} d^{2} B \,a^{3} c \,x^{3}}{30}-\frac {7 i^{3} g^{3} b^{2} B a \,c^{3} x^{3}}{30}+\frac {3 i^{3} g^{3} d A \,a^{3} c^{2} x^{2}}{2}+\frac {3 i^{3} g^{3} b A \,a^{2} c^{3} x^{2}}{2}+\frac {i^{3} g^{3} d^{2} B \,a^{4} c \,x^{2}}{40 b}+\frac {3 i^{3} g^{3} d B \,a^{3} c^{2} x^{2}}{10}-\frac {i^{3} g^{3} d^{2} B \,a^{5} c x}{20 b^{2}}+\frac {3 i^{3} g^{3} d B \,a^{4} c^{2} x}{20 b}-\frac {3 i^{3} g^{3} b B \,a^{2} c^{4} x}{20 d}+\frac {i^{3} g^{3} b^{2} B a \,c^{5} x}{20 d^{2}}+\frac {i^{3} g^{3} d^{2} B \ln \left (b x +a \right ) a^{6} c}{20 b^{3}}-\frac {3 i^{3} g^{3} d B \ln \left (b x +a \right ) a^{5} c^{2}}{20 b^{2}}+\frac {3 i^{3} g^{3} b B \ln \left (-d x -c \right ) a^{2} c^{5}}{20 d^{2}}-\frac {i^{3} g^{3} b^{2} B \ln \left (-d x -c \right ) a \,c^{6}}{20 d^{3}}+\frac {i^{3} g^{3} B x \left (20 d^{3} b^{3} x^{6}+70 a \,b^{2} d^{3} x^{5}+70 b^{3} c \,d^{2} x^{5}+84 a^{2} b \,d^{3} x^{4}+252 a \,b^{2} c \,d^{2} x^{4}+84 b^{3} c^{2} d \,x^{4}+35 a^{3} d^{3} x^{3}+315 a^{2} b c \,d^{2} x^{3}+315 a \,b^{2} c^{2} d \,x^{3}+35 b^{3} c^{3} x^{3}+140 a^{3} c \,d^{2} x^{2}+420 a^{2} b \,c^{2} d \,x^{2}+140 a \,b^{2} c^{3} x^{2}+210 a^{3} c^{2} d x +210 a^{2} b \,c^{3} x +140 c^{3} a^{3}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{140}+\frac {i^{3} g^{3} d^{3} b^{3} A \,x^{7}}{7}+\frac {i^{3} g^{3} d^{3} b^{2} A a \,x^{6}}{2}+\frac {i^{3} g^{3} d^{2} b^{3} A c \,x^{6}}{2}+\frac {i^{3} g^{3} d^{3} b^{2} B a \,x^{6}}{42}-\frac {i^{3} g^{3} d^{2} b^{3} B c \,x^{6}}{42}+\frac {3 i^{3} g^{3} d^{3} b A \,a^{2} x^{5}}{5}+\frac {3 i^{3} g^{3} d \,b^{3} A \,c^{2} x^{5}}{5}+\frac {i^{3} g^{3} d^{3} b B \,a^{2} x^{5}}{14}-\frac {i^{3} g^{3} d \,b^{3} B \,c^{2} x^{5}}{14}+\frac {i^{3} g^{3} d^{3} A \,a^{3} x^{4}}{4}+\frac {i^{3} g^{3} b^{3} A \,c^{3} x^{4}}{4}+\frac {17 i^{3} g^{3} d^{3} B \,a^{3} x^{4}}{280}-\frac {17 i^{3} g^{3} b^{3} B \,c^{3} x^{4}}{280}+\frac {i^{3} g^{3} d^{3} B \,a^{4} x^{3}}{420 b}-\frac {i^{3} g^{3} b^{3} B \,c^{4} x^{3}}{420 d}-\frac {i^{3} g^{3} d^{3} B \,a^{5} x^{2}}{280 b^{2}}+\frac {i^{3} g^{3} b^{3} B \,c^{5} x^{2}}{280 d^{2}}+\frac {i^{3} g^{3} d^{3} B \,a^{6} x}{140 b^{3}}-\frac {i^{3} g^{3} b^{3} B \,c^{6} x}{140 d^{3}}+\frac {i^{3} g^{3} B \ln \left (b x +a \right ) a^{4} c^{3}}{4 b}-\frac {i^{3} g^{3} B \ln \left (-d x -c \right ) a^{3} c^{4}}{4 d}-\frac {i^{3} g^{3} d^{3} B \ln \left (b x +a \right ) a^{7}}{140 b^{4}}+\frac {i^{3} g^{3} b^{3} B \ln \left (-d x -c \right ) c^{7}}{140 d^{4}}\)	\(1194\)
derivativedivides	\(\text {Expression too large to display}\)	\(20505\)
default	\(\text {Expression too large to display}\)	\(20505\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2577 vs. \(2 (401) = 802\).
time = 0.36, size = 2577, normalized size = 5.64 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*I*A*b^3*d^3*g^3*x^7 - 1/2*I*A*b^3*c*d^2*g^3*x^6 - 1/2*I*A*a*b^2*d^3*g^3*x^6 - 3/5*I*A*b^3*c^2*d*g^3*x^5 -
9/5*I*A*a*b^2*c*d^2*g^3*x^5 - 3/5*I*A*a^2*b*d^3*g^3*x^5 - 1/4*I*A*b^3*c^3*g^3*x^4 - 9/4*I*A*a*b^2*c^2*d*g^3*x
^4 - 9/4*I*A*a^2*b*c*d^2*g^3*x^4 - 1/4*I*A*a^3*d^3*g^3*x^4 - I*A*a*b^2*c^3*g^3*x^3 - 3*I*A*a^2*b*c^2*d*g^3*x^3
- I*A*a^3*c*d^2*g^3*x^3 - 3/2*I*A*a^2*b*c^3*g^3*x^2 - 3/2*I*A*a^3*c^2*d*g^3*x^2 - I*(x*log(b*x*e/(d*x + c) +
a*e/(d*x + c)) + a*log(b*x + a)/b - c*log(d*x + c)/d)*B*a^3*c^3*g^3 - 3/2*I*(x^2*log(b*x*e/(d*x + c) + a*e/(d*
x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*B*a^2*b*c^3*g^3 - 1/2*I*(2*x^3*lo
g(b*x*e/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^
2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*a*b^2*c^3*g^3 - 1/24*I*(6*x^4*log(b*x*e/(d*x + c) + a*e/(d*x + c)) -
6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*
x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*b^3*c^3*g^3 - 3/2*I*(x^2*log(b*x*e/(d*x + c) + a*e/(d*x + c)) - a^
2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*B*a^3*c^2*d*g^3 - 3/2*I*(2*x^3*log(b*x*e/(d*x
+ c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c
^2 - a^2*d^2)*x)/(b^2*d^2))*B*a^2*b*c^2*d*g^3 - 3/8*I*(6*x^4*log(b*x*e/(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^2*d*g^3 - 1/20*I*(12*x^5*log(b*x*e/(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^2*d*g^3 - 1/2*I*(2*x^3*log(
b*x*e/(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*c*d^2*g^3 - 3/8*I*(6*x^4*log(b*x*e/(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*c*d^2*g^3 - 3/20*I*(12*x^5*log(b*x*e/(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*c*d^2*g^3 - 1/120*I
*(60*x^6*log(b*x*e/(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*c*d^2*g^3 - 1/24*I*(6*x^4*log(b*x*e/(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^3*d^3*g^3 - 1/20*I*(12*x^5*log(b*x*e/(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^2*b*d^
3*g^3 - 1/120*I*(60*x^6*log(b*x*e/(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*a*b^2*d^3*g^3 - 1/420*I*(60*x^7*log
(b*x*e/(d*x + c) + a*e/(d*x + c)) + 60*a^7*log(b*x + a)/b^7 - 60*c^7*log(d*x + c)/d^7 - (10*(b^6*c*d^5 - a*b^5
*d^6)*x^6 - 12*(b^6*c^2*d^4 - a^2*b^4*d^6)*x^5 + 15*(b^6*c^3*d^3 - a^3*b^3*d^6)*x^4 - 20*(b^6*c^4*d^2 - a^4*b^
2*d^6)*x^3 + 30*(b^6*c^5*d - a^5*b*d^6)*x^2 - 60*(b^6*c^6 - a^6*d^6)*x)/(b^6*d^6))*B*b^3*d^3*g^3 - I*A*a^3*c^3
*g^3*x

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 889 vs. \(2 (401) = 802\).
time = 0.61, size = 889, normalized size = 1.95 \begin {gather*} \frac {-120 i \, A b^{7} d^{7} g^{3} x^{7} - 20 \, {\left ({\left (21 i \, A - i \, B\right )} b^{7} c d^{6} + {\left (21 i \, A + i \, B\right )} a b^{6} d^{7}\right )} g^{3} x^{6} - 12 \, {\left ({\left (42 i \, A - 5 i \, B\right )} b^{7} c^{2} d^{5} + 126 i \, A a b^{6} c d^{6} + {\left (42 i \, A + 5 i \, B\right )} a^{2} b^{5} d^{7}\right )} g^{3} x^{5} - 3 \, {\left ({\left (70 i \, A - 17 i \, B\right )} b^{7} c^{3} d^{4} + 7 \, {\left (90 i \, A - 7 i \, B\right )} a b^{6} c^{2} d^{5} + 7 \, {\left (90 i \, A + 7 i \, B\right )} a^{2} b^{5} c d^{6} + {\left (70 i \, A + 17 i \, B\right )} a^{3} b^{4} d^{7}\right )} g^{3} x^{4} - 2 \, {\left (-i \, B b^{7} c^{4} d^{3} + 14 \, {\left (30 i \, A - 7 i \, B\right )} a b^{6} c^{3} d^{4} + 1260 i \, A a^{2} b^{5} c^{2} d^{5} + 14 \, {\left (30 i \, A + 7 i \, B\right )} a^{3} b^{4} c d^{6} + i \, B a^{4} b^{3} d^{7}\right )} g^{3} x^{3} - 3 \, {\left (i \, B b^{7} c^{5} d^{2} - 7 i \, B a b^{6} c^{4} d^{3} + 84 \, {\left (5 i \, A - i \, B\right )} a^{2} b^{5} c^{3} d^{4} + 84 \, {\left (5 i \, A + i \, B\right )} a^{3} b^{4} c^{2} d^{5} + 7 i \, B a^{4} b^{3} c d^{6} - i \, B a^{5} b^{2} d^{7}\right )} g^{3} x^{2} - 6 \, {\left (-i \, B b^{7} c^{6} d + 7 i \, B a b^{6} c^{5} d^{2} - 21 i \, B a^{2} b^{5} c^{4} d^{3} + 140 i \, A a^{3} b^{4} c^{3} d^{4} + 21 i \, B a^{4} b^{3} c^{2} d^{5} - 7 i \, B a^{5} b^{2} c d^{6} + i \, B a^{6} b d^{7}\right )} g^{3} x - 6 \, {\left (35 i \, B a^{4} b^{3} c^{3} d^{4} - 21 i \, B a^{5} b^{2} c^{2} d^{5} + 7 i \, B a^{6} b c d^{6} - i \, B a^{7} d^{7}\right )} g^{3} \log \left (\frac {b x + a}{b}\right ) - 6 \, {\left (i \, B b^{7} c^{7} - 7 i \, B a b^{6} c^{6} d + 21 i \, B a^{2} b^{5} c^{5} d^{2} - 35 i \, B a^{3} b^{4} c^{4} d^{3}\right )} g^{3} \log \left (\frac {d x + c}{d}\right ) - 6 \, {\left (20 i \, B b^{7} d^{7} g^{3} x^{7} + 140 i \, B a^{3} b^{4} c^{3} d^{4} g^{3} x + 70 \, {\left (i \, B b^{7} c d^{6} + i \, B a b^{6} d^{7}\right )} g^{3} x^{6} + 84 \, {\left (i \, B b^{7} c^{2} d^{5} + 3 i \, B a b^{6} c d^{6} + i \, B a^{2} b^{5} d^{7}\right )} g^{3} x^{5} + 35 \, {\left (i \, B b^{7} c^{3} d^{4} + 9 i \, B a b^{6} c^{2} d^{5} + 9 i \, B a^{2} b^{5} c d^{6} + i \, B a^{3} b^{4} d^{7}\right )} g^{3} x^{4} + 140 \, {\left (i \, B a b^{6} c^{3} d^{4} + 3 i \, B a^{2} b^{5} c^{2} d^{5} + i \, B a^{3} b^{4} c d^{6}\right )} g^{3} x^{3} + 210 \, {\left (i \, B a^{2} b^{5} c^{3} d^{4} + i \, B a^{3} b^{4} c^{2} d^{5}\right )} g^{3} x^{2}\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{840 \, b^{4} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2161 vs. \(2 (427) = 854\).
time = 18.01, size = 2161, normalized size = 4.73 \begin {gather*} \frac {A b^{3} d^{3} g^{3} i^{3} x^{7}}{7} - \frac {B a^{4} g^{3} i^{3} \left (a^{3} d^{3} - 7 a^{2} b c d^{2} + 21 a b^{2} c^{2} d - 35 b^{3} c^{3}\right ) \log {\left (x + \frac {B a^{7} c d^{6} g^{3} i^{3} - 7 B a^{6} b c^{2} d^{5} g^{3} i^{3} + 21 B a^{5} b^{2} c^{3} d^{4} g^{3} i^{3} + \frac {B a^{5} d^{4} g^{3} i^{3} \left (a^{3} d^{3} - 7 a^{2} b c d^{2} + 21 a b^{2} c^{2} d - 35 b^{3} c^{3}\right )}{b} - 70 B a^{4} b^{3} c^{4} d^{3} g^{3} i^{3} - B a^{4} c d^{3} g^{3} i^{3} \left (a^{3} d^{3} - 7 a^{2} b c d^{2} + 21 a b^{2} c^{2} d - 35 b^{3} c^{3}\right ) + 21 B a^{3} b^{4} c^{5} d^{2} g^{3} i^{3} - 7 B a^{2} b^{5} c^{6} d g^{3} i^{3} + B a b^{6} c^{7} g^{3} i^{3}}{B a^{7} d^{7} g^{3} i^{3} - 7 B a^{6} b c d^{6} g^{3} i^{3} + 21 B a^{5} b^{2} c^{2} d^{5} g^{3} i^{3} - 35 B a^{4} b^{3} c^{3} d^{4} g^{3} i^{3} - 35 B a^{3} b^{4} c^{4} d^{3} g^{3} i^{3} + 21 B a^{2} b^{5} c^{5} d^{2} g^{3} i^{3} - 7 B a b^{6} c^{6} d g^{3} i^{3} + B b^{7} c^{7} g^{3} i^{3}} \right )}}{140 b^{4}} - \frac {B c^{4} g^{3} i^{3} \cdot \left (35 a^{3} d^{3} - 21 a^{2} b c d^{2} + 7 a b^{2} c^{2} d - b^{3} c^{3}\right ) \log {\left (x + \frac {B a^{7} c d^{6} g^{3} i^{3} - 7 B a^{6} b c^{2} d^{5} g^{3} i^{3} + 21 B a^{5} b^{2} c^{3} d^{4} g^{3} i^{3} - 70 B a^{4} b^{3} c^{4} d^{3} g^{3} i^{3} + 21 B a^{3} b^{4} c^{5} d^{2} g^{3} i^{3} - 7 B a^{2} b^{5} c^{6} d g^{3} i^{3} + B a b^{6} c^{7} g^{3} i^{3} + B a b^{3} c^{4} g^{3} i^{3} \cdot \left (35 a^{3} d^{3} - 21 a^{2} b c d^{2} + 7 a b^{2} c^{2} d - b^{3} c^{3}\right ) - \frac {B b^{4} c^{5} g^{3} i^{3} \cdot \left (35 a^{3} d^{3} - 21 a^{2} b c d^{2} + 7 a b^{2} c^{2} d - b^{3} c^{3}\right )}{d}}{B a^{7} d^{7} g^{3} i^{3} - 7 B a^{6} b c d^{6} g^{3} i^{3} + 21 B a^{5} b^{2} c^{2} d^{5} g^{3} i^{3} - 35 B a^{4} b^{3} c^{3} d^{4} g^{3} i^{3} - 35 B a^{3} b^{4} c^{4} d^{3} g^{3} i^{3} + 21 B a^{2} b^{5} c^{5} d^{2} g^{3} i^{3} - 7 B a b^{6} c^{6} d g^{3} i^{3} + B b^{7} c^{7} g^{3} i^{3}} \right )}}{140 d^{4}} + x^{6} \left (\frac {A a b^{2} d^{3} g^{3} i^{3}}{2} + \frac {A b^{3} c d^{2} g^{3} i^{3}}{2} + \frac {B a b^{2} d^{3} g^{3} i^{3}}{42} - \frac {B b^{3} c d^{2} g^{3} i^{3}}{42}\right ) + x^{5} \cdot \left (\frac {3 A a^{2} b d^{3} g^{3} i^{3}}{5} + \frac {9 A a b^{2} c d^{2} g^{3} i^{3}}{5} + \frac {3 A b^{3} c^{2} d g^{3} i^{3}}{5} + \frac {B a^{2} b d^{3} g^{3} i^{3}}{14} - \frac {B b^{3} c^{2} d g^{3} i^{3}}{14}\right ) + x^{4} \left (\frac {A a^{3} d^{3} g^{3} i^{3}}{4} + \frac {9 A a^{2} b c d^{2} g^{3} i^{3}}{4} + \frac {9 A a b^{2} c^{2} d g^{3} i^{3}}{4} + \frac {A b^{3} c^{3} g^{3} i^{3}}{4} + \frac {17 B a^{3} d^{3} g^{3} i^{3}}{280} + \frac {7 B a^{2} b c d^{2} g^{3} i^{3}}{40} - \frac {7 B a b^{2} c^{2} d g^{3} i^{3}}{40} - \frac {17 B b^{3} c^{3} g^{3} i^{3}}{280}\right ) + x^{3} \left (A a^{3} c d^{2} g^{3} i^{3} + 3 A a^{2} b c^{2} d g^{3} i^{3} + A a b^{2} c^{3} g^{3} i^{3} + \frac {B a^{4} d^{3} g^{3} i^{3}}{420 b} + \frac {7 B a^{3} c d^{2} g^{3} i^{3}}{30} - \frac {7 B a b^{2} c^{3} g^{3} i^{3}}{30} - \frac {B b^{3} c^{4} g^{3} i^{3}}{420 d}\right ) + x^{2} \cdot \left (\frac {3 A a^{3} c^{2} d g^{3} i^{3}}{2} + \frac {3 A a^{2} b c^{3} g^{3} i^{3}}{2} - \frac {B a^{5} d^{3} g^{3} i^{3}}{280 b^{2}} + \frac {B a^{4} c d^{2} g^{3} i^{3}}{40 b} + \frac {3 B a^{3} c^{2} d g^{3} i^{3}}{10} - \frac {3 B a^{2} b c^{3} g^{3} i^{3}}{10} - \frac {B a b^{2} c^{4} g^{3} i^{3}}{40 d} + \frac {B b^{3} c^{5} g^{3} i^{3}}{280 d^{2}}\right ) + x \left (A a^{3} c^{3} g^{3} i^{3} + \frac {B a^{6} d^{3} g^{3} i^{3}}{140 b^{3}} - \frac {B a^{5} c d^{2} g^{3} i^{3}}{20 b^{2}} + \frac {3 B a^{4} c^{2} d g^{3} i^{3}}{20 b} - \frac {3 B a^{2} b c^{4} g^{3} i^{3}}{20 d} + \frac {B a b^{2} c^{5} g^{3} i^{3}}{20 d^{2}} - \frac {B b^{3} c^{6} g^{3} i^{3}}{140 d^{3}}\right ) + \left (B a^{3} c^{3} g^{3} i^{3} x + \frac {3 B a^{3} c^{2} d g^{3} i^{3} x^{2}}{2} + B a^{3} c d^{2} g^{3} i^{3} x^{3} + \frac {B a^{3} d^{3} g^{3} i^{3} x^{4}}{4} + \frac {3 B a^{2} b c^{3} g^{3} i^{3} x^{2}}{2} + 3 B a^{2} b c^{2} d g^{3} i^{3} x^{3} + \frac {9 B a^{2} b c d^{2} g^{3} i^{3} x^{4}}{4} + \frac {3 B a^{2} b d^{3} g^{3} i^{3} x^{5}}{5} + B a b^{2} c^{3} g^{3} i^{3} x^{3} + \frac {9 B a b^{2} c^{2} d g^{3} i^{3} x^{4}}{4} + \frac {9 B a b^{2} c d^{2} g^{3} i^{3} x^{5}}{5} + \frac {B a b^{2} d^{3} g^{3} i^{3} x^{6}}{2} + \frac {B b^{3} c^{3} g^{3} i^{3} x^{4}}{4} + \frac {3 B b^{3} c^{2} d g^{3} i^{3} x^{5}}{5} + \frac {B b^{3} c d^{2} g^{3} i^{3} x^{6}}{2} + \frac {B b^{3} d^{3} g^{3} i^{3} x^{7}}{7}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*b**3*d**3*g**3*i**3*x**7/7 - B*a**4*g**3*i**3*(a**3*d**3 - 7*a**2*b*c*d**2 + 21*a*b**2*c**2*d - 35*b**3*c**3
)*log(x + (B*a**7*c*d**6*g**3*i**3 - 7*B*a**6*b*c**2*d**5*g**3*i**3 + 21*B*a**5*b**2*c**3*d**4*g**3*i**3 + B*a
**5*d**4*g**3*i**3*(a**3*d**3 - 7*a**2*b*c*d**2 + 21*a*b**2*c**2*d - 35*b**3*c**3)/b - 70*B*a**4*b**3*c**4*d**
3*g**3*i**3 - B*a**4*c*d**3*g**3*i**3*(a**3*d**3 - 7*a**2*b*c*d**2 + 21*a*b**2*c**2*d - 35*b**3*c**3) + 21*B*a
**3*b**4*c**5*d**2*g**3*i**3 - 7*B*a**2*b**5*c**6*d*g**3*i**3 + B*a*b**6*c**7*g**3*i**3)/(B*a**7*d**7*g**3*i**
3 - 7*B*a**6*b*c*d**6*g**3*i**3 + 21*B*a**5*b**2*c**2*d**5*g**3*i**3 - 35*B*a**4*b**3*c**3*d**4*g**3*i**3 - 35
*B*a**3*b**4*c**4*d**3*g**3*i**3 + 21*B*a**2*b**5*c**5*d**2*g**3*i**3 - 7*B*a*b**6*c**6*d*g**3*i**3 + B*b**7*c
**7*g**3*i**3))/(140*b**4) - B*c**4*g**3*i**3*(35*a**3*d**3 - 21*a**2*b*c*d**2 + 7*a*b**2*c**2*d - b**3*c**3)*
log(x + (B*a**7*c*d**6*g**3*i**3 - 7*B*a**6*b*c**2*d**5*g**3*i**3 + 21*B*a**5*b**2*c**3*d**4*g**3*i**3 - 70*B*
a**4*b**3*c**4*d**3*g**3*i**3 + 21*B*a**3*b**4*c**5*d**2*g**3*i**3 - 7*B*a**2*b**5*c**6*d*g**3*i**3 + B*a*b**6
*c**7*g**3*i**3 + B*a*b**3*c**4*g**3*i**3*(35*a**3*d**3 - 21*a**2*b*c*d**2 + 7*a*b**2*c**2*d - b**3*c**3) - B*
b**4*c**5*g**3*i**3*(35*a**3*d**3 - 21*a**2*b*c*d**2 + 7*a*b**2*c**2*d - b**3*c**3)/d)/(B*a**7*d**7*g**3*i**3
- 7*B*a**6*b*c*d**6*g**3*i**3 + 21*B*a**5*b**2*c**2*d**5*g**3*i**3 - 35*B*a**4*b**3*c**3*d**4*g**3*i**3 - 35*B
*a**3*b**4*c**4*d**3*g**3*i**3 + 21*B*a**2*b**5*c**5*d**2*g**3*i**3 - 7*B*a*b**6*c**6*d*g**3*i**3 + B*b**7*c**
7*g**3*i**3))/(140*d**4) + x**6*(A*a*b**2*d**3*g**3*i**3/2 + A*b**3*c*d**2*g**3*i**3/2 + B*a*b**2*d**3*g**3*i*
*3/42 - B*b**3*c*d**2*g**3*i**3/42) + x**5*(3*A*a**2*b*d**3*g**3*i**3/5 + 9*A*a*b**2*c*d**2*g**3*i**3/5 + 3*A*
b**3*c**2*d*g**3*i**3/5 + B*a**2*b*d**3*g**3*i**3/14 - B*b**3*c**2*d*g**3*i**3/14) + x**4*(A*a**3*d**3*g**3*i*
*3/4 + 9*A*a**2*b*c*d**2*g**3*i**3/4 + 9*A*a*b**2*c**2*d*g**3*i**3/4 + A*b**3*c**3*g**3*i**3/4 + 17*B*a**3*d**
3*g**3*i**3/280 + 7*B*a**2*b*c*d**2*g**3*i**3/40 - 7*B*a*b**2*c**2*d*g**3*i**3/40 - 17*B*b**3*c**3*g**3*i**3/2
80) + x**3*(A*a**3*c*d**2*g**3*i**3 + 3*A*a**2*b*c**2*d*g**3*i**3 + A*a*b**2*c**3*g**3*i**3 + B*a**4*d**3*g**3
*i**3/(420*b) + 7*B*a**3*c*d**2*g**3*i**3/30 - 7*B*a*b**2*c**3*g**3*i**3/30 - B*b**3*c**4*g**3*i**3/(420*d)) +
x**2*(3*A*a**3*c**2*d*g**3*i**3/2 + 3*A*a**2*b*c**3*g**3*i**3/2 - B*a**5*d**3*g**3*i**3/(280*b**2) + B*a**4*c
*d**2*g**3*i**3/(40*b) + 3*B*a**3*c**2*d*g**3*i**3/10 - 3*B*a**2*b*c**3*g**3*i**3/10 - B*a*b**2*c**4*g**3*i**3
/(40*d) + B*b**3*c**5*g**3*i**3/(280*d**2)) + x*(A*a**3*c**3*g**3*i**3 + B*a**6*d**3*g**3*i**3/(140*b**3) - B*
a**5*c*d**2*g**3*i**3/(20*b**2) + 3*B*a**4*c**2*d*g**3*i**3/(20*b) - 3*B*a**2*b*c**4*g**3*i**3/(20*d) + B*a*b*
*2*c**5*g**3*i**3/(20*d**2) - B*b**3*c**6*g**3*i**3/(140*d**3)) + (B*a**3*c**3*g**3*i**3*x + 3*B*a**3*c**2*d*g
**3*i**3*x**2/2 + B*a**3*c*d**2*g**3*i**3*x**3 + B*a**3*d**3*g**3*i**3*x**4/4 + 3*B*a**2*b*c**3*g**3*i**3*x**2
/2 + 3*B*a**2*b*c**2*d*g**3*i**3*x**3 + 9*B*a**2*b*c*d**2*g**3*i**3*x**4/4 + 3*B*a**2*b*d**3*g**3*i**3*x**5/5
+ B*a*b**2*c**3*g**3*i**3*x**3 + 9*B*a*b**2*c**2*d*g**3*i**3*x**4/4 + 9*B*a*b**2*c*d**2*g**3*i**3*x**5/5 + B*a
*b**2*d**3*g**3*i**3*x**6/2 + B*b**3*c**3*g**3*i**3*x**4/4 + 3*B*b**3*c**2*d*g**3*i**3*x**5/5 + B*b**3*c*d**2*
g**3*i**3*x**6/2 + B*b**3*d**3*g**3*i**3*x**7/7)*log(e*(a + b*x)/(c + d*x))

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 9900 vs. \(2 (401) = 802\).
time = 4.14, size = 9900, normalized size = 21.66 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(6*I*B*b^15*c^8*g^3*e^8*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 48*I*B*a*b^14*c^7*d*g^3*e^8*log(-b*e + (
b*x*e + a*e)*d/(d*x + c)) + 168*I*B*a^2*b^13*c^6*d^2*g^3*e^8*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 336*I*B*a
^3*b^12*c^5*d^3*g^3*e^8*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 420*I*B*a^4*b^11*c^4*d^4*g^3*e^8*log(-b*e + (b
*x*e + a*e)*d/(d*x + c)) - 336*I*B*a^5*b^10*c^3*d^5*g^3*e^8*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 168*I*B*a^
6*b^9*c^2*d^6*g^3*e^8*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 48*I*B*a^7*b^8*c*d^7*g^3*e^8*log(-b*e + (b*x*e +
a*e)*d/(d*x + c)) + 6*I*B*a^8*b^7*d^8*g^3*e^8*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 42*I*(b*x*e + a*e)*B*b^
14*c^8*d*g^3*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 336*I*(b*x*e + a*e)*B*a*b^13*c^7*d^2*g^3*e^
7*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 1176*I*(b*x*e + a*e)*B*a^2*b^12*c^6*d^3*g^3*e^7*log(-b*e +
(b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 2352*I*(b*x*e + a*e)*B*a^3*b^11*c^5*d^4*g^3*e^7*log(-b*e + (b*x*e + a*
e)*d/(d*x + c))/(d*x + c) - 2940*I*(b*x*e + a*e)*B*a^4*b^10*c^4*d^5*g^3*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x +
c))/(d*x + c) + 2352*I*(b*x*e + a*e)*B*a^5*b^9*c^3*d^6*g^3*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)
- 1176*I*(b*x*e + a*e)*B*a^6*b^8*c^2*d^7*g^3*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 336*I*(b*x
*e + a*e)*B*a^7*b^7*c*d^8*g^3*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 42*I*(b*x*e + a*e)*B*a^8*b
^6*d^9*g^3*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 126*I*(b*x*e + a*e)^2*B*b^13*c^8*d^2*g^3*e^6*
log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 1008*I*(b*x*e + a*e)^2*B*a*b^12*c^7*d^3*g^3*e^6*log(-b*e +
(b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 3528*I*(b*x*e + a*e)^2*B*a^2*b^11*c^6*d^4*g^3*e^6*log(-b*e + (b*x*e
+ a*e)*d/(d*x + c))/(d*x + c)^2 - 7056*I*(b*x*e + a*e)^2*B*a^3*b^10*c^5*d^5*g^3*e^6*log(-b*e + (b*x*e + a*e)*d
/(d*x + c))/(d*x + c)^2 + 8820*I*(b*x*e + a*e)^2*B*a^4*b^9*c^4*d^6*g^3*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c
))/(d*x + c)^2 - 7056*I*(b*x*e + a*e)^2*B*a^5*b^8*c^3*d^7*g^3*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x +
c)^2 + 3528*I*(b*x*e + a*e)^2*B*a^6*b^7*c^2*d^8*g^3*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 1
008*I*(b*x*e + a*e)^2*B*a^7*b^6*c*d^9*g^3*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 126*I*(b*x*e
+ a*e)^2*B*a^8*b^5*d^10*g^3*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 210*I*(b*x*e + a*e)^3*B*b
^12*c^8*d^3*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 1680*I*(b*x*e + a*e)^3*B*a*b^11*c^7*d^
4*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 5880*I*(b*x*e + a*e)^3*B*a^2*b^10*c^6*d^5*g^3*e^
5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 11760*I*(b*x*e + a*e)^3*B*a^3*b^9*c^5*d^6*g^3*e^5*log(-b
*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 14700*I*(b*x*e + a*e)^3*B*a^4*b^8*c^4*d^7*g^3*e^5*log(-b*e + (b*
x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 11760*I*(b*x*e + a*e)^3*B*a^5*b^7*c^3*d^8*g^3*e^5*log(-b*e + (b*x*e + a*
e)*d/(d*x + c))/(d*x + c)^3 - 5880*I*(b*x*e + a*e)^3*B*a^6*b^6*c^2*d^9*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x
+ c))/(d*x + c)^3 + 1680*I*(b*x*e + a*e)^3*B*a^7*b^5*c*d^10*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*
x + c)^3 - 210*I*(b*x*e + a*e)^3*B*a^8*b^4*d^11*g^3*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 21
0*I*(b*x*e + a*e)^4*B*b^11*c^8*d^4*g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 1680*I*(b*x*e +
a*e)^4*B*a*b^10*c^7*d^5*g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 5880*I*(b*x*e + a*e)^4*B*
a^2*b^9*c^6*d^6*g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 11760*I*(b*x*e + a*e)^4*B*a^3*b^8*
c^5*d^7*g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 14700*I*(b*x*e + a*e)^4*B*a^4*b^7*c^4*d^8*
g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 11760*I*(b*x*e + a*e)^4*B*a^5*b^6*c^3*d^9*g^3*e^4*
log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 5880*I*(b*x*e + a*e)^4*B*a^6*b^5*c^2*d^10*g^3*e^4*log(-b*e
+ (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 1680*I*(b*x*e + a*e)^4*B*a^7*b^4*c*d^11*g^3*e^4*log(-b*e + (b*x*e
+ a*e)*d/(d*x + c))/(d*x + c)^4 + 210*I*(b*x*e + a*e)^4*B*a^8*b^3*d^12*g^3*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x
+ c))/(d*x + c)^4 - 126*I*(b*x*e + a*e)^5*B*b^10*c^8*d^5*g^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x +
c)^5 + 1008*I*(b*x*e + a*e)^5*B*a*b^9*c^7*d^6*g^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 - 352
8*I*(b*x*e + a*e)^5*B*a^2*b^8*c^6*d^7*g^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 + 7056*I*(b*x*
e + a*e)^5*B*a^3*b^7*c^5*d^8*g^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 - 8820*I*(b*x*e + a*e)^
5*B*a^4*b^6*c^4*d^9*g^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 + 7056*I*(b*x*e + a*e)^5*B*a^5*b
^5*c^3*d^10*g^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 - 3528*I*(b*x*e + a*e)^5*B*a^6*b^4*c^2*d
^11*g^3*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 + 1008*I*(b*x*e + a*e)^5*B*a^7*b^3*c*d^12*g^3*e^
3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + ...

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Mupad [B]
time = 6.58, size = 2500, normalized size = 5.47 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(((140*a*d + 140*b*c)*(((140*a*d + 140*b*c)*((a*c*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d - B*b*c))
/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12*A*a^2*d^2
+ 12*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*i^3))/(b*d) - ((140*a*d + 140*b*
c)*((g^3*i^3*(20*A*a^3*d^3 + 20*A*b^3*c^3 + 3*B*a^3*d^3 - 3*B*b^3*c^3 + 120*A*a*b^2*c^2*d + 120*A*a^2*b*c*d^2
- 6*B*a*b^2*c^2*d + 6*B*a^2*b*c*d^2))/5 + ((140*a*d + 140*b*c)*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*
d - B*b*c))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12
*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*i^3))/(140*b*d) - (a*
c*((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d - B*b*c))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140))/(
b*d)))/(140*b*d) + (g^3*i^3*(4*A*a^4*d^4 + 4*A*b^4*c^4 + B*a^4*d^4 - B*b^4*c^4 + 144*A*a^2*b^2*c^2*d^2 + 64*A*
a*b^3*c^3*d + 64*A*a^3*b*c*d^3 - 8*B*a*b^3*c^3*d + 8*B*a^3*b*c*d^3))/(4*b*d)))/(140*b*d) + (a*c*((g^3*i^3*(20*
A*a^3*d^3 + 20*A*b^3*c^3 + 3*B*a^3*d^3 - 3*B*b^3*c^3 + 120*A*a*b^2*c^2*d + 120*A*a^2*b*c*d^2 - 6*B*a*b^2*c^2*d
+ 6*B*a^2*b*c*d^2))/5 + ((140*a*d + 140*b*c)*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d - B*b*c))/7 - (
A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12*A*a^2*d^2 + 12*A
*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*i^3))/(140*b*d) - (a*c*((b^2*d^2*g^3*i
^3*(28*A*a*d + 28*A*b*c + B*a*d - B*b*c))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140))/(b*d)))/(b*d) - (a
*c*g^3*i^3*(4*A*a^3*d^3 + 4*A*b^3*c^3 + B*a^3*d^3 - B*b^3*c^3 + 24*A*a*b^2*c^2*d + 24*A*a^2*b*c*d^2 - 2*B*a*b^
2*c^2*d + 2*B*a^2*b*c*d^2))/(b*d)))/(140*b*d) - (a*c*((a*c*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d -
B*b*c))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12*A*a
^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*i^3))/(b*d) - ((140*a*d +
140*b*c)*((g^3*i^3*(20*A*a^3*d^3 + 20*A*b^3*c^3 + 3*B*a^3*d^3 - 3*B*b^3*c^3 + 120*A*a*b^2*c^2*d + 120*A*a^2*b
*c*d^2 - 6*B*a*b^2*c^2*d + 6*B*a^2*b*c*d^2))/5 + ((140*a*d + 140*b*c)*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c
+ B*a*d - B*b*c))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*
i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*i^3))/(140*b*d
) - (a*c*((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d - B*b*c))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/
140))/(b*d)))/(140*b*d) + (g^3*i^3*(4*A*a^4*d^4 + 4*A*b^4*c^4 + B*a^4*d^4 - B*b^4*c^4 + 144*A*a^2*b^2*c^2*d^2
+ 64*A*a*b^3*c^3*d + 64*A*a^3*b*c*d^3 - 8*B*a*b^3*c^3*d + 8*B*a^3*b*c*d^3))/(4*b*d)))/(b*d) + (a^2*c^2*g^3*i^3
*(12*A*a^2*d^2 + 12*A*b^2*c^2 + 3*B*a^2*d^2 - 3*B*b^2*c^2 + 32*A*a*b*c*d))/(2*b*d)) + x^6*((b^2*d^2*g^3*i^3*(2
8*A*a*d + 28*A*b*c + B*a*d - B*b*c))/42 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/840) + x^3*((a*c*((((b^2*d^2
*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d - B*b*c))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 14
0*b*c))/(140*b*d) - (b*d*g^3*i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*c*d))/2 + A*a
*b^2*c*d^2*g^3*i^3))/(3*b*d) - ((140*a*d + 140*b*c)*((g^3*i^3*(20*A*a^3*d^3 + 20*A*b^3*c^3 + 3*B*a^3*d^3 - 3*B
*b^3*c^3 + 120*A*a*b^2*c^2*d + 120*A*a^2*b*c*d^2 - 6*B*a*b^2*c^2*d + 6*B*a^2*b*c*d^2))/5 + ((140*a*d + 140*b*c
)*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d - B*b*c))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*
(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*
c*d))/2 + A*a*b^2*c*d^2*g^3*i^3))/(140*b*d) - (a*c*((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d - B*b*c))/7
- (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140))/(b*d)))/(420*b*d) + (g^3*i^3*(4*A*a^4*d^4 + 4*A*b^4*c^4 + B*a^
4*d^4 - B*b^4*c^4 + 144*A*a^2*b^2*c^2*d^2 + 64*A*a*b^3*c^3*d + 64*A*a^3*b*c*d^3 - 8*B*a*b^3*c^3*d + 8*B*a^3*b*
c*d^3))/(12*b*d)) - x^2*(((140*a*d + 140*b*c)*((a*c*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d - B*b*c))
/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12*A*a^2*d^2
+ 12*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*i^3))/(b*d) - ((140*a*d + 140*b*
c)*((g^3*i^3*(20*A*a^3*d^3 + 20*A*b^3*c^3 + 3*B*a^3*d^3 - 3*B*b^3*c^3 + 120*A*a*b^2*c^2*d + 120*A*a^2*b*c*d^2
- 6*B*a*b^2*c^2*d + 6*B*a^2*b*c*d^2))/5 + ((140*a*d + 140*b*c)*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*
d - B*b*c))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12
*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*i^3))/(140*b*d) - (a*
c*((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d - B*b*c))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140))/(
b*d)))/(140*b*d) + (g^3*i^3*(4*A*a^4*d^4 + 4*A*...

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