∫ (ag + bgx)3(ci + dix)3(A + B log(e(a+bx) c+dx )) dx

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

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

[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

________________________________________________________________________________________

Rubi [A] time = 0.94, antiderivative size = 416, normalized size of antiderivative = 0.91, number of steps used = 18, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2528, 2525, 12, 43} \[ \frac {d^2 g^3 i^3 (a+b x)^6 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b^4}+\frac {d^3 g^3 i^3 (a+b x)^7 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{7 b^4}+\frac {g^3 i^3 (a+b x)^4 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 b^4}+\frac {3 d g^3 i^3 (a+b x)^5 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 b^4}-\frac {B g^3 i^3 x (b c-a d)^6}{140 b^3 d^3}+\frac {B g^3 i^3 (a+b x)^2 (b c-a d)^5}{280 b^4 d^2}-\frac {B d^2 g^3 i^3 (a+b x)^6 (b c-a d)}{42 b^4}+\frac {B g^3 i^3 (b c-a d)^7 \log (c+d x)}{140 b^4 d^4}-\frac {B g^3 i^3 (a+b x)^3 (b c-a d)^4}{420 b^4 d}-\frac {17 B g^3 i^3 (a+b x)^4 (b c-a d)^3}{280 b^4}-\frac {B d g^3 i^3 (a+b x)^5 (b c-a d)^2}{14 b^4} \]

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*(a + b*x)^2)/(280*b^4*d^2) - (B*(b*c - a

*d)^4*g^3*i^3*(a + b*x)^3)/(420*b^4*d) - (17*B*(b*c - a*d)^3*g^3*i^3*(a + b*x)^4)/(280*b^4) - (B*d*(b*c - a*d)

^2*g^3*i^3*(a + b*x)^5)/(14*b^4) - (B*d^2*(b*c - a*d)*g^3*i^3*(a + b*x)^6)/(42*b^4) + ((b*c - a*d)^3*g^3*i^3*(

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

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

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

og[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 43

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 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m

+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +

1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc

tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*

RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF

unctionQ[RGx, x] && IGtQ[n, 0]

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*}

________________________________________________________________________________________

Mathematica [A] time = 0.59, size = 586, normalized size = 1.28 \[ \frac {g^3 i^3 \left (120 d^3 (a+b x)^7 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+420 d^2 (a+b x)^6 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+210 (a+b x)^4 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+504 d (a+b x)^5 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {120 b^2 B c x (b c-a d)^5}{d^3}+\frac {126 B (b c-a d)^7 \log (c+d x)}{d^4}-\frac {120 b B c (b c-a d)^6 \log (c+d x)}{d^4}-\frac {126 b B x (b c-a d)^6}{d^3}+\frac {120 a B (b c-a d)^6 \log (c+d x)}{d^3}+\frac {63 B (a+b x)^2 (b c-a d)^5}{d^2}-\frac {60 b B c (a+b x)^2 (b c-a d)^4}{d^2}-20 b B c d^2 (a+b x)^6+24 a B d^2 (a+b x)^5 (a d-b c)+\frac {120 a b B x (a d-b c)^5}{d^2}-\frac {42 B (a+b x)^3 (b c-a d)^4}{d}+\frac {60 a B (a+b x)^2 (b c-a d)^4}{d}+\frac {40 b B c (a+b x)^3 (b c-a d)^3}{d}-84 B d (a+b x)^5 (b c-a d)^2-30 b B c (a+b x)^4 (b c-a d)^2+30 a B d (a+b x)^4 (b c-a d)^2+24 b B c d (a+b x)^5 (b c-a d)+21 B (a+b x)^4 (a d-b c)^3+40 a B (a+b x)^3 (a d-b c)^3+20 a B d^3 (a+b x)^6\right )}{840 b^4} \]

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)

________________________________________________________________________________________

fricas [B] time = 1.54, size = 912, normalized size = 2.00 \[ \frac {120 \, A b^{7} d^{7} g^{3} i^{3} x^{7} + 20 \, {\left ({\left (21 \, A - B\right )} b^{7} c d^{6} + {\left (21 \, A + B\right )} a b^{6} d^{7}\right )} g^{3} i^{3} x^{6} + 12 \, {\left ({\left (42 \, A - 5 \, B\right )} b^{7} c^{2} d^{5} + 126 \, A a b^{6} c d^{6} + {\left (42 \, A + 5 \, B\right )} a^{2} b^{5} d^{7}\right )} g^{3} i^{3} x^{5} + 3 \, {\left ({\left (70 \, A - 17 \, B\right )} b^{7} c^{3} d^{4} + 7 \, {\left (90 \, A - 7 \, B\right )} a b^{6} c^{2} d^{5} + 7 \, {\left (90 \, A + 7 \, B\right )} a^{2} b^{5} c d^{6} + {\left (70 \, A + 17 \, B\right )} a^{3} b^{4} d^{7}\right )} g^{3} i^{3} x^{4} - 2 \, {\left (B b^{7} c^{4} d^{3} - 14 \, {\left (30 \, A - 7 \, B\right )} a b^{6} c^{3} d^{4} - 1260 \, A a^{2} b^{5} c^{2} d^{5} - 14 \, {\left (30 \, A + 7 \, B\right )} a^{3} b^{4} c d^{6} - B a^{4} b^{3} d^{7}\right )} g^{3} i^{3} x^{3} + 3 \, {\left (B b^{7} c^{5} d^{2} - 7 \, B a b^{6} c^{4} d^{3} + 84 \, {\left (5 \, A - B\right )} a^{2} b^{5} c^{3} d^{4} + 84 \, {\left (5 \, A + B\right )} a^{3} b^{4} c^{2} d^{5} + 7 \, B a^{4} b^{3} c d^{6} - B a^{5} b^{2} d^{7}\right )} g^{3} i^{3} x^{2} - 6 \, {\left (B b^{7} c^{6} d - 7 \, B a b^{6} c^{5} d^{2} + 21 \, B a^{2} b^{5} c^{4} d^{3} - 140 \, A a^{3} b^{4} c^{3} d^{4} - 21 \, B a^{4} b^{3} c^{2} d^{5} + 7 \, B a^{5} b^{2} c d^{6} - B a^{6} b d^{7}\right )} g^{3} i^{3} x + 6 \, {\left (35 \, B a^{4} b^{3} c^{3} d^{4} - 21 \, B a^{5} b^{2} c^{2} d^{5} + 7 \, B a^{6} b c d^{6} - B a^{7} d^{7}\right )} g^{3} i^{3} \log \left (b x + a\right ) + 6 \, {\left (B b^{7} c^{7} - 7 \, B a b^{6} c^{6} d + 21 \, B a^{2} b^{5} c^{5} d^{2} - 35 \, B a^{3} b^{4} c^{4} d^{3}\right )} g^{3} i^{3} \log \left (d x + c\right ) + 6 \, {\left (20 \, B b^{7} d^{7} g^{3} i^{3} x^{7} + 140 \, B a^{3} b^{4} c^{3} d^{4} g^{3} i^{3} x + 70 \, {\left (B b^{7} c d^{6} + B a b^{6} d^{7}\right )} g^{3} i^{3} x^{6} + 84 \, {\left (B b^{7} c^{2} d^{5} + 3 \, B a b^{6} c d^{6} + B a^{2} b^{5} d^{7}\right )} g^{3} i^{3} x^{5} + 35 \, {\left (B b^{7} c^{3} d^{4} + 9 \, B a b^{6} c^{2} d^{5} + 9 \, B a^{2} b^{5} c d^{6} + B a^{3} b^{4} d^{7}\right )} g^{3} i^{3} x^{4} + 140 \, {\left (B a b^{6} c^{3} d^{4} + 3 \, B a^{2} b^{5} c^{2} d^{5} + B a^{3} b^{4} c d^{6}\right )} g^{3} i^{3} x^{3} + 210 \, {\left (B a^{2} b^{5} c^{3} d^{4} + B a^{3} b^{4} c^{2} d^{5}\right )} g^{3} i^{3} x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{840 \, b^{4} d^{4}} \]

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*A*b^7*d^7*g^3*i^3*x^7 + 20*((21*A - B)*b^7*c*d^6 + (21*A + B)*a*b^6*d^7)*g^3*i^3*x^6 + 12*((42*A -

5*B)*b^7*c^2*d^5 + 126*A*a*b^6*c*d^6 + (42*A + 5*B)*a^2*b^5*d^7)*g^3*i^3*x^5 + 3*((70*A - 17*B)*b^7*c^3*d^4 +

7*(90*A - 7*B)*a*b^6*c^2*d^5 + 7*(90*A + 7*B)*a^2*b^5*c*d^6 + (70*A + 17*B)*a^3*b^4*d^7)*g^3*i^3*x^4 - 2*(B*b^

7*c^4*d^3 - 14*(30*A - 7*B)*a*b^6*c^3*d^4 - 1260*A*a^2*b^5*c^2*d^5 - 14*(30*A + 7*B)*a^3*b^4*c*d^6 - B*a^4*b^3

*d^7)*g^3*i^3*x^3 + 3*(B*b^7*c^5*d^2 - 7*B*a*b^6*c^4*d^3 + 84*(5*A - B)*a^2*b^5*c^3*d^4 + 84*(5*A + B)*a^3*b^4

*c^2*d^5 + 7*B*a^4*b^3*c*d^6 - B*a^5*b^2*d^7)*g^3*i^3*x^2 - 6*(B*b^7*c^6*d - 7*B*a*b^6*c^5*d^2 + 21*B*a^2*b^5*

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

a^4*b^3*c^3*d^4 - 21*B*a^5*b^2*c^2*d^5 + 7*B*a^6*b*c*d^6 - B*a^7*d^7)*g^3*i^3*log(b*x + a) + 6*(B*b^7*c^7 - 7*

B*a*b^6*c^6*d + 21*B*a^2*b^5*c^5*d^2 - 35*B*a^3*b^4*c^4*d^3)*g^3*i^3*log(d*x + c) + 6*(20*B*b^7*d^7*g^3*i^3*x^

7 + 140*B*a^3*b^4*c^3*d^4*g^3*i^3*x + 70*(B*b^7*c*d^6 + B*a*b^6*d^7)*g^3*i^3*x^6 + 84*(B*b^7*c^2*d^5 + 3*B*a*b

^6*c*d^6 + B*a^2*b^5*d^7)*g^3*i^3*x^5 + 35*(B*b^7*c^3*d^4 + 9*B*a*b^6*c^2*d^5 + 9*B*a^2*b^5*c*d^6 + B*a^3*b^4*

d^7)*g^3*i^3*x^4 + 140*(B*a*b^6*c^3*d^4 + 3*B*a^2*b^5*c^2*d^5 + B*a^3*b^4*c*d^6)*g^3*i^3*x^3 + 210*(B*a^2*b^5*

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

________________________________________________________________________________________

giac [B] time = 2.58, size = 10098, normalized size = 22.10 \[ \text {result too large to display} \]

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*B*b^15*c^8*g^3*i*e^8*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 48*B*a*b^14*c^7*d*g^3*i*e^8*log(-b*e + (

b*x*e + a*e)*d/(d*x + c)) + 168*B*a^2*b^13*c^6*d^2*g^3*i*e^8*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 336*B*a^3

*b^12*c^5*d^3*g^3*i*e^8*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 420*B*a^4*b^11*c^4*d^4*g^3*i*e^8*log(-b*e + (b

*x*e + a*e)*d/(d*x + c)) - 336*B*a^5*b^10*c^3*d^5*g^3*i*e^8*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + 168*B*a^6*

b^9*c^2*d^6*g^3*i*e^8*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 48*B*a^7*b^8*c*d^7*g^3*i*e^8*log(-b*e + (b*x*e +

a*e)*d/(d*x + c)) + 6*B*a^8*b^7*d^8*g^3*i*e^8*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 42*(b*x*e + a*e)*B*b^14

*c^8*d*g^3*i*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 336*(b*x*e + a*e)*B*a*b^13*c^7*d^2*g^3*i*e^

7*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 1176*(b*x*e + a*e)*B*a^2*b^12*c^6*d^3*g^3*i*e^7*log(-b*e +

(b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 2352*(b*x*e + a*e)*B*a^3*b^11*c^5*d^4*g^3*i*e^7*log(-b*e + (b*x*e + a*

e)*d/(d*x + c))/(d*x + c) - 2940*(b*x*e + a*e)*B*a^4*b^10*c^4*d^5*g^3*i*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x +

c))/(d*x + c) + 2352*(b*x*e + a*e)*B*a^5*b^9*c^3*d^6*g^3*i*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)

- 1176*(b*x*e + a*e)*B*a^6*b^8*c^2*d^7*g^3*i*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 336*(b*x*e

+ a*e)*B*a^7*b^7*c*d^8*g^3*i*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 42*(b*x*e + a*e)*B*a^8*b^6

*d^9*g^3*i*e^7*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 126*(b*x*e + a*e)^2*B*b^13*c^8*d^2*g^3*i*e^6*

log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 1008*(b*x*e + a*e)^2*B*a*b^12*c^7*d^3*g^3*i*e^6*log(-b*e +

(b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 3528*(b*x*e + a*e)^2*B*a^2*b^11*c^6*d^4*g^3*i*e^6*log(-b*e + (b*x*e

+ a*e)*d/(d*x + c))/(d*x + c)^2 - 7056*(b*x*e + a*e)^2*B*a^3*b^10*c^5*d^5*g^3*i*e^6*log(-b*e + (b*x*e + a*e)*d

/(d*x + c))/(d*x + c)^2 + 8820*(b*x*e + a*e)^2*B*a^4*b^9*c^4*d^6*g^3*i*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c

))/(d*x + c)^2 - 7056*(b*x*e + a*e)^2*B*a^5*b^8*c^3*d^7*g^3*i*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x +

c)^2 + 3528*(b*x*e + a*e)^2*B*a^6*b^7*c^2*d^8*g^3*i*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 1

008*(b*x*e + a*e)^2*B*a^7*b^6*c*d^9*g^3*i*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 126*(b*x*e +

a*e)^2*B*a^8*b^5*d^10*g^3*i*e^6*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 210*(b*x*e + a*e)^3*B*b^1

2*c^8*d^3*g^3*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 1680*(b*x*e + a*e)^3*B*a*b^11*c^7*d^4*

g^3*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 5880*(b*x*e + a*e)^3*B*a^2*b^10*c^6*d^5*g^3*i*e^

5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 11760*(b*x*e + a*e)^3*B*a^3*b^9*c^5*d^6*g^3*i*e^5*log(-b

*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 14700*(b*x*e + a*e)^3*B*a^4*b^8*c^4*d^7*g^3*i*e^5*log(-b*e + (b*

x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 11760*(b*x*e + a*e)^3*B*a^5*b^7*c^3*d^8*g^3*i*e^5*log(-b*e + (b*x*e + a*

e)*d/(d*x + c))/(d*x + c)^3 - 5880*(b*x*e + a*e)^3*B*a^6*b^6*c^2*d^9*g^3*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x

+ c))/(d*x + c)^3 + 1680*(b*x*e + a*e)^3*B*a^7*b^5*c*d^10*g^3*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*

x + c)^3 - 210*(b*x*e + a*e)^3*B*a^8*b^4*d^11*g^3*i*e^5*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 21

0*(b*x*e + a*e)^4*B*b^11*c^8*d^4*g^3*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 1680*(b*x*e + a

*e)^4*B*a*b^10*c^7*d^5*g^3*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 5880*(b*x*e + a*e)^4*B*a^

2*b^9*c^6*d^6*g^3*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 11760*(b*x*e + a*e)^4*B*a^3*b^8*c^

5*d^7*g^3*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 14700*(b*x*e + a*e)^4*B*a^4*b^7*c^4*d^8*g^

3*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 11760*(b*x*e + a*e)^4*B*a^5*b^6*c^3*d^9*g^3*i*e^4*

log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 + 5880*(b*x*e + a*e)^4*B*a^6*b^5*c^2*d^10*g^3*i*e^4*log(-b*e

+ (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^4 - 1680*(b*x*e + a*e)^4*B*a^7*b^4*c*d^11*g^3*i*e^4*log(-b*e + (b*x*e

+ a*e)*d/(d*x + c))/(d*x + c)^4 + 210*(b*x*e + a*e)^4*B*a^8*b^3*d^12*g^3*i*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x

+ c))/(d*x + c)^4 - 126*(b*x*e + a*e)^5*B*b^10*c^8*d^5*g^3*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x +

c)^5 + 1008*(b*x*e + a*e)^5*B*a*b^9*c^7*d^6*g^3*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 - 352

8*(b*x*e + a*e)^5*B*a^2*b^8*c^6*d^7*g^3*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 + 7056*(b*x*e

+ a*e)^5*B*a^3*b^7*c^5*d^8*g^3*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 - 8820*(b*x*e + a*e)^5*

B*a^4*b^6*c^4*d^9*g^3*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 + 7056*(b*x*e + a*e)^5*B*a^5*b^5

*c^3*d^10*g^3*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 - 3528*(b*x*e + a*e)^5*B*a^6*b^4*c^2*d^1

1*g^3*i*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 + 1008*(b*x*e + a*e)^5*B*a^7*b^3*c*d^12*g^3*i*e^

3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 - 126*(b*x*e + a*e)^5*B*a^8*b^2*d^13*g^3*i*e^3*log(-b*e +

(b*x*e + a*e)*d/(d*x + c))/(d*x + c)^5 + 42*(b*x*e + a*e)^6*B*b^9*c^8*d^6*g^3*i*e^2*log(-b*e + (b*x*e + a*e)*d

/(d*x + c))/(d*x + c)^6 - 336*(b*x*e + a*e)^6*B*a*b^8*c^7*d^7*g^3*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/

(d*x + c)^6 + 1176*(b*x*e + a*e)^6*B*a^2*b^7*c^6*d^8*g^3*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)

^6 - 2352*(b*x*e + a*e)^6*B*a^3*b^6*c^5*d^9*g^3*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^6 + 2940

*(b*x*e + a*e)^6*B*a^4*b^5*c^4*d^10*g^3*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^6 - 2352*(b*x*e

+ a*e)^6*B*a^5*b^4*c^3*d^11*g^3*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^6 + 1176*(b*x*e + a*e)^6

*B*a^6*b^3*c^2*d^12*g^3*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^6 - 336*(b*x*e + a*e)^6*B*a^7*b^

2*c*d^13*g^3*i*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^6 + 42*(b*x*e + a*e)^6*B*a^8*b*d^14*g^3*i*e

^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^6 - 6*(b*x*e + a*e)^7*B*b^8*c^8*d^7*g^3*i*e*log(-b*e + (b*x

*e + a*e)*d/(d*x + c))/(d*x + c)^7 + 48*(b*x*e + a*e)^7*B*a*b^7*c^7*d^8*g^3*i*e*log(-b*e + (b*x*e + a*e)*d/(d*

x + c))/(d*x + c)^7 - 168*(b*x*e + a*e)^7*B*a^2*b^6*c^6*d^9*g^3*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x

+ c)^7 + 336*(b*x*e + a*e)^7*B*a^3*b^5*c^5*d^10*g^3*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^7 - 4

20*(b*x*e + a*e)^7*B*a^4*b^4*c^4*d^11*g^3*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^7 + 336*(b*x*e +

a*e)^7*B*a^5*b^3*c^3*d^12*g^3*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^7 - 168*(b*x*e + a*e)^7*B*a

^6*b^2*c^2*d^13*g^3*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^7 + 48*(b*x*e + a*e)^7*B*a^7*b*c*d^14*

g^3*i*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^7 - 6*(b*x*e + a*e)^7*B*a^8*d^15*g^3*i*e*log(-b*e + (b

*x*e + a*e)*d/(d*x + c))/(d*x + c)^7 - 210*(b*x*e + a*e)^4*B*b^11*c^8*d^4*g^3*i*e^4*log((b*x*e + a*e)/(d*x + c

))/(d*x + c)^4 + 1680*(b*x*e + a*e)^4*B*a*b^10*c^7*d^5*g^3*i*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 58

80*(b*x*e + a*e)^4*B*a^2*b^9*c^6*d^6*g^3*i*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 11760*(b*x*e + a*e)^

4*B*a^3*b^8*c^5*d^7*g^3*i*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 14700*(b*x*e + a*e)^4*B*a^4*b^7*c^4*d

^8*g^3*i*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 11760*(b*x*e + a*e)^4*B*a^5*b^6*c^3*d^9*g^3*i*e^4*log(

(b*x*e + a*e)/(d*x + c))/(d*x + c)^4 - 5880*(b*x*e + a*e)^4*B*a^6*b^5*c^2*d^10*g^3*i*e^4*log((b*x*e + a*e)/(d*

x + c))/(d*x + c)^4 + 1680*(b*x*e + a*e)^4*B*a^7*b^4*c*d^11*g^3*i*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4

- 210*(b*x*e + a*e)^4*B*a^8*b^3*d^12*g^3*i*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^4 + 126*(b*x*e + a*e)^5

*B*b^10*c^8*d^5*g^3*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 - 1008*(b*x*e + a*e)^5*B*a*b^9*c^7*d^6*g^3*

i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 + 3528*(b*x*e + a*e)^5*B*a^2*b^8*c^6*d^7*g^3*i*e^3*log((b*x*e +

a*e)/(d*x + c))/(d*x + c)^5 - 7056*(b*x*e + a*e)^5*B*a^3*b^7*c^5*d^8*g^3*i*e^3*log((b*x*e + a*e)/(d*x + c))/(

d*x + c)^5 + 8820*(b*x*e + a*e)^5*B*a^4*b^6*c^4*d^9*g^3*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 - 7056*

(b*x*e + a*e)^5*B*a^5*b^5*c^3*d^10*g^3*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 + 3528*(b*x*e + a*e)^5*B

*a^6*b^4*c^2*d^11*g^3*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 - 1008*(b*x*e + a*e)^5*B*a^7*b^3*c*d^12*g

^3*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^5 + 126*(b*x*e + a*e)^5*B*a^8*b^2*d^13*g^3*i*e^3*log((b*x*e +

a*e)/(d*x + c))/(d*x + c)^5 - 42*(b*x*e + a*e)^6*B*b^9*c^8*d^6*g^3*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c

)^6 + 336*(b*x*e + a*e)^6*B*a*b^8*c^7*d^7*g^3*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^6 - 1176*(b*x*e + a

*e)^6*B*a^2*b^7*c^6*d^8*g^3*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^6 + 2352*(b*x*e + a*e)^6*B*a^3*b^6*c^

5*d^9*g^3*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^6 - 2940*(b*x*e + a*e)^6*B*a^4*b^5*c^4*d^10*g^3*i*e^2*l

og((b*x*e + a*e)/(d*x + c))/(d*x + c)^6 + 2352*(b*x*e + a*e)^6*B*a^5*b^4*c^3*d^11*g^3*i*e^2*log((b*x*e + a*e)/

(d*x + c))/(d*x + c)^6 - 1176*(b*x*e + a*e)^6*B*a^6*b^3*c^2*d^12*g^3*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x +

c)^6 + 336*(b*x*e + a*e)^6*B*a^7*b^2*c*d^13*g^3*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^6 - 42*(b*x*e +

a*e)^6*B*a^8*b*d^14*g^3*i*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^6 + 6*(b*x*e + a*e)^7*B*b^8*c^8*d^7*g^3*i

*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^7 - 48*(b*x*e + a*e)^7*B*a*b^7*c^7*d^8*g^3*i*e*log((b*x*e + a*e)/(d*

x + c))/(d*x + c)^7 + 168*(b*x*e + a*e)^7*B*a^2*b^6*c^6*d^9*g^3*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^7 -

336*(b*x*e + a*e)^7*B*a^3*b^5*c^5*d^10*g^3*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^7 + 420*(b*x*e + a*e)^7

*B*a^4*b^4*c^4*d^11*g^3*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^7 - 336*(b*x*e + a*e)^7*B*a^5*b^3*c^3*d^12*

g^3*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^7 + 168*(b*x*e + a*e)^7*B*a^6*b^2*c^2*d^13*g^3*i*e*log((b*x*e +

a*e)/(d*x + c))/(d*x + c)^7 - 48*(b*x*e + a*e)^7*B*a^7*b*c*d^14*g^3*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c

)^7 + 6*(b*x*e + a*e)^7*B*a^8*d^15*g^3*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^7 + 6*A*b^15*c^8*g^3*i*e^8 -

48*A*a*b^14*c^7*d*g^3*i*e^8 + 168*A*a^2*b^13*c^6*d^2*g^3*i*e^8 - 336*A*a^3*b^12*c^5*d^3*g^3*i*e^8 + 420*A*a^4

*b^11*c^4*d^4*g^3*i*e^8 - 336*A*a^5*b^10*c^3*d^5*g^3*i*e^8 + 168*A*a^6*b^9*c^2*d^6*g^3*i*e^8 - 48*A*a^7*b^8*c*

d^7*g^3*i*e^8 + 6*A*a^8*b^7*d^8*g^3*i*e^8 - 42*(b*x*e + a*e)*A*b^14*c^8*d*g^3*i*e^7/(d*x + c) + 6*(b*x*e + a*e

)*B*b^14*c^8*d*g^3*i*e^7/(d*x + c) + 336*(b*x*e + a*e)*A*a*b^13*c^7*d^2*g^3*i*e^7/(d*x + c) - 48*(b*x*e + a*e)

*B*a*b^13*c^7*d^2*g^3*i*e^7/(d*x + c) - 1176*(b*x*e + a*e)*A*a^2*b^12*c^6*d^3*g^3*i*e^7/(d*x + c) + 168*(b*x*e

+ a*e)*B*a^2*b^12*c^6*d^3*g^3*i*e^7/(d*x + c) + 2352*(b*x*e + a*e)*A*a^3*b^11*c^5*d^4*g^3*i*e^7/(d*x + c) - 3

36*(b*x*e + a*e)*B*a^3*b^11*c^5*d^4*g^3*i*e^7/(d*x + c) - 2940*(b*x*e + a*e)*A*a^4*b^10*c^4*d^5*g^3*i*e^7/(d*x

+ c) + 420*(b*x*e + a*e)*B*a^4*b^10*c^4*d^5*g^3*i*e^7/(d*x + c) + 2352*(b*x*e + a*e)*A*a^5*b^9*c^3*d^6*g^3*i*

e^7/(d*x + c) - 336*(b*x*e + a*e)*B*a^5*b^9*c^3*d^6*g^3*i*e^7/(d*x + c) - 1176*(b*x*e + a*e)*A*a^6*b^8*c^2*d^7

*g^3*i*e^7/(d*x + c) + 168*(b*x*e + a*e)*B*a^6*b^8*c^2*d^7*g^3*i*e^7/(d*x + c) + 336*(b*x*e + a*e)*A*a^7*b^7*c

*d^8*g^3*i*e^7/(d*x + c) - 48*(b*x*e + a*e)*B*a^7*b^7*c*d^8*g^3*i*e^7/(d*x + c) - 42*(b*x*e + a*e)*A*a^8*b^6*d

^9*g^3*i*e^7/(d*x + c) + 6*(b*x*e + a*e)*B*a^8*b^6*d^9*g^3*i*e^7/(d*x + c) + 126*(b*x*e + a*e)^2*A*b^13*c^8*d^

2*g^3*i*e^6/(d*x + c)^2 - 39*(b*x*e + a*e)^2*B*b^13*c^8*d^2*g^3*i*e^6/(d*x + c)^2 - 1008*(b*x*e + a*e)^2*A*a*b

^12*c^7*d^3*g^3*i*e^6/(d*x + c)^2 + 312*(b*x*e + a*e)^2*B*a*b^12*c^7*d^3*g^3*i*e^6/(d*x + c)^2 + 3528*(b*x*e +

a*e)^2*A*a^2*b^11*c^6*d^4*g^3*i*e^6/(d*x + c)^2 - 1092*(b*x*e + a*e)^2*B*a^2*b^11*c^6*d^4*g^3*i*e^6/(d*x + c)

^2 - 7056*(b*x*e + a*e)^2*A*a^3*b^10*c^5*d^5*g^3*i*e^6/(d*x + c)^2 + 2184*(b*x*e + a*e)^2*B*a^3*b^10*c^5*d^5*g

^3*i*e^6/(d*x + c)^2 + 8820*(b*x*e + a*e)^2*A*a^4*b^9*c^4*d^6*g^3*i*e^6/(d*x + c)^2 - 2730*(b*x*e + a*e)^2*B*a

^4*b^9*c^4*d^6*g^3*i*e^6/(d*x + c)^2 - 7056*(b*x*e + a*e)^2*A*a^5*b^8*c^3*d^7*g^3*i*e^6/(d*x + c)^2 + 2184*(b*

x*e + a*e)^2*B*a^5*b^8*c^3*d^7*g^3*i*e^6/(d*x + c)^2 + 3528*(b*x*e + a*e)^2*A*a^6*b^7*c^2*d^8*g^3*i*e^6/(d*x +

c)^2 - 1092*(b*x*e + a*e)^2*B*a^6*b^7*c^2*d^8*g^3*i*e^6/(d*x + c)^2 - 1008*(b*x*e + a*e)^2*A*a^7*b^6*c*d^9*g^

3*i*e^6/(d*x + c)^2 + 312*(b*x*e + a*e)^2*B*a^7*b^6*c*d^9*g^3*i*e^6/(d*x + c)^2 + 126*(b*x*e + a*e)^2*A*a^8*b^

5*d^10*g^3*i*e^6/(d*x + c)^2 - 39*(b*x*e + a*e)^2*B*a^8*b^5*d^10*g^3*i*e^6/(d*x + c)^2 - 210*(b*x*e + a*e)^3*A

*b^12*c^8*d^3*g^3*i*e^5/(d*x + c)^3 + 107*(b*x*e + a*e)^3*B*b^12*c^8*d^3*g^3*i*e^5/(d*x + c)^3 + 1680*(b*x*e +

a*e)^3*A*a*b^11*c^7*d^4*g^3*i*e^5/(d*x + c)^3 - 856*(b*x*e + a*e)^3*B*a*b^11*c^7*d^4*g^3*i*e^5/(d*x + c)^3 -

5880*(b*x*e + a*e)^3*A*a^2*b^10*c^6*d^5*g^3*i*e^5/(d*x + c)^3 + 2996*(b*x*e + a*e)^3*B*a^2*b^10*c^6*d^5*g^3*i*

e^5/(d*x + c)^3 + 11760*(b*x*e + a*e)^3*A*a^3*b^9*c^5*d^6*g^3*i*e^5/(d*x + c)^3 - 5992*(b*x*e + a*e)^3*B*a^3*b

^9*c^5*d^6*g^3*i*e^5/(d*x + c)^3 - 14700*(b*x*e + a*e)^3*A*a^4*b^8*c^4*d^7*g^3*i*e^5/(d*x + c)^3 + 7490*(b*x*e

+ a*e)^3*B*a^4*b^8*c^4*d^7*g^3*i*e^5/(d*x + c)^3 + 11760*(b*x*e + a*e)^3*A*a^5*b^7*c^3*d^8*g^3*i*e^5/(d*x + c

)^3 - 5992*(b*x*e + a*e)^3*B*a^5*b^7*c^3*d^8*g^3*i*e^5/(d*x + c)^3 - 5880*(b*x*e + a*e)^3*A*a^6*b^6*c^2*d^9*g^

3*i*e^5/(d*x + c)^3 + 2996*(b*x*e + a*e)^3*B*a^6*b^6*c^2*d^9*g^3*i*e^5/(d*x + c)^3 + 1680*(b*x*e + a*e)^3*A*a^

7*b^5*c*d^10*g^3*i*e^5/(d*x + c)^3 - 856*(b*x*e + a*e)^3*B*a^7*b^5*c*d^10*g^3*i*e^5/(d*x + c)^3 - 210*(b*x*e +

a*e)^3*A*a^8*b^4*d^11*g^3*i*e^5/(d*x + c)^3 + 107*(b*x*e + a*e)^3*B*a^8*b^4*d^11*g^3*i*e^5/(d*x + c)^3 - 107*

(b*x*e + a*e)^4*B*b^11*c^8*d^4*g^3*i*e^4/(d*x + c)^4 + 856*(b*x*e + a*e)^4*B*a*b^10*c^7*d^5*g^3*i*e^4/(d*x + c

)^4 - 2996*(b*x*e + a*e)^4*B*a^2*b^9*c^6*d^6*g^3*i*e^4/(d*x + c)^4 + 5992*(b*x*e + a*e)^4*B*a^3*b^8*c^5*d^7*g^

3*i*e^4/(d*x + c)^4 - 7490*(b*x*e + a*e)^4*B*a^4*b^7*c^4*d^8*g^3*i*e^4/(d*x + c)^4 + 5992*(b*x*e + a*e)^4*B*a^

5*b^6*c^3*d^9*g^3*i*e^4/(d*x + c)^4 - 2996*(b*x*e + a*e)^4*B*a^6*b^5*c^2*d^10*g^3*i*e^4/(d*x + c)^4 + 856*(b*x

*e + a*e)^4*B*a^7*b^4*c*d^11*g^3*i*e^4/(d*x + c)^4 - 107*(b*x*e + a*e)^4*B*a^8*b^3*d^12*g^3*i*e^4/(d*x + c)^4

+ 39*(b*x*e + a*e)^5*B*b^10*c^8*d^5*g^3*i*e^3/(d*x + c)^5 - 312*(b*x*e + a*e)^5*B*a*b^9*c^7*d^6*g^3*i*e^3/(d*x

+ c)^5 + 1092*(b*x*e + a*e)^5*B*a^2*b^8*c^6*d^7*g^3*i*e^3/(d*x + c)^5 - 2184*(b*x*e + a*e)^5*B*a^3*b^7*c^5*d^

8*g^3*i*e^3/(d*x + c)^5 + 2730*(b*x*e + a*e)^5*B*a^4*b^6*c^4*d^9*g^3*i*e^3/(d*x + c)^5 - 2184*(b*x*e + a*e)^5*

B*a^5*b^5*c^3*d^10*g^3*i*e^3/(d*x + c)^5 + 1092*(b*x*e + a*e)^5*B*a^6*b^4*c^2*d^11*g^3*i*e^3/(d*x + c)^5 - 312

*(b*x*e + a*e)^5*B*a^7*b^3*c*d^12*g^3*i*e^3/(d*x + c)^5 + 39*(b*x*e + a*e)^5*B*a^8*b^2*d^13*g^3*i*e^3/(d*x + c

)^5 - 6*(b*x*e + a*e)^6*B*b^9*c^8*d^6*g^3*i*e^2/(d*x + c)^6 + 48*(b*x*e + a*e)^6*B*a*b^8*c^7*d^7*g^3*i*e^2/(d*

x + c)^6 - 168*(b*x*e + a*e)^6*B*a^2*b^7*c^6*d^8*g^3*i*e^2/(d*x + c)^6 + 336*(b*x*e + a*e)^6*B*a^3*b^6*c^5*d^9

*g^3*i*e^2/(d*x + c)^6 - 420*(b*x*e + a*e)^6*B*a^4*b^5*c^4*d^10*g^3*i*e^2/(d*x + c)^6 + 336*(b*x*e + a*e)^6*B*

a^5*b^4*c^3*d^11*g^3*i*e^2/(d*x + c)^6 - 168*(b*x*e + a*e)^6*B*a^6*b^3*c^2*d^12*g^3*i*e^2/(d*x + c)^6 + 48*(b*

x*e + a*e)^6*B*a^7*b^2*c*d^13*g^3*i*e^2/(d*x + c)^6 - 6*(b*x*e + a*e)^6*B*a^8*b*d^14*g^3*i*e^2/(d*x + c)^6)*(b

*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))/(b^11*d^4*e^7 - 7*(b*x*e + a*e)*b^10*d^5

*e^6/(d*x + c) + 21*(b*x*e + a*e)^2*b^9*d^6*e^5/(d*x + c)^2 - 35*(b*x*e + a*e)^3*b^8*d^7*e^4/(d*x + c)^3 + 35*

(b*x*e + a*e)^4*b^7*d^8*e^3/(d*x + c)^4 - 21*(b*x*e + a*e)^5*b^6*d^9*e^2/(d*x + c)^5 + 7*(b*x*e + a*e)^6*b^5*d

^10*e/(d*x + c)^6 - (b*x*e + a*e)^7*b^4*d^11/(d*x + c)^7)

________________________________________________________________________________________

maple [B] time = 0.20, size = 11172, normalized size = 24.45 \[ \text {output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

maxima [B] time = 1.84, size = 2637, normalized size = 5.77 \[ \text {result too large to display} \]

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*A*b^3*d^3*g^3*i^3*x^7 + 1/2*A*b^3*c*d^2*g^3*i^3*x^6 + 1/2*A*a*b^2*d^3*g^3*i^3*x^6 + 3/5*A*b^3*c^2*d*g^3*i^

3*x^5 + 9/5*A*a*b^2*c*d^2*g^3*i^3*x^5 + 3/5*A*a^2*b*d^3*g^3*i^3*x^5 + 1/4*A*b^3*c^3*g^3*i^3*x^4 + 9/4*A*a*b^2*

c^2*d*g^3*i^3*x^4 + 9/4*A*a^2*b*c*d^2*g^3*i^3*x^4 + 1/4*A*a^3*d^3*g^3*i^3*x^4 + A*a*b^2*c^3*g^3*i^3*x^3 + 3*A*

a^2*b*c^2*d*g^3*i^3*x^3 + A*a^3*c*d^2*g^3*i^3*x^3 + 3/2*A*a^2*b*c^3*g^3*i^3*x^2 + 3/2*A*a^3*c^2*d*g^3*i^3*x^2

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

log(b*e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*B*a^

2*b*c^3*g^3*i^3 + 1/2*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c

)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*a*b^2*c^3*g^3*i^3 + 1/24*(6*x^4*log(b

*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)

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

e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*B*a^3*c^2*

d*g^3*i^3 + 3/2*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3

- ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*a^2*b*c^2*d*g^3*i^3 + 3/8*(6*x^4*log(b*e*x/

(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/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*i^3 + 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*b^3*c^2*d*g^3*i^3 + 1/2*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*l

og(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*i^3 + 3/8*(6*

x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/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*i^3 + 3/

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*c*d^2*g^3*i^3 + 1/120*(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*c*d^2*g^3*i^3 + 1/24*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x +

c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))

*B*a^3*d^3*g^3*i^3 + 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^2*b*d^3*g^3*i^3 + 1/120*(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*a*b^2*d^3*g^3*i^3 + 1/420*(60*x^7*log(b*e*x/(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^3 + A*a^3*c^3*g^3*i^3*x

________________________________________________________________________________________

mupad [B] time = 6.58, size = 4347, normalized size = 9.51 \[ \text {result too large to display} \]

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))