3.1.64 \(\int (a g+b g x)^3 (c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2 \, dx\) [64]

Optimal. Leaf size=711 \[ \frac {3 B^2 (b c-a d)^5 g^3 i^2 x}{20 b^2 d^3}+\frac {B^2 (b c-a d)^2 g^3 i^2 (a+b x)^4}{60 b^3}-\frac {3 B^2 (b c-a d)^4 g^3 i^2 (c+d x)^2}{40 b d^4}+\frac {B^2 (b c-a d)^3 g^3 i^2 (c+d x)^3}{60 d^4}-\frac {B (b c-a d)^3 g^3 i^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{90 b^3 d}-\frac {B (b c-a d)^2 g^3 i^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^3}-\frac {B (b c-a d) g^3 i^2 (a+b x)^4 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^2}+\frac {(b c-a d)^2 g^3 i^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{60 b^3}+\frac {(b c-a d) g^3 i^2 (a+b x)^4 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{15 b^2}+\frac {g^3 i^2 (a+b x)^4 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 b}+\frac {B (b c-a d)^4 g^3 i^2 (a+b x)^2 \left (3 A+B+3 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{180 b^3 d^2}-\frac {B (b c-a d)^5 g^3 i^2 (a+b x) \left (6 A+5 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{180 b^3 d^3}-\frac {B (b c-a d)^6 g^3 i^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 A+11 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{180 b^3 d^4}-\frac {B^2 (b c-a d)^6 g^3 i^2 \log (c+d x)}{20 b^3 d^4}-\frac {B^2 (b c-a d)^6 g^3 i^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{30 b^3 d^4} \]

[Out]

3/20*B^2*(-a*d+b*c)^5*g^3*i^2*x/b^2/d^3+1/60*B^2*(-a*d+b*c)^2*g^3*i^2*(b*x+a)^4/b^3-3/40*B^2*(-a*d+b*c)^4*g^3*
i^2*(d*x+c)^2/b/d^4+1/60*B^2*(-a*d+b*c)^3*g^3*i^2*(d*x+c)^3/d^4-1/90*B*(-a*d+b*c)^3*g^3*i^2*(b*x+a)^3*(A+B*ln(
e*(b*x+a)/(d*x+c)))/b^3/d-1/20*B*(-a*d+b*c)^2*g^3*i^2*(b*x+a)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^3-1/15*B*(-a*d+b
*c)*g^3*i^2*(b*x+a)^4*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^2+1/60*(-a*d+b*c)^2*g^3*i^2*(b*x+a)^4*(A+B*ln(e*(b
*x+a)/(d*x+c)))^2/b^3+1/15*(-a*d+b*c)*g^3*i^2*(b*x+a)^4*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b^2+1/6*g^3*i^2*
(b*x+a)^4*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b+1/180*B*(-a*d+b*c)^4*g^3*i^2*(b*x+a)^2*(3*A+B+3*B*ln(e*(b*
x+a)/(d*x+c)))/b^3/d^2-1/180*B*(-a*d+b*c)^5*g^3*i^2*(b*x+a)*(6*A+5*B+6*B*ln(e*(b*x+a)/(d*x+c)))/b^3/d^3-1/180*
B*(-a*d+b*c)^6*g^3*i^2*ln((-a*d+b*c)/b/(d*x+c))*(6*A+11*B+6*B*ln(e*(b*x+a)/(d*x+c)))/b^3/d^4-1/20*B^2*(-a*d+b*
c)^6*g^3*i^2*ln(d*x+c)/b^3/d^4-1/30*B^2*(-a*d+b*c)^6*g^3*i^2*polylog(2,d*(b*x+a)/b/(d*x+c))/b^3/d^4

________________________________________________________________________________________

Rubi [A]
time = 0.67, antiderivative size = 711, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2562, 2383, 2381, 2384, 2354, 2438, 2373, 45, 47, 37, 2382, 12, 79} \begin {gather*} -\frac {B^2 g^3 i^2 (b c-a d)^6 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{30 b^3 d^4}-\frac {B g^3 i^2 (b c-a d)^6 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+11 B\right )}{180 b^3 d^4}-\frac {B g^3 i^2 (a+b x) (b c-a d)^5 \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+5 B\right )}{180 b^3 d^3}+\frac {B g^3 i^2 (a+b x)^2 (b c-a d)^4 \left (3 B \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A+B\right )}{180 b^3 d^2}-\frac {B g^3 i^2 (a+b x)^3 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{90 b^3 d}+\frac {g^3 i^2 (a+b x)^4 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{60 b^3}-\frac {B g^3 i^2 (a+b x)^4 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{20 b^3}+\frac {g^3 i^2 (a+b x)^4 (c+d x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{15 b^2}-\frac {B g^3 i^2 (a+b x)^4 (c+d x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{15 b^2}+\frac {g^3 i^2 (a+b x)^4 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{6 b}-\frac {B^2 g^3 i^2 (b c-a d)^6 \log (c+d x)}{20 b^3 d^4}+\frac {B^2 g^3 i^2 (a+b x)^4 (b c-a d)^2}{60 b^3}+\frac {3 B^2 g^3 i^2 x (b c-a d)^5}{20 b^2 d^3}-\frac {3 B^2 g^3 i^2 (c+d x)^2 (b c-a d)^4}{40 b d^4}+\frac {B^2 g^3 i^2 (c+d x)^3 (b c-a d)^3}{60 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*B^2*(b*c - a*d)^5*g^3*i^2*x)/(20*b^2*d^3) + (B^2*(b*c - a*d)^2*g^3*i^2*(a + b*x)^4)/(60*b^3) - (3*B^2*(b*c
- a*d)^4*g^3*i^2*(c + d*x)^2)/(40*b*d^4) + (B^2*(b*c - a*d)^3*g^3*i^2*(c + d*x)^3)/(60*d^4) - (B*(b*c - a*d)^3
*g^3*i^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(90*b^3*d) - (B*(b*c - a*d)^2*g^3*i^2*(a + b*x)^4*(
A + B*Log[(e*(a + b*x))/(c + d*x)]))/(20*b^3) - (B*(b*c - a*d)*g^3*i^2*(a + b*x)^4*(c + d*x)*(A + B*Log[(e*(a
+ b*x))/(c + d*x)]))/(15*b^2) + ((b*c - a*d)^2*g^3*i^2*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(60
*b^3) + ((b*c - a*d)*g^3*i^2*(a + b*x)^4*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(15*b^2) + (g^3*i^2
*(a + b*x)^4*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(6*b) + (B*(b*c - a*d)^4*g^3*i^2*(a + b*x)^2*
(3*A + B + 3*B*Log[(e*(a + b*x))/(c + d*x)]))/(180*b^3*d^2) - (B*(b*c - a*d)^5*g^3*i^2*(a + b*x)*(6*A + 5*B +
6*B*Log[(e*(a + b*x))/(c + d*x)]))/(180*b^3*d^3) - (B*(b*c - a*d)^6*g^3*i^2*Log[(b*c - a*d)/(b*(c + d*x))]*(6*
A + 11*B + 6*B*Log[(e*(a + b*x))/(c + d*x)]))/(180*b^3*d^4) - (B^2*(b*c - a*d)^6*g^3*i^2*Log[c + d*x])/(20*b^3
*d^4) - (B^2*(b*c - a*d)^6*g^3*i^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(30*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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2373

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/(d*f*(m + 1))), x] - Dist[b*(n/(d*(m + 1))), Int[(f*x)^
m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m + r*(q + 1) + 1, 0] && NeQ[
m, -1]

Rule 2381

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp
[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + Dist[b*n*(p/(d*(q + 1))), Int[(
f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[m
+ q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]

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 2383

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp
[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + (Dist[(m + q + 2)/(d*(q + 1)),
Int[(f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p, x], x] + Dist[b*n*(p/(d*(q + 1))), Int[(f*x)^m*(d + e*x)^(
q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[p,
0] && LtQ[q, -1] && GtQ[m, 0]

Rule 2384

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(f*x
)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])/(e*(q + 1))), x] - Dist[f/(e*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x)^(
q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 (64 c+64 d x)^2 (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx &=\int \left (\frac {4096 (b c-a d)^2 (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2}+\frac {8192 d (b c-a d) (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}+\frac {4096 d^2 (a g+b g x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}\right ) \, dx\\ &=\frac {\left (4096 (b c-a d)^2\right ) \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{b^2}+\frac {\left (4096 d^2\right ) \int (a g+b g x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{b^2 g^2}+\frac {(8192 d (b c-a d)) \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{b^2 g}\\ &=\frac {1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac {8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac {2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}-\frac {\left (4096 B d^2\right ) \int \frac {(b c-a d) g^6 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{3 b^3 g^3}-\frac {(16384 B d (b c-a d)) \int \frac {(b c-a d) g^5 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{5 b^3 g^2}-\frac {\left (2048 B (b c-a d)^2\right ) \int \frac {(b c-a d) g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{b^3 g}\\ &=\frac {1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac {8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac {2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}-\frac {\left (4096 B d^2 (b c-a d) g^3\right ) \int \frac {(a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{3 b^3}-\frac {\left (16384 B d (b c-a d)^2 g^3\right ) \int \frac {(a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{5 b^3}-\frac {\left (2048 B (b c-a d)^3 g^3\right ) \int \frac {(a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x} \, dx}{b^3}\\ &=\frac {1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac {8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac {2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}-\frac {\left (4096 B d^2 (b c-a d) g^3\right ) \int \left (\frac {b (b c-a d)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^5}-\frac {b (b c-a d)^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac {b (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {b (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {b (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d}+\frac {(-b c+a d)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^5 (c+d x)}\right ) \, dx}{3 b^3}-\frac {\left (16384 B d (b c-a d)^2 g^3\right ) \int \left (-\frac {b (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4}+\frac {b (b c-a d)^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {b (b c-a d) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {b (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d}+\frac {(-b c+a d)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4 (c+d x)}\right ) \, dx}{5 b^3}-\frac {\left (2048 B (b c-a d)^3 g^3\right ) \int \left (\frac {b (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {b (b c-a d) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {b (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d}+\frac {(-b c+a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3 (c+d x)}\right ) \, dx}{b^3}\\ &=\frac {1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac {8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac {2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}-\frac {\left (4096 B d (b c-a d) g^3\right ) \int (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{3 b^2}+\frac {\left (4096 B (b c-a d)^2 g^3\right ) \int (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{3 b^2}-\frac {\left (16384 B (b c-a d)^2 g^3\right ) \int (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b^2}-\frac {\left (4096 B (b c-a d)^3 g^3\right ) \int (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{3 b^2 d}-\frac {\left (2048 B (b c-a d)^3 g^3\right ) \int (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d}+\frac {\left (16384 B (b c-a d)^3 g^3\right ) \int (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b^2 d}+\frac {\left (4096 B (b c-a d)^4 g^3\right ) \int (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{3 b^2 d^2}+\frac {\left (2048 B (b c-a d)^4 g^3\right ) \int (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d^2}-\frac {\left (16384 B (b c-a d)^4 g^3\right ) \int (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b^2 d^2}-\frac {\left (4096 B (b c-a d)^5 g^3\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{3 b^2 d^3}-\frac {\left (2048 B (b c-a d)^5 g^3\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d^3}+\frac {\left (16384 B (b c-a d)^5 g^3\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b^2 d^3}+\frac {\left (4096 B (b c-a d)^6 g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{3 b^3 d^3}+\frac {\left (2048 B (b c-a d)^6 g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^3 d^3}-\frac {\left (16384 B (b c-a d)^6 g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{5 b^3 d^3}\\ &=-\frac {2048 A B (b c-a d)^5 g^3 x}{15 b^2 d^3}+\frac {1024 B (b c-a d)^4 g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 d^2}-\frac {2048 B (b c-a d)^3 g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b^3 d}-\frac {7168 B (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3}-\frac {4096 B d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3}+\frac {1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac {8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac {2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}+\frac {2048 B (b c-a d)^6 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 d^4}+\frac {\left (4096 B^2 d (b c-a d) g^3\right ) \int \frac {(b c-a d) (a+b x)^4}{c+d x} \, dx}{15 b^3}-\frac {\left (1024 B^2 (b c-a d)^2 g^3\right ) \int \frac {(b c-a d) (a+b x)^3}{c+d x} \, dx}{3 b^3}+\frac {\left (4096 B^2 (b c-a d)^2 g^3\right ) \int \frac {(b c-a d) (a+b x)^3}{c+d x} \, dx}{5 b^3}+\frac {\left (4096 B^2 (b c-a d)^3 g^3\right ) \int \frac {(b c-a d) (a+b x)^2}{c+d x} \, dx}{9 b^3 d}+\frac {\left (2048 B^2 (b c-a d)^3 g^3\right ) \int \frac {(b c-a d) (a+b x)^2}{c+d x} \, dx}{3 b^3 d}-\frac {\left (16384 B^2 (b c-a d)^3 g^3\right ) \int \frac {(b c-a d) (a+b x)^2}{c+d x} \, dx}{15 b^3 d}-\frac {\left (2048 B^2 (b c-a d)^4 g^3\right ) \int \frac {(b c-a d) (a+b x)}{c+d x} \, dx}{3 b^3 d^2}-\frac {\left (1024 B^2 (b c-a d)^4 g^3\right ) \int \frac {(b c-a d) (a+b x)}{c+d x} \, dx}{b^3 d^2}+\frac {\left (8192 B^2 (b c-a d)^4 g^3\right ) \int \frac {(b c-a d) (a+b x)}{c+d x} \, dx}{5 b^3 d^2}-\frac {\left (4096 B^2 (b c-a d)^5 g^3\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{3 b^2 d^3}-\frac {\left (2048 B^2 (b c-a d)^5 g^3\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{b^2 d^3}+\frac {\left (16384 B^2 (b c-a d)^5 g^3\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{5 b^2 d^3}-\frac {\left (4096 B^2 (b c-a d)^6 g^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{3 b^3 d^4}-\frac {\left (2048 B^2 (b c-a d)^6 g^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^3 d^4}+\frac {\left (16384 B^2 (b c-a d)^6 g^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{5 b^3 d^4}\\ &=-\frac {2048 A B (b c-a d)^5 g^3 x}{15 b^2 d^3}-\frac {2048 B^2 (b c-a d)^5 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{15 b^3 d^3}+\frac {1024 B (b c-a d)^4 g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 d^2}-\frac {2048 B (b c-a d)^3 g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b^3 d}-\frac {7168 B (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3}-\frac {4096 B d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3}+\frac {1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac {8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac {2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}+\frac {2048 B (b c-a d)^6 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 d^4}+\frac {\left (4096 B^2 d (b c-a d)^2 g^3\right ) \int \frac {(a+b x)^4}{c+d x} \, dx}{15 b^3}-\frac {\left (1024 B^2 (b c-a d)^3 g^3\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{3 b^3}+\frac {\left (4096 B^2 (b c-a d)^3 g^3\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{5 b^3}+\frac {\left (4096 B^2 (b c-a d)^4 g^3\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{9 b^3 d}+\frac {\left (2048 B^2 (b c-a d)^4 g^3\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{3 b^3 d}-\frac {\left (16384 B^2 (b c-a d)^4 g^3\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{15 b^3 d}-\frac {\left (2048 B^2 (b c-a d)^5 g^3\right ) \int \frac {a+b x}{c+d x} \, dx}{3 b^3 d^2}-\frac {\left (1024 B^2 (b c-a d)^5 g^3\right ) \int \frac {a+b x}{c+d x} \, dx}{b^3 d^2}+\frac {\left (8192 B^2 (b c-a d)^5 g^3\right ) \int \frac {a+b x}{c+d x} \, dx}{5 b^3 d^2}+\frac {\left (4096 B^2 (b c-a d)^6 g^3\right ) \int \frac {1}{c+d x} \, dx}{3 b^3 d^3}+\frac {\left (2048 B^2 (b c-a d)^6 g^3\right ) \int \frac {1}{c+d x} \, dx}{b^3 d^3}-\frac {\left (16384 B^2 (b c-a d)^6 g^3\right ) \int \frac {1}{c+d x} \, dx}{5 b^3 d^3}-\frac {\left (4096 B^2 (b c-a d)^6 g^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{3 b^3 d^4 e}-\frac {\left (2048 B^2 (b c-a d)^6 g^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^3 d^4 e}+\frac {\left (16384 B^2 (b c-a d)^6 g^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{5 b^3 d^4 e}\\ &=-\frac {2048 A B (b c-a d)^5 g^3 x}{15 b^2 d^3}-\frac {2048 B^2 (b c-a d)^5 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{15 b^3 d^3}+\frac {1024 B (b c-a d)^4 g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 d^2}-\frac {2048 B (b c-a d)^3 g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b^3 d}-\frac {7168 B (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3}-\frac {4096 B d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3}+\frac {1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac {8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac {2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}+\frac {2048 B^2 (b c-a d)^6 g^3 \log (c+d x)}{15 b^3 d^4}+\frac {2048 B (b c-a d)^6 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 d^4}+\frac {\left (4096 B^2 d (b c-a d)^2 g^3\right ) \int \left (-\frac {b (b c-a d)^3}{d^4}+\frac {b (b c-a d)^2 (a+b x)}{d^3}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (a+b x)^3}{d}+\frac {(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{15 b^3}-\frac {\left (1024 B^2 (b c-a d)^3 g^3\right ) \int \left (\frac {b (b c-a d)^2}{d^3}-\frac {b (b c-a d) (a+b x)}{d^2}+\frac {b (a+b x)^2}{d}+\frac {(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{3 b^3}+\frac {\left (4096 B^2 (b c-a d)^3 g^3\right ) \int \left (\frac {b (b c-a d)^2}{d^3}-\frac {b (b c-a d) (a+b x)}{d^2}+\frac {b (a+b x)^2}{d}+\frac {(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{5 b^3}+\frac {\left (4096 B^2 (b c-a d)^4 g^3\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{9 b^3 d}+\frac {\left (2048 B^2 (b c-a d)^4 g^3\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b^3 d}-\frac {\left (16384 B^2 (b c-a d)^4 g^3\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{15 b^3 d}-\frac {\left (2048 B^2 (b c-a d)^5 g^3\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{3 b^3 d^2}-\frac {\left (1024 B^2 (b c-a d)^5 g^3\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{b^3 d^2}+\frac {\left (8192 B^2 (b c-a d)^5 g^3\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{5 b^3 d^2}-\frac {\left (4096 B^2 (b c-a d)^6 g^3\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{3 b^3 d^4 e}-\frac {\left (2048 B^2 (b c-a d)^6 g^3\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{b^3 d^4 e}+\frac {\left (16384 B^2 (b c-a d)^6 g^3\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{5 b^3 d^4 e}\\ &=-\frac {2048 A B (b c-a d)^5 g^3 x}{15 b^2 d^3}+\frac {4096 B^2 (b c-a d)^5 g^3 x}{45 b^2 d^3}-\frac {3584 B^2 (b c-a d)^4 g^3 (a+b x)^2}{45 b^3 d^2}+\frac {1024 B^2 (b c-a d)^3 g^3 (a+b x)^3}{15 b^3 d}+\frac {1024 B^2 (b c-a d)^2 g^3 (a+b x)^4}{15 b^3}-\frac {2048 B^2 (b c-a d)^5 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{15 b^3 d^3}+\frac {1024 B (b c-a d)^4 g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 d^2}-\frac {2048 B (b c-a d)^3 g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b^3 d}-\frac {7168 B (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3}-\frac {4096 B d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3}+\frac {1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac {8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac {2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}+\frac {2048 B^2 (b c-a d)^6 g^3 \log (c+d x)}{45 b^3 d^4}+\frac {2048 B (b c-a d)^6 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 d^4}-\frac {\left (4096 B^2 (b c-a d)^6 g^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 b^2 d^4}-\frac {\left (2048 B^2 (b c-a d)^6 g^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 d^4}+\frac {\left (16384 B^2 (b c-a d)^6 g^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{5 b^2 d^4}+\frac {\left (4096 B^2 (b c-a d)^6 g^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b^3 d^3}+\frac {\left (2048 B^2 (b c-a d)^6 g^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 d^3}-\frac {\left (16384 B^2 (b c-a d)^6 g^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{5 b^3 d^3}\\ &=-\frac {2048 A B (b c-a d)^5 g^3 x}{15 b^2 d^3}+\frac {4096 B^2 (b c-a d)^5 g^3 x}{45 b^2 d^3}-\frac {3584 B^2 (b c-a d)^4 g^3 (a+b x)^2}{45 b^3 d^2}+\frac {1024 B^2 (b c-a d)^3 g^3 (a+b x)^3}{15 b^3 d}+\frac {1024 B^2 (b c-a d)^2 g^3 (a+b x)^4}{15 b^3}-\frac {2048 B^2 (b c-a d)^5 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{15 b^3 d^3}+\frac {1024 B (b c-a d)^4 g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 d^2}-\frac {2048 B (b c-a d)^3 g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b^3 d}-\frac {7168 B (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3}-\frac {4096 B d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3}+\frac {1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac {8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac {2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}+\frac {2048 B^2 (b c-a d)^6 g^3 \log (c+d x)}{45 b^3 d^4}-\frac {2048 B^2 (b c-a d)^6 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{15 b^3 d^4}+\frac {2048 B (b c-a d)^6 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 d^4}+\frac {\left (4096 B^2 (b c-a d)^6 g^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b^3 d^4}+\frac {\left (2048 B^2 (b c-a d)^6 g^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 d^4}-\frac {\left (16384 B^2 (b c-a d)^6 g^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{5 b^3 d^4}+\frac {\left (4096 B^2 (b c-a d)^6 g^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b^3 d^3}+\frac {\left (2048 B^2 (b c-a d)^6 g^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 d^3}-\frac {\left (16384 B^2 (b c-a d)^6 g^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{5 b^3 d^3}\\ &=-\frac {2048 A B (b c-a d)^5 g^3 x}{15 b^2 d^3}+\frac {4096 B^2 (b c-a d)^5 g^3 x}{45 b^2 d^3}-\frac {3584 B^2 (b c-a d)^4 g^3 (a+b x)^2}{45 b^3 d^2}+\frac {1024 B^2 (b c-a d)^3 g^3 (a+b x)^3}{15 b^3 d}+\frac {1024 B^2 (b c-a d)^2 g^3 (a+b x)^4}{15 b^3}-\frac {2048 B^2 (b c-a d)^5 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{15 b^3 d^3}+\frac {1024 B (b c-a d)^4 g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 d^2}-\frac {2048 B (b c-a d)^3 g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b^3 d}-\frac {7168 B (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3}-\frac {4096 B d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3}+\frac {1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac {8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac {2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}+\frac {2048 B^2 (b c-a d)^6 g^3 \log (c+d x)}{45 b^3 d^4}-\frac {2048 B^2 (b c-a d)^6 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{15 b^3 d^4}+\frac {2048 B (b c-a d)^6 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 d^4}+\frac {1024 B^2 (b c-a d)^6 g^3 \log ^2(c+d x)}{15 b^3 d^4}+\frac {\left (4096 B^2 (b c-a d)^6 g^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b^3 d^4}+\frac {\left (2048 B^2 (b c-a d)^6 g^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^3 d^4}-\frac {\left (16384 B^2 (b c-a d)^6 g^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{5 b^3 d^4}\\ &=-\frac {2048 A B (b c-a d)^5 g^3 x}{15 b^2 d^3}+\frac {4096 B^2 (b c-a d)^5 g^3 x}{45 b^2 d^3}-\frac {3584 B^2 (b c-a d)^4 g^3 (a+b x)^2}{45 b^3 d^2}+\frac {1024 B^2 (b c-a d)^3 g^3 (a+b x)^3}{15 b^3 d}+\frac {1024 B^2 (b c-a d)^2 g^3 (a+b x)^4}{15 b^3}-\frac {2048 B^2 (b c-a d)^5 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{15 b^3 d^3}+\frac {1024 B (b c-a d)^4 g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 d^2}-\frac {2048 B (b c-a d)^3 g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b^3 d}-\frac {7168 B (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3}-\frac {4096 B d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3}+\frac {1024 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3}+\frac {8192 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3}+\frac {2048 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3}+\frac {2048 B^2 (b c-a d)^6 g^3 \log (c+d x)}{45 b^3 d^4}-\frac {2048 B^2 (b c-a d)^6 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{15 b^3 d^4}+\frac {2048 B (b c-a d)^6 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 d^4}+\frac {1024 B^2 (b c-a d)^6 g^3 \log ^2(c+d x)}{15 b^3 d^4}-\frac {2048 B^2 (b c-a d)^6 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{15 b^3 d^4}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1559\) vs. \(2(711)=1422\).
time = 0.91, size = 1559, normalized size = 2.19 \begin {gather*} \frac {g^3 i^2 \left (15 (b c-a d)^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+24 d (b c-a d) (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+10 d^2 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2-\frac {5 B (b c-a d)^3 \left (6 A b d (b c-a d)^2 x+6 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 d^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 B (b c-a d)^3 \log (c+d x)-6 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+B (b c-a d) \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+3 B (b c-a d)^2 (b d x+(-b c+a d) \log (c+d x))+3 B (b c-a d)^3 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^4}+\frac {2 B (b c-a d)^2 \left (24 A b d (b c-a d)^3 x+24 B d (b c-a d)^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-12 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+8 d^3 (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 d^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-24 B (b c-a d)^4 \log (c+d x)-24 (b c-a d)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+4 B (b c-a d)^2 \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+B (b c-a d) \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)^3 (b d x+(-b c+a d) \log (c+d x))+12 B (b c-a d)^4 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^4}-\frac {B (b c-a d) \left (24 b^2 B c d (b c-a d)^3 x+120 A b d (b c-a d)^4 x+130 b B d (b c-a d)^4 x+24 a b B d^2 (-b c+a d)^3 x-12 b B c d^2 (b c-a d)^2 (a+b x)^2+12 a B d^3 (b c-a d)^2 (a+b x)^2+35 B d^2 (-b c+a d)^3 (a+b x)^2+8 b B c d^3 (b c-a d) (a+b x)^3+10 B d^3 (b c-a d)^2 (a+b x)^3+8 a B d^4 (-b c+a d) (a+b x)^3-6 b B c d^4 (a+b x)^4+6 a B d^5 (a+b x)^4+120 B d (b c-a d)^4 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+60 d^2 (-b c+a d)^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+40 d^3 (b c-a d)^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+30 d^4 (-b c+a d) (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+24 d^5 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-24 b B c (b c-a d)^4 \log (c+d x)+24 a B d (b c-a d)^4 \log (c+d x)-250 B (b c-a d)^5 \log (c+d x)-120 (b c-a d)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+60 B (b c-a d)^5 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{6 d^4}\right )}{60 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(g^3*i^2*(15*(b*c - a*d)^2*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 24*d*(b*c - a*d)*(a + b*x)^5*(
A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 10*d^2*(a + b*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - (5*B*(b*c
- a*d)^3*(6*A*b*d*(b*c - a*d)^2*x + 6*B*d*(b*c - a*d)^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + 3*d^2*(-(b*c)
+ a*d)*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2*d^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x
)]) - 6*B*(b*c - a*d)^3*Log[c + d*x] - 6*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] + B*(
b*c - a*d)*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]) + 3*B*(b*c - a*d)^2*(b*d*x +
(-(b*c) + a*d)*Log[c + d*x]) + 3*B*(b*c - a*d)^3*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c
+ d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/d^4 + (2*B*(b*c - a*d)^2*(24*A*b*d*(b*c - a*d)^3*x + 24*B*
d*(b*c - a*d)^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] - 12*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(a + b
*x))/(c + d*x)]) + 8*d^3*(b*c - a*d)*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 6*d^4*(a + b*x)^4*(A +
B*Log[(e*(a + b*x))/(c + d*x)]) - 24*B*(b*c - a*d)^4*Log[c + d*x] - 24*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x))
/(c + d*x)])*Log[c + d*x] + 4*B*(b*c - a*d)^2*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c +
d*x]) + B*(b*c - a*d)*(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)^3*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 12*B*(b*c - a*d)^4*((2*Log
[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/d^4 -
(B*(b*c - a*d)*(24*b^2*B*c*d*(b*c - a*d)^3*x + 120*A*b*d*(b*c - a*d)^4*x + 130*b*B*d*(b*c - a*d)^4*x + 24*a*b
*B*d^2*(-(b*c) + a*d)^3*x - 12*b*B*c*d^2*(b*c - a*d)^2*(a + b*x)^2 + 12*a*B*d^3*(b*c - a*d)^2*(a + b*x)^2 + 35
*B*d^2*(-(b*c) + a*d)^3*(a + b*x)^2 + 8*b*B*c*d^3*(b*c - a*d)*(a + b*x)^3 + 10*B*d^3*(b*c - a*d)^2*(a + b*x)^3
+ 8*a*B*d^4*(-(b*c) + a*d)*(a + b*x)^3 - 6*b*B*c*d^4*(a + b*x)^4 + 6*a*B*d^5*(a + b*x)^4 + 120*B*d*(b*c - a*d
)^4*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + 60*d^2*(-(b*c) + a*d)^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c +
d*x)]) + 40*d^3*(b*c - a*d)^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 30*d^4*(-(b*c) + a*d)*(a + b
*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 24*d^5*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 24*b*B*
c*(b*c - a*d)^4*Log[c + d*x] + 24*a*B*d*(b*c - a*d)^4*Log[c + d*x] - 250*B*(b*c - a*d)^5*Log[c + d*x] - 120*(b
*c - a*d)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] + 60*B*(b*c - a*d)^5*((2*Log[(d*(a + b*x))/(-(b*
c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/(6*d^4)))/(60*b^3)

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Maple [F]
time = 0.55, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right )^{3} \left (d i x +c i \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 4015 vs. \(2 (644) = 1288\).
time = 0.44, size = 4015, normalized size = 5.65 \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)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="maxima")

[Out]

-1/6*A^2*b^3*d^2*g^3*x^6 - 2/5*A^2*b^3*c*d*g^3*x^5 - 3/5*A^2*a*b^2*d^2*g^3*x^5 - 1/4*A^2*b^3*c^2*g^3*x^4 - 3/2
*A^2*a*b^2*c*d*g^3*x^4 - 3/4*A^2*a^2*b*d^2*g^3*x^4 - A^2*a*b^2*c^2*g^3*x^3 - 2*A^2*a^2*b*c*d*g^3*x^3 - 1/3*A^2
*a^3*d^2*g^3*x^3 - 3/2*A^2*a^2*b*c^2*g^3*x^2 - A^2*a^3*c*d*g^3*x^2 - 2*(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)*A*B*a^3*c^2*g^3 - 3*(x^2*log(b*x*e/(d*x + c) + a*e/(d*x + c)) - a^2*lo
g(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*A*B*a^2*b*c^2*g^3 - (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))*A*B*a*b^2*c^2*g^3 - 1/12*(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))*A*B*b^3*c^2*g^3 - 2*(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))*A*B*a^3*c*d*g^3 - 2*(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)
)*A*B*a^2*b*c*d*g^3 - 1/2*(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
))*A*B*a*b^2*c*d*g^3 - 1/15*(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*lo
g(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))*A*B*b^3*c*d*g^3 - 1/3*(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))*A*B*a^3*d^2*g^3 - 1/4*(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))*A*B*a^2*b*d^2*g^3 - 1/10*(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))*A*B*a*b^2*d^2*g^3 - 1/180*(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))*A*B*b^3*d^2*g^3 - A^2*a^3*c^2*g^3*x - 1/180*(8*b^5*c^6*g^3 - 42*a*b^4*c^5*d*g^3 + 87*a^2*b
^3*c^4*d^2*g^3 - 86*a^3*b^2*c^3*d^3*g^3 - 33*a^4*b*c^2*d^4*g^3 + 6*a^5*c*d^5*g^3)*B^2*log(d*x + c)/(b^2*d^4) -
1/30*(b^6*c^6*g^3 - 6*a*b^5*c^5*d*g^3 + 15*a^2*b^4*c^4*d^2*g^3 - 20*a^3*b^3*c^3*d^3*g^3 + 15*a^4*b^2*c^2*d^4*
g^3 - 6*a^5*b*c*d^5*g^3 + a^6*d^6*g^3)*(log(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)
/(b*c - a*d)))*B^2/(b^3*d^4) - 1/360*(60*B^2*b^6*d^6*g^3*x^6 + 120*(b^6*c*d^5*g^3 + 2*a*b^5*d^6*g^3)*B^2*x^5 +
6*(9*b^6*c^2*d^4*g^3 + 82*a*b^5*c*d^5*g^3 + 59*a^2*b^4*d^6*g^3)*B^2*x^4 + 2*(b^6*c^3*d^3*g^3 + 105*a*b^5*c^2*
d^4*g^3 + 387*a^2*b^4*c*d^5*g^3 + 107*a^3*b^3*d^6*g^3)*B^2*x^3 - (b^6*c^4*d^2*g^3 - 10*a*b^5*c^3*d^3*g^3 - 300
*a^2*b^4*c^2*d^4*g^3 - 574*a^3*b^3*c*d^5*g^3 - 17*a^4*b^2*d^6*g^3)*B^2*x^2 - 2*(2*b^6*c^5*d*g^3 - 9*a*b^5*c^4*
d^2*g^3 + 13*a^2*b^4*c^3*d^3*g^3 - 113*a^3*b^3*c^2*d^4*g^3 - 87*a^4*b^2*c*d^5*g^3 + 14*a^5*b*d^6*g^3)*B^2*x +
6*(10*B^2*b^6*d^6*g^3*x^6 + 60*B^2*a^3*b^3*c^2*d^4*g^3*x + 12*(2*b^6*c*d^5*g^3 + 3*a*b^5*d^6*g^3)*B^2*x^5 + 15
*(b^6*c^2*d^4*g^3 + 6*a*b^5*c*d^5*g^3 + 3*a^2*b^4*d^6*g^3)*B^2*x^4 + 20*(3*a*b^5*c^2*d^4*g^3 + 6*a^2*b^4*c*d^5
*g^3 + a^3*b^3*d^6*g^3)*B^2*x^3 + 30*(3*a^2*b^4*c^2*d^4*g^3 + 2*a^3*b^3*c*d^5*g^3)*B^2*x^2 + (15*a^4*b^2*c^2*d
^4*g^3 - 6*a^5*b*c*d^5*g^3 + a^6*d^6*g^3)*B^2)*log(b*x + a)^2 + 6*(10*B^2*b^6*d^6*g^3*x^6 + 60*B^2*a^3*b^3*c^2
*d^4*g^3*x + 12*(2*b^6*c*d^5*g^3 + 3*a*b^5*d^6*g^3)*B^2*x^5 + 15*(b^6*c^2*d^4*g^3 + 6*a*b^5*c*d^5*g^3 + 3*a^2*
b^4*d^6*g^3)*B^2*x^4 + 20*(3*a*b^5*c^2*d^4*g^3 + 6*a^2*b^4*c*d^5*g^3 + a^3*b^3*d^6*g^3)*B^2*x^3 + 30*(3*a^2*b^
4*c^2*d^4*g^3 + 2*a^3*b^3*c*d^5*g^3)*B^2*x^2 - (b^6*c^6*g^3 - 6*a*b^5*c^5*d*g^3 + 15*a^2*b^4*c^4*d^2*g^3 - 20*
a^3*b^3*c^3*d^3*g^3)*B^2)*log(d*x + c)^2 + 2*(60*B^2*b^6*d^6*g^3*x^6 + 12*(11*b^6*c*d^5*g^3 + 19*a*b^5*d^6*g^3
)*B^2*x^5 + 3*(23*b^6*c^2*d^4*g^3 + 174*a*b^5*c*d^5*g^3 + 103*a^2*b^4*d^6*g^3)*B^2*x^4 - 2*(b^6*c^3*d^3*g^3 -
141*a*b^5*c^2*d^4*g^3 - 381*a^2*b^4*c*d^5*g^3 - 79*a^3*b^3*d^6*g^3)*B^2*x^3 + 3*(b^6*c^4*d^2*g^3 - 6*a*b^5*c^3
*d^3*g^3 + 150*a^2*b^4*c^2*d^4*g^3 + 154*a^3*b^3*c*d^5*g^3 + a^4*b^2*d^6*g^3)*B^2*x^2 - 6*(b^6*c^5*d*g^3 - 6*a
*b^5*c^4*d^2*g^3 + 15*a^2*b^4*c^3*d^3*g^3 - 65*a^3*b^3*c^2*d^4*g^3 - 6*a^4*b^2*c*d^5*g^3 + a^5*b*d^6*g^3)*B^2*
x - (6*a*b^5*c^5*d*g^3 - 33*a^2*b^4*c^4*d^2*g^3...

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="fricas")

[Out]

-1/60*(10*B^2*b^3*d^2*g^3*x^6 + 60*B^2*a^3*c^2*g^3*x + 12*(2*B^2*b^3*c*d + 3*B^2*a*b^2*d^2)*g^3*x^5 + 15*(B^2*
b^3*c^2 + 6*B^2*a*b^2*c*d + 3*B^2*a^2*b*d^2)*g^3*x^4 + 20*(3*B^2*a*b^2*c^2 + 6*B^2*a^2*b*c*d + B^2*a^3*d^2)*g^
3*x^3 + 30*(3*B^2*a^2*b*c^2 + 2*B^2*a^3*c*d)*g^3*x^2)*log((b*x + a)*e/(d*x + c))^2 + integral(-1/30*(30*A^2*b^
4*d^3*g^3*x^7 + 30*A^2*a^4*c^3*g^3 + 30*(3*A^2*b^4*c*d^2 + 4*A^2*a*b^3*d^3)*g^3*x^6 + 90*(A^2*b^4*c^2*d + 4*A^
2*a*b^3*c*d^2 + 2*A^2*a^2*b^2*d^3)*g^3*x^5 + 30*(A^2*b^4*c^3 + 12*A^2*a*b^3*c^2*d + 18*A^2*a^2*b^2*c*d^2 + 4*A
^2*a^3*b*d^3)*g^3*x^4 + 30*(4*A^2*a*b^3*c^3 + 18*A^2*a^2*b^2*c^2*d + 12*A^2*a^3*b*c*d^2 + A^2*a^4*d^3)*g^3*x^3
+ 90*(2*A^2*a^2*b^2*c^3 + 4*A^2*a^3*b*c^2*d + A^2*a^4*c*d^2)*g^3*x^2 + 30*(4*A^2*a^3*b*c^3 + 3*A^2*a^4*c^2*d)
*g^3*x + (60*A*B*b^4*d^3*g^3*x^7 + 60*A*B*a^4*c^3*g^3 + 10*((18*A*B - B^2)*b^4*c*d^2 + (24*A*B + B^2)*a*b^3*d^
3)*g^3*x^6 + 12*((15*A*B - 2*B^2)*b^4*c^2*d + (60*A*B - B^2)*a*b^3*c*d^2 + 3*(10*A*B + B^2)*a^2*b^2*d^3)*g^3*x
^5 + 15*((4*A*B - B^2)*b^4*c^3 + (48*A*B - 5*B^2)*a*b^3*c^2*d + 3*(24*A*B + B^2)*a^2*b^2*c*d^2 + (16*A*B + 3*B
^2)*a^3*b*d^3)*g^3*x^4 + 20*(3*(4*A*B - B^2)*a*b^3*c^3 + 3*(18*A*B - B^2)*a^2*b^2*c^2*d + (36*A*B + 5*B^2)*a^3
*b*c*d^2 + (3*A*B + B^2)*a^4*d^3)*g^3*x^3 + 30*(3*(4*A*B - B^2)*a^2*b^2*c^3 + (24*A*B + B^2)*a^3*b*c^2*d + 2*(
3*A*B + B^2)*a^4*c*d^2)*g^3*x^2 + 60*((4*A*B - B^2)*a^3*b*c^3 + (3*A*B + B^2)*a^4*c^2*d)*g^3*x)*log((b*x + a)*
e/(d*x + c)))/(b*d*x^2 + a*c + (b*c + a*d)*x), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**2*(A+B*ln(e*(b*x+a)/(d*x+c)))**2,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a\,g+b\,g\,x\right )}^3\,{\left (c\,i+d\,i\,x\right )}^2\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2 \,d x \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)^2*(A + B*log((e*(a + b*x))/(c + d*x)))^2,x)

[Out]

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

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