3.1.66 \(\int (a g+b g x) (c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2 \, dx\) [66]

Optimal. Leaf size=589 \[ \frac {B^2 (b c-a d)^3 g i^2 x}{12 b^2 d}+\frac {B^2 (b c-a d)^2 g i^2 (c+d x)^2}{12 b d^2}-\frac {B^2 (b c-a d)^4 g i^2 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^3 d^2}-\frac {B (b c-a d)^3 g i^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b^3 d}-\frac {B (b c-a d)^2 g i^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b^3}+\frac {B (b c-a d)^2 g i^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b d^2}-\frac {B (b c-a d) g i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^2}+\frac {(b c-a d)^2 g i^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{12 b^3}+\frac {(b c-a d) g i^2 (a+b x)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 b^2}+\frac {g i^2 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 b}-\frac {B (b c-a d)^4 g i^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 \log (c+d x)}{4 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^3 d^2} \]

[Out]

1/12*B^2*(-a*d+b*c)^3*g*i^2*x/b^2/d+1/12*B^2*(-a*d+b*c)^2*g*i^2*(d*x+c)^2/b/d^2-1/12*B^2*(-a*d+b*c)^4*g*i^2*ln
((b*x+a)/(d*x+c))/b^3/d^2-1/6*B*(-a*d+b*c)^3*g*i^2*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^3/d-1/6*B*(-a*d+b*c)^
2*g*i^2*(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^3+1/4*B*(-a*d+b*c)^2*g*i^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)
))/b/d^2-1/6*B*(-a*d+b*c)*g*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^2+1/12*(-a*d+b*c)^2*g*i^2*(b*x+a)^2*(A
+B*ln(e*(b*x+a)/(d*x+c)))^2/b^3+1/6*(-a*d+b*c)*g*i^2*(b*x+a)^2*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b^2+1/4*g
*i^2*(b*x+a)^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b-1/6*B*(-a*d+b*c)^4*g*i^2*ln((-a*d+b*c)/b/(d*x+c))*(A+
B+B*ln(e*(b*x+a)/(d*x+c)))/b^3/d^2-1/4*B^2*(-a*d+b*c)^4*g*i^2*ln(d*x+c)/b^3/d^2-1/6*B^2*(-a*d+b*c)^4*g*i^2*pol
ylog(2,d*(b*x+a)/b/(d*x+c))/b^3/d^2

________________________________________________________________________________________

Rubi [A]
time = 0.42, antiderivative size = 589, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {2562, 2383, 2381, 2384, 2354, 2438, 2373, 45, 2382, 12, 78} \begin {gather*} -\frac {B^2 g i^2 (b c-a d)^4 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^3 d^2}-\frac {B g i^2 (b c-a d)^4 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A+B\right )}{6 b^3 d^2}-\frac {B g i^2 (a+b x) (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 b^3 d}+\frac {g i^2 (a+b x)^2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{12 b^3}-\frac {B g i^2 (a+b x)^2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 b^3}+\frac {g i^2 (a+b x)^2 (c+d x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{6 b^2}+\frac {B g i^2 (c+d x)^2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 b d^2}-\frac {B g i^2 (c+d x)^3 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 d^2}+\frac {g i^2 (a+b x)^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b}-\frac {B^2 g i^2 (b c-a d)^4 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^3 d^2}-\frac {B^2 g i^2 (b c-a d)^4 \log (c+d x)}{4 b^3 d^2}+\frac {B^2 g i^2 x (b c-a d)^3}{12 b^2 d}+\frac {B^2 g i^2 (c+d x)^2 (b c-a d)^2}{12 b d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B^2*(b*c - a*d)^3*g*i^2*x)/(12*b^2*d) + (B^2*(b*c - a*d)^2*g*i^2*(c + d*x)^2)/(12*b*d^2) - (B^2*(b*c - a*d)^4
*g*i^2*Log[(a + b*x)/(c + d*x)])/(12*b^3*d^2) - (B*(b*c - a*d)^3*g*i^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c +
d*x)]))/(6*b^3*d) - (B*(b*c - a*d)^2*g*i^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^3) + (B*(b*
c - a*d)^2*g*i^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b*d^2) - (B*(b*c - a*d)*g*i^2*(c + d*x)^
3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*d^2) + ((b*c - a*d)^2*g*i^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c
+ d*x)])^2)/(12*b^3) + ((b*c - a*d)*g*i^2*(a + b*x)^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(6*b^
2) + (g*i^2*(a + b*x)^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*b) - (B*(b*c - a*d)^4*g*i^2*Log
[(b*c - a*d)/(b*(c + d*x))]*(A + B + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^3*d^2) - (B^2*(b*c - a*d)^4*g*i^2*L
og[c + d*x])/(4*b^3*d^2) - (B^2*(b*c - a*d)^4*g*i^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(6*b^3*d^2)

Rule 12

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 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 (66 c+66 d x)^2 (a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx &=\int \left (\frac {(-b c+a d) g (66 c+66 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d}+\frac {b g (66 c+66 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{66 d}\right ) \, dx\\ &=\frac {(b g) \int (66 c+66 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{66 d}+\frac {((-b c+a d) g) \int (66 c+66 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{d}\\ &=-\frac {1452 (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {1089 b g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}-\frac {(b B g) \int \frac {18974736 (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{8712 d^2}+\frac {(B (b c-a d) g) \int \frac {287496 (b c-a d) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{99 d^2}\\ &=-\frac {1452 (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {1089 b g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}-\frac {(2178 b B (b c-a d) g) \int \frac {(c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{d^2}+\frac {\left (2904 B (b c-a d)^2 g\right ) \int \frac {(c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{d^2}\\ &=-\frac {1452 (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {1089 b g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}-\frac {(2178 b B (b c-a d) g) \int \left (\frac {d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {(b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac {d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b}\right ) \, dx}{d^2}+\frac {\left (2904 B (b c-a d)^2 g\right ) \int \left (\frac {d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac {(b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (a+b x)}+\frac {d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b}\right ) \, dx}{d^2}\\ &=-\frac {1452 (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {1089 b g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}-\frac {(2178 B (b c-a d) g) \int (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{d}-\frac {\left (2178 B (b c-a d)^2 g\right ) \int (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b d}+\frac {\left (2904 B (b c-a d)^2 g\right ) \int (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b d}-\frac {\left (2178 B (b c-a d)^3 g\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d}+\frac {\left (2904 B (b c-a d)^3 g\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d}-\frac {\left (2178 B (b c-a d)^4 g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 d^2}+\frac {\left (2904 B (b c-a d)^4 g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 d^2}\\ &=\frac {726 A B (b c-a d)^3 g x}{b^2 d}+\frac {363 B (b c-a d)^2 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b d^2}-\frac {726 B (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {726 B (b c-a d)^4 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 d^2}-\frac {1452 (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {1089 b g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {\left (726 B^2 (b c-a d) g\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{d^2}+\frac {\left (1089 B^2 (b c-a d)^2 g\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{b d^2}-\frac {\left (1452 B^2 (b c-a d)^2 g\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{b d^2}-\frac {\left (2178 B^2 (b c-a d)^3 g\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{b^2 d}+\frac {\left (2904 B^2 (b c-a d)^3 g\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{b^2 d}+\frac {\left (2178 B^2 (b c-a d)^4 g\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 (a+b x)}{e (a+b x)} \, dx}{b^3 d^2}-\frac {\left (2904 B^2 (b c-a d)^4 g\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 (a+b x)}{e (a+b x)} \, dx}{b^3 d^2}\\ &=\frac {726 A B (b c-a d)^3 g x}{b^2 d}+\frac {726 B^2 (b c-a d)^3 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 d}+\frac {363 B (b c-a d)^2 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b d^2}-\frac {726 B (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {726 B (b c-a d)^4 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 d^2}-\frac {1452 (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {1089 b g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {\left (726 B^2 (b c-a d)^2 g\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{d^2}+\frac {\left (1089 B^2 (b c-a d)^3 g\right ) \int \frac {c+d x}{a+b x} \, dx}{b d^2}-\frac {\left (1452 B^2 (b c-a d)^3 g\right ) \int \frac {c+d x}{a+b x} \, dx}{b d^2}+\frac {\left (2178 B^2 (b c-a d)^4 g\right ) \int \frac {1}{c+d x} \, dx}{b^3 d}-\frac {\left (2904 B^2 (b c-a d)^4 g\right ) \int \frac {1}{c+d x} \, dx}{b^3 d}+\frac {\left (2178 B^2 (b c-a d)^4 g\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 (a+b x)}{a+b x} \, dx}{b^3 d^2 e}-\frac {\left (2904 B^2 (b c-a d)^4 g\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 (a+b x)}{a+b x} \, dx}{b^3 d^2 e}\\ &=\frac {726 A B (b c-a d)^3 g x}{b^2 d}+\frac {726 B^2 (b c-a d)^3 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 d}+\frac {363 B (b c-a d)^2 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b d^2}-\frac {726 B (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {726 B (b c-a d)^4 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 d^2}-\frac {1452 (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {1089 b g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}-\frac {726 B^2 (b c-a d)^4 g \log (c+d x)}{b^3 d^2}+\frac {\left (726 B^2 (b c-a d)^2 g\right ) \int \left (\frac {d (b c-a d)}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)}+\frac {d (c+d x)}{b}\right ) \, dx}{d^2}+\frac {\left (1089 B^2 (b c-a d)^3 g\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{b d^2}-\frac {\left (1452 B^2 (b c-a d)^3 g\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{b d^2}+\frac {\left (2178 B^2 (b c-a d)^4 g\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^3 d^2 e}-\frac {\left (2904 B^2 (b c-a d)^4 g\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^3 d^2 e}\\ &=\frac {726 A B (b c-a d)^3 g x}{b^2 d}+\frac {363 B^2 (b c-a d)^3 g x}{b^2 d}+\frac {363 B^2 (b c-a d)^2 g (c+d x)^2}{b d^2}+\frac {363 B^2 (b c-a d)^4 g \log (a+b x)}{b^3 d^2}+\frac {726 B^2 (b c-a d)^3 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 d}+\frac {363 B (b c-a d)^2 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b d^2}-\frac {726 B (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {726 B (b c-a d)^4 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 d^2}-\frac {1452 (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {1089 b g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}-\frac {726 B^2 (b c-a d)^4 g \log (c+d x)}{b^3 d^2}+\frac {\left (2178 B^2 (b c-a d)^4 g\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 d^2}-\frac {\left (2904 B^2 (b c-a d)^4 g\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 d^2}-\frac {\left (2178 B^2 (b c-a d)^4 g\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 d}+\frac {\left (2904 B^2 (b c-a d)^4 g\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 d}\\ &=\frac {726 A B (b c-a d)^3 g x}{b^2 d}+\frac {363 B^2 (b c-a d)^3 g x}{b^2 d}+\frac {363 B^2 (b c-a d)^2 g (c+d x)^2}{b d^2}+\frac {363 B^2 (b c-a d)^4 g \log (a+b x)}{b^3 d^2}+\frac {726 B^2 (b c-a d)^3 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 d}+\frac {363 B (b c-a d)^2 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b d^2}-\frac {726 B (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {726 B (b c-a d)^4 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 d^2}-\frac {1452 (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {1089 b g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}-\frac {726 B^2 (b c-a d)^4 g \log (c+d x)}{b^3 d^2}+\frac {726 B^2 (b c-a d)^4 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 d^2}+\frac {\left (2178 B^2 (b c-a d)^4 g\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 d^2}-\frac {\left (2904 B^2 (b c-a d)^4 g\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 d^2}+\frac {\left (2178 B^2 (b c-a d)^4 g\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 d^2}-\frac {\left (2904 B^2 (b c-a d)^4 g\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 d^2}\\ &=\frac {726 A B (b c-a d)^3 g x}{b^2 d}+\frac {363 B^2 (b c-a d)^3 g x}{b^2 d}+\frac {363 B^2 (b c-a d)^2 g (c+d x)^2}{b d^2}+\frac {363 B^2 (b c-a d)^4 g \log (a+b x)}{b^3 d^2}-\frac {363 B^2 (b c-a d)^4 g \log ^2(a+b x)}{b^3 d^2}+\frac {726 B^2 (b c-a d)^3 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 d}+\frac {363 B (b c-a d)^2 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b d^2}-\frac {726 B (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {726 B (b c-a d)^4 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 d^2}-\frac {1452 (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {1089 b g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}-\frac {726 B^2 (b c-a d)^4 g \log (c+d x)}{b^3 d^2}+\frac {726 B^2 (b c-a d)^4 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 d^2}+\frac {\left (2178 B^2 (b c-a d)^4 g\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 d^2}-\frac {\left (2904 B^2 (b c-a d)^4 g\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 d^2}\\ &=\frac {726 A B (b c-a d)^3 g x}{b^2 d}+\frac {363 B^2 (b c-a d)^3 g x}{b^2 d}+\frac {363 B^2 (b c-a d)^2 g (c+d x)^2}{b d^2}+\frac {363 B^2 (b c-a d)^4 g \log (a+b x)}{b^3 d^2}-\frac {363 B^2 (b c-a d)^4 g \log ^2(a+b x)}{b^3 d^2}+\frac {726 B^2 (b c-a d)^3 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 d}+\frac {363 B (b c-a d)^2 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b d^2}-\frac {726 B (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2}+\frac {726 B (b c-a d)^4 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 d^2}-\frac {1452 (b c-a d) g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {1089 b g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}-\frac {726 B^2 (b c-a d)^4 g \log (c+d x)}{b^3 d^2}+\frac {726 B^2 (b c-a d)^4 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 d^2}+\frac {726 B^2 (b c-a d)^4 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 d^2}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 677, normalized size = 1.15 \begin {gather*} \frac {g i^2 \left (-4 (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+3 b (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {4 B (b c-a d)^2 \left (2 A b d (b c-a d) x-B (b c-a d) (b d x+(b c-a d) \log (a+b x))+2 B d (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+b^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 B (b c-a d)^2 \log (c+d x)-B (b c-a d)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b^3}-\frac {B (b c-a d) \left (6 A b d (b c-a d)^2 x-3 B (b c-a d)^2 (b d x+(b c-a d) \log (a+b x))-B (b c-a d) \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )+6 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 b^2 (b c-a d) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 b^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 (b c-a d)^3 \log (a+b x) \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)-3 B (b c-a d)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b^3}\right )}{12 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(g*i^2*(-4*(b*c - a*d)*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 3*b*(c + d*x)^4*(A + B*Log[(e*(a +
b*x))/(c + d*x)])^2 + (4*B*(b*c - a*d)^2*(2*A*b*d*(b*c - a*d)*x - B*(b*c - a*d)*(b*d*x + (b*c - a*d)*Log[a +
b*x]) + 2*B*d*(b*c - a*d)*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + b^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]) + 2*(b*c - a*d)^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 2*B*(b*c - a*d)^2*Log[c + d*x]
- B*(b*c - a*d)^2*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x)
)/(-(b*c) + a*d)])))/b^3 - (B*(b*c - a*d)*(6*A*b*d*(b*c - a*d)^2*x - 3*B*(b*c - a*d)^2*(b*d*x + (b*c - a*d)*Lo
g[a + b*x]) - B*(b*c - a*d)*(2*b*d*(b*c - a*d)*x + b^2*(c + d*x)^2 + 2*(b*c - a*d)^2*Log[a + b*x]) + 6*B*d*(b*
c - a*d)^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + 3*b^2*(b*c - a*d)*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]) + 2*b^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 6*(b*c - a*d)^3*Log[a + b*x]*(A + B*Log[(e
*(a + b*x))/(c + d*x)]) - 6*B*(b*c - a*d)^3*Log[c + d*x] - 3*B*(b*c - a*d)^3*(Log[a + b*x]*(Log[a + b*x] - 2*L
og[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])))/b^3))/(12*d^2)

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Maple [F]
time = 0.29, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right ) \left (d i x +c i \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)*(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)*(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. 1736 vs. \(2 (531) = 1062\).
time = 0.39, size = 1736, normalized size = 2.95 \begin {gather*} -\frac {1}{4} \, A^{2} b d^{2} g x^{4} - \frac {2}{3} \, A^{2} b c d g x^{3} - \frac {1}{3} \, A^{2} a d^{2} g x^{3} - \frac {1}{2} \, A^{2} b c^{2} g x^{2} - A^{2} a c d g x^{2} - 2 \, {\left (x \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} A B a c^{2} g - {\left (x^{2} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} A B b c^{2} g - 2 \, {\left (x^{2} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} A B a c d g - \frac {2}{3} \, {\left (2 \, x^{3} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} A B b c d g - \frac {1}{3} \, {\left (2 \, x^{3} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} A B a d^{2} g - \frac {1}{12} \, {\left (6 \, x^{4} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} A B b d^{2} g - A^{2} a c^{2} g x - \frac {{\left (b^{3} c^{4} g - 2 \, a b^{2} c^{3} d g - 7 \, a^{2} b c^{2} d^{2} g + 2 \, a^{3} c d^{3} g\right )} B^{2} \log \left (d x + c\right )}{12 \, b^{2} d^{2}} - \frac {{\left (b^{4} c^{4} g - 4 \, a b^{3} c^{3} d g + 6 \, a^{2} b^{2} c^{2} d^{2} g - 4 \, a^{3} b c d^{3} g + a^{4} d^{4} g\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B^{2}}{6 \, b^{3} d^{2}} - \frac {3 \, B^{2} b^{4} d^{4} g x^{4} + 6 \, {\left (b^{4} c d^{3} g + a b^{3} d^{4} g\right )} B^{2} x^{3} + 2 \, {\left (b^{4} c^{2} d^{2} g + 7 \, a b^{3} c d^{3} g + a^{2} b^{2} d^{4} g\right )} B^{2} x^{2} + {\left (b^{4} c^{3} d g + a b^{3} c^{2} d^{2} g + 13 \, a^{2} b^{2} c d^{3} g - 3 \, a^{3} b d^{4} g\right )} B^{2} x + {\left (3 \, B^{2} b^{4} d^{4} g x^{4} + 12 \, B^{2} a b^{3} c^{2} d^{2} g x + 4 \, {\left (2 \, b^{4} c d^{3} g + a b^{3} d^{4} g\right )} B^{2} x^{3} + 6 \, {\left (b^{4} c^{2} d^{2} g + 2 \, a b^{3} c d^{3} g\right )} B^{2} x^{2} + {\left (6 \, a^{2} b^{2} c^{2} d^{2} g - 4 \, a^{3} b c d^{3} g + a^{4} d^{4} g\right )} B^{2}\right )} \log \left (b x + a\right )^{2} + {\left (3 \, B^{2} b^{4} d^{4} g x^{4} + 12 \, B^{2} a b^{3} c^{2} d^{2} g x + 4 \, {\left (2 \, b^{4} c d^{3} g + a b^{3} d^{4} g\right )} B^{2} x^{3} + 6 \, {\left (b^{4} c^{2} d^{2} g + 2 \, a b^{3} c d^{3} g\right )} B^{2} x^{2} - {\left (b^{4} c^{4} g - 4 \, a b^{3} c^{3} d g\right )} B^{2}\right )} \log \left (d x + c\right )^{2} + {\left (6 \, B^{2} b^{4} d^{4} g x^{4} + 2 \, {\left (7 \, b^{4} c d^{3} g + 5 \, a b^{3} d^{4} g\right )} B^{2} x^{3} + {\left (7 \, b^{4} c^{2} d^{2} g + 28 \, a b^{3} c d^{3} g + a^{2} b^{2} d^{4} g\right )} B^{2} x^{2} - 2 \, {\left (b^{4} c^{3} d g - 10 \, a b^{3} c^{2} d^{2} g - 4 \, a^{2} b^{2} c d^{3} g + a^{3} b d^{4} g\right )} B^{2} x - {\left (2 \, a b^{3} c^{3} d g - 13 \, a^{2} b^{2} c^{2} d^{2} g + 6 \, a^{3} b c d^{3} g - a^{4} d^{4} g\right )} B^{2}\right )} \log \left (b x + a\right ) - {\left (6 \, B^{2} b^{4} d^{4} g x^{4} + 2 \, {\left (7 \, b^{4} c d^{3} g + 5 \, a b^{3} d^{4} g\right )} B^{2} x^{3} + {\left (7 \, b^{4} c^{2} d^{2} g + 28 \, a b^{3} c d^{3} g + a^{2} b^{2} d^{4} g\right )} B^{2} x^{2} - 2 \, {\left (b^{4} c^{3} d g - 10 \, a b^{3} c^{2} d^{2} g - 4 \, a^{2} b^{2} c d^{3} g + a^{3} b d^{4} g\right )} B^{2} x + 2 \, {\left (3 \, B^{2} b^{4} d^{4} g x^{4} + 12 \, B^{2} a b^{3} c^{2} d^{2} g x + 4 \, {\left (2 \, b^{4} c d^{3} g + a b^{3} d^{4} g\right )} B^{2} x^{3} + 6 \, {\left (b^{4} c^{2} d^{2} g + 2 \, a b^{3} c d^{3} g\right )} B^{2} x^{2} + {\left (6 \, a^{2} b^{2} c^{2} d^{2} g - 4 \, a^{3} b c d^{3} g + a^{4} d^{4} g\right )} B^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{12 \, b^{3} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*A^2*b*d^2*g*x^4 - 2/3*A^2*b*c*d*g*x^3 - 1/3*A^2*a*d^2*g*x^3 - 1/2*A^2*b*c^2*g*x^2 - A^2*a*c*d*g*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*c^2*g - (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*b*c^2*g - 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*c*d*g - 2/3*(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*b*c*d*g - 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*d^2*g - 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*d^2*g - A^2*a*c^2*g*x - 1/12*(b^3*c^4*g - 2*a*b^2*c^3*d*g - 7*a^2*b*c^2*d^2*g + 2*a^
3*c*d^3*g)*B^2*log(d*x + c)/(b^2*d^2) - 1/6*(b^4*c^4*g - 4*a*b^3*c^3*d*g + 6*a^2*b^2*c^2*d^2*g - 4*a^3*b*c*d^3
*g + a^4*d^4*g)*(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^2) - 1/12*(3*B^2*b^4*d^4*g*x^4 + 6*(b^4*c*d^3*g + a*b^3*d^4*g)*B^2*x^3 + 2*(b^4*c^2*d^2*g + 7*a*b^3*c*d^3*g
+ a^2*b^2*d^4*g)*B^2*x^2 + (b^4*c^3*d*g + a*b^3*c^2*d^2*g + 13*a^2*b^2*c*d^3*g - 3*a^3*b*d^4*g)*B^2*x + (3*B^
2*b^4*d^4*g*x^4 + 12*B^2*a*b^3*c^2*d^2*g*x + 4*(2*b^4*c*d^3*g + a*b^3*d^4*g)*B^2*x^3 + 6*(b^4*c^2*d^2*g + 2*a*
b^3*c*d^3*g)*B^2*x^2 + (6*a^2*b^2*c^2*d^2*g - 4*a^3*b*c*d^3*g + a^4*d^4*g)*B^2)*log(b*x + a)^2 + (3*B^2*b^4*d^
4*g*x^4 + 12*B^2*a*b^3*c^2*d^2*g*x + 4*(2*b^4*c*d^3*g + a*b^3*d^4*g)*B^2*x^3 + 6*(b^4*c^2*d^2*g + 2*a*b^3*c*d^
3*g)*B^2*x^2 - (b^4*c^4*g - 4*a*b^3*c^3*d*g)*B^2)*log(d*x + c)^2 + (6*B^2*b^4*d^4*g*x^4 + 2*(7*b^4*c*d^3*g + 5
*a*b^3*d^4*g)*B^2*x^3 + (7*b^4*c^2*d^2*g + 28*a*b^3*c*d^3*g + a^2*b^2*d^4*g)*B^2*x^2 - 2*(b^4*c^3*d*g - 10*a*b
^3*c^2*d^2*g - 4*a^2*b^2*c*d^3*g + a^3*b*d^4*g)*B^2*x - (2*a*b^3*c^3*d*g - 13*a^2*b^2*c^2*d^2*g + 6*a^3*b*c*d^
3*g - a^4*d^4*g)*B^2)*log(b*x + a) - (6*B^2*b^4*d^4*g*x^4 + 2*(7*b^4*c*d^3*g + 5*a*b^3*d^4*g)*B^2*x^3 + (7*b^4
*c^2*d^2*g + 28*a*b^3*c*d^3*g + a^2*b^2*d^4*g)*B^2*x^2 - 2*(b^4*c^3*d*g - 10*a*b^3*c^2*d^2*g - 4*a^2*b^2*c*d^3
*g + a^3*b*d^4*g)*B^2*x + 2*(3*B^2*b^4*d^4*g*x^4 + 12*B^2*a*b^3*c^2*d^2*g*x + 4*(2*b^4*c*d^3*g + a*b^3*d^4*g)*
B^2*x^3 + 6*(b^4*c^2*d^2*g + 2*a*b^3*c*d^3*g)*B^2*x^2 + (6*a^2*b^2*c^2*d^2*g - 4*a^3*b*c*d^3*g + a^4*d^4*g)*B^
2)*log(b*x + a))*log(d*x + c))/(b^3*d^2)

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

[Out]

-1/12*(3*B^2*b*d^2*g*x^4 + 12*B^2*a*c^2*g*x + 4*(2*B^2*b*c*d + B^2*a*d^2)*g*x^3 + 6*(B^2*b*c^2 + 2*B^2*a*c*d)*
g*x^2)*log((b*x + a)*e/(d*x + c))^2 + integral(-1/6*(6*A^2*b^2*d^3*g*x^5 + 6*A^2*a^2*c^3*g + 6*(3*A^2*b^2*c*d^
2 + 2*A^2*a*b*d^3)*g*x^4 + 6*(3*A^2*b^2*c^2*d + 6*A^2*a*b*c*d^2 + A^2*a^2*d^3)*g*x^3 + 6*(A^2*b^2*c^3 + 6*A^2*
a*b*c^2*d + 3*A^2*a^2*c*d^2)*g*x^2 + 6*(2*A^2*a*b*c^3 + 3*A^2*a^2*c^2*d)*g*x + (12*A*B*b^2*d^3*g*x^5 + 12*A*B*
a^2*c^3*g + 3*((12*A*B - B^2)*b^2*c*d^2 + (8*A*B + B^2)*a*b*d^3)*g*x^4 + 4*((9*A*B - 2*B^2)*b^2*c^2*d + (18*A*
B + B^2)*a*b*c*d^2 + (3*A*B + B^2)*a^2*d^3)*g*x^3 + 6*((2*A*B - B^2)*b^2*c^3 + (12*A*B - B^2)*a*b*c^2*d + 2*(3
*A*B + B^2)*a^2*c*d^2)*g*x^2 + 12*((2*A*B - B^2)*a*b*c^3 + (3*A*B + B^2)*a^2*c^2*d)*g*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)*(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)*(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)*(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 )\,{\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)*(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)*(c*i + d*i*x)^2*(A + B*log((e*(a + b*x))/(c + d*x)))^2, x)

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