3.1.76 \(\int (a g+b g x) (c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))^2 \, dx\) [76]

Optimal. Leaf size=730 \[ \frac {B^2 (b c-a d)^4 g i^3 x}{60 b^3 d}+\frac {B^2 (b c-a d)^3 g i^3 (c+d x)^2}{30 b^2 d^2}+\frac {B^2 (b c-a d)^2 g i^3 (c+d x)^3}{30 b d^2}-\frac {B^2 (b c-a d)^5 g i^3 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^4 d^2}-\frac {B (b c-a d)^4 g i^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4 d}-\frac {B (b c-a d)^3 g i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4}+\frac {3 B (b c-a d)^3 g i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^2 d^2}+\frac {B (b c-a d)^2 g i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^2}-\frac {B (b c-a d) g i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 d^2}+\frac {(b c-a d)^3 g i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^4}+\frac {(b c-a d)^2 g i^3 (a+b x)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^3}+\frac {3 (b c-a d) g i^3 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^2}+\frac {g i^3 (a+b x)^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}-\frac {B (b c-a d)^5 g i^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4 d^2}-\frac {11 B^2 (b c-a d)^5 g i^3 \log (c+d x)}{60 b^4 d^2}-\frac {B^2 (b c-a d)^5 g i^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{10 b^4 d^2} \]

[Out]

1/60*B^2*(-a*d+b*c)^4*g*i^3*x/b^3/d+1/30*B^2*(-a*d+b*c)^3*g*i^3*(d*x+c)^2/b^2/d^2+1/30*B^2*(-a*d+b*c)^2*g*i^3*
(d*x+c)^3/b/d^2-1/12*B^2*(-a*d+b*c)^5*g*i^3*ln((b*x+a)/(d*x+c))/b^4/d^2-1/10*B*(-a*d+b*c)^4*g*i^3*(b*x+a)*(A+B
*ln(e*(b*x+a)/(d*x+c)))/b^4/d-1/10*B*(-a*d+b*c)^3*g*i^3*(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^4+3/20*B*(-a*d
+b*c)^3*g*i^3*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^2/d^2+1/30*B*(-a*d+b*c)^2*g*i^3*(d*x+c)^3*(A+B*ln(e*(b*x
+a)/(d*x+c)))/b/d^2-1/10*B*(-a*d+b*c)*g*i^3*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^2+1/20*(-a*d+b*c)^3*g*i^3*
(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b^4+1/10*(-a*d+b*c)^2*g*i^3*(b*x+a)^2*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c
)))^2/b^3+3/20*(-a*d+b*c)*g*i^3*(b*x+a)^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b^2+1/5*g*i^3*(b*x+a)^2*(d*x
+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b-1/10*B*(-a*d+b*c)^5*g*i^3*ln((-a*d+b*c)/b/(d*x+c))*(A+B+B*ln(e*(b*x+a)/(
d*x+c)))/b^4/d^2-11/60*B^2*(-a*d+b*c)^5*g*i^3*ln(d*x+c)/b^4/d^2-1/10*B^2*(-a*d+b*c)^5*g*i^3*polylog(2,d*(b*x+a
)/b/(d*x+c))/b^4/d^2

________________________________________________________________________________________

Rubi [A]
time = 0.56, antiderivative size = 730, normalized size of antiderivative = 1.00, number of steps used = 19, 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^3 (b c-a d)^5 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{10 b^4 d^2}-\frac {B g i^3 (b c-a d)^5 \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 )}{10 b^4 d^2}-\frac {B g i^3 (a+b x) (b c-a d)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{10 b^4 d}+\frac {g i^3 (a+b x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{20 b^4}-\frac {B g i^3 (a+b x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{10 b^4}+\frac {g i^3 (a+b x)^2 (c+d x) (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{10 b^3}+\frac {3 B g i^3 (c+d x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{20 b^2 d^2}+\frac {3 g i^3 (a+b x)^2 (c+d x)^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{20 b^2}+\frac {B g i^3 (c+d x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{30 b d^2}-\frac {B g i^3 (c+d x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{10 d^2}+\frac {g i^3 (a+b x)^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 b}-\frac {B^2 g i^3 (b c-a d)^5 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^4 d^2}-\frac {11 B^2 g i^3 (b c-a d)^5 \log (c+d x)}{60 b^4 d^2}+\frac {B^2 g i^3 x (b c-a d)^4}{60 b^3 d}+\frac {B^2 g i^3 (c+d x)^2 (b c-a d)^3}{30 b^2 d^2}+\frac {B^2 g i^3 (c+d x)^3 (b c-a d)^2}{30 b d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B^2*(b*c - a*d)^4*g*i^3*x)/(60*b^3*d) + (B^2*(b*c - a*d)^3*g*i^3*(c + d*x)^2)/(30*b^2*d^2) + (B^2*(b*c - a*d)
^2*g*i^3*(c + d*x)^3)/(30*b*d^2) - (B^2*(b*c - a*d)^5*g*i^3*Log[(a + b*x)/(c + d*x)])/(12*b^4*d^2) - (B*(b*c -
a*d)^4*g*i^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*b^4*d) - (B*(b*c - a*d)^3*g*i^3*(a + b*x)^2*
(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*b^4) + (3*B*(b*c - a*d)^3*g*i^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))
/(c + d*x)]))/(20*b^2*d^2) + (B*(b*c - a*d)^2*g*i^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(30*b*d^
2) - (B*(b*c - a*d)*g*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*d^2) + ((b*c - a*d)^3*g*i^3*(a
+ b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(20*b^4) + ((b*c - a*d)^2*g*i^3*(a + b*x)^2*(c + d*x)*(A + B
*Log[(e*(a + b*x))/(c + d*x)])^2)/(10*b^3) + (3*(b*c - a*d)*g*i^3*(a + b*x)^2*(c + d*x)^2*(A + B*Log[(e*(a + b
*x))/(c + d*x)])^2)/(20*b^2) + (g*i^3*(a + b*x)^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(5*b) -
(B*(b*c - a*d)^5*g*i^3*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*b^4*d^2) -
(11*B^2*(b*c - a*d)^5*g*i^3*Log[c + d*x])/(60*b^4*d^2) - (B^2*(b*c - a*d)^5*g*i^3*PolyLog[2, (d*(a + b*x))/(b
*(c + d*x))])/(10*b^4*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 (76 c+76 d x)^3 (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 (76 c+76 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d}+\frac {b g (76 c+76 d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{76 d}\right ) \, dx\\ &=\frac {(b g) \int (76 c+76 d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{76 d}+\frac {((-b c+a d) g) \int (76 c+76 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{d}\\ &=-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {(b B g) \int \frac {2535525376 (b c-a d) (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{14440 d^2}+\frac {(B (b c-a d) g) \int \frac {33362176 (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}{152 d^2}\\ &=-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {(877952 b B (b c-a d) g) \int \frac {(c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{5 d^2}+\frac {\left (219488 B (b c-a d)^2 g\right ) \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 {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {(877952 b B (b c-a d) g) \int \left (\frac {d (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4}+\frac {(b c-a d)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 (a+b x)}+\frac {d (b c-a d)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {d (b c-a d) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac {d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b}\right ) \, dx}{5 d^2}+\frac {\left (219488 B (b c-a d)^2 g\right ) \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 {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {(877952 B (b c-a d) g) \int (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{5 d}-\frac {\left (877952 B (b c-a d)^2 g\right ) \int (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b d}+\frac {\left (219488 B (b c-a d)^2 g\right ) \int (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b d}-\frac {\left (877952 B (b c-a d)^3 g\right ) \int (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b^2 d}+\frac {\left (219488 B (b c-a d)^3 g\right ) \int (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d}-\frac {\left (877952 B (b c-a d)^4 g\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{5 b^3 d}+\frac {\left (219488 B (b c-a d)^4 g\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^3 d}-\frac {\left (877952 B (b c-a d)^5 g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{5 b^3 d^2}+\frac {\left (219488 B (b c-a d)^5 g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^3 d^2}\\ &=\frac {219488 A B (b c-a d)^4 g x}{5 b^3 d}+\frac {109744 B (b c-a d)^3 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2 d^2}+\frac {219488 B (b c-a d)^2 g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b d^2}-\frac {219488 B (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}+\frac {219488 B (b c-a d)^5 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 d^2}-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}+\frac {\left (219488 B^2 (b c-a d) g\right ) \int \frac {(b c-a d) (c+d x)^3}{a+b x} \, dx}{5 d^2}+\frac {\left (877952 B^2 (b c-a d)^2 g\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{15 b d^2}-\frac {\left (219488 B^2 (b c-a d)^2 g\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{3 b d^2}+\frac {\left (438976 B^2 (b c-a d)^3 g\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{5 b^2 d^2}-\frac {\left (109744 B^2 (b c-a d)^3 g\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{b^2 d^2}-\frac {\left (877952 B^2 (b c-a d)^4 g\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{5 b^3 d}+\frac {\left (219488 B^2 (b c-a d)^4 g\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{b^3 d}+\frac {\left (877952 B^2 (b c-a d)^5 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}{5 b^4 d^2}-\frac {\left (219488 B^2 (b c-a d)^5 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^4 d^2}\\ &=\frac {219488 A B (b c-a d)^4 g x}{5 b^3 d}+\frac {219488 B^2 (b c-a d)^4 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{5 b^4 d}+\frac {109744 B (b c-a d)^3 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2 d^2}+\frac {219488 B (b c-a d)^2 g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b d^2}-\frac {219488 B (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}+\frac {219488 B (b c-a d)^5 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 d^2}-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}+\frac {\left (219488 B^2 (b c-a d)^2 g\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{5 d^2}+\frac {\left (877952 B^2 (b c-a d)^3 g\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{15 b d^2}-\frac {\left (219488 B^2 (b c-a d)^3 g\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{3 b d^2}+\frac {\left (438976 B^2 (b c-a d)^4 g\right ) \int \frac {c+d x}{a+b x} \, dx}{5 b^2 d^2}-\frac {\left (109744 B^2 (b c-a d)^4 g\right ) \int \frac {c+d x}{a+b x} \, dx}{b^2 d^2}+\frac {\left (877952 B^2 (b c-a d)^5 g\right ) \int \frac {1}{c+d x} \, dx}{5 b^4 d}-\frac {\left (219488 B^2 (b c-a d)^5 g\right ) \int \frac {1}{c+d x} \, dx}{b^4 d}+\frac {\left (877952 B^2 (b c-a d)^5 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}{5 b^4 d^2 e}-\frac {\left (219488 B^2 (b c-a d)^5 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^4 d^2 e}\\ &=\frac {219488 A B (b c-a d)^4 g x}{5 b^3 d}+\frac {219488 B^2 (b c-a d)^4 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{5 b^4 d}+\frac {109744 B (b c-a d)^3 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2 d^2}+\frac {219488 B (b c-a d)^2 g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b d^2}-\frac {219488 B (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}+\frac {219488 B (b c-a d)^5 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 d^2}-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {219488 B^2 (b c-a d)^5 g \log (c+d x)}{5 b^4 d^2}+\frac {\left (219488 B^2 (b c-a d)^2 g\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}{5 d^2}+\frac {\left (877952 B^2 (b c-a d)^3 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}{15 b d^2}-\frac {\left (219488 B^2 (b c-a d)^3 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}{3 b d^2}+\frac {\left (438976 B^2 (b c-a d)^4 g\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{5 b^2 d^2}-\frac {\left (109744 B^2 (b c-a d)^4 g\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{b^2 d^2}+\frac {\left (877952 B^2 (b c-a d)^5 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}{5 b^4 d^2 e}-\frac {\left (219488 B^2 (b c-a d)^5 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^4 d^2 e}\\ &=\frac {219488 A B (b c-a d)^4 g x}{5 b^3 d}+\frac {109744 B^2 (b c-a d)^4 g x}{15 b^3 d}+\frac {219488 B^2 (b c-a d)^3 g (c+d x)^2}{15 b^2 d^2}+\frac {219488 B^2 (b c-a d)^2 g (c+d x)^3}{15 b d^2}+\frac {109744 B^2 (b c-a d)^5 g \log (a+b x)}{15 b^4 d^2}+\frac {219488 B^2 (b c-a d)^4 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{5 b^4 d}+\frac {109744 B (b c-a d)^3 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2 d^2}+\frac {219488 B (b c-a d)^2 g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b d^2}-\frac {219488 B (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}+\frac {219488 B (b c-a d)^5 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 d^2}-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {219488 B^2 (b c-a d)^5 g \log (c+d x)}{5 b^4 d^2}+\frac {\left (877952 B^2 (b c-a d)^5 g\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{5 b^3 d^2}-\frac {\left (219488 B^2 (b c-a d)^5 g\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 d^2}-\frac {\left (877952 B^2 (b c-a d)^5 g\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{5 b^4 d}+\frac {\left (219488 B^2 (b c-a d)^5 g\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 d}\\ &=\frac {219488 A B (b c-a d)^4 g x}{5 b^3 d}+\frac {109744 B^2 (b c-a d)^4 g x}{15 b^3 d}+\frac {219488 B^2 (b c-a d)^3 g (c+d x)^2}{15 b^2 d^2}+\frac {219488 B^2 (b c-a d)^2 g (c+d x)^3}{15 b d^2}+\frac {109744 B^2 (b c-a d)^5 g \log (a+b x)}{15 b^4 d^2}+\frac {219488 B^2 (b c-a d)^4 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{5 b^4 d}+\frac {109744 B (b c-a d)^3 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2 d^2}+\frac {219488 B (b c-a d)^2 g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b d^2}-\frac {219488 B (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}+\frac {219488 B (b c-a d)^5 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 d^2}-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {219488 B^2 (b c-a d)^5 g \log (c+d x)}{5 b^4 d^2}+\frac {219488 B^2 (b c-a d)^5 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{5 b^4 d^2}+\frac {\left (877952 B^2 (b c-a d)^5 g\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{5 b^4 d^2}-\frac {\left (219488 B^2 (b c-a d)^5 g\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 d^2}+\frac {\left (877952 B^2 (b c-a d)^5 g\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{5 b^3 d^2}-\frac {\left (219488 B^2 (b c-a d)^5 g\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 d^2}\\ &=\frac {219488 A B (b c-a d)^4 g x}{5 b^3 d}+\frac {109744 B^2 (b c-a d)^4 g x}{15 b^3 d}+\frac {219488 B^2 (b c-a d)^3 g (c+d x)^2}{15 b^2 d^2}+\frac {219488 B^2 (b c-a d)^2 g (c+d x)^3}{15 b d^2}+\frac {109744 B^2 (b c-a d)^5 g \log (a+b x)}{15 b^4 d^2}-\frac {109744 B^2 (b c-a d)^5 g \log ^2(a+b x)}{5 b^4 d^2}+\frac {219488 B^2 (b c-a d)^4 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{5 b^4 d}+\frac {109744 B (b c-a d)^3 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2 d^2}+\frac {219488 B (b c-a d)^2 g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b d^2}-\frac {219488 B (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}+\frac {219488 B (b c-a d)^5 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 d^2}-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {219488 B^2 (b c-a d)^5 g \log (c+d x)}{5 b^4 d^2}+\frac {219488 B^2 (b c-a d)^5 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{5 b^4 d^2}+\frac {\left (877952 B^2 (b c-a d)^5 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 )}{5 b^4 d^2}-\frac {\left (219488 B^2 (b c-a d)^5 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^4 d^2}\\ &=\frac {219488 A B (b c-a d)^4 g x}{5 b^3 d}+\frac {109744 B^2 (b c-a d)^4 g x}{15 b^3 d}+\frac {219488 B^2 (b c-a d)^3 g (c+d x)^2}{15 b^2 d^2}+\frac {219488 B^2 (b c-a d)^2 g (c+d x)^3}{15 b d^2}+\frac {109744 B^2 (b c-a d)^5 g \log (a+b x)}{15 b^4 d^2}-\frac {109744 B^2 (b c-a d)^5 g \log ^2(a+b x)}{5 b^4 d^2}+\frac {219488 B^2 (b c-a d)^4 g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{5 b^4 d}+\frac {109744 B (b c-a d)^3 g (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^2 d^2}+\frac {219488 B (b c-a d)^2 g (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b d^2}-\frac {219488 B (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^2}+\frac {219488 B (b c-a d)^5 g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b^4 d^2}-\frac {109744 (b c-a d) g (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}+\frac {438976 b g (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 d^2}-\frac {219488 B^2 (b c-a d)^5 g \log (c+d x)}{5 b^4 d^2}+\frac {219488 B^2 (b c-a d)^5 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{5 b^4 d^2}+\frac {219488 B^2 (b c-a d)^5 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{5 b^4 d^2}\\ \end {align*}

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Mathematica [A]
time = 0.47, size = 901, normalized size = 1.23 \begin {gather*} \frac {g i^3 \left (-5 (b c-a d) (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+4 b (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {5 B (b c-a d)^2 \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 )}{3 b^4}-\frac {B (b c-a d) \left (24 A b d (b c-a d)^3 x-12 B (b c-a d)^3 (b d x+(b c-a d) \log (a+b x))-4 B (b c-a d)^2 \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 )-B (b c-a d) \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )+24 B d (b c-a d)^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+12 b^2 (b c-a d)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+8 b^3 (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 b^4 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+24 (b c-a d)^4 \log (a+b x) \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)-12 B (b c-a d)^4 \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 )}{3 b^4}\right )}{20 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(g*i^3*(-5*(b*c - a*d)*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 4*b*(c + d*x)^5*(A + B*Log[(e*(a +
b*x))/(c + d*x)])^2 + (5*B*(b*c - a*d)^2*(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)])))/(3*b^4) - (B*(b*c - a*d)*(24*A*
b*d*(b*c - a*d)^3*x - 12*B*(b*c - a*d)^3*(b*d*x + (b*c - a*d)*Log[a + b*x]) - 4*B*(b*c - a*d)^2*(2*b*d*(b*c -
a*d)*x + b^2*(c + d*x)^2 + 2*(b*c - a*d)^2*Log[a + b*x]) - B*(b*c - a*d)*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c -
a*d)*(c + d*x)^2 + 2*b^3*(c + d*x)^3 + 6*(b*c - a*d)^3*Log[a + b*x]) + 24*B*d*(b*c - a*d)^3*(a + b*x)*Log[(e*
(a + b*x))/(c + d*x)] + 12*b^2*(b*c - a*d)^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 8*b^3*(b*c - a
*d)*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 6*b^4*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])
+ 24*(b*c - a*d)^4*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 24*B*(b*c - a*d)^4*Log[c + d*x] - 12*B*
(b*c - a*d)^4*(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)])))/(3*b^4)))/(20*d^2)

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Maple [F]
time = 0.34, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right ) \left (d i x +c i \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2508 vs. \(2 (659) = 1318\).
time = 0.46, size = 2508, normalized size = 3.44 \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)*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="maxima")

[Out]

-1/5*I*A^2*b*d^3*g*x^5 - 3/4*I*A^2*b*c*d^2*g*x^4 - 1/4*I*A^2*a*d^3*g*x^4 - I*A^2*b*c^2*d*g*x^3 - I*A^2*a*c*d^2
*g*x^3 - 1/2*I*A^2*b*c^3*g*x^2 - 3/2*I*A^2*a*c^2*d*g*x^2 - 2*I*(x*log(b*x*e/(d*x + c) + a*e/(d*x + c)) + a*log
(b*x + a)/b - c*log(d*x + c)/d)*A*B*a*c^3*g - I*(x^2*log(b*x*e/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b
^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*A*B*b*c^3*g - 3*I*(x^2*log(b*x*e/(d*x + c) + a*e/(d*x + c)) -
a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*A*B*a*c^2*d*g - I*(2*x^3*log(b*x*e/(d*x +
c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2
- a^2*d^2)*x)/(b^2*d^2))*A*B*b*c^2*d*g - I*(2*x^3*log(b*x*e/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^
3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*A*B*a*c*d^2*g - 1/
4*I*(6*x^4*log(b*x*e/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*
d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*A*B*b*c*d^2*g - 1/1
2*I*(6*x^4*log(b*x*e/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*
d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*A*B*a*d^3*g - 1/30*
I*(12*x^5*log(b*x*e/(d*x + c) + a*e/(d*x + c)) + 12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c
*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*
d^4)*x)/(b^4*d^4))*A*B*b*d^3*g - I*A^2*a*c^3*g*x + 1/60*(-I*b^4*c^5*g - I*a*b^3*c^4*d*g + 47*I*a^2*b^2*c^3*d^2
*g - 27*I*a^3*b*c^2*d^3*g + 6*I*a^4*c*d^4*g)*B^2*log(d*x + c)/(b^3*d^2) + 1/10*(-I*b^5*c^5*g + 5*I*a*b^4*c^4*d
*g - 10*I*a^2*b^3*c^3*d^2*g + 10*I*a^3*b^2*c^2*d^3*g - 5*I*a^4*b*c*d^4*g + I*a^5*d^5*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^4*d^2) - 1/60*(12*I*B^2*b^5*d^5*g*x^5 -
3*(-13*I*b^5*c*d^4*g - 7*I*a*b^4*d^5*g)*B^2*x^4 - 4*(-10*I*b^5*c^2*d^3*g - 19*I*a*b^4*c*d^4*g - I*a^2*b^3*d^5
*g)*B^2*x^3 + (11*I*b^5*c^3*d^2*g + 87*I*a*b^4*c^2*d^3*g + 27*I*a^2*b^3*c*d^4*g - 5*I*a^3*b^2*d^5*g)*B^2*x^2 +
(5*I*b^5*c^4*d*g + 2*I*a*b^4*c^3*d^2*g + 84*I*a^2*b^3*c^2*d^3*g - 38*I*a^3*b^2*c*d^4*g + 7*I*a^4*b*d^5*g)*B^2
*x - 3*(-4*I*B^2*b^5*d^5*g*x^5 - 20*I*B^2*a*b^4*c^3*d^2*g*x + 5*(-3*I*b^5*c*d^4*g - I*a*b^4*d^5*g)*B^2*x^4 + 2
0*(-I*b^5*c^2*d^3*g - I*a*b^4*c*d^4*g)*B^2*x^3 + 10*(-I*b^5*c^3*d^2*g - 3*I*a*b^4*c^2*d^3*g)*B^2*x^2 + (-10*I*
a^2*b^3*c^3*d^2*g + 10*I*a^3*b^2*c^2*d^3*g - 5*I*a^4*b*c*d^4*g + I*a^5*d^5*g)*B^2)*log(b*x + a)^2 - 3*(-4*I*B^
2*b^5*d^5*g*x^5 - 20*I*B^2*a*b^4*c^3*d^2*g*x + 5*(-3*I*b^5*c*d^4*g - I*a*b^4*d^5*g)*B^2*x^4 + 20*(-I*b^5*c^2*d
^3*g - I*a*b^4*c*d^4*g)*B^2*x^3 + 10*(-I*b^5*c^3*d^2*g - 3*I*a*b^4*c^2*d^3*g)*B^2*x^2 + (I*b^5*c^5*g - 5*I*a*b
^4*c^4*d*g)*B^2)*log(d*x + c)^2 + (24*I*B^2*b^5*d^5*g*x^5 - 12*(-7*I*b^5*c*d^4*g - 3*I*a*b^4*d^5*g)*B^2*x^4 -
2*(-49*I*b^5*c^2*d^3*g - 70*I*a*b^4*c*d^4*g - I*a^2*b^3*d^5*g)*B^2*x^3 - 3*(-11*I*b^5*c^3*d^2*g - 65*I*a*b^4*c
^2*d^3*g - 5*I*a^2*b^3*c*d^4*g + I*a^3*b^2*d^5*g)*B^2*x^2 - 6*(I*b^5*c^4*d*g - 15*I*a*b^4*c^3*d^2*g - 10*I*a^2
*b^3*c^2*d^3*g + 5*I*a^3*b^2*c*d^4*g - I*a^4*b*d^5*g)*B^2*x + (-6*I*a*b^4*c^4*d*g + 57*I*a^2*b^3*c^3*d^2*g - 3
7*I*a^3*b^2*c^2*d^3*g + 11*I*a^4*b*c*d^4*g - I*a^5*d^5*g)*B^2)*log(b*x + a) + (-24*I*B^2*b^5*d^5*g*x^5 - 12*(7
*I*b^5*c*d^4*g + 3*I*a*b^4*d^5*g)*B^2*x^4 - 2*(49*I*b^5*c^2*d^3*g + 70*I*a*b^4*c*d^4*g + I*a^2*b^3*d^5*g)*B^2*
x^3 - 3*(11*I*b^5*c^3*d^2*g + 65*I*a*b^4*c^2*d^3*g + 5*I*a^2*b^3*c*d^4*g - I*a^3*b^2*d^5*g)*B^2*x^2 - 6*(-I*b^
5*c^4*d*g + 15*I*a*b^4*c^3*d^2*g + 10*I*a^2*b^3*c^2*d^3*g - 5*I*a^3*b^2*c*d^4*g + I*a^4*b*d^5*g)*B^2*x - 6*(4*
I*B^2*b^5*d^5*g*x^5 + 20*I*B^2*a*b^4*c^3*d^2*g*x + 5*(3*I*b^5*c*d^4*g + I*a*b^4*d^5*g)*B^2*x^4 + 20*(I*b^5*c^2
*d^3*g + I*a*b^4*c*d^4*g)*B^2*x^3 + 10*(I*b^5*c^3*d^2*g + 3*I*a*b^4*c^2*d^3*g)*B^2*x^2 + (10*I*a^2*b^3*c^3*d^2
*g - 10*I*a^3*b^2*c^2*d^3*g + 5*I*a^4*b*c*d^4*g - I*a^5*d^5*g)*B^2)*log(b*x + a))*log(d*x + c))/(b^4*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)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="fricas")

[Out]

1/20*(-4*I*B^2*b*d^3*g*x^5 - 20*I*B^2*a*c^3*g*x - 5*(3*I*B^2*b*c*d^2 + I*B^2*a*d^3)*g*x^4 - 20*(I*B^2*b*c^2*d
+ I*B^2*a*c*d^2)*g*x^3 - 10*(I*B^2*b*c^3 + 3*I*B^2*a*c^2*d)*g*x^2)*log((b*x + a)*e/(d*x + c))^2 + integral(1/1
0*(-10*I*A^2*b^2*d^4*g*x^6 - 10*I*A^2*a^2*c^4*g - 20*(2*I*A^2*b^2*c*d^3 + I*A^2*a*b*d^4)*g*x^5 - 10*(6*I*A^2*b
^2*c^2*d^2 + 8*I*A^2*a*b*c*d^3 + I*A^2*a^2*d^4)*g*x^4 - 40*(I*A^2*b^2*c^3*d + 3*I*A^2*a*b*c^2*d^2 + I*A^2*a^2*
c*d^3)*g*x^3 - 10*(I*A^2*b^2*c^4 + 8*I*A^2*a*b*c^3*d + 6*I*A^2*a^2*c^2*d^2)*g*x^2 - 20*(I*A^2*a*b*c^4 + 2*I*A^
2*a^2*c^3*d)*g*x + (-20*I*A*B*b^2*d^4*g*x^6 - 20*I*A*B*a^2*c^4*g - 4*((20*I*A*B - I*B^2)*b^2*c*d^3 + (10*I*A*B
+ I*B^2)*a*b*d^4)*g*x^5 - 5*(3*(8*I*A*B - I*B^2)*b^2*c^2*d^2 + 2*(16*I*A*B + I*B^2)*a*b*c*d^3 + (4*I*A*B + I*
B^2)*a^2*d^4)*g*x^4 - 20*(12*I*A*B*a*b*c^2*d^2 + (4*I*A*B - I*B^2)*b^2*c^3*d + (4*I*A*B + I*B^2)*a^2*c*d^3)*g*
x^3 - 10*((2*I*A*B - I*B^2)*b^2*c^4 + 2*(8*I*A*B - I*B^2)*a*b*c^3*d + 3*(4*I*A*B + I*B^2)*a^2*c^2*d^2)*g*x^2 -
20*((2*I*A*B - I*B^2)*a*b*c^4 + (4*I*A*B + I*B^2)*a^2*c^3*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)**3*(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)^3*(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)^3*(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 )}^3\,{\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)^3*(A + B*log((e*(a + b*x))/(c + d*x)))^2,x)

[Out]

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

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