3.183 \(\int (a g+b g x)^3 (A+B \log (\frac{e (c+d x)}{a+b x}))^2 \, dx\)

Optimal. Leaf size=420 \[ -\frac{B^2 g^3 (b c-a d)^4 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{2 b d^4}+\frac{B g^3 (b c-a d)^4 \log \left (1-\frac{d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{2 b d^4}+\frac{B g^3 (c+d x) (b c-a d)^3 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{2 d^4}-\frac{B g^3 (a+b x)^2 (b c-a d)^2 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{4 b d^2}+\frac{B g^3 (a+b x)^3 (b c-a d) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{6 b d}+\frac{g^3 (a+b x)^4 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )^2}{4 b}-\frac{5 B^2 g^3 x (b c-a d)^3}{12 d^3}+\frac{B^2 g^3 (a+b x)^2 (b c-a d)^2}{12 b d^2}+\frac{11 B^2 g^3 (b c-a d)^4 \log (a+b x)}{12 b d^4}+\frac{5 B^2 g^3 (b c-a d)^4 \log \left (\frac{c+d x}{a+b x}\right )}{12 b d^4} \]

[Out]

(-5*B^2*(b*c - a*d)^3*g^3*x)/(12*d^3) + (B^2*(b*c - a*d)^2*g^3*(a + b*x)^2)/(12*b*d^2) + (11*B^2*(b*c - a*d)^4
*g^3*Log[a + b*x])/(12*b*d^4) + (5*B^2*(b*c - a*d)^4*g^3*Log[(c + d*x)/(a + b*x)])/(12*b*d^4) - (B*(b*c - a*d)
^2*g^3*(a + b*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(4*b*d^2) + (B*(b*c - a*d)*g^3*(a + b*x)^3*(A + B*Log
[(e*(c + d*x))/(a + b*x)]))/(6*b*d) + (B*(b*c - a*d)^3*g^3*(c + d*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(2*
d^4) + (g^3*(a + b*x)^4*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2)/(4*b) + (B*(b*c - a*d)^4*g^3*(A + B*Log[(e*(c
+ d*x))/(a + b*x)])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/(2*b*d^4) - (B^2*(b*c - a*d)^4*g^3*PolyLog[2, (d*(a
+ b*x))/(b*(c + d*x))])/(2*b*d^4)

________________________________________________________________________________________

Rubi [A]  time = 0.648144, antiderivative size = 474, normalized size of antiderivative = 1.13, number of steps used = 24, number of rules used = 13, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.406, Rules used = {2525, 12, 2528, 2486, 31, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac{B^2 g^3 (b c-a d)^4 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{2 b d^4}-\frac{B g^3 (b c-a d)^4 \log (c+d x) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{2 b d^4}-\frac{B g^3 (a+b x)^2 (b c-a d)^2 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{4 b d^2}+\frac{A B g^3 x (b c-a d)^3}{2 d^3}+\frac{B g^3 (a+b x)^3 (b c-a d) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{6 b d}+\frac{g^3 (a+b x)^4 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )^2}{4 b}+\frac{B^2 g^3 (a+b x) (b c-a d)^3 \log \left (\frac{e (c+d x)}{a+b x}\right )}{2 b d^3}-\frac{5 B^2 g^3 x (b c-a d)^3}{12 d^3}+\frac{B^2 g^3 (a+b x)^2 (b c-a d)^2}{12 b d^2}+\frac{B^2 g^3 (b c-a d)^4 \log ^2(c+d x)}{4 b d^4}+\frac{11 B^2 g^3 (b c-a d)^4 \log (c+d x)}{12 b d^4}-\frac{B^2 g^3 (b c-a d)^4 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{2 b d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*B*(b*c - a*d)^3*g^3*x)/(2*d^3) - (5*B^2*(b*c - a*d)^3*g^3*x)/(12*d^3) + (B^2*(b*c - a*d)^2*g^3*(a + b*x)^2)
/(12*b*d^2) + (11*B^2*(b*c - a*d)^4*g^3*Log[c + d*x])/(12*b*d^4) - (B^2*(b*c - a*d)^4*g^3*Log[-((d*(a + b*x))/
(b*c - a*d))]*Log[c + d*x])/(2*b*d^4) + (B^2*(b*c - a*d)^4*g^3*Log[c + d*x]^2)/(4*b*d^4) + (B^2*(b*c - a*d)^3*
g^3*(a + b*x)*Log[(e*(c + d*x))/(a + b*x)])/(2*b*d^3) - (B*(b*c - a*d)^2*g^3*(a + b*x)^2*(A + B*Log[(e*(c + d*
x))/(a + b*x)]))/(4*b*d^2) + (B*(b*c - a*d)*g^3*(a + b*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(6*b*d) - (B
*(b*c - a*d)^4*g^3*Log[c + d*x]*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(2*b*d^4) + (g^3*(a + b*x)^4*(A + B*Log[
(e*(c + d*x))/(a + b*x)])^2)/(4*b) - (B^2*(b*c - a*d)^4*g^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(2*b*d^4)

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 12

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

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2486

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 43

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

Rule 2524

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2394

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

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

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 2390

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin{align*} \int (a g+b g x)^3 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2 \, dx &=\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{4 b}-\frac{B \int \frac{(b c-a d) g^4 (a+b x)^3 \left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{c+d x} \, dx}{2 b g}\\ &=\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{4 b}-\frac{\left (B (b c-a d) g^3\right ) \int \frac{(a+b x)^3 \left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{c+d x} \, dx}{2 b}\\ &=\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{4 b}-\frac{\left (B (b c-a d) g^3\right ) \int \left (\frac{b (b c-a d)^2 \left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{d^3}-\frac{b (b c-a d) (a+b x) \left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{d^2}+\frac{b (a+b x)^2 \left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{d}+\frac{(-b c+a d)^3 \left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{d^3 (c+d x)}\right ) \, dx}{2 b}\\ &=\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{4 b}-\frac{\left (B (b c-a d) g^3\right ) \int (a+b x)^2 \left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \, dx}{2 d}+\frac{\left (B (b c-a d)^2 g^3\right ) \int (a+b x) \left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \, dx}{2 d^2}-\frac{\left (B (b c-a d)^3 g^3\right ) \int \left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )\right ) \, dx}{2 d^3}+\frac{\left (B (b c-a d)^4 g^3\right ) \int \frac{-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )}{c+d x} \, dx}{2 b d^3}\\ &=\frac{A B (b c-a d)^3 g^3 x}{2 d^3}-\frac{B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{4 b d^2}+\frac{B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{6 b d}-\frac{B (b c-a d)^4 g^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{2 b d^4}+\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{4 b}-\frac{\left (B^2 (b c-a d) g^3\right ) \int \frac{(-b c+a d) (a+b x)^2}{c+d x} \, dx}{6 b d}+\frac{\left (B^2 (b c-a d)^2 g^3\right ) \int \frac{(b c-a d) (-a-b x)}{c+d x} \, dx}{4 b d^2}+\frac{\left (B^2 (b c-a d)^3 g^3\right ) \int \log \left (\frac{e (c+d x)}{a+b x}\right ) \, dx}{2 d^3}+\frac{\left (B^2 (b c-a d)^4 g^3\right ) \int \frac{(a+b x) \left (\frac{d e}{a+b x}-\frac{b e (c+d x)}{(a+b x)^2}\right ) \log (c+d x)}{e (c+d x)} \, dx}{2 b d^4}\\ &=\frac{A B (b c-a d)^3 g^3 x}{2 d^3}+\frac{B^2 (b c-a d)^3 g^3 (a+b x) \log \left (\frac{e (c+d x)}{a+b x}\right )}{2 b d^3}-\frac{B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{4 b d^2}+\frac{B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{6 b d}-\frac{B (b c-a d)^4 g^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{2 b d^4}+\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{4 b}+\frac{\left (B^2 (b c-a d)^2 g^3\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{6 b d}+\frac{\left (B^2 (b c-a d)^3 g^3\right ) \int \frac{-a-b x}{c+d x} \, dx}{4 b d^2}+\frac{\left (B^2 (b c-a d)^4 g^3\right ) \int \frac{1}{c+d x} \, dx}{2 b d^3}+\frac{\left (B^2 (b c-a d)^4 g^3\right ) \int \frac{(a+b x) \left (\frac{d e}{a+b x}-\frac{b e (c+d x)}{(a+b x)^2}\right ) \log (c+d x)}{c+d x} \, dx}{2 b d^4 e}\\ &=\frac{A B (b c-a d)^3 g^3 x}{2 d^3}+\frac{B^2 (b c-a d)^4 g^3 \log (c+d x)}{2 b d^4}+\frac{B^2 (b c-a d)^3 g^3 (a+b x) \log \left (\frac{e (c+d x)}{a+b x}\right )}{2 b d^3}-\frac{B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{4 b d^2}+\frac{B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{6 b d}-\frac{B (b c-a d)^4 g^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{2 b d^4}+\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{4 b}+\frac{\left (B^2 (b c-a d)^2 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}{6 b d}+\frac{\left (B^2 (b c-a d)^3 g^3\right ) \int \left (-\frac{b}{d}+\frac{b c-a d}{d (c+d x)}\right ) \, dx}{4 b d^2}+\frac{\left (B^2 (b c-a d)^4 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}{2 b d^4 e}\\ &=\frac{A B (b c-a d)^3 g^3 x}{2 d^3}-\frac{5 B^2 (b c-a d)^3 g^3 x}{12 d^3}+\frac{B^2 (b c-a d)^2 g^3 (a+b x)^2}{12 b d^2}+\frac{11 B^2 (b c-a d)^4 g^3 \log (c+d x)}{12 b d^4}+\frac{B^2 (b c-a d)^3 g^3 (a+b x) \log \left (\frac{e (c+d x)}{a+b x}\right )}{2 b d^3}-\frac{B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{4 b d^2}+\frac{B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{6 b d}-\frac{B (b c-a d)^4 g^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{2 b d^4}+\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{4 b}-\frac{\left (B^2 (b c-a d)^4 g^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{2 d^4}+\frac{\left (B^2 (b c-a d)^4 g^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{2 b d^3}\\ &=\frac{A B (b c-a d)^3 g^3 x}{2 d^3}-\frac{5 B^2 (b c-a d)^3 g^3 x}{12 d^3}+\frac{B^2 (b c-a d)^2 g^3 (a+b x)^2}{12 b d^2}+\frac{11 B^2 (b c-a d)^4 g^3 \log (c+d x)}{12 b d^4}-\frac{B^2 (b c-a d)^4 g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b d^4}+\frac{B^2 (b c-a d)^3 g^3 (a+b x) \log \left (\frac{e (c+d x)}{a+b x}\right )}{2 b d^3}-\frac{B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{4 b d^2}+\frac{B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{6 b d}-\frac{B (b c-a d)^4 g^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{2 b d^4}+\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{4 b}+\frac{\left (B^2 (b c-a d)^4 g^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{2 b d^4}+\frac{\left (B^2 (b c-a d)^4 g^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2 b d^3}\\ &=\frac{A B (b c-a d)^3 g^3 x}{2 d^3}-\frac{5 B^2 (b c-a d)^3 g^3 x}{12 d^3}+\frac{B^2 (b c-a d)^2 g^3 (a+b x)^2}{12 b d^2}+\frac{11 B^2 (b c-a d)^4 g^3 \log (c+d x)}{12 b d^4}-\frac{B^2 (b c-a d)^4 g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b d^4}+\frac{B^2 (b c-a d)^4 g^3 \log ^2(c+d x)}{4 b d^4}+\frac{B^2 (b c-a d)^3 g^3 (a+b x) \log \left (\frac{e (c+d x)}{a+b x}\right )}{2 b d^3}-\frac{B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{4 b d^2}+\frac{B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{6 b d}-\frac{B (b c-a d)^4 g^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{2 b d^4}+\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{4 b}+\frac{\left (B^2 (b c-a d)^4 g^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2 b d^4}\\ &=\frac{A B (b c-a d)^3 g^3 x}{2 d^3}-\frac{5 B^2 (b c-a d)^3 g^3 x}{12 d^3}+\frac{B^2 (b c-a d)^2 g^3 (a+b x)^2}{12 b d^2}+\frac{11 B^2 (b c-a d)^4 g^3 \log (c+d x)}{12 b d^4}-\frac{B^2 (b c-a d)^4 g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b d^4}+\frac{B^2 (b c-a d)^4 g^3 \log ^2(c+d x)}{4 b d^4}+\frac{B^2 (b c-a d)^3 g^3 (a+b x) \log \left (\frac{e (c+d x)}{a+b x}\right )}{2 b d^3}-\frac{B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{4 b d^2}+\frac{B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{6 b d}-\frac{B (b c-a d)^4 g^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{2 b d^4}+\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{4 b}-\frac{B^2 (b c-a d)^4 g^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2 b d^4}\\ \end{align*}

Mathematica [A]  time = 0.35497, size = 392, normalized size = 0.93 \[ \frac{g^3 \left (\frac{B (b c-a d) \left (-3 B (b c-a d)^3 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 d^3 (a+b x)^3 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )+3 d^2 (a+b x)^2 (a d-b c) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )-6 (b c-a d)^3 \log (c+d x) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )+6 A b d x (b c-a d)^2-B (b c-a d) \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )+6 b B (c+d x) (b c-a d)^2 \log \left (\frac{e (c+d x)}{a+b x}\right )+6 B (b c-a d)^3 \log (a+b x)-3 B (b c-a d)^2 ((a d-b c) \log (c+d x)+b d x)\right )}{3 d^4}+(a+b x)^4 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )^2\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.958, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{3} \left ( A+B\ln \left ({\frac{e \left ( dx+c \right ) }{bx+a}} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.68998, size = 2342, normalized size = 5.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*A^2*b^3*g^3*x^4 + A^2*a*b^2*g^3*x^3 + 3/2*A^2*a^2*b*g^3*x^2 + 2*(x*log(d*e*x/(b*x + a) + c*e/(b*x + a)) -
a*log(b*x + a)/b + c*log(d*x + c)/d)*A*B*a^3*g^3 + 3*(x^2*log(d*e*x/(b*x + a) + c*e/(b*x + a)) + 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^2*b*g^3 + (2*x^3*log(d*e*x/(b*x + a) + c*e/(b*x +
a)) - 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*g^3 + 1/12*(6*x^4*log(d*e*x/(b*x + a) + c*e/(b*x + a)) + 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*g^3 + A^2*a^3*g^3*x - 1/12*((6*g^3*log(e) - 11*g^3)*b^3*c^4 - 2*(12*g^3*log(e) - 19*g^3)*a*b^2*
c^3*d + 9*(4*g^3*log(e) - 5*g^3)*a^2*b*c^2*d^2 - 6*(4*g^3*log(e) - 3*g^3)*a^3*c*d^3)*B^2*log(d*x + c)/d^4 + 1/
2*(b^4*c^4*g^3 - 4*a*b^3*c^3*d*g^3 + 6*a^2*b^2*c^2*d^2*g^3 - 4*a^3*b*c*d^3*g^3 + a^4*d^4*g^3)*(log(b*x + a)*lo
g((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B^2/(b*d^4) + 1/12*(3*B^2*b^4*d^4*g^3*x^
4*log(e)^2 + 2*(b^4*c*d^3*g^3*log(e) + (6*g^3*log(e)^2 - g^3*log(e))*a*b^3*d^4)*B^2*x^3 - ((3*g^3*log(e) - g^3
)*b^4*c^2*d^2 - 2*(6*g^3*log(e) - g^3)*a*b^3*c*d^3 - (18*g^3*log(e)^2 - 9*g^3*log(e) + g^3)*a^2*b^2*d^4)*B^2*x
^2 + ((6*g^3*log(e) - 5*g^3)*b^4*c^3*d - (24*g^3*log(e) - 17*g^3)*a*b^3*c^2*d^2 + (36*g^3*log(e) - 19*g^3)*a^2
*b^2*c*d^3 + (12*g^3*log(e)^2 - 18*g^3*log(e) + 7*g^3)*a^3*b*d^4)*B^2*x + 3*(B^2*b^4*d^4*g^3*x^4 + 4*B^2*a*b^3
*d^4*g^3*x^3 + 6*B^2*a^2*b^2*d^4*g^3*x^2 + 4*B^2*a^3*b*d^4*g^3*x + B^2*a^4*d^4*g^3)*log(b*x + a)^2 + 3*(B^2*b^
4*d^4*g^3*x^4 + 4*B^2*a*b^3*d^4*g^3*x^3 + 6*B^2*a^2*b^2*d^4*g^3*x^2 + 4*B^2*a^3*b*d^4*g^3*x - (b^4*c^4*g^3 - 4
*a*b^3*c^3*d*g^3 + 6*a^2*b^2*c^2*d^2*g^3 - 4*a^3*b*c*d^3*g^3)*B^2)*log(d*x + c)^2 - (6*B^2*b^4*d^4*g^3*x^4*log
(e) + 2*(b^4*c*d^3*g^3 + (12*g^3*log(e) - g^3)*a*b^3*d^4)*B^2*x^3 - 3*(b^4*c^2*d^2*g^3 - 4*a*b^3*c*d^3*g^3 - 3
*(4*g^3*log(e) - g^3)*a^2*b^2*d^4)*B^2*x^2 + 6*(b^4*c^3*d*g^3 - 4*a*b^3*c^2*d^2*g^3 + 6*a^2*b^2*c*d^3*g^3 + (4
*g^3*log(e) - 3*g^3)*a^3*b*d^4)*B^2*x + (6*a*b^3*c^3*d*g^3 - 21*a^2*b^2*c^2*d^2*g^3 + 26*a^3*b*c*d^3*g^3 + (6*
g^3*log(e) - 11*g^3)*a^4*d^4)*B^2)*log(b*x + a) + (6*B^2*b^4*d^4*g^3*x^4*log(e) + 2*(b^4*c*d^3*g^3 + (12*g^3*l
og(e) - g^3)*a*b^3*d^4)*B^2*x^3 - 3*(b^4*c^2*d^2*g^3 - 4*a*b^3*c*d^3*g^3 - 3*(4*g^3*log(e) - g^3)*a^2*b^2*d^4)
*B^2*x^2 + 6*(b^4*c^3*d*g^3 - 4*a*b^3*c^2*d^2*g^3 + 6*a^2*b^2*c*d^3*g^3 + (4*g^3*log(e) - 3*g^3)*a^3*b*d^4)*B^
2*x - 6*(B^2*b^4*d^4*g^3*x^4 + 4*B^2*a*b^3*d^4*g^3*x^3 + 6*B^2*a^2*b^2*d^4*g^3*x^2 + 4*B^2*a^3*b*d^4*g^3*x + B
^2*a^4*d^4*g^3)*log(b*x + a))*log(d*x + c))/(b*d^4)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} b^{3} g^{3} x^{3} + 3 \, A^{2} a b^{2} g^{3} x^{2} + 3 \, A^{2} a^{2} b g^{3} x + A^{2} a^{3} g^{3} +{\left (B^{2} b^{3} g^{3} x^{3} + 3 \, B^{2} a b^{2} g^{3} x^{2} + 3 \, B^{2} a^{2} b g^{3} x + B^{2} a^{3} g^{3}\right )} \log \left (\frac{d e x + c e}{b x + a}\right )^{2} + 2 \,{\left (A B b^{3} g^{3} x^{3} + 3 \, A B a b^{2} g^{3} x^{2} + 3 \, A B a^{2} b g^{3} x + A B a^{3} g^{3}\right )} \log \left (\frac{d e x + c e}{b x + a}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b g x + a g\right )}^{3}{\left (B \log \left (\frac{{\left (d x + c\right )} e}{b x + a}\right ) + A\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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