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

Optimal. Leaf size=361 \[ \frac{3 b^2 B g^3 (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d^4 i^3}+\frac{b^2 g^3 (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (3 B \log \left (\frac{e (a+b x)}{c+d x}\right )+3 A+B\right )}{d^4 i^3}+\frac{g^3 (a+b x)^2 (b c-a d) \left (3 B \log \left (\frac{e (a+b x)}{c+d x}\right )+3 A+B\right )}{2 d^2 i^3 (c+d x)^2}+\frac{b g^3 (3 A+B) (a+b x) (b c-a d)}{d^3 i^3 (c+d x)}+\frac{g^3 (a+b x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d i^3 (c+d x)^2}+\frac{3 b B g^3 (a+b x) (b c-a d) \log \left (\frac{e (a+b x)}{c+d x}\right )}{d^3 i^3 (c+d x)}-\frac{3 b B g^3 (a+b x) (b c-a d)}{d^3 i^3 (c+d x)}-\frac{3 B g^3 (a+b x)^2 (b c-a d)}{4 d^2 i^3 (c+d x)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.73376, antiderivative size = 442, normalized size of antiderivative = 1.22, number of steps used = 22, number of rules used = 13, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.325, Rules used = {2528, 2486, 31, 2525, 12, 44, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac{3 b^2 B g^3 (b c-a d) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^4 i^3}-\frac{3 b^2 g^3 (b c-a d) \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^4 i^3}-\frac{3 b g^3 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^4 i^3 (c+d x)}+\frac{g^3 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 d^4 i^3 (c+d x)^2}+\frac{b^2 B g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{d^3 i^3}-\frac{3 b^2 B g^3 (b c-a d) \log ^2(c+d x)}{2 d^4 i^3}+\frac{5 b^2 B g^3 (b c-a d) \log (a+b x)}{2 d^4 i^3}+\frac{3 b^2 B g^3 (b c-a d) \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d^4 i^3}-\frac{7 b^2 B g^3 (b c-a d) \log (c+d x)}{2 d^4 i^3}+\frac{5 b B g^3 (b c-a d)^2}{2 d^4 i^3 (c+d x)}-\frac{B g^3 (b c-a d)^3}{4 d^4 i^3 (c+d x)^2}+\frac{A b^3 g^3 x}{d^3 i^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b^3*g^3*x)/(d^3*i^3) - (B*(b*c - a*d)^3*g^3)/(4*d^4*i^3*(c + d*x)^2) + (5*b*B*(b*c - a*d)^2*g^3)/(2*d^4*i^3
*(c + d*x)) + (5*b^2*B*(b*c - a*d)*g^3*Log[a + b*x])/(2*d^4*i^3) + (b^2*B*g^3*(a + b*x)*Log[(e*(a + b*x))/(c +
d*x)])/(d^3*i^3) + ((b*c - a*d)^3*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d^4*i^3*(c + d*x)^2) - (3*b*(b
*c - a*d)^2*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^4*i^3*(c + d*x)) - (7*b^2*B*(b*c - a*d)*g^3*Log[c + d
*x])/(2*d^4*i^3) + (3*b^2*B*(b*c - a*d)*g^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(d^4*i^3) - (3*b^2
*(b*c - a*d)*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(d^4*i^3) - (3*b^2*B*(b*c - a*d)*g^3*Log[c
+ d*x]^2)/(2*d^4*i^3) + (3*b^2*B*(b*c - a*d)*g^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(d^4*i^3)

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 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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[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 \frac{(a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(47 c+47 d x)^3} \, dx &=\int \left (\frac{b^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{103823 d^3}+\frac{(-b c+a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{103823 d^3 (c+d x)^3}+\frac{3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{103823 d^3 (c+d x)^2}-\frac{3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{103823 d^3 (c+d x)}\right ) \, dx\\ &=\frac{\left (b^3 g^3\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{103823 d^3}-\frac{\left (3 b^2 (b c-a d) g^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{103823 d^3}+\frac{\left (3 b (b c-a d)^2 g^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{103823 d^3}-\frac{\left ((b c-a d)^3 g^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{103823 d^3}\\ &=\frac{A b^3 g^3 x}{103823 d^3}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{207646 d^4 (c+d x)^2}-\frac{3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{103823 d^4 (c+d x)}-\frac{3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{103823 d^4}+\frac{\left (b^3 B g^3\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{103823 d^3}+\frac{\left (3 b^2 B (b c-a d) g^3\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{103823 d^4}+\frac{\left (3 b B (b c-a d)^2 g^3\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{103823 d^4}-\frac{\left (B (b c-a d)^3 g^3\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^3} \, dx}{207646 d^4}\\ &=\frac{A b^3 g^3 x}{103823 d^3}+\frac{b^2 B g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{103823 d^3}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{207646 d^4 (c+d x)^2}-\frac{3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{103823 d^4 (c+d x)}-\frac{3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{103823 d^4}-\frac{\left (b^2 B (b c-a d) g^3\right ) \int \frac{1}{c+d x} \, dx}{103823 d^3}+\frac{\left (3 b B (b c-a d)^3 g^3\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{103823 d^4}-\frac{\left (B (b c-a d)^4 g^3\right ) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{207646 d^4}+\frac{\left (3 b^2 B (b c-a d) g^3\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{103823 d^4 e}\\ &=\frac{A b^3 g^3 x}{103823 d^3}+\frac{b^2 B g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{103823 d^3}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{207646 d^4 (c+d x)^2}-\frac{3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{103823 d^4 (c+d x)}-\frac{b^2 B (b c-a d) g^3 \log (c+d x)}{103823 d^4}-\frac{3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{103823 d^4}+\frac{\left (3 b B (b c-a d)^3 g^3\right ) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{103823 d^4}-\frac{\left (B (b c-a d)^4 g^3\right ) \int \left (\frac{b^3}{(b c-a d)^3 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^3}-\frac{b d}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{207646 d^4}+\frac{\left (3 b^2 B (b c-a d) 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}{103823 d^4 e}\\ &=\frac{A b^3 g^3 x}{103823 d^3}-\frac{B (b c-a d)^3 g^3}{415292 d^4 (c+d x)^2}+\frac{5 b B (b c-a d)^2 g^3}{207646 d^4 (c+d x)}+\frac{5 b^2 B (b c-a d) g^3 \log (a+b x)}{207646 d^4}+\frac{b^2 B g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{103823 d^3}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{207646 d^4 (c+d x)^2}-\frac{3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{103823 d^4 (c+d x)}-\frac{7 b^2 B (b c-a d) g^3 \log (c+d x)}{207646 d^4}-\frac{3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{103823 d^4}+\frac{\left (3 b^3 B (b c-a d) g^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{103823 d^4}-\frac{\left (3 b^2 B (b c-a d) g^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{103823 d^3}\\ &=\frac{A b^3 g^3 x}{103823 d^3}-\frac{B (b c-a d)^3 g^3}{415292 d^4 (c+d x)^2}+\frac{5 b B (b c-a d)^2 g^3}{207646 d^4 (c+d x)}+\frac{5 b^2 B (b c-a d) g^3 \log (a+b x)}{207646 d^4}+\frac{b^2 B g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{103823 d^3}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{207646 d^4 (c+d x)^2}-\frac{3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{103823 d^4 (c+d x)}-\frac{7 b^2 B (b c-a d) g^3 \log (c+d x)}{207646 d^4}+\frac{3 b^2 B (b c-a d) g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{103823 d^4}-\frac{3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{103823 d^4}-\frac{\left (3 b^2 B (b c-a d) g^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{103823 d^4}-\frac{\left (3 b^2 B (b c-a d) g^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{103823 d^3}\\ &=\frac{A b^3 g^3 x}{103823 d^3}-\frac{B (b c-a d)^3 g^3}{415292 d^4 (c+d x)^2}+\frac{5 b B (b c-a d)^2 g^3}{207646 d^4 (c+d x)}+\frac{5 b^2 B (b c-a d) g^3 \log (a+b x)}{207646 d^4}+\frac{b^2 B g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{103823 d^3}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{207646 d^4 (c+d x)^2}-\frac{3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{103823 d^4 (c+d x)}-\frac{7 b^2 B (b c-a d) g^3 \log (c+d x)}{207646 d^4}+\frac{3 b^2 B (b c-a d) g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{103823 d^4}-\frac{3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{103823 d^4}-\frac{3 b^2 B (b c-a d) g^3 \log ^2(c+d x)}{207646 d^4}-\frac{\left (3 b^2 B (b c-a d) 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 )}{103823 d^4}\\ &=\frac{A b^3 g^3 x}{103823 d^3}-\frac{B (b c-a d)^3 g^3}{415292 d^4 (c+d x)^2}+\frac{5 b B (b c-a d)^2 g^3}{207646 d^4 (c+d x)}+\frac{5 b^2 B (b c-a d) g^3 \log (a+b x)}{207646 d^4}+\frac{b^2 B g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{103823 d^3}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{207646 d^4 (c+d x)^2}-\frac{3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{103823 d^4 (c+d x)}-\frac{7 b^2 B (b c-a d) g^3 \log (c+d x)}{207646 d^4}+\frac{3 b^2 B (b c-a d) g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{103823 d^4}-\frac{3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{103823 d^4}-\frac{3 b^2 B (b c-a d) g^3 \log ^2(c+d x)}{207646 d^4}+\frac{3 b^2 B (b c-a d) g^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{103823 d^4}\\ \end{align*}

Mathematica [A]  time = 0.466764, size = 317, normalized size = 0.88 \[ \frac{g^3 \left (6 b^2 B (b c-a d) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-12 b^2 (b c-a d) \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-\frac{12 b (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{c+d x}+\frac{2 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{(c+d x)^2}+4 b^2 B d (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )+10 b^2 B (b c-a d) \log (a+b x)-14 b^2 B (b c-a d) \log (c+d x)+\frac{10 b B (b c-a d)^2}{c+d x}-\frac{B (b c-a d)^3}{(c+d x)^2}+4 A b^3 d x\right )}{4 d^4 i^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^3*(4*A*b^3*d*x - (B*(b*c - a*d)^3)/(c + d*x)^2 + (10*b*B*(b*c - a*d)^2)/(c + d*x) + 10*b^2*B*(b*c - a*d)*Lo
g[a + b*x] + 4*b^2*B*d*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + (2*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c +
d*x)]))/(c + d*x)^2 - (12*b*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x) - 14*b^2*B*(b*c - a
*d)*Log[c + d*x] - 12*b^2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] + 6*b^2*B*(b*c - a*d)*
((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)]))
)/(4*d^4*i^3)

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Maple [B]  time = 0.157, size = 1815, normalized size = 5. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/d*g^3/i^3*A/(d*x+c)^2*a^3+1/4/d*g^3/i^3*B/(d*x+c)^2*a^3+5/2/d^4*g^3/i^3*A*b^3*c-9/4/d^4*g^3/i^3*B*b^3*c-5
/2/d^3*g^3/i^3*A*b^2*a+6/d^3*g^3/i^3*B*b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)*c*a+3/2/d^2*g^3/i^3*B*ln(b*
e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)^2*a^2*b*c+e/d^3*g^3/i^3*B*b^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*
a-e/(d*x+c)*b*c)*a-e/d^4*g^3/i^3*B*b^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*c-3/2/d^3
*g^3/i^3*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)^2*a*b^2*c^2+e/d^4*g^3/i^3*B*b^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x
+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)/(d*x+c)*c^2-3/4/d^2*g^3/i^3*B/(d*x+c)^2*a^2*b*c+6/d^3*g^3/i^3*A*b^2/(d*x+c)
*a*c-5/d^3*g^3/i^3*B*b^2/(d*x+c)*a*c+3/4/d^3*g^3/i^3*B/(d*x+c)^2*a*b^2*c^2+3/2/d^2*g^3/i^3*A/(d*x+c)^2*a^2*b*c
-3/d^2*g^3/i^3*B*b*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)*a^2+1/2/d^4*g^3/i^3*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c
))/(d*x+c)^2*b^3*c^3-3/d^3*g^3/i^3*B*b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-
b*e)/b/e)*a-3/2/d^3*g^3/i^3*A/(d*x+c)^2*a*b^2*c^2+e/d^3*g^3/i^3*A*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a+3/d^4*g^
3/i^3*B*b^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)/b/e)*c-3/d^4*g^3/i^3*B*b
^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)*c^2-e/d^4*g^3/i^3*A*b^4/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*c+9/4/d^3*g^3
/i^3*B*b^2*a+e/d^2*g^3/i^3*B*b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)/(d*x+c)*a^2-3/d
^2*g^3/i^3*A*b/(d*x+c)*a^2+5/2/d^2*g^3/i^3*B*b/(d*x+c)*a^2-3/d^4*g^3/i^3*A*b^3/(d*x+c)*c^2+1/2/d^4*g^3/i^3*A/(
d*x+c)^2*b^3*c^3-3/d^3*g^3/i^3*B*b^2*dilog(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)/b/e)*a+3/d^4*g^3/i^3*B*b^3*d
ilog(-(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)/b/e)*c-3/d^3*g^3/i^3*A*b^2*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)
*a+3/d^4*g^3/i^3*A*b^3*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*c-1/d^3*g^3/i^3*B*b^2*ln(d*(b*e/d+(a*d-b*c)*e/d
/(d*x+c))-b*e)*a+1/d^4*g^3/i^3*B*b^3*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*c-5/2/d^3*g^3/i^3*B*b^2*ln(b*e/d+
(a*d-b*c)*e/d/(d*x+c))*a+5/2/d^4*g^3/i^3*B*b^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c+5/2/d^4*g^3/i^3*B*b^3/(d*x+c)
*c^2-1/4/d^4*g^3/i^3*B/(d*x+c)^2*b^3*c^3-1/2/d*g^3/i^3*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)^2*a^3-2*e/d^3
*g^3/i^3*B*b^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)/(d*x+c)*c*a

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Maxima [B]  time = 1.81583, size = 2750, normalized size = 7.62 \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*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^3,x, algorithm="maxima")

[Out]

-3/4*B*a^2*b*g^3*(2*(2*d*x + c)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(d^4*i^3*x^2 + 2*c*d^3*i^3*x + c^2*d^2*i^
3) - (b*c^2 - 3*a*c*d + 2*(b*c*d - 2*a*d^2)*x)/((b*c*d^4 - a*d^5)*i^3*x^2 + 2*(b*c^2*d^3 - a*c*d^4)*i^3*x + (b
*c^3*d^2 - a*c^2*d^3)*i^3) - 2*(b^2*c - 2*a*b*d)*log(b*x + a)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*i^3) + 2*
(b^2*c - 2*a*b*d)*log(d*x + c)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*i^3)) - 1/2*A*b^3*g^3*((6*c^2*d*x + 5*c^
3)/(d^6*i^3*x^2 + 2*c*d^5*i^3*x + c^2*d^4*i^3) - 2*x/(d^3*i^3) + 6*c*log(d*x + c)/(d^4*i^3)) + 1/4*B*a^3*g^3*(
(2*b*d*x + 3*b*c - a*d)/((b*c*d^3 - a*d^4)*i^3*x^2 + 2*(b*c^2*d^2 - a*c*d^3)*i^3*x + (b*c^3*d - a*c^2*d^2)*i^3
) - 2*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(d^3*i^3*x^2 + 2*c*d^2*i^3*x + c^2*d*i^3) + 2*b^2*log(b*x + a)/((b^
2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*i^3) - 2*b^2*log(d*x + c)/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*i^3)) + 3/2*A*
a*b^2*g^3*((4*c*d*x + 3*c^2)/(d^5*i^3*x^2 + 2*c*d^4*i^3*x + c^2*d^3*i^3) + 2*log(d*x + c)/(d^3*i^3)) - 3/2*(2*
d*x + c)*A*a^2*b*g^3/(d^4*i^3*x^2 + 2*c*d^3*i^3*x + c^2*d^2*i^3) - 1/2*A*a^3*g^3/(d^3*i^3*x^2 + 2*c*d^2*i^3*x
+ c^2*d*i^3) + 1/2*(6*a^3*b^2*d^3*g^3*log(e) - (6*g^3*log(e) + 7*g^3)*b^5*c^3 + (18*g^3*log(e) + 19*g^3)*a*b^4
*c^2*d - 2*(9*g^3*log(e) + 7*g^3)*a^2*b^3*c*d^2)*B*log(d*x + c)/(b^2*c^2*d^4*i^3 - 2*a*b*c*d^5*i^3 + a^2*d^6*i
^3) + 1/4*(4*(b^5*c^2*d^3*g^3*log(e) - 2*a*b^4*c*d^4*g^3*log(e) + a^2*b^3*d^5*g^3*log(e))*B*x^3 + 8*(b^5*c^3*d
^2*g^3*log(e) - 2*a*b^4*c^2*d^3*g^3*log(e) + a^2*b^3*c*d^4*g^3*log(e))*B*x^2 - 2*((4*g^3*log(e) - 5*g^3)*b^5*c
^4*d - 20*(g^3*log(e) - g^3)*a*b^4*c^3*d^2 + (28*g^3*log(e) - 27*g^3)*a^2*b^3*c^2*d^3 - 12*(g^3*log(e) - g^3)*
a^3*b^2*c*d^4)*B*x + 6*((b^5*c^3*d^2*g^3 - 3*a*b^4*c^2*d^3*g^3 + 3*a^2*b^3*c*d^4*g^3 - a^3*b^2*d^5*g^3)*B*x^2
+ 2*(b^5*c^4*d*g^3 - 3*a*b^4*c^3*d^2*g^3 + 3*a^2*b^3*c^2*d^3*g^3 - a^3*b^2*c*d^4*g^3)*B*x + (b^5*c^5*g^3 - 3*a
*b^4*c^4*d*g^3 + 3*a^2*b^3*c^3*d^2*g^3 - a^3*b^2*c^2*d^3*g^3)*B)*log(d*x + c)^2 - ((10*g^3*log(e) - 9*g^3)*b^5
*c^5 - (38*g^3*log(e) - 35*g^3)*a*b^4*c^4*d + (46*g^3*log(e) - 47*g^3)*a^2*b^3*c^3*d^2 - 3*(6*g^3*log(e) - 7*g
^3)*a^3*b^2*c^2*d^3)*B + 2*(2*(b^5*c^2*d^3*g^3 - 2*a*b^4*c*d^4*g^3 + a^2*b^3*d^5*g^3)*B*x^3 + (9*b^5*c^3*d^2*g
^3 - 21*a*b^4*c^2*d^3*g^3 + 12*a^2*b^3*c*d^4*g^3 + 2*a^3*b^2*d^5*g^3)*B*x^2 + 2*(3*b^5*c^4*d*g^3 - 3*a*b^4*c^3
*d^2*g^3 - 6*a^2*b^3*c^2*d^3*g^3 + 8*a^3*b^2*c*d^4*g^3)*B*x + (6*a*b^4*c^4*d*g^3 - 15*a^2*b^3*c^3*d^2*g^3 + 11
*a^3*b^2*c^2*d^3*g^3)*B)*log(b*x + a) - 2*(2*(b^5*c^2*d^3*g^3 - 2*a*b^4*c*d^4*g^3 + a^2*b^3*d^5*g^3)*B*x^3 + 4
*(b^5*c^3*d^2*g^3 - 2*a*b^4*c^2*d^3*g^3 + a^2*b^3*c*d^4*g^3)*B*x^2 - 4*(b^5*c^4*d*g^3 - 5*a*b^4*c^3*d^2*g^3 +
7*a^2*b^3*c^2*d^3*g^3 - 3*a^3*b^2*c*d^4*g^3)*B*x - (5*b^5*c^5*g^3 - 19*a*b^4*c^4*d*g^3 + 23*a^2*b^3*c^3*d^2*g^
3 - 9*a^3*b^2*c^2*d^3*g^3)*B)*log(d*x + c))/(b^2*c^4*d^4*i^3 - 2*a*b*c^3*d^5*i^3 + a^2*c^2*d^6*i^3 + (b^2*c^2*
d^6*i^3 - 2*a*b*c*d^7*i^3 + a^2*d^8*i^3)*x^2 + 2*(b^2*c^3*d^5*i^3 - 2*a*b*c^2*d^6*i^3 + a^2*c*d^7*i^3)*x) - 3*
(b^3*c*g^3 - a*b^2*d*g^3)*(log(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d))
)*B/(d^4*i^3)

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

[Out]

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

[Out]

Timed out

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

[Out]

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