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

Optimal. Leaf size=345 \[ \frac{3 B d^2 i^3 (b c-a d) \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{b^4 g^3}+\frac{d^3 i^3 (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^3}-\frac{3 d^2 i^3 (b c-a d) \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^3}-\frac{i^3 (c+d x)^2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^2 g^3 (a+b x)^2}-\frac{2 d i^3 (c+d x) (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^3 (a+b x)}-\frac{B d^2 i^3 (b c-a d) \log (c+d x)}{b^4 g^3}-\frac{B i^3 (c+d x)^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac{2 B d i^3 (c+d x) (b c-a d)}{b^3 g^3 (a+b x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.717528, antiderivative size = 442, normalized size of antiderivative = 1.28, 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, 2390, 2301, 2394, 2393, 2391} \[ \frac{3 B d^2 i^3 (b c-a d) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g^3}+\frac{3 d^2 i^3 (b c-a d) \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^3}-\frac{3 d i^3 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^3 (a+b x)}-\frac{i^3 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^4 g^3 (a+b x)^2}+\frac{B d^3 i^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac{3 B d^2 i^3 (b c-a d) \log ^2(a+b x)}{2 b^4 g^3}-\frac{5 B d^2 i^3 (b c-a d) \log (a+b x)}{2 b^4 g^3}+\frac{3 B d^2 i^3 (b c-a d) \log (c+d x)}{2 b^4 g^3}+\frac{3 B d^2 i^3 (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^3}-\frac{5 B d i^3 (b c-a d)^2}{2 b^4 g^3 (a+b x)}-\frac{B i^3 (b c-a d)^3}{4 b^4 g^3 (a+b x)^2}+\frac{A d^3 i^3 x}{b^3 g^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

\begin{align*} \int \frac{(26 c+26 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx &=\int \left (\frac{17576 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac{17576 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^3}+\frac{52728 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}+\frac{52728 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}\right ) \, dx\\ &=\frac{\left (17576 d^3\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b^3 g^3}+\frac{\left (52728 d^2 (b c-a d)\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^3 g^3}+\frac{\left (52728 d (b c-a d)^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^3 g^3}+\frac{\left (17576 (b c-a d)^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^3 g^3}\\ &=\frac{17576 A d^3 x}{b^3 g^3}-\frac{8788 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac{52728 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac{52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}+\frac{\left (17576 B d^3\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{b^3 g^3}-\frac{\left (52728 B d^2 (b c-a d)\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 g^3}+\frac{\left (52728 B d (b c-a d)^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^3}+\frac{\left (8788 B (b c-a d)^3\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^3}\\ &=\frac{17576 A d^3 x}{b^3 g^3}+\frac{17576 B d^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac{8788 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac{52728 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac{52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}-\frac{\left (17576 B d^3 (b c-a d)\right ) \int \frac{1}{c+d x} \, dx}{b^4 g^3}+\frac{\left (52728 B d (b c-a d)^3\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^3}+\frac{\left (8788 B (b c-a d)^4\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^3}-\frac{\left (52728 B d^2 (b c-a d)\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 e g^3}\\ &=\frac{17576 A d^3 x}{b^3 g^3}+\frac{17576 B d^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac{8788 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac{52728 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac{52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}-\frac{17576 B d^2 (b c-a d) \log (c+d x)}{b^4 g^3}+\frac{\left (52728 B d (b c-a d)^3\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^4 g^3}+\frac{\left (8788 B (b c-a d)^4\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^4 g^3}-\frac{\left (52728 B d^2 (b c-a d)\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 e g^3}\\ &=\frac{17576 A d^3 x}{b^3 g^3}-\frac{4394 B (b c-a d)^3}{b^4 g^3 (a+b x)^2}-\frac{43940 B d (b c-a d)^2}{b^4 g^3 (a+b x)}-\frac{43940 B d^2 (b c-a d) \log (a+b x)}{b^4 g^3}+\frac{17576 B d^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac{8788 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac{52728 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac{52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}+\frac{26364 B d^2 (b c-a d) \log (c+d x)}{b^4 g^3}-\frac{\left (52728 B d^2 (b c-a d)\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 g^3}+\frac{\left (52728 B d^3 (b c-a d)\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 g^3}\\ &=\frac{17576 A d^3 x}{b^3 g^3}-\frac{4394 B (b c-a d)^3}{b^4 g^3 (a+b x)^2}-\frac{43940 B d (b c-a d)^2}{b^4 g^3 (a+b x)}-\frac{43940 B d^2 (b c-a d) \log (a+b x)}{b^4 g^3}+\frac{17576 B d^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac{8788 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac{52728 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac{52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}+\frac{26364 B d^2 (b c-a d) \log (c+d x)}{b^4 g^3}+\frac{52728 B d^2 (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^3}-\frac{\left (52728 B d^2 (b c-a d)\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^3}-\frac{\left (52728 B d^2 (b c-a d)\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^3}\\ &=\frac{17576 A d^3 x}{b^3 g^3}-\frac{4394 B (b c-a d)^3}{b^4 g^3 (a+b x)^2}-\frac{43940 B d (b c-a d)^2}{b^4 g^3 (a+b x)}-\frac{43940 B d^2 (b c-a d) \log (a+b x)}{b^4 g^3}-\frac{26364 B d^2 (b c-a d) \log ^2(a+b x)}{b^4 g^3}+\frac{17576 B d^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac{8788 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac{52728 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac{52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}+\frac{26364 B d^2 (b c-a d) \log (c+d x)}{b^4 g^3}+\frac{52728 B d^2 (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^3}-\frac{\left (52728 B d^2 (b c-a d)\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 g^3}\\ &=\frac{17576 A d^3 x}{b^3 g^3}-\frac{4394 B (b c-a d)^3}{b^4 g^3 (a+b x)^2}-\frac{43940 B d (b c-a d)^2}{b^4 g^3 (a+b x)}-\frac{43940 B d^2 (b c-a d) \log (a+b x)}{b^4 g^3}-\frac{26364 B d^2 (b c-a d) \log ^2(a+b x)}{b^4 g^3}+\frac{17576 B d^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac{8788 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac{52728 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac{52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}+\frac{26364 B d^2 (b c-a d) \log (c+d x)}{b^4 g^3}+\frac{52728 B d^2 (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^3}+\frac{52728 B d^2 (b c-a d) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g^3}\\ \end{align*}

Mathematica [A]  time = 0.439761, size = 314, normalized size = 0.91 \[ \frac{i^3 \left (6 B d^2 (a d-b c) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+12 d^2 (b c-a d) \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-\frac{12 d (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac{2 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{(a+b x)^2}+4 B d^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )+10 B d^2 (a d-b c) \log (a+b x)+6 B d^2 (b c-a d) \log (c+d x)-\frac{10 B d (b c-a d)^2}{a+b x}-\frac{B (b c-a d)^3}{(a+b x)^2}+4 A b d^3 x\right )}{4 b^4 g^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.164, size = 1855, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.88724, size = 3108, normalized size = 9.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*B*c^2*d*i^3*(2*(2*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^4*g^3*x^2 + 2*a*b^3*g^3*x + a^2*b^2*g^
3) + (3*a*b*c - a^2*d + 2*(2*b^2*c - a*b*d)*x)/((b^5*c - a*b^4*d)*g^3*x^2 + 2*(a*b^4*c - a^2*b^3*d)*g^3*x + (a
^2*b^3*c - a^3*b^2*d)*g^3) + 2*(2*b*c*d - a*d^2)*log(b*x + a)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*g^3) - 2*
(2*b*c*d - a*d^2)*log(d*x + c)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*g^3)) - 1/2*A*d^3*i^3*((6*a^2*b*x + 5*a^
3)/(b^6*g^3*x^2 + 2*a*b^5*g^3*x + a^2*b^4*g^3) - 2*x/(b^3*g^3) + 6*a*log(b*x + a)/(b^4*g^3)) + 3/2*A*c*d^2*i^3
*((4*a*b*x + 3*a^2)/(b^5*g^3*x^2 + 2*a*b^4*g^3*x + a^2*b^3*g^3) + 2*log(b*x + a)/(b^3*g^3)) + 1/4*B*c^3*i^3*((
2*b*d*x - b*c + 3*a*d)/((b^4*c - a*b^3*d)*g^3*x^2 + 2*(a*b^3*c - a^2*b^2*d)*g^3*x + (a^2*b^2*c - a^3*b*d)*g^3)
- 2*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^3*g^3*x^2 + 2*a*b^2*g^3*x + a^2*b*g^3) + 2*d^2*log(b*x + a)/((b^3
*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3) - 2*d^2*log(d*x + c)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3)) - 3/2*(2*
b*x + a)*A*c^2*d*i^3/(b^4*g^3*x^2 + 2*a*b^3*g^3*x + a^2*b^2*g^3) - 1/2*A*c^3*i^3/(b^3*g^3*x^2 + 2*a*b^2*g^3*x
+ a^2*b*g^3) - 1/2*(2*b^3*c^3*d^2*i^3 + 8*a*b^2*c^2*d^3*i^3 - 13*a^2*b*c*d^4*i^3 + 5*a^3*d^5*i^3)*B*log(d*x +
c)/(b^6*c^2*g^3 - 2*a*b^5*c*d*g^3 + a^2*b^4*d^2*g^3) + 1/4*(4*(b^5*c^2*d^3*i^3*log(e) - 2*a*b^4*c*d^4*i^3*log(
e) + a^2*b^3*d^5*i^3*log(e))*B*x^3 + 8*(a*b^4*c^2*d^3*i^3*log(e) - 2*a^2*b^3*c*d^4*i^3*log(e) + a^3*b^2*d^5*i^
3*log(e))*B*x^2 + 2*(12*(i^3*log(e) + i^3)*a*b^4*c^3*d^2 - (28*i^3*log(e) + 27*i^3)*a^2*b^3*c^2*d^3 + 20*(i^3*
log(e) + i^3)*a^3*b^2*c*d^4 - (4*i^3*log(e) + 5*i^3)*a^4*b*d^5)*B*x + 6*((b^5*c^3*d^2*i^3 - 3*a*b^4*c^2*d^3*i^
3 + 3*a^2*b^3*c*d^4*i^3 - a^3*b^2*d^5*i^3)*B*x^2 + 2*(a*b^4*c^3*d^2*i^3 - 3*a^2*b^3*c^2*d^3*i^3 + 3*a^3*b^2*c*
d^4*i^3 - a^4*b*d^5*i^3)*B*x + (a^2*b^3*c^3*d^2*i^3 - 3*a^3*b^2*c^2*d^3*i^3 + 3*a^4*b*c*d^4*i^3 - a^5*d^5*i^3)
*B)*log(b*x + a)^2 + (3*(6*i^3*log(e) + 7*i^3)*a^2*b^3*c^3*d^2 - (46*i^3*log(e) + 47*i^3)*a^3*b^2*c^2*d^3 + (3
8*i^3*log(e) + 35*i^3)*a^4*b*c*d^4 - (10*i^3*log(e) + 9*i^3)*a^5*d^5)*B + 2*(2*(b^5*c^2*d^3*i^3 - 2*a*b^4*c*d^
4*i^3 + a^2*b^3*d^5*i^3)*B*x^3 + (6*b^5*c^3*d^2*i^3*log(e) - 18*(i^3*log(e) - i^3)*a*b^4*c^2*d^3 + 9*(2*i^3*lo
g(e) - 3*i^3)*a^2*b^3*c*d^4 - (6*i^3*log(e) - 11*i^3)*a^3*b^2*d^5)*B*x^2 - 2*(18*a^2*b^3*c^2*d^3*i^3*log(e) -
6*(i^3*log(e) + i^3)*a*b^4*c^3*d^2 - 9*(2*i^3*log(e) - i^3)*a^3*b^2*c*d^4 + (6*i^3*log(e) - 5*i^3)*a^4*b*d^5)*
B*x + (18*a^4*b*c*d^4*i^3*log(e) + 3*(2*i^3*log(e) + 3*i^3)*a^2*b^3*c^3*d^2 - 9*(2*i^3*log(e) + i^3)*a^3*b^2*c
^2*d^3 - 2*(3*i^3*log(e) - i^3)*a^5*d^5)*B)*log(b*x + a) - 2*(2*(b^5*c^2*d^3*i^3 - 2*a*b^4*c*d^4*i^3 + a^2*b^3
*d^5*i^3)*B*x^3 + 4*(a*b^4*c^2*d^3*i^3 - 2*a^2*b^3*c*d^4*i^3 + a^3*b^2*d^5*i^3)*B*x^2 + 4*(3*a*b^4*c^3*d^2*i^3
- 7*a^2*b^3*c^2*d^3*i^3 + 5*a^3*b^2*c*d^4*i^3 - a^4*b*d^5*i^3)*B*x + (9*a^2*b^3*c^3*d^2*i^3 - 23*a^3*b^2*c^2*
d^3*i^3 + 19*a^4*b*c*d^4*i^3 - 5*a^5*d^5*i^3)*B + 6*((b^5*c^3*d^2*i^3 - 3*a*b^4*c^2*d^3*i^3 + 3*a^2*b^3*c*d^4*
i^3 - a^3*b^2*d^5*i^3)*B*x^2 + 2*(a*b^4*c^3*d^2*i^3 - 3*a^2*b^3*c^2*d^3*i^3 + 3*a^3*b^2*c*d^4*i^3 - a^4*b*d^5*
i^3)*B*x + (a^2*b^3*c^3*d^2*i^3 - 3*a^3*b^2*c^2*d^3*i^3 + 3*a^4*b*c*d^4*i^3 - a^5*d^5*i^3)*B)*log(b*x + a))*lo
g(d*x + c))/(a^2*b^6*c^2*g^3 - 2*a^3*b^5*c*d*g^3 + a^4*b^4*d^2*g^3 + (b^8*c^2*g^3 - 2*a*b^7*c*d*g^3 + a^2*b^6*
d^2*g^3)*x^2 + 2*(a*b^7*c^2*g^3 - 2*a^2*b^6*c*d*g^3 + a^3*b^5*d^2*g^3)*x) + 3*(b*c*d^2*i^3 - a*d^3*i^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/(b^4*g^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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