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

Optimal. Leaf size=373 \[ \frac{3 B d i^3 (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}+\frac{2 d^2 i^3 (a+b x) (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^2}+\frac{d i^3 (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^2 g^2}-\frac{i^3 (c+d x) (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^2 (a+b x)}-\frac{3 d i^3 (b c-a d)^2 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^2}-\frac{B d^2 i^3 x (b c-a d)}{2 b^3 g^2}-\frac{B i^3 (c+d x) (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac{B d i^3 (b c-a d)^2 \log \left (\frac{a+b x}{c+d x}\right )}{2 b^4 g^2}-\frac{5 B d i^3 (b c-a d)^2 \log (c+d x)}{2 b^4 g^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.695625, antiderivative size = 521, normalized size of antiderivative = 1.4, number of steps used = 22, number of rules used = 14, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {2528, 2486, 31, 2525, 12, 72, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{3 B d i^3 (b c-a d)^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g^2}-\frac{a^2 B d^3 i^3 \log (a+b x)}{2 b^4 g^2}+\frac{d^3 i^3 x^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^2 g^2}+\frac{3 d i^3 (b c-a d)^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^2}-\frac{i^3 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^2 (a+b x)}+\frac{A d^2 i^3 x (3 b c-2 a d)}{b^3 g^2}+\frac{B d^2 i^3 (a+b x) (3 b c-2 a d) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac{B d^2 i^3 x (b c-a d)}{2 b^3 g^2}-\frac{B i^3 (b c-a d)^3}{b^4 g^2 (a+b x)}-\frac{3 B d i^3 (b c-a d)^2 \log ^2(a+b x)}{2 b^4 g^2}-\frac{B d i^3 (b c-a d)^2 \log (a+b x)}{b^4 g^2}+\frac{B d i^3 (b c-a d)^2 \log (c+d x)}{b^4 g^2}-\frac{B d i^3 (3 b c-2 a d) (b c-a d) \log (c+d x)}{b^4 g^2}+\frac{3 B d i^3 (b c-a d)^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac{B c^2 d i^3 \log (c+d x)}{2 b^2 g^2} \]

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)^2,x]

[Out]

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

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 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

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{(25 c+25 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx &=\int \left (\frac{15625 d^2 (3 b c-2 a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac{15625 d^3 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}+\frac{15625 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)^2}+\frac{46875 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}\right ) \, dx\\ &=\frac{\left (15625 d^3\right ) \int x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 g^2}+\frac{\left (15625 d^2 (3 b c-2 a d)\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b^3 g^2}+\frac{\left (46875 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} \, dx}{b^3 g^2}+\frac{\left (15625 (b c-a d)^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^3 g^2}\\ &=\frac{15625 A d^2 (3 b c-2 a d) x}{b^3 g^2}+\frac{15625 d^3 x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^2}-\frac{15625 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^2 (a+b x)}+\frac{46875 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}-\frac{\left (15625 B d^3\right ) \int \frac{(b c-a d) x^2}{(a+b x) (c+d x)} \, dx}{2 b^2 g^2}+\frac{\left (15625 B d^2 (3 b c-2 a d)\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{b^3 g^2}-\frac{\left (46875 B d (b c-a d)^2\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^2}+\frac{\left (15625 B (b c-a d)^3\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^2}\\ &=\frac{15625 A d^2 (3 b c-2 a d) x}{b^3 g^2}+\frac{15625 B d^2 (3 b c-2 a d) (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^4 g^2}+\frac{15625 d^3 x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^2}-\frac{15625 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^2 (a+b x)}+\frac{46875 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}-\frac{\left (15625 B d^3 (b c-a d)\right ) \int \frac{x^2}{(a+b x) (c+d x)} \, dx}{2 b^2 g^2}-\frac{\left (15625 B d^2 (3 b c-2 a d) (b c-a d)\right ) \int \frac{1}{c+d x} \, dx}{b^4 g^2}+\frac{\left (15625 B (b c-a d)^4\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^2}-\frac{\left (46875 B d (b c-a d)^2\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^2}\\ &=\frac{15625 A d^2 (3 b c-2 a d) x}{b^3 g^2}+\frac{15625 B d^2 (3 b c-2 a d) (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^4 g^2}+\frac{15625 d^3 x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^2}-\frac{15625 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^2 (a+b x)}+\frac{46875 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}-\frac{15625 B d (3 b c-2 a d) (b c-a d) \log (c+d x)}{b^4 g^2}-\frac{\left (15625 B d^3 (b c-a d)\right ) \int \left (\frac{1}{b d}+\frac{a^2}{b (b c-a d) (a+b x)}+\frac{c^2}{d (-b c+a d) (c+d x)}\right ) \, dx}{2 b^2 g^2}+\frac{\left (15625 B (b c-a d)^4\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^2}-\frac{\left (46875 B d (b c-a d)^2\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^2}\\ &=\frac{15625 A d^2 (3 b c-2 a d) x}{b^3 g^2}-\frac{15625 B d^2 (b c-a d) x}{2 b^3 g^2}-\frac{15625 B (b c-a d)^3}{b^4 g^2 (a+b x)}-\frac{15625 a^2 B d^3 \log (a+b x)}{2 b^4 g^2}-\frac{15625 B d (b c-a d)^2 \log (a+b x)}{b^4 g^2}+\frac{15625 B d^2 (3 b c-2 a d) (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^4 g^2}+\frac{15625 d^3 x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^2}-\frac{15625 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^2 (a+b x)}+\frac{46875 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac{15625 B c^2 d \log (c+d x)}{2 b^2 g^2}-\frac{15625 B d (3 b c-2 a d) (b c-a d) \log (c+d x)}{b^4 g^2}+\frac{15625 B d (b c-a d)^2 \log (c+d x)}{b^4 g^2}-\frac{\left (46875 B d (b c-a d)^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 g^2}+\frac{\left (46875 B d^2 (b c-a d)^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 g^2}\\ &=\frac{15625 A d^2 (3 b c-2 a d) x}{b^3 g^2}-\frac{15625 B d^2 (b c-a d) x}{2 b^3 g^2}-\frac{15625 B (b c-a d)^3}{b^4 g^2 (a+b x)}-\frac{15625 a^2 B d^3 \log (a+b x)}{2 b^4 g^2}-\frac{15625 B d (b c-a d)^2 \log (a+b x)}{b^4 g^2}+\frac{15625 B d^2 (3 b c-2 a d) (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^4 g^2}+\frac{15625 d^3 x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^2}-\frac{15625 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^2 (a+b x)}+\frac{46875 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac{15625 B c^2 d \log (c+d x)}{2 b^2 g^2}-\frac{15625 B d (3 b c-2 a d) (b c-a d) \log (c+d x)}{b^4 g^2}+\frac{15625 B d (b c-a d)^2 \log (c+d x)}{b^4 g^2}+\frac{46875 B d (b c-a d)^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac{\left (46875 B d (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^2}-\frac{\left (46875 B d (b c-a d)^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^2}\\ &=\frac{15625 A d^2 (3 b c-2 a d) x}{b^3 g^2}-\frac{15625 B d^2 (b c-a d) x}{2 b^3 g^2}-\frac{15625 B (b c-a d)^3}{b^4 g^2 (a+b x)}-\frac{15625 a^2 B d^3 \log (a+b x)}{2 b^4 g^2}-\frac{15625 B d (b c-a d)^2 \log (a+b x)}{b^4 g^2}-\frac{46875 B d (b c-a d)^2 \log ^2(a+b x)}{2 b^4 g^2}+\frac{15625 B d^2 (3 b c-2 a d) (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^4 g^2}+\frac{15625 d^3 x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^2}-\frac{15625 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^2 (a+b x)}+\frac{46875 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac{15625 B c^2 d \log (c+d x)}{2 b^2 g^2}-\frac{15625 B d (3 b c-2 a d) (b c-a d) \log (c+d x)}{b^4 g^2}+\frac{15625 B d (b c-a d)^2 \log (c+d x)}{b^4 g^2}+\frac{46875 B d (b c-a d)^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac{\left (46875 B d (b c-a d)^2\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^2}\\ &=\frac{15625 A d^2 (3 b c-2 a d) x}{b^3 g^2}-\frac{15625 B d^2 (b c-a d) x}{2 b^3 g^2}-\frac{15625 B (b c-a d)^3}{b^4 g^2 (a+b x)}-\frac{15625 a^2 B d^3 \log (a+b x)}{2 b^4 g^2}-\frac{15625 B d (b c-a d)^2 \log (a+b x)}{b^4 g^2}-\frac{46875 B d (b c-a d)^2 \log ^2(a+b x)}{2 b^4 g^2}+\frac{15625 B d^2 (3 b c-2 a d) (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^4 g^2}+\frac{15625 d^3 x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^2}-\frac{15625 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^2 (a+b x)}+\frac{46875 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac{15625 B c^2 d \log (c+d x)}{2 b^2 g^2}-\frac{15625 B d (3 b c-2 a d) (b c-a d) \log (c+d x)}{b^4 g^2}+\frac{15625 B d (b c-a d)^2 \log (c+d x)}{b^4 g^2}+\frac{46875 B d (b c-a d)^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac{46875 B d (b c-a d)^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g^2}\\ \end{align*}

Mathematica [A]  time = 0.406583, size = 374, normalized size = 1. \[ \frac{i^3 \left (-3 B d (b c-a d)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-a^2 B d^3 \log (a+b x)+b^2 d^3 x^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+6 d (b c-a d)^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-\frac{2 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+2 A b d^2 x (3 b c-2 a d)+2 B d^2 (a+b x) (3 b c-2 a d) \log \left (\frac{e (a+b x)}{c+d x}\right )-b B d^2 x (b c-a d)-\frac{2 B (b c-a d)^3}{a+b x}-2 B d (b c-a d)^2 \log (a+b x)+2 B d (b c-a d)^2 \log (c+d x)-2 B d (a d-b c) (2 a d-3 b c) \log (c+d x)+b^2 B c^2 d \log (c+d x)\right )}{2 b^4 g^2} \]

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)^2,x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.179, size = 3141, normalized size = 8.4 \begin{align*} \text{output 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)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.70177, size = 2026, normalized size = 5.43 \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)^2,x, algorithm="maxima")

[Out]

-3*A*(a^2/(b^4*g^2*x + a*b^3*g^2) - x/(b^2*g^2) + 2*a*log(b*x + a)/(b^3*g^2))*c*d^2*i^3 + 1/2*(2*a^3/(b^5*g^2*
x + a*b^4*g^2) + 6*a^2*log(b*x + a)/(b^4*g^2) + (b*x^2 - 4*a*x)/(b^3*g^2))*A*d^3*i^3 + 3*A*c^2*d*i^3*(a/(b^3*g
^2*x + a*b^2*g^2) + log(b*x + a)/(b^2*g^2)) - B*c^3*i^3*(log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^2*g^2*x + a*b
*g^2) + 1/(b^2*g^2*x + a*b*g^2) + d*log(b*x + a)/((b^2*c - a*b*d)*g^2) - d*log(d*x + c)/((b^2*c - a*b*d)*g^2))
- A*c^3*i^3/(b^2*g^2*x + a*b*g^2) - 1/2*(5*b^3*c^3*d*i^3 - 3*a*b^2*c^2*d^2*i^3 - 2*a^2*b*c*d^3*i^3 + 2*a^3*d^
4*i^3)*B*log(d*x + c)/(b^5*c*g^2 - a*b^4*d*g^2) + 1/2*((b^4*c*d^3*i^3*log(e) - a*b^3*d^4*i^3*log(e))*B*x^3 + (
(6*i^3*log(e) - i^3)*b^4*c^2*d^2 - (9*i^3*log(e) - 2*i^3)*a*b^3*c*d^3 + (3*i^3*log(e) - i^3)*a^2*b^2*d^4)*B*x^
2 + ((6*i^3*log(e) - i^3)*a*b^3*c^2*d^2 - 2*(5*i^3*log(e) - i^3)*a^2*b^2*c*d^3 + (4*i^3*log(e) - i^3)*a^3*b*d^
4)*B*x + 3*((b^4*c^3*d*i^3 - 3*a*b^3*c^2*d^2*i^3 + 3*a^2*b^2*c*d^3*i^3 - a^3*b*d^4*i^3)*B*x + (a*b^3*c^3*d*i^3
- 3*a^2*b^2*c^2*d^2*i^3 + 3*a^3*b*c*d^3*i^3 - a^4*d^4*i^3)*B)*log(b*x + a)^2 + 2*(3*(i^3*log(e) + i^3)*a*b^3*
c^3*d - 6*(i^3*log(e) + i^3)*a^2*b^2*c^2*d^2 + 4*(i^3*log(e) + i^3)*a^3*b*c*d^3 - (i^3*log(e) + i^3)*a^4*d^4)*
B + ((b^4*c*d^3*i^3 - a*b^3*d^4*i^3)*B*x^3 + 3*(2*b^4*c^2*d^2*i^3 - 3*a*b^3*c*d^3*i^3 + a^2*b^2*d^4*i^3)*B*x^2
+ (6*b^4*c^3*d*i^3*log(e) - 18*(i^3*log(e) - i^3)*a*b^3*c^2*d^2 + 9*(2*i^3*log(e) - 3*i^3)*a^2*b^2*c*d^3 - (6
*i^3*log(e) - 11*i^3)*a^3*b*d^4)*B*x - (18*a^2*b^2*c^2*d^2*i^3*log(e) - 6*(i^3*log(e) + i^3)*a*b^3*c^3*d - 9*(
2*i^3*log(e) - i^3)*a^3*b*c*d^3 + (6*i^3*log(e) - 5*i^3)*a^4*d^4)*B)*log(b*x + a) - ((b^4*c*d^3*i^3 - a*b^3*d^
4*i^3)*B*x^3 + 3*(2*b^4*c^2*d^2*i^3 - 3*a*b^3*c*d^3*i^3 + a^2*b^2*d^4*i^3)*B*x^2 + 2*(3*a*b^3*c^2*d^2*i^3 - 5*
a^2*b^2*c*d^3*i^3 + 2*a^3*b*d^4*i^3)*B*x + 2*(3*a*b^3*c^3*d*i^3 - 6*a^2*b^2*c^2*d^2*i^3 + 4*a^3*b*c*d^3*i^3 -
a^4*d^4*i^3)*B + 6*((b^4*c^3*d*i^3 - 3*a*b^3*c^2*d^2*i^3 + 3*a^2*b^2*c*d^3*i^3 - a^3*b*d^4*i^3)*B*x + (a*b^3*c
^3*d*i^3 - 3*a^2*b^2*c^2*d^2*i^3 + 3*a^3*b*c*d^3*i^3 - a^4*d^4*i^3)*B)*log(b*x + a))*log(d*x + c))/(a*b^5*c*g^
2 - a^2*b^4*d*g^2 + (b^6*c*g^2 - a*b^5*d*g^2)*x) + 3*(b^2*c^2*d*i^3 - 2*a*b*c*d^2*i^3 + a^2*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^2)

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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^{2} g^{2} x^{2} + 2 \, a b g^{2} x + a^{2} g^{2}}, 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)^2,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^2*g^2*x^2 + 2*a*b*g^2*x + a^2*g^2), 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)**2,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 )}^{2}}\,{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)^2,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)^2, x)