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

Optimal. Leaf size=359 \[ -\frac{3 b B g^3 n (b c-a d)^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac{g^3 (a+b x)^2 (b c-a d) \left (3 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{2 d^2 i^2 (c+d x)}-\frac{b g^3 (b c-a d)^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+6 A+5 B n\right )}{2 d^4 i^2}-\frac{g^3 (a+b x) (6 A+5 B n) (b c-a d)^2}{2 d^3 i^2 (c+d x)}+\frac{g^3 (a+b x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 d i^2 (c+d x)}-\frac{3 B g^3 (a+b x) (b c-a d)^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{d^3 i^2 (c+d x)}+\frac{3 B g^3 n (a+b x) (b c-a d)^2}{d^3 i^2 (c+d x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.727243, antiderivative size = 541, normalized size of antiderivative = 1.51, number of steps used = 21, number of rules used = 14, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.326, Rules used = {2528, 2486, 31, 2525, 12, 72, 44, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac{3 b B g^3 n (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^4 i^2}-\frac{a^2 b B g^3 n \log (a+b x)}{2 d^2 i^2}+\frac{b^3 g^3 x^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^2 i^2}-\frac{A b^2 g^3 x (2 b c-3 a d)}{d^3 i^2}+\frac{g^3 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^4 i^2 (c+d x)}+\frac{3 b g^3 (b c-a d)^2 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^4 i^2}-\frac{b^2 B g^3 n x (b c-a d)}{2 d^3 i^2}-\frac{b B g^3 (a+b x) (2 b c-3 a d) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{d^3 i^2}-\frac{B g^3 n (b c-a d)^3}{d^4 i^2 (c+d x)}+\frac{3 b B g^3 n (b c-a d)^2 \log ^2(c+d x)}{2 d^4 i^2}-\frac{b B g^3 n (b c-a d)^2 \log (a+b x)}{d^4 i^2}+\frac{b B g^3 n (b c-a d)^2 \log (c+d x)}{d^4 i^2}+\frac{b B g^3 n (2 b c-3 a d) (b c-a d) \log (c+d x)}{d^4 i^2}-\frac{3 b B g^3 n (b c-a d)^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d^4 i^2}+\frac{b^3 B c^2 g^3 n \log (c+d x)}{2 d^4 i^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((A*b^2*(2*b*c - 3*a*d)*g^3*x)/(d^3*i^2)) - (b^2*B*(b*c - a*d)*g^3*n*x)/(2*d^3*i^2) - (B*(b*c - a*d)^3*g^3*n)
/(d^4*i^2*(c + d*x)) - (a^2*b*B*g^3*n*Log[a + b*x])/(2*d^2*i^2) - (b*B*(b*c - a*d)^2*g^3*n*Log[a + b*x])/(d^4*
i^2) - (b*B*(2*b*c - 3*a*d)*g^3*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(d^3*i^2) + (b^3*g^3*x^2*(A + B*Log[
e*((a + b*x)/(c + d*x))^n]))/(2*d^2*i^2) + ((b*c - a*d)^3*g^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(d^4*i^2
*(c + d*x)) + (b^3*B*c^2*g^3*n*Log[c + d*x])/(2*d^4*i^2) + (b*B*(2*b*c - 3*a*d)*(b*c - a*d)*g^3*n*Log[c + d*x]
)/(d^4*i^2) + (b*B*(b*c - a*d)^2*g^3*n*Log[c + d*x])/(d^4*i^2) - (3*b*B*(b*c - a*d)^2*g^3*n*Log[-((d*(a + b*x)
)/(b*c - a*d))]*Log[c + d*x])/(d^4*i^2) + (3*b*(b*c - a*d)^2*g^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c
+ d*x])/(d^4*i^2) + (3*b*B*(b*c - a*d)^2*g^3*n*Log[c + d*x]^2)/(2*d^4*i^2) - (3*b*B*(b*c - a*d)^2*g^3*n*PolyLo
g[2, (b*(c + d*x))/(b*c - a*d)])/(d^4*i^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 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 (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(143 c+143 d x)^2} \, dx &=\int \left (-\frac{b^2 (2 b c-3 a d) g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20449 d^3}+\frac{b^3 g^3 x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20449 d^2}+\frac{(-b c+a d)^3 g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20449 d^3 (c+d x)^2}+\frac{3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20449 d^3 (c+d x)}\right ) \, dx\\ &=\frac{\left (b^3 g^3\right ) \int x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{20449 d^2}-\frac{\left (b^2 (2 b c-3 a d) g^3\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{20449 d^3}+\frac{\left (3 b (b c-a d)^2 g^3\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{20449 d^3}-\frac{\left ((b c-a d)^3 g^3\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{20449 d^3}\\ &=-\frac{A b^2 (2 b c-3 a d) g^3 x}{20449 d^3}+\frac{b^3 g^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{40898 d^2}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20449 d^4 (c+d x)}+\frac{3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{20449 d^4}-\frac{\left (b^2 B (2 b c-3 a d) g^3\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{20449 d^3}-\frac{\left (b^3 B g^3 n\right ) \int \frac{(b c-a d) x^2}{(a+b x) (c+d x)} \, dx}{40898 d^2}-\frac{\left (3 b B (b c-a d)^2 g^3 n\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{20449 d^4}-\frac{\left (B (b c-a d)^3 g^3 n\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{20449 d^4}\\ &=-\frac{A b^2 (2 b c-3 a d) g^3 x}{20449 d^3}-\frac{b B (2 b c-3 a d) g^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{20449 d^3}+\frac{b^3 g^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{40898 d^2}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20449 d^4 (c+d x)}+\frac{3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{20449 d^4}-\frac{\left (b^3 B (b c-a d) g^3 n\right ) \int \frac{x^2}{(a+b x) (c+d x)} \, dx}{40898 d^2}+\frac{\left (b B (2 b c-3 a d) (b c-a d) g^3 n\right ) \int \frac{1}{c+d x} \, dx}{20449 d^3}-\frac{\left (3 b B (b c-a d)^2 g^3 n\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{20449 d^4}-\frac{\left (B (b c-a d)^4 g^3 n\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{20449 d^4}\\ &=-\frac{A b^2 (2 b c-3 a d) g^3 x}{20449 d^3}-\frac{b B (2 b c-3 a d) g^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{20449 d^3}+\frac{b^3 g^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{40898 d^2}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20449 d^4 (c+d x)}+\frac{b B (2 b c-3 a d) (b c-a d) g^3 n \log (c+d x)}{20449 d^4}+\frac{3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{20449 d^4}-\frac{\left (b^3 B (b c-a d) g^3 n\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}{40898 d^2}-\frac{\left (3 b^2 B (b c-a d)^2 g^3 n\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{20449 d^4}+\frac{\left (3 b B (b c-a d)^2 g^3 n\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{20449 d^3}-\frac{\left (B (b c-a d)^4 g^3 n\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}{20449 d^4}\\ &=-\frac{A b^2 (2 b c-3 a d) g^3 x}{20449 d^3}-\frac{b^2 B (b c-a d) g^3 n x}{40898 d^3}-\frac{B (b c-a d)^3 g^3 n}{20449 d^4 (c+d x)}-\frac{a^2 b B g^3 n \log (a+b x)}{40898 d^2}-\frac{b B (b c-a d)^2 g^3 n \log (a+b x)}{20449 d^4}-\frac{b B (2 b c-3 a d) g^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{20449 d^3}+\frac{b^3 g^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{40898 d^2}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20449 d^4 (c+d x)}+\frac{b^3 B c^2 g^3 n \log (c+d x)}{40898 d^4}+\frac{b B (2 b c-3 a d) (b c-a d) g^3 n \log (c+d x)}{20449 d^4}+\frac{b B (b c-a d)^2 g^3 n \log (c+d x)}{20449 d^4}-\frac{3 b B (b c-a d)^2 g^3 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{20449 d^4}+\frac{3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{20449 d^4}+\frac{\left (3 b B (b c-a d)^2 g^3 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{20449 d^4}+\frac{\left (3 b B (b c-a d)^2 g^3 n\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{20449 d^3}\\ &=-\frac{A b^2 (2 b c-3 a d) g^3 x}{20449 d^3}-\frac{b^2 B (b c-a d) g^3 n x}{40898 d^3}-\frac{B (b c-a d)^3 g^3 n}{20449 d^4 (c+d x)}-\frac{a^2 b B g^3 n \log (a+b x)}{40898 d^2}-\frac{b B (b c-a d)^2 g^3 n \log (a+b x)}{20449 d^4}-\frac{b B (2 b c-3 a d) g^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{20449 d^3}+\frac{b^3 g^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{40898 d^2}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20449 d^4 (c+d x)}+\frac{b^3 B c^2 g^3 n \log (c+d x)}{40898 d^4}+\frac{b B (2 b c-3 a d) (b c-a d) g^3 n \log (c+d x)}{20449 d^4}+\frac{b B (b c-a d)^2 g^3 n \log (c+d x)}{20449 d^4}-\frac{3 b B (b c-a d)^2 g^3 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{20449 d^4}+\frac{3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{20449 d^4}+\frac{3 b B (b c-a d)^2 g^3 n \log ^2(c+d x)}{40898 d^4}+\frac{\left (3 b B (b c-a d)^2 g^3 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{20449 d^4}\\ &=-\frac{A b^2 (2 b c-3 a d) g^3 x}{20449 d^3}-\frac{b^2 B (b c-a d) g^3 n x}{40898 d^3}-\frac{B (b c-a d)^3 g^3 n}{20449 d^4 (c+d x)}-\frac{a^2 b B g^3 n \log (a+b x)}{40898 d^2}-\frac{b B (b c-a d)^2 g^3 n \log (a+b x)}{20449 d^4}-\frac{b B (2 b c-3 a d) g^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{20449 d^3}+\frac{b^3 g^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{40898 d^2}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{20449 d^4 (c+d x)}+\frac{b^3 B c^2 g^3 n \log (c+d x)}{40898 d^4}+\frac{b B (2 b c-3 a d) (b c-a d) g^3 n \log (c+d x)}{20449 d^4}+\frac{b B (b c-a d)^2 g^3 n \log (c+d x)}{20449 d^4}-\frac{3 b B (b c-a d)^2 g^3 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{20449 d^4}+\frac{3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{20449 d^4}+\frac{3 b B (b c-a d)^2 g^3 n \log ^2(c+d x)}{40898 d^4}-\frac{3 b B (b c-a d)^2 g^3 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{20449 d^4}\\ \end{align*}

Mathematica [A]  time = 0.440093, size = 375, normalized size = 1.04 \[ \frac{g^3 \left (-3 b B n (b c-a d)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+b B n \left (b \left (d x (a d-b c)+b c^2 \log (c+d x)\right )-a^2 d^2 \log (a+b x)\right )+b^3 d^2 x^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-2 A b^2 d x (2 b c-3 a d)+6 b (b c-a d)^2 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+\frac{2 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}-2 b B d (a+b x) (2 b c-3 a d) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+2 b B n (2 b c-3 a d) (b c-a d) \log (c+d x)-2 B n (b c-a d)^2 \left (\frac{b c-a d}{c+d x}+b \log (a+b x)-b \log (c+d x)\right )\right )}{2 d^4 i^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.66, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bgx+ag \right ) ^{3}}{ \left ( dix+ci \right ) ^{2}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, 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*((b*x+a)/(d*x+c))^n))/(d*i*x+c*i)^2,x)

[Out]

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

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Maxima [B]  time = 2.63196, size = 2554, normalized size = 7.11 \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))^n))/(d*i*x+c*i)^2,x, algorithm="maxima")

[Out]

B*a^3*g^3*n*(1/(d^2*i^2*x + c*d*i^2) + b*log(b*x + a)/((b*c*d - a*d^2)*i^2) - b*log(d*x + c)/((b*c*d - a*d^2)*
i^2)) + 1/2*(2*c^3/(d^5*i^2*x + c*d^4*i^2) + 6*c^2*log(d*x + c)/(d^4*i^2) + (d*x^2 - 4*c*x)/(d^3*i^2))*A*b^3*g
^3 - 3*A*a*b^2*(c^2/(d^4*i^2*x + c*d^3*i^2) - x/(d^2*i^2) + 2*c*log(d*x + c)/(d^3*i^2))*g^3 + 3*A*a^2*b*g^3*(c
/(d^3*i^2*x + c*d^2*i^2) + log(d*x + c)/(d^2*i^2)) - B*a^3*g^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(d^2*i^2
*x + c*d*i^2) - A*a^3*g^3/(d^2*i^2*x + c*d*i^2) - 1/2*(6*a^3*b*d^3*g^3*log(e) - (7*g^3*n + 6*g^3*log(e))*b^4*c
^3 + (17*g^3*n + 18*g^3*log(e))*a*b^3*c^2*d - 6*(2*g^3*n + 3*g^3*log(e))*a^2*b^2*c*d^2)*B*log(d*x + c)/(b*c*d^
4*i^2 - a*d^5*i^2) + 1/2*((b^4*c*d^3*g^3*log(e) - a*b^3*d^4*g^3*log(e))*B*x^3 - ((g^3*n + 3*g^3*log(e))*b^4*c^
2*d^2 - (2*g^3*n + 9*g^3*log(e))*a*b^3*c*d^3 + (g^3*n + 6*g^3*log(e))*a^2*b^2*d^4)*B*x^2 - ((g^3*n + 4*g^3*log
(e))*b^4*c^3*d - 2*(g^3*n + 5*g^3*log(e))*a*b^3*c^2*d^2 + (g^3*n + 6*g^3*log(e))*a^2*b^2*c*d^3)*B*x - 6*((b^4*
c^3*d*g^3*n - 3*a*b^3*c^2*d^2*g^3*n + 3*a^2*b^2*c*d^3*g^3*n - a^3*b*d^4*g^3*n)*B*x + (b^4*c^4*g^3*n - 3*a*b^3*
c^3*d*g^3*n + 3*a^2*b^2*c^2*d^2*g^3*n - a^3*b*c*d^3*g^3*n)*B)*log(b*x + a)*log(d*x + c) + 3*((b^4*c^3*d*g^3*n
- 3*a*b^3*c^2*d^2*g^3*n + 3*a^2*b^2*c*d^3*g^3*n - a^3*b*d^4*g^3*n)*B*x + (b^4*c^4*g^3*n - 3*a*b^3*c^3*d*g^3*n
+ 3*a^2*b^2*c^2*d^2*g^3*n - a^3*b*c*d^3*g^3*n)*B)*log(d*x + c)^2 - 2*((g^3*n - g^3*log(e))*b^4*c^4 - 4*(g^3*n
- g^3*log(e))*a*b^3*c^3*d + 6*(g^3*n - g^3*log(e))*a^2*b^2*c^2*d^2 - 3*(g^3*n - g^3*log(e))*a^3*b*c*d^3)*B - (
(2*b^4*c^3*d*g^3*n - 2*a*b^3*c^2*d^2*g^3*n - 3*a^2*b^2*c*d^3*g^3*n + 5*a^3*b*d^4*g^3*n)*B*x + (2*b^4*c^4*g^3*n
- 2*a*b^3*c^3*d*g^3*n - 3*a^2*b^2*c^2*d^2*g^3*n + 5*a^3*b*c*d^3*g^3*n)*B)*log(b*x + a) + ((b^4*c*d^3*g^3 - a*
b^3*d^4*g^3)*B*x^3 - 3*(b^4*c^2*d^2*g^3 - 3*a*b^3*c*d^3*g^3 + 2*a^2*b^2*d^4*g^3)*B*x^2 - 2*(2*b^4*c^3*d*g^3 -
5*a*b^3*c^2*d^2*g^3 + 3*a^2*b^2*c*d^3*g^3)*B*x + 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 -
3*a^3*b*c*d^3*g^3)*B + 6*((b^4*c^3*d*g^3 - 3*a*b^3*c^2*d^2*g^3 + 3*a^2*b^2*c*d^3*g^3 - a^3*b*d^4*g^3)*B*x + (b
^4*c^4*g^3 - 3*a*b^3*c^3*d*g^3 + 3*a^2*b^2*c^2*d^2*g^3 - a^3*b*c*d^3*g^3)*B)*log(d*x + c))*log((b*x + a)^n) -
((b^4*c*d^3*g^3 - a*b^3*d^4*g^3)*B*x^3 - 3*(b^4*c^2*d^2*g^3 - 3*a*b^3*c*d^3*g^3 + 2*a^2*b^2*d^4*g^3)*B*x^2 - 2
*(2*b^4*c^3*d*g^3 - 5*a*b^3*c^2*d^2*g^3 + 3*a^2*b^2*c*d^3*g^3)*B*x + 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 - 3*a^3*b*c*d^3*g^3)*B + 6*((b^4*c^3*d*g^3 - 3*a*b^3*c^2*d^2*g^3 + 3*a^2*b^2*c*d^3*g^3 - a^3
*b*d^4*g^3)*B*x + (b^4*c^4*g^3 - 3*a*b^3*c^3*d*g^3 + 3*a^2*b^2*c^2*d^2*g^3 - a^3*b*c*d^3*g^3)*B)*log(d*x + c))
*log((d*x + c)^n))/(b*c^2*d^4*i^2 - a*c*d^5*i^2 + (b*c*d^5*i^2 - a*d^6*i^2)*x) + 3*(b^3*c^2*g^3*n - 2*a*b^2*c*
d*g^3*n + a^2*b*d^2*g^3*n)*(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^2)

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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 (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{d^{2} i^{2} x^{2} + 2 \, c d i^{2} x + c^{2} i^{2}}, 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))^n))/(d*i*x+c*i)^2,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(e*((b*x + a)/(d*x + c))^n))/(d^2*i^2*x^2 + 2*c*d*i^2*x + c^2*i^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((b*g*x+a*g)**3*(A+B*ln(e*((b*x+a)/(d*x+c))**n))/(d*i*x+c*i)**2,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 (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}}{{\left (d i x + c i\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*((b*x+a)/(d*x+c))^n))/(d*i*x+c*i)^2,x, algorithm="giac")

[Out]

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