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

Optimal. Leaf size=269 \[ \frac{B g^3 n (b c-a d)^3 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d^4 i}-\frac{g^3 (a+b x)^2 (b c-a d) \left (3 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{6 d^2 i}+\frac{g^3 (a+b x) (b c-a d)^2 \left (6 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+6 A+5 B n\right )}{6 d^3 i}+\frac{g^3 (b c-a d)^3 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+6 A+11 B n\right )}{6 d^4 i}+\frac{g^3 (a+b x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 d i} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.645896, antiderivative size = 426, normalized size of antiderivative = 1.58, number of steps used = 22, number of rules used = 13, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.302, Rules used = {2528, 2486, 31, 2525, 12, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac{B g^3 n (b c-a d)^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^4 i}-\frac{g^3 (a+b x)^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^2 i}-\frac{g^3 (b c-a d)^3 \log (c i+d i x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^4 i}+\frac{g^3 (a+b x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 d i}+\frac{A b g^3 x (b c-a d)^2}{d^3 i}+\frac{B g^3 (a+b x) (b c-a d)^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{d^3 i}-\frac{B g^3 n (a+b x)^2 (b c-a d)}{6 d^2 i}+\frac{5 b B g^3 n x (b c-a d)^2}{6 d^3 i}-\frac{B g^3 n (b c-a d)^3 \log ^2(i (c+d x))}{2 d^4 i}-\frac{11 B g^3 n (b c-a d)^3 \log (c+d x)}{6 d^4 i}+\frac{B g^3 n (b c-a d)^3 \log (c i+d i x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d^4 i} \]

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),x]

[Out]

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

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 43

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

Rule 2524

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

Rule 2418

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

Rule 2394

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

Rule 2393

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2390

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

Rule 2301

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

Rubi steps

\begin{align*} \int \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 )}{135 c+135 d x} \, dx &=\int \left (\frac{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 )}{135 d^3}+\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 )}{d^3 (135 c+135 d x)}-\frac{b (b c-a d) g^2 (a g+b g x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{135 d^2}+\frac{b g (a g+b g x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{135 d}\right ) \, dx\\ &=\frac{(b g) \int (a g+b g x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{135 d}-\frac{\left (b (b c-a d) g^2\right ) \int (a g+b g x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{135 d^2}+\frac{\left (b (b c-a d)^2 g^3\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{135 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 )}{135 c+135 d x} \, dx}{d^3}\\ &=\frac{A b (b c-a d)^2 g^3 x}{135 d^3}-\frac{(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{270 d^2}+\frac{g^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{405 d}-\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 ) \log (135 c+135 d x)}{135 d^4}+\frac{\left (b B (b c-a d)^2 g^3\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{135 d^3}-\frac{(B n) \int \frac{(b c-a d) g^3 (a+b x)^2}{c+d x} \, dx}{405 d}+\frac{(B (b c-a d) g n) \int \frac{(b c-a d) g^2 (a+b x)}{c+d x} \, dx}{270 d^2}+\frac{\left (B (b c-a d)^3 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 (135 c+135 d x)}{a+b x} \, dx}{135 d^4}\\ &=\frac{A b (b c-a d)^2 g^3 x}{135 d^3}+\frac{B (b c-a d)^2 g^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{135 d^3}-\frac{(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{270 d^2}+\frac{g^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{405 d}-\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 ) \log (135 c+135 d x)}{135 d^4}-\frac{\left (B (b c-a d) g^3 n\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{405 d}+\frac{\left (B (b c-a d)^2 g^3 n\right ) \int \frac{a+b x}{c+d x} \, dx}{270 d^2}+\frac{\left (B (b c-a d)^3 g^3 n\right ) \int \left (\frac{b \log (135 c+135 d x)}{a+b x}-\frac{d \log (135 c+135 d x)}{c+d x}\right ) \, dx}{135 d^4}-\frac{\left (B (b c-a d)^3 g^3 n\right ) \int \frac{1}{c+d x} \, dx}{135 d^3}\\ &=\frac{A b (b c-a d)^2 g^3 x}{135 d^3}+\frac{B (b c-a d)^2 g^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{135 d^3}-\frac{(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{270 d^2}+\frac{g^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{405 d}-\frac{B (b c-a d)^3 g^3 n \log (c+d x)}{135 d^4}-\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 ) \log (135 c+135 d x)}{135 d^4}-\frac{\left (B (b c-a d) g^3 n\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{405 d}+\frac{\left (B (b c-a d)^2 g^3 n\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{270 d^2}+\frac{\left (b B (b c-a d)^3 g^3 n\right ) \int \frac{\log (135 c+135 d x)}{a+b x} \, dx}{135 d^4}-\frac{\left (B (b c-a d)^3 g^3 n\right ) \int \frac{\log (135 c+135 d x)}{c+d x} \, dx}{135 d^3}\\ &=\frac{A b (b c-a d)^2 g^3 x}{135 d^3}+\frac{b B (b c-a d)^2 g^3 n x}{162 d^3}-\frac{B (b c-a d) g^3 n (a+b x)^2}{810 d^2}+\frac{B (b c-a d)^2 g^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{135 d^3}-\frac{(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{270 d^2}+\frac{g^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{405 d}-\frac{11 B (b c-a d)^3 g^3 n \log (c+d x)}{810 d^4}+\frac{B (b c-a d)^3 g^3 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (135 c+135 d x)}{135 d^4}-\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 ) \log (135 c+135 d x)}{135 d^4}-\frac{\left (B (b c-a d)^3 g^3 n\right ) \operatorname{Subst}\left (\int \frac{135 \log (x)}{x} \, dx,x,135 c+135 d x\right )}{18225 d^4}-\frac{\left (B (b c-a d)^3 g^3 n\right ) \int \frac{\log \left (\frac{135 d (a+b x)}{-135 b c+135 a d}\right )}{135 c+135 d x} \, dx}{d^3}\\ &=\frac{A b (b c-a d)^2 g^3 x}{135 d^3}+\frac{b B (b c-a d)^2 g^3 n x}{162 d^3}-\frac{B (b c-a d) g^3 n (a+b x)^2}{810 d^2}+\frac{B (b c-a d)^2 g^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{135 d^3}-\frac{(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{270 d^2}+\frac{g^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{405 d}-\frac{11 B (b c-a d)^3 g^3 n \log (c+d x)}{810 d^4}+\frac{B (b c-a d)^3 g^3 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (135 c+135 d x)}{135 d^4}-\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 ) \log (135 c+135 d x)}{135 d^4}-\frac{\left (B (b c-a d)^3 g^3 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,135 c+135 d x\right )}{135 d^4}-\frac{\left (B (b c-a d)^3 g^3 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-135 b c+135 a d}\right )}{x} \, dx,x,135 c+135 d x\right )}{135 d^4}\\ &=\frac{A b (b c-a d)^2 g^3 x}{135 d^3}+\frac{b B (b c-a d)^2 g^3 n x}{162 d^3}-\frac{B (b c-a d) g^3 n (a+b x)^2}{810 d^2}+\frac{B (b c-a d)^2 g^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{135 d^3}-\frac{(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{270 d^2}+\frac{g^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{405 d}-\frac{11 B (b c-a d)^3 g^3 n \log (c+d x)}{810 d^4}-\frac{B (b c-a d)^3 g^3 n \log ^2(135 (c+d x))}{270 d^4}+\frac{B (b c-a d)^3 g^3 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (135 c+135 d x)}{135 d^4}-\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 ) \log (135 c+135 d x)}{135 d^4}+\frac{B (b c-a d)^3 g^3 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{135 d^4}\\ \end{align*}

Mathematica [A]  time = 0.277999, size = 370, normalized size = 1.38 \[ \frac{g^3 \left (3 B n (b c-a d)^3 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (i (c+d x)) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (i (c+d x))\right )\right )+2 d^3 (a+b x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+3 d^2 (a+b x)^2 (a d-b c) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-6 (b c-a d)^3 \log (i (c+d x)) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+6 A b d x (b c-a d)^2+B n (b c-a d) \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )+6 B d (a+b x) (b c-a d)^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-6 B n (b c-a d)^3 \log (c+d x)+3 B n (b c-a d)^2 ((a d-b c) \log (c+d x)+b d x)\right )}{6 d^4 i} \]

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),x]

[Out]

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

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Maple [F]  time = 0.685, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bgx+ag \right ) ^{3}}{dix+ci} \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),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),x)

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Maxima [B]  time = 2.76182, size = 1354, normalized size = 5.03 \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),x, algorithm="maxima")

[Out]

3*A*a^2*b*g^3*(x/(d*i) - c*log(d*x + c)/(d^2*i)) - 1/6*A*b^3*g^3*(6*c^3*log(d*x + c)/(d^4*i) - (2*d^2*x^3 - 3*
c*d*x^2 + 6*c^2*x)/(d^3*i)) + 3/2*A*a*b^2*g^3*(2*c^2*log(d*x + c)/(d^3*i) + (d*x^2 - 2*c*x)/(d^2*i)) + A*a^3*g
^3*log(d*i*x + c*i)/(d*i) - (b^3*c^3*g^3*n - 3*a*b^2*c^2*d*g^3*n + 3*a^2*b*c*d^2*g^3*n - a^3*d^3*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) + 1/6*(6*a^3*d^3*g^3
*log(e) - (11*g^3*n + 6*g^3*log(e))*b^3*c^3 + 9*(3*g^3*n + 2*g^3*log(e))*a*b^2*c^2*d - 18*(g^3*n + g^3*log(e))
*a^2*b*c*d^2)*B*log(d*x + c)/(d^4*i) + 1/6*(2*B*b^3*d^3*g^3*x^3*log(e) - ((g^3*n + 3*g^3*log(e))*b^3*c*d^2 - (
g^3*n + 9*g^3*log(e))*a*b^2*d^3)*B*x^2 + 6*(b^3*c^3*g^3*n - 3*a*b^2*c^2*d*g^3*n + 3*a^2*b*c*d^2*g^3*n - a^3*d^
3*g^3*n)*B*log(b*x + a)*log(d*x + c) - 3*(b^3*c^3*g^3*n - 3*a*b^2*c^2*d*g^3*n + 3*a^2*b*c*d^2*g^3*n - a^3*d^3*
g^3*n)*B*log(d*x + c)^2 + ((5*g^3*n + 6*g^3*log(e))*b^3*c^2*d - 6*(2*g^3*n + 3*g^3*log(e))*a*b^2*c*d^2 + (7*g^
3*n + 18*g^3*log(e))*a^2*b*d^3)*B*x + (6*a*b^2*c^2*d*g^3*n - 15*a^2*b*c*d^2*g^3*n + 11*a^3*d^3*g^3*n)*B*log(b*
x + a) + (2*B*b^3*d^3*g^3*x^3 - 3*(b^3*c*d^2*g^3 - 3*a*b^2*d^3*g^3)*B*x^2 + 6*(b^3*c^2*d*g^3 - 3*a*b^2*c*d^2*g
^3 + 3*a^2*b*d^3*g^3)*B*x - 6*(b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^3)*B*log(d*x +
c))*log((b*x + a)^n) - (2*B*b^3*d^3*g^3*x^3 - 3*(b^3*c*d^2*g^3 - 3*a*b^2*d^3*g^3)*B*x^2 + 6*(b^3*c^2*d*g^3 - 3
*a*b^2*c*d^2*g^3 + 3*a^2*b*d^3*g^3)*B*x - 6*(b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^3
)*B*log(d*x + c))*log((d*x + c)^n))/(d^4*i)

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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 i x + c i}, 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),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*i*x + c*i), 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),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 )}}{d i x + c i}\,{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),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), x)