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

Optimal. Leaf size=382 \[ \frac {b^2 g^3 (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{d^4 i^3}+\frac {b g^3 (a+b x) (3 A+B n) (b c-a d)}{d^3 i^3 (c+d x)}+\frac {g^3 (a+b x)^2 (b c-a d) \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{2 d^2 i^3 (c+d x)^2}+\frac {g^3 (a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d i^3 (c+d x)^2}+\frac {3 b^2 B g^3 n (b c-a d) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^3}+\frac {3 b B g^3 (a+b x) (b c-a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^3 i^3 (c+d x)}-\frac {3 b B g^3 n (a+b x) (b c-a d)}{d^3 i^3 (c+d x)}-\frac {3 B g^3 n (a+b x)^2 (b c-a d)}{4 d^2 i^3 (c+d x)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.75, antiderivative size = 461, normalized size of antiderivative = 1.21, number of steps used = 21, 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, 44, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac {3 b^2 B g^3 n (b c-a d) \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^4 i^3}-\frac {3 b^2 g^3 (b c-a d) \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^4 i^3}-\frac {3 b g^3 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^4 i^3 (c+d x)}+\frac {g^3 (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^4 i^3 (c+d x)^2}+\frac {b^2 B g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^3 i^3}-\frac {3 b^2 B g^3 n (b c-a d) \log ^2(c+d x)}{2 d^4 i^3}+\frac {5 b^2 B g^3 n (b c-a d) \log (a+b x)}{2 d^4 i^3}-\frac {7 b^2 B g^3 n (b c-a d) \log (c+d x)}{2 d^4 i^3}+\frac {3 b^2 B g^3 n (b c-a d) \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d^4 i^3}+\frac {5 b B g^3 n (b c-a d)^2}{2 d^4 i^3 (c+d x)}-\frac {B g^3 n (b c-a d)^3}{4 d^4 i^3 (c+d x)^2}+\frac {A b^3 g^3 x}{d^3 i^3} \]

Antiderivative was successfully verified.

[In]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, 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 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 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 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 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 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 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 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 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 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 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]

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 )}{(151 c+151 d x)^3} \, dx &=\int \left (\frac {b^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3442951 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 )}{3442951 d^3 (c+d x)^3}+\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 )}{3442951 d^3 (c+d x)^2}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3442951 d^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (b^3 g^3\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3442951 d^3}-\frac {\left (3 b^2 (b c-a d) 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}{3442951 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)^2} \, dx}{3442951 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)^3} \, dx}{3442951 d^3}\\ &=\frac {A b^3 g^3 x}{3442951 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 )}{6885902 d^4 (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 )}{3442951 d^4 (c+d x)}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3442951 d^4}+\frac {\left (b^3 B g^3\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{3442951 d^3}+\frac {\left (3 b^2 B (b c-a d) 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}{3442951 d^4}+\frac {\left (3 b B (b c-a d)^2 g^3 n\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{3442951 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)^3} \, dx}{6885902 d^4}\\ &=\frac {A b^3 g^3 x}{3442951 d^3}+\frac {b^2 B g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3442951 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 )}{6885902 d^4 (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 )}{3442951 d^4 (c+d x)}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3442951 d^4}+\frac {\left (3 b^2 B (b c-a d) 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}{3442951 d^4}-\frac {\left (b^2 B (b c-a d) g^3 n\right ) \int \frac {1}{c+d x} \, dx}{3442951 d^3}+\frac {\left (3 b B (b c-a d)^3 g^3 n\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{3442951 d^4}-\frac {\left (B (b c-a d)^4 g^3 n\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{6885902 d^4}\\ &=\frac {A b^3 g^3 x}{3442951 d^3}+\frac {b^2 B g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3442951 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 )}{6885902 d^4 (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 )}{3442951 d^4 (c+d x)}-\frac {b^2 B (b c-a d) g^3 n \log (c+d x)}{3442951 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3442951 d^4}+\frac {\left (3 b^3 B (b c-a d) g^3 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3442951 d^4}-\frac {\left (3 b^2 B (b c-a d) g^3 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3442951 d^3}+\frac {\left (3 b B (b c-a d)^3 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}{3442951 d^4}-\frac {\left (B (b c-a d)^4 g^3 n\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}{6885902 d^4}\\ &=\frac {A b^3 g^3 x}{3442951 d^3}-\frac {B (b c-a d)^3 g^3 n}{13771804 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n}{6885902 d^4 (c+d x)}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x)}{6885902 d^4}+\frac {b^2 B g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3442951 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 )}{6885902 d^4 (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 )}{3442951 d^4 (c+d x)}-\frac {7 b^2 B (b c-a d) g^3 n \log (c+d x)}{6885902 d^4}+\frac {3 b^2 B (b c-a d) g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3442951 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3442951 d^4}-\frac {\left (3 b^2 B (b c-a d) g^3 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3442951 d^4}-\frac {\left (3 b^2 B (b c-a d) g^3 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3442951 d^3}\\ &=\frac {A b^3 g^3 x}{3442951 d^3}-\frac {B (b c-a d)^3 g^3 n}{13771804 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n}{6885902 d^4 (c+d x)}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x)}{6885902 d^4}+\frac {b^2 B g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3442951 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 )}{6885902 d^4 (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 )}{3442951 d^4 (c+d x)}-\frac {7 b^2 B (b c-a d) g^3 n \log (c+d x)}{6885902 d^4}+\frac {3 b^2 B (b c-a d) g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3442951 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3442951 d^4}-\frac {3 b^2 B (b c-a d) g^3 n \log ^2(c+d x)}{6885902 d^4}-\frac {\left (3 b^2 B (b c-a d) 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 )}{3442951 d^4}\\ &=\frac {A b^3 g^3 x}{3442951 d^3}-\frac {B (b c-a d)^3 g^3 n}{13771804 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n}{6885902 d^4 (c+d x)}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x)}{6885902 d^4}+\frac {b^2 B g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3442951 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 )}{6885902 d^4 (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 )}{3442951 d^4 (c+d x)}-\frac {7 b^2 B (b c-a d) g^3 n \log (c+d x)}{6885902 d^4}+\frac {3 b^2 B (b c-a d) g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3442951 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3442951 d^4}-\frac {3 b^2 B (b c-a d) g^3 n \log ^2(c+d x)}{6885902 d^4}+\frac {3 b^2 B (b c-a d) g^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3442951 d^4}\\ \end {align*}

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Mathematica [A]  time = 0.45, size = 334, normalized size = 0.87 \[ \frac {g^3 \left (-12 b^2 (b c-a d) \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-\frac {12 b (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}+\frac {2 (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(c+d x)^2}+4 b^2 B d (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 b^2 B n (b c-a d) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+10 b^2 B n (b c-a d) \log (a+b x)-14 b^2 B n (b c-a d) \log (c+d x)+\frac {10 b B n (b c-a d)^2}{c+d x}-\frac {B n (b c-a d)^3}{(c+d x)^2}+4 A b^3 d x\right )}{4 d^4 i^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A b^{3} g^{3} x^{3} + 3 \, A a b^{2} g^{3} x^{2} + 3 \, A a^{2} b g^{3} x + A a^{3} g^{3} + {\left (B b^{3} g^{3} x^{3} + 3 \, B a b^{2} g^{3} x^{2} + 3 \, B a^{2} b g^{3} x + B a^{3} g^{3}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{d^{3} i^{3} x^{3} + 3 \, c d^{2} i^{3} x^{2} + 3 \, c^{2} d i^{3} x + c^{3} i^{3}}, x\right ) \]

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)^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(e*((b*x + a)/(d*x + c))^n))/(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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)^3,x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right )^{3} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}{\left (d i x +c i \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 5.52, size = 2894, normalized size = 7.58 \[ \text {result too large to display} \]

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)^3,x, algorithm="maxima")

[Out]

3/4*B*a^2*b*g^3*n*((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)) + 1/4*B*a^3*g^3*n*((
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*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)) - 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)) + 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)*B*a^2*b*g^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(d^4*i^
3*x^2 + 2*c*d^3*i^3*x + c^2*d^2*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*B*a^3*g^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(d^3*i^3*x^2 + 2*c*d^2*i^3*x + c^2*d*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) - (7*g^3*n + 6*g^3*log(e))*b^5*c
^3 + (19*g^3*n + 18*g^3*log(e))*a*b^4*c^2*d - 2*(7*g^3*n + 9*g^3*log(e))*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*((5*g^3*n - 4*g^3*log(e))*b^5*c^4*d - 20*(g^3*n - g^3*log(e))*a*b^4*c^3*d^2 + (27*g^3*n - 28*g^3*log
(e))*a^2*b^3*c^2*d^3 - 12*(g^3*n - g^3*log(e))*a^3*b^2*c*d^4)*B*x + 12*((b^5*c^3*d^2*g^3*n - 3*a*b^4*c^2*d^3*g
^3*n + 3*a^2*b^3*c*d^4*g^3*n - a^3*b^2*d^5*g^3*n)*B*x^2 + 2*(b^5*c^4*d*g^3*n - 3*a*b^4*c^3*d^2*g^3*n + 3*a^2*b
^3*c^2*d^3*g^3*n - a^3*b^2*c*d^4*g^3*n)*B*x + (b^5*c^5*g^3*n - 3*a*b^4*c^4*d*g^3*n + 3*a^2*b^3*c^3*d^2*g^3*n -
a^3*b^2*c^2*d^3*g^3*n)*B)*log(b*x + a)*log(d*x + c) - 6*((b^5*c^3*d^2*g^3*n - 3*a*b^4*c^2*d^3*g^3*n + 3*a^2*b
^3*c*d^4*g^3*n - a^3*b^2*d^5*g^3*n)*B*x^2 + 2*(b^5*c^4*d*g^3*n - 3*a*b^4*c^3*d^2*g^3*n + 3*a^2*b^3*c^2*d^3*g^3
*n - a^3*b^2*c*d^4*g^3*n)*B*x + (b^5*c^5*g^3*n - 3*a*b^4*c^4*d*g^3*n + 3*a^2*b^3*c^3*d^2*g^3*n - a^3*b^2*c^2*d
^3*g^3*n)*B)*log(d*x + c)^2 + ((9*g^3*n - 10*g^3*log(e))*b^5*c^5 - (35*g^3*n - 38*g^3*log(e))*a*b^4*c^4*d + (4
7*g^3*n - 46*g^3*log(e))*a^2*b^3*c^3*d^2 - 3*(7*g^3*n - 6*g^3*log(e))*a^3*b^2*c^2*d^3)*B + 2*((5*b^5*c^3*d^2*g
^3*n - 13*a*b^4*c^2*d^3*g^3*n + 8*a^2*b^3*c*d^4*g^3*n + 2*a^3*b^2*d^5*g^3*n)*B*x^2 + 2*(5*b^5*c^4*d*g^3*n - 13
*a*b^4*c^3*d^2*g^3*n + 8*a^2*b^3*c^2*d^3*g^3*n + 2*a^3*b^2*c*d^4*g^3*n)*B*x + (5*b^5*c^5*g^3*n - 13*a*b^4*c^4*
d*g^3*n + 8*a^2*b^3*c^3*d^2*g^3*n + 2*a^3*b^2*c^2*d^3*g^3*n)*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 - 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))*log((b*x + a)^n) - 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 - 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))*log((d*x + c)^n))/(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*n - a*b^2*d*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^3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a\,g+b\,g\,x\right )}^3\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{{\left (c\,i+d\,i\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**3,x)

[Out]

Timed out

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