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

Optimal. Leaf size=266 \[ -\frac {b^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i (a+b x)^2 (b c-a d)^3}+\frac {d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (b c-a d)^3}+\frac {2 b d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (a+b x) (b c-a d)^3}-\frac {B d^2 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^3 i (b c-a d)^3}-\frac {B n (c+d x)^2 \left (b-\frac {4 d (a+b x)}{c+d x}\right )^2}{4 g^3 i (a+b x)^2 (b c-a d)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 0.83, antiderivative size = 557, normalized size of antiderivative = 2.09, number of steps used = 26, number of rules used = 11, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {B d^2 n \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac {B d^2 n \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac {d^2 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (b c-a d)^3}-\frac {d^2 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (b c-a d)^3}+\frac {d \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (a+b x) (b c-a d)^2}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 g^3 i (a+b x)^2 (b c-a d)}-\frac {B d^2 n \log ^2(a+b x)}{2 g^3 i (b c-a d)^3}-\frac {B d^2 n \log ^2(c+d x)}{2 g^3 i (b c-a d)^3}+\frac {3 B d^2 n \log (a+b x)}{2 g^3 i (b c-a d)^3}-\frac {3 B d^2 n \log (c+d x)}{2 g^3 i (b c-a d)^3}+\frac {B d^2 n \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac {B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac {3 B d n}{2 g^3 i (a+b x) (b c-a d)^2}-\frac {B n}{4 g^3 i (a+b x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(141 c+141 d x) (a g+b g x)^3} \, dx &=\int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d) g^3 (a+b x)^3}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (b d^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{141 (b c-a d)^3 g^3}-\frac {d^3 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{141 (b c-a d)^3 g^3}-\frac {(b d) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{141 (b c-a d)^2 g^3}+\frac {b \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{141 (b c-a d) g^3}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac {\left (B d^2 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{141 (b c-a d)^3 g^3}+\frac {\left (B d^2 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}{141 (b c-a d)^3 g^3}-\frac {(B d n) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{141 (b c-a d)^2 g^3}+\frac {(B n) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{282 (b c-a d) g^3}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}+\frac {(B n) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{282 g^3}-\frac {\left (B d^2 n\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{141 (b c-a d)^3 g^3}+\frac {\left (B d^2 n\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{141 (b c-a d)^3 g^3}-\frac {(B d n) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{141 (b c-a d) g^3}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}+\frac {(B n) \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}{282 g^3}-\frac {\left (b B d^2 n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{141 (b c-a d)^3 g^3}+\frac {\left (b B d^2 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{141 (b c-a d)^3 g^3}+\frac {\left (B d^3 n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{141 (b c-a d)^3 g^3}-\frac {\left (B d^3 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{141 (b c-a d)^3 g^3}-\frac {(B d n) \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}{141 (b c-a d) g^3}\\ &=-\frac {B n}{564 (b c-a d) g^3 (a+b x)^2}+\frac {B d n}{94 (b c-a d)^2 g^3 (a+b x)}+\frac {B d^2 n \log (a+b x)}{94 (b c-a d)^3 g^3}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac {B d^2 n \log (c+d x)}{94 (b c-a d)^3 g^3}+\frac {B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}+\frac {B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{141 (b c-a d)^3 g^3}-\frac {\left (B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{141 (b c-a d)^3 g^3}-\frac {\left (B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{141 (b c-a d)^3 g^3}-\frac {\left (b B d^2 n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{141 (b c-a d)^3 g^3}-\frac {\left (B d^3 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{141 (b c-a d)^3 g^3}\\ &=-\frac {B n}{564 (b c-a d) g^3 (a+b x)^2}+\frac {B d n}{94 (b c-a d)^2 g^3 (a+b x)}+\frac {B d^2 n \log (a+b x)}{94 (b c-a d)^3 g^3}-\frac {B d^2 n \log ^2(a+b x)}{282 (b c-a d)^3 g^3}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac {B d^2 n \log (c+d x)}{94 (b c-a d)^3 g^3}+\frac {B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac {B d^2 n \log ^2(c+d x)}{282 (b c-a d)^3 g^3}+\frac {B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{141 (b c-a d)^3 g^3}-\frac {\left (B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{141 (b c-a d)^3 g^3}-\frac {\left (B d^2 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 )}{141 (b c-a d)^3 g^3}\\ &=-\frac {B n}{564 (b c-a d) g^3 (a+b x)^2}+\frac {B d n}{94 (b c-a d)^2 g^3 (a+b x)}+\frac {B d^2 n \log (a+b x)}{94 (b c-a d)^3 g^3}-\frac {B d^2 n \log ^2(a+b x)}{282 (b c-a d)^3 g^3}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac {B d^2 n \log (c+d x)}{94 (b c-a d)^3 g^3}+\frac {B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac {B d^2 n \log ^2(c+d x)}{282 (b c-a d)^3 g^3}+\frac {B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{141 (b c-a d)^3 g^3}+\frac {B d^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{141 (b c-a d)^3 g^3}+\frac {B d^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{141 (b c-a d)^3 g^3}\\ \end {align*}

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Mathematica [C]  time = 0.41, size = 434, normalized size = 1.63 \[ \frac {4 d^2 (a+b x)^2 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-4 d^2 (a+b x)^2 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+4 d (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-2 B d^2 n (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+2 B d^2 n (a+b x)^2 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-B n \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )+4 B d n (a+b x) (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)}{4 g^3 i (a+b x)^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.93, size = 483, normalized size = 1.82 \[ -\frac {2 \, A b^{2} c^{2} - 8 \, A a b c d + 6 \, A a^{2} d^{2} - 2 \, {\left (B b^{2} d^{2} n x^{2} + 2 \, B a b d^{2} n x + B a^{2} d^{2} n\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (B b^{2} c^{2} - 8 \, B a b c d + 7 \, B a^{2} d^{2}\right )} n - 2 \, {\left (2 \, A b^{2} c d - 2 \, A a b d^{2} + 3 \, {\left (B b^{2} c d - B a b d^{2}\right )} n\right )} x + 2 \, {\left (B b^{2} c^{2} - 4 \, B a b c d + 3 \, B a^{2} d^{2} - 2 \, {\left (B b^{2} c d - B a b d^{2}\right )} x - 2 \, {\left (B b^{2} d^{2} x^{2} + 2 \, B a b d^{2} x + B a^{2} d^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e) - 2 \, {\left (2 \, A a^{2} d^{2} + {\left (3 \, B b^{2} d^{2} n + 2 \, A b^{2} d^{2}\right )} x^{2} - {\left (B b^{2} c^{2} - 4 \, B a b c d\right )} n + 2 \, {\left (2 \, A a b d^{2} + {\left (B b^{2} c d + 2 \, B a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} g^{3} i x + {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} g^{3} i\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*A*b^2*c^2 - 8*A*a*b*c*d + 6*A*a^2*d^2 - 2*(B*b^2*d^2*n*x^2 + 2*B*a*b*d^2*n*x + B*a^2*d^2*n)*log((b*x +
a)/(d*x + c))^2 + (B*b^2*c^2 - 8*B*a*b*c*d + 7*B*a^2*d^2)*n - 2*(2*A*b^2*c*d - 2*A*a*b*d^2 + 3*(B*b^2*c*d - B
*a*b*d^2)*n)*x + 2*(B*b^2*c^2 - 4*B*a*b*c*d + 3*B*a^2*d^2 - 2*(B*b^2*c*d - B*a*b*d^2)*x - 2*(B*b^2*d^2*x^2 + 2
*B*a*b*d^2*x + B*a^2*d^2)*log((b*x + a)/(d*x + c)))*log(e) - 2*(2*A*a^2*d^2 + (3*B*b^2*d^2*n + 2*A*b^2*d^2)*x^
2 - (B*b^2*c^2 - 4*B*a*b*c*d)*n + 2*(2*A*a*b*d^2 + (B*b^2*c*d + 2*B*a*b*d^2)*n)*x)*log((b*x + a)/(d*x + c)))/(
(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^3*i*x^2 + 2*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b
^2*c*d^2 - a^4*b*d^3)*g^3*i*x + (a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*g^3*i)

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

[Out]

Timed out

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maple [F]  time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{\left (b g x +a g \right )^{3} \left (d i x +c i \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.73, size = 888, normalized size = 3.34 \[ \frac {1}{2} \, B {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} g^{3} i x + {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} g^{3} i} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i}\right )} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) - \frac {{\left (b^{2} c^{2} - 8 \, a b c d + 7 \, a^{2} d^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (d x + c\right )^{2} - 6 \, {\left (b^{2} c d - a b d^{2}\right )} x - 6 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 6 \, a b d^{2} x + 3 \, a^{2} d^{2} - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B n}{4 \, {\left (a^{2} b^{3} c^{3} g^{3} i - 3 \, a^{3} b^{2} c^{2} d g^{3} i + 3 \, a^{4} b c d^{2} g^{3} i - a^{5} d^{3} g^{3} i + {\left (b^{5} c^{3} g^{3} i - 3 \, a b^{4} c^{2} d g^{3} i + 3 \, a^{2} b^{3} c d^{2} g^{3} i - a^{3} b^{2} d^{3} g^{3} i\right )} x^{2} + 2 \, {\left (a b^{4} c^{3} g^{3} i - 3 \, a^{2} b^{3} c^{2} d g^{3} i + 3 \, a^{3} b^{2} c d^{2} g^{3} i - a^{4} b d^{3} g^{3} i\right )} x\right )}} + \frac {1}{2} \, A {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} g^{3} i x + {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} g^{3} i} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*B*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*g^3*i*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d
+ a^3*b*d^2)*g^3*i*x + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*g^3*i) + 2*d^2*log(b*x + a)/((b^3*c^3 - 3*a*b^2*
c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^2*log(d*x + c)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d
^3)*g^3*i))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) - 1/4*(b^2*c^2 - 8*a*b*c*d + 7*a^2*d^2 + 2*(b^2*d^2*x^2 + 2
*a*b*d^2*x + a^2*d^2)*log(b*x + a)^2 + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(d*x + c)^2 - 6*(b^2*c*d - a
*b*d^2)*x - 6*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a) + 2*(3*b^2*d^2*x^2 + 6*a*b*d^2*x + 3*a^2*d^2
- 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a))*log(d*x + c))*B*n/(a^2*b^3*c^3*g^3*i - 3*a^3*b^2*c^2*d
*g^3*i + 3*a^4*b*c*d^2*g^3*i - a^5*d^3*g^3*i + (b^5*c^3*g^3*i - 3*a*b^4*c^2*d*g^3*i + 3*a^2*b^3*c*d^2*g^3*i -
a^3*b^2*d^3*g^3*i)*x^2 + 2*(a*b^4*c^3*g^3*i - 3*a^2*b^3*c^2*d*g^3*i + 3*a^3*b^2*c*d^2*g^3*i - a^4*b*d^3*g^3*i)
*x) + 1/2*A*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*g^3*i*x^2 + 2*(a*b^3*c^2 - 2*a^2*b
^2*c*d + a^3*b*d^2)*g^3*i*x + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*g^3*i) + 2*d^2*log(b*x + a)/((b^3*c^3 - 3*
a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^2*log(d*x + c)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 -
a^3*d^3)*g^3*i))

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mupad [B]  time = 6.34, size = 573, normalized size = 2.15 \[ \frac {B\,d^2\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {g^3\,i\,n\,\left (a\,d-b\,c\right )\,\left (2\,a\,d-b\,c\right )}{2\,d^2}+\frac {a\,g^3\,i\,n\,\left (a\,d-b\,c\right )}{2\,d}+\frac {b\,g^3\,i\,n\,x\,\left (a\,d-b\,c\right )}{d}\right )}{g^3\,i\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (i\,a^2\,g^3+2\,i\,a\,b\,g^3\,x+i\,b^2\,g^3\,x^2\right )}-\frac {B\,d^2\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{2\,g^3\,i\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {\frac {6\,A\,a\,d-2\,A\,b\,c+7\,B\,a\,d\,n-B\,b\,c\,n}{2\,\left (a\,d-b\,c\right )}+\frac {d\,x\,\left (2\,A\,b+3\,B\,b\,n\right )}{a\,d-b\,c}}{x^2\,\left (2\,b^3\,c\,g^3\,i-2\,a\,b^2\,d\,g^3\,i\right )+x\,\left (4\,a\,b^2\,c\,g^3\,i-4\,a^2\,b\,d\,g^3\,i\right )-2\,a^3\,d\,g^3\,i+2\,a^2\,b\,c\,g^3\,i}+\frac {d^2\,\mathrm {atan}\left (\frac {d^2\,\left (A+\frac {3\,B\,n}{2}\right )\,\left (2\,i\,a^3\,d^3\,g^3-2\,i\,a^2\,b\,c\,d^2\,g^3-2\,i\,a\,b^2\,c^2\,d\,g^3+2\,i\,b^3\,c^3\,g^3\right )\,1{}\mathrm {i}}{g^3\,i\,\left (2\,A\,d^2+3\,B\,d^2\,n\right )\,{\left (a\,d-b\,c\right )}^3}+\frac {b\,d^3\,x\,\left (A+\frac {3\,B\,n}{2}\right )\,\left (i\,a^2\,d^2\,g^3-2\,i\,a\,b\,c\,d\,g^3+i\,b^2\,c^2\,g^3\right )\,4{}\mathrm {i}}{g^3\,i\,\left (2\,A\,d^2+3\,B\,d^2\,n\right )\,{\left (a\,d-b\,c\right )}^3}\right )\,\left (A+\frac {3\,B\,n}{2}\right )\,2{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d^2*atan((d^2*(A + (3*B*n)/2)*(2*a^3*d^3*g^3*i + 2*b^3*c^3*g^3*i - 2*a*b^2*c^2*d*g^3*i - 2*a^2*b*c*d^2*g^3*i)
*1i)/(g^3*i*(2*A*d^2 + 3*B*d^2*n)*(a*d - b*c)^3) + (b*d^3*x*(A + (3*B*n)/2)*(a^2*d^2*g^3*i + b^2*c^2*g^3*i - 2
*a*b*c*d*g^3*i)*4i)/(g^3*i*(2*A*d^2 + 3*B*d^2*n)*(a*d - b*c)^3))*(A + (3*B*n)/2)*2i)/(g^3*i*(a*d - b*c)^3) - (
(6*A*a*d - 2*A*b*c + 7*B*a*d*n - B*b*c*n)/(2*(a*d - b*c)) + (d*x*(2*A*b + 3*B*b*n))/(a*d - b*c))/(x^2*(2*b^3*c
*g^3*i - 2*a*b^2*d*g^3*i) + x*(4*a*b^2*c*g^3*i - 4*a^2*b*d*g^3*i) - 2*a^3*d*g^3*i + 2*a^2*b*c*g^3*i) - (B*d^2*
log(e*((a + b*x)/(c + d*x))^n)^2)/(2*g^3*i*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (B*d^2*log(e*((a +
b*x)/(c + d*x))^n)*((g^3*i*n*(a*d - b*c)*(2*a*d - b*c))/(2*d^2) + (a*g^3*i*n*(a*d - b*c))/(2*d) + (b*g^3*i*n*
x*(a*d - b*c))/d))/(g^3*i*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(a^2*g^3*i + b^2*g^3*i*x^2 + 2*a*b*g^3
*i*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((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**3/(d*i*x+c*i),x)

[Out]

Timed out

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