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

Optimal. Leaf size=364 \[ -\frac {b^3 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 i^2 (a+b x)^2 (b c-a d)^4}+\frac {3 b^2 d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^2 (a+b x) (b c-a d)^4}-\frac {d^3 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^2 (c+d x) (b c-a d)^4}+\frac {3 b d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^2 (b c-a d)^4}-\frac {b^3 B (c+d x)^2}{4 g^3 i^2 (a+b x)^2 (b c-a d)^4}+\frac {3 b^2 B d (c+d x)}{g^3 i^2 (a+b x) (b c-a d)^4}+\frac {B d^3 (a+b x)}{g^3 i^2 (c+d x) (b c-a d)^4}-\frac {3 b B d^2 \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^3 i^2 (b c-a d)^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 1.12, antiderivative size = 630, normalized size of antiderivative = 1.73, number of steps used = 32, number of rules used = 11, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {3 b B d^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g^3 i^2 (b c-a d)^4}+\frac {3 b B d^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g^3 i^2 (b c-a d)^4}+\frac {3 b d^2 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^2 (b c-a d)^4}+\frac {d^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^2 (c+d x) (b c-a d)^3}-\frac {3 b d^2 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^2 (b c-a d)^4}+\frac {2 b d \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^2 (a+b x) (b c-a d)^3}-\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 i^2 (a+b x)^2 (b c-a d)^2}-\frac {B d^2}{g^3 i^2 (c+d x) (b c-a d)^3}-\frac {3 b B d^2 \log ^2(a+b x)}{2 g^3 i^2 (b c-a d)^4}-\frac {3 b B d^2 \log ^2(c+d x)}{2 g^3 i^2 (b c-a d)^4}+\frac {3 b B d^2 \log (a+b x)}{2 g^3 i^2 (b c-a d)^4}+\frac {3 b B d^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g^3 i^2 (b c-a d)^4}-\frac {3 b B d^2 \log (c+d x)}{2 g^3 i^2 (b c-a d)^4}+\frac {3 b B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g^3 i^2 (b c-a d)^4}+\frac {5 b B d}{2 g^3 i^2 (a+b x) (b c-a d)^3}-\frac {b B}{4 g^3 i^2 (a+b x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 (\frac {e (a+b x)}{c+d x}\right )}{(45 c+45 d x)^2 (a g+b g x)^3} \, dx &=\int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^2 g^3 (a+b x)^3}-\frac {2 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (a+b x)^2}+\frac {b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{675 (b c-a d)^4 g^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (c+d x)^2}-\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{675 (b c-a d)^4 g^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (b^2 d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{675 (b c-a d)^4 g^3}-\frac {\left (b d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{675 (b c-a d)^4 g^3}-\frac {\left (2 b^2 d\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{2025 (b c-a d)^3 g^3}-\frac {d^3 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{2025 (b c-a d)^3 g^3}+\frac {b^2 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{2025 (b c-a d)^2 g^3}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4050 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (c+d x)}+\frac {b d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{675 (b c-a d)^4 g^3}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{675 (b c-a d)^4 g^3}-\frac {\left (b B 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}{675 (b c-a d)^4 g^3}+\frac {\left (b B 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 (c+d x)}{e (a+b x)} \, dx}{675 (b c-a d)^4 g^3}-\frac {(2 b B d) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{2025 (b c-a d)^3 g^3}-\frac {\left (B d^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{2025 (b c-a d)^3 g^3}+\frac {(b B) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{4050 (b c-a d)^2 g^3}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4050 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (c+d x)}+\frac {b d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{675 (b c-a d)^4 g^3}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{675 (b c-a d)^4 g^3}-\frac {(2 b B d) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{2025 (b c-a d)^2 g^3}-\frac {\left (B d^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{2025 (b c-a d)^2 g^3}+\frac {(b B) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{4050 (b c-a d) g^3}-\frac {\left (b B 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}{675 (b c-a d)^4 e g^3}+\frac {\left (b B 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 (c+d x)}{a+b x} \, dx}{675 (b c-a d)^4 e g^3}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4050 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (c+d x)}+\frac {b d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{675 (b c-a d)^4 g^3}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{675 (b c-a d)^4 g^3}-\frac {(2 b B d) \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}{2025 (b c-a d)^2 g^3}-\frac {\left (B d^2\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}{2025 (b c-a d)^2 g^3}+\frac {(b B) \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}{4050 (b c-a d) g^3}-\frac {\left (b B 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}{675 (b c-a d)^4 e g^3}+\frac {\left (b B d^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{675 (b c-a d)^4 e g^3}\\ &=-\frac {b B}{8100 (b c-a d)^2 g^3 (a+b x)^2}+\frac {b B d}{810 (b c-a d)^3 g^3 (a+b x)}-\frac {B d^2}{2025 (b c-a d)^3 g^3 (c+d x)}+\frac {b B d^2 \log (a+b x)}{1350 (b c-a d)^4 g^3}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4050 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (c+d x)}+\frac {b d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{675 (b c-a d)^4 g^3}-\frac {b B d^2 \log (c+d x)}{1350 (b c-a d)^4 g^3}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{675 (b c-a d)^4 g^3}-\frac {\left (b^2 B d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{675 (b c-a d)^4 g^3}+\frac {\left (b^2 B d^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{675 (b c-a d)^4 g^3}+\frac {\left (b B d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{675 (b c-a d)^4 g^3}-\frac {\left (b B d^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{675 (b c-a d)^4 g^3}\\ &=-\frac {b B}{8100 (b c-a d)^2 g^3 (a+b x)^2}+\frac {b B d}{810 (b c-a d)^3 g^3 (a+b x)}-\frac {B d^2}{2025 (b c-a d)^3 g^3 (c+d x)}+\frac {b B d^2 \log (a+b x)}{1350 (b c-a d)^4 g^3}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4050 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (c+d x)}+\frac {b d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{675 (b c-a d)^4 g^3}-\frac {b B d^2 \log (c+d x)}{1350 (b c-a d)^4 g^3}+\frac {b B d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{675 (b c-a d)^4 g^3}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{675 (b c-a d)^4 g^3}+\frac {b B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{675 (b c-a d)^4 g^3}-\frac {\left (b B d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{675 (b c-a d)^4 g^3}-\frac {\left (b B d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{675 (b c-a d)^4 g^3}-\frac {\left (b^2 B d^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{675 (b c-a d)^4 g^3}-\frac {\left (b B d^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{675 (b c-a d)^4 g^3}\\ &=-\frac {b B}{8100 (b c-a d)^2 g^3 (a+b x)^2}+\frac {b B d}{810 (b c-a d)^3 g^3 (a+b x)}-\frac {B d^2}{2025 (b c-a d)^3 g^3 (c+d x)}+\frac {b B d^2 \log (a+b x)}{1350 (b c-a d)^4 g^3}-\frac {b B d^2 \log ^2(a+b x)}{1350 (b c-a d)^4 g^3}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4050 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (c+d x)}+\frac {b d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{675 (b c-a d)^4 g^3}-\frac {b B d^2 \log (c+d x)}{1350 (b c-a d)^4 g^3}+\frac {b B d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{675 (b c-a d)^4 g^3}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{675 (b c-a d)^4 g^3}-\frac {b B d^2 \log ^2(c+d x)}{1350 (b c-a d)^4 g^3}+\frac {b B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{675 (b c-a d)^4 g^3}-\frac {\left (b B 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 )}{675 (b c-a d)^4 g^3}-\frac {\left (b B d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{675 (b c-a d)^4 g^3}\\ &=-\frac {b B}{8100 (b c-a d)^2 g^3 (a+b x)^2}+\frac {b B d}{810 (b c-a d)^3 g^3 (a+b x)}-\frac {B d^2}{2025 (b c-a d)^3 g^3 (c+d x)}+\frac {b B d^2 \log (a+b x)}{1350 (b c-a d)^4 g^3}-\frac {b B d^2 \log ^2(a+b x)}{1350 (b c-a d)^4 g^3}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4050 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2025 (b c-a d)^3 g^3 (c+d x)}+\frac {b d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{675 (b c-a d)^4 g^3}-\frac {b B d^2 \log (c+d x)}{1350 (b c-a d)^4 g^3}+\frac {b B d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{675 (b c-a d)^4 g^3}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{675 (b c-a d)^4 g^3}-\frac {b B d^2 \log ^2(c+d x)}{1350 (b c-a d)^4 g^3}+\frac {b B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{675 (b c-a d)^4 g^3}+\frac {b B d^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{675 (b c-a d)^4 g^3}+\frac {b B d^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{675 (b c-a d)^4 g^3}\\ \end {align*}

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Mathematica [C]  time = 0.73, size = 453, normalized size = 1.24 \[ \frac {12 b d^2 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {4 d^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-12 b d^2 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {8 b d (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {2 b (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a+b x)^2}+\frac {8 b^2 B c d}{a+b x}-6 b B d^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+6 b B d^2 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+\frac {2 b B d (b c-a d)}{a+b x}-\frac {b B (b c-a d)^2}{(a+b x)^2}-\frac {8 a b B d^2}{a+b x}+6 b B d^2 \log (a+b x)+\frac {4 a B d^3}{c+d x}-\frac {4 b B c d^2}{c+d x}-6 b B d^2 \log (c+d x)}{4 g^3 i^2 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((b*B*(b*c - a*d)^2)/(a + b*x)^2) + (8*b^2*B*c*d)/(a + b*x) - (8*a*b*B*d^2)/(a + b*x) + (2*b*B*d*(b*c - a*d)
)/(a + b*x) - (4*b*B*c*d^2)/(c + d*x) + (4*a*B*d^3)/(c + d*x) + 6*b*B*d^2*Log[a + b*x] - (2*b*(b*c - a*d)^2*(A
+ B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x)^2 + (8*b*d*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a
+ b*x) + (4*d^2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x) + 12*b*d^2*Log[a + b*x]*(A + B*Log
[(e*(a + b*x))/(c + d*x)]) - 6*b*B*d^2*Log[c + d*x] - 12*b*d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*
x] - 6*b*B*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)]) + 6*b*B*d^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)^4*g^3*i^2)

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fricas [A]  time = 0.88, size = 664, normalized size = 1.82 \[ -\frac {{\left (2 \, A + B\right )} b^{3} c^{3} - 12 \, {\left (A + B\right )} a b^{2} c^{2} d + 3 \, {\left (2 \, A + 5 \, B\right )} a^{2} b c d^{2} + 4 \, {\left (A - B\right )} a^{3} d^{3} - 6 \, {\left ({\left (2 \, A + B\right )} b^{3} c d^{2} - {\left (2 \, A + B\right )} a b^{2} d^{3}\right )} x^{2} - 6 \, {\left (B b^{3} d^{3} x^{3} + B a^{2} b c d^{2} + {\left (B b^{3} c d^{2} + 2 \, B a b^{2} d^{3}\right )} x^{2} + {\left (2 \, B a b^{2} c d^{2} + B a^{2} b d^{3}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 3 \, {\left ({\left (2 \, A + 3 \, B\right )} b^{3} c^{2} d + 2 \, {\left (2 \, A - B\right )} a b^{2} c d^{2} - {\left (6 \, A + B\right )} a^{2} b d^{3}\right )} x - 2 \, {\left (3 \, {\left (2 \, A + B\right )} b^{3} d^{3} x^{3} - B b^{3} c^{3} + 6 \, B a b^{2} c^{2} d + 6 \, A a^{2} b c d^{2} - 2 \, B a^{3} d^{3} + 3 \, {\left ({\left (2 \, A + 3 \, B\right )} b^{3} c d^{2} + 4 \, A a b^{2} d^{3}\right )} x^{2} + 3 \, {\left (B b^{3} c^{2} d + 4 \, {\left (A + B\right )} a b^{2} c d^{2} + 2 \, {\left (A - B\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{6} c^{4} d - 4 \, a b^{5} c^{3} d^{2} + 6 \, a^{2} b^{4} c^{2} d^{3} - 4 \, a^{3} b^{3} c d^{4} + a^{4} b^{2} d^{5}\right )} g^{3} i^{2} x^{3} + {\left (b^{6} c^{5} - 2 \, a b^{5} c^{4} d - 2 \, a^{2} b^{4} c^{3} d^{2} + 8 \, a^{3} b^{3} c^{2} d^{3} - 7 \, a^{4} b^{2} c d^{4} + 2 \, a^{5} b d^{5}\right )} g^{3} i^{2} x^{2} + {\left (2 \, a b^{5} c^{5} - 7 \, a^{2} b^{4} c^{4} d + 8 \, a^{3} b^{3} c^{3} d^{2} - 2 \, a^{4} b^{2} c^{2} d^{3} - 2 \, a^{5} b c d^{4} + a^{6} d^{5}\right )} g^{3} i^{2} x + {\left (a^{2} b^{4} c^{5} - 4 \, a^{3} b^{3} c^{4} d + 6 \, a^{4} b^{2} c^{3} d^{2} - 4 \, a^{5} b c^{2} d^{3} + a^{6} c d^{4}\right )} g^{3} i^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((2*A + B)*b^3*c^3 - 12*(A + B)*a*b^2*c^2*d + 3*(2*A + 5*B)*a^2*b*c*d^2 + 4*(A - B)*a^3*d^3 - 6*((2*A + B
)*b^3*c*d^2 - (2*A + B)*a*b^2*d^3)*x^2 - 6*(B*b^3*d^3*x^3 + B*a^2*b*c*d^2 + (B*b^3*c*d^2 + 2*B*a*b^2*d^3)*x^2
+ (2*B*a*b^2*c*d^2 + B*a^2*b*d^3)*x)*log((b*e*x + a*e)/(d*x + c))^2 - 3*((2*A + 3*B)*b^3*c^2*d + 2*(2*A - B)*a
*b^2*c*d^2 - (6*A + B)*a^2*b*d^3)*x - 2*(3*(2*A + B)*b^3*d^3*x^3 - B*b^3*c^3 + 6*B*a*b^2*c^2*d + 6*A*a^2*b*c*d
^2 - 2*B*a^3*d^3 + 3*((2*A + 3*B)*b^3*c*d^2 + 4*A*a*b^2*d^3)*x^2 + 3*(B*b^3*c^2*d + 4*(A + B)*a*b^2*c*d^2 + 2*
(A - B)*a^2*b*d^3)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^6*c^4*d - 4*a*b^5*c^3*d^2 + 6*a^2*b^4*c^2*d^3 - 4*a^3*
b^3*c*d^4 + a^4*b^2*d^5)*g^3*i^2*x^3 + (b^6*c^5 - 2*a*b^5*c^4*d - 2*a^2*b^4*c^3*d^2 + 8*a^3*b^3*c^2*d^3 - 7*a^
4*b^2*c*d^4 + 2*a^5*b*d^5)*g^3*i^2*x^2 + (2*a*b^5*c^5 - 7*a^2*b^4*c^4*d + 8*a^3*b^3*c^3*d^2 - 2*a^4*b^2*c^2*d^
3 - 2*a^5*b*c*d^4 + a^6*d^5)*g^3*i^2*x + (a^2*b^4*c^5 - 4*a^3*b^3*c^4*d + 6*a^4*b^2*c^3*d^2 - 4*a^5*b*c^2*d^3
+ a^6*c*d^4)*g^3*i^2)

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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)))/(b*g*x+a*g)^3/(d*i*x+c*i)^2,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.05, size = 1635, normalized size = 4.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 2.37, size = 1721, normalized size = 4.73 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*B*((6*b^2*d^2*x^2 - b^2*c^2 + 5*a*b*c*d + 2*a^2*d^2 + 3*(b^2*c*d + 3*a*b*d^2)*x)/((b^5*c^3*d - 3*a*b^4*c^2
*d^2 + 3*a^2*b^3*c*d^3 - a^3*b^2*d^4)*g^3*i^2*x^3 + (b^5*c^4 - a*b^4*c^3*d - 3*a^2*b^3*c^2*d^2 + 5*a^3*b^2*c*d
^3 - 2*a^4*b*d^4)*g^3*i^2*x^2 + (2*a*b^4*c^4 - 5*a^2*b^3*c^3*d + 3*a^3*b^2*c^2*d^2 + a^4*b*c*d^3 - a^5*d^4)*g^
3*i^2*x + (a^2*b^3*c^4 - 3*a^3*b^2*c^3*d + 3*a^4*b*c^2*d^2 - a^5*c*d^3)*g^3*i^2) + 6*b*d^2*log(b*x + a)/((b^4*
c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^3*i^2) - 6*b*d^2*log(d*x + c)/((b^4*c^4 -
4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^3*i^2))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) +
1/2*A*((6*b^2*d^2*x^2 - b^2*c^2 + 5*a*b*c*d + 2*a^2*d^2 + 3*(b^2*c*d + 3*a*b*d^2)*x)/((b^5*c^3*d - 3*a*b^4*c^
2*d^2 + 3*a^2*b^3*c*d^3 - a^3*b^2*d^4)*g^3*i^2*x^3 + (b^5*c^4 - a*b^4*c^3*d - 3*a^2*b^3*c^2*d^2 + 5*a^3*b^2*c*
d^3 - 2*a^4*b*d^4)*g^3*i^2*x^2 + (2*a*b^4*c^4 - 5*a^2*b^3*c^3*d + 3*a^3*b^2*c^2*d^2 + a^4*b*c*d^3 - a^5*d^4)*g
^3*i^2*x + (a^2*b^3*c^4 - 3*a^3*b^2*c^3*d + 3*a^4*b*c^2*d^2 - a^5*c*d^3)*g^3*i^2) + 6*b*d^2*log(b*x + a)/((b^4
*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^3*i^2) - 6*b*d^2*log(d*x + c)/((b^4*c^4
- 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^3*i^2)) - 1/4*(b^3*c^3 - 12*a*b^2*c^2*d + 15*
a^2*b*c*d^2 - 4*a^3*d^3 - 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 + 6*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*
d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log(b*x + a)^2 + 6*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2
*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log(d*x + c)^2 - 3*(3*b^3*c^2*d - 2*a*b^2*c*d^2 - a^2*b*d^3)*x - 6*
(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log(b*x + a) + 6*(
b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x - 2*(b^3*d^3*x^3 + a
^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log(b*x + a))*log(d*x + c))*B/(a^2
*b^4*c^5*g^3*i^2 - 4*a^3*b^3*c^4*d*g^3*i^2 + 6*a^4*b^2*c^3*d^2*g^3*i^2 - 4*a^5*b*c^2*d^3*g^3*i^2 + a^6*c*d^4*g
^3*i^2 + (b^6*c^4*d*g^3*i^2 - 4*a*b^5*c^3*d^2*g^3*i^2 + 6*a^2*b^4*c^2*d^3*g^3*i^2 - 4*a^3*b^3*c*d^4*g^3*i^2 +
a^4*b^2*d^5*g^3*i^2)*x^3 + (b^6*c^5*g^3*i^2 - 2*a*b^5*c^4*d*g^3*i^2 - 2*a^2*b^4*c^3*d^2*g^3*i^2 + 8*a^3*b^3*c^
2*d^3*g^3*i^2 - 7*a^4*b^2*c*d^4*g^3*i^2 + 2*a^5*b*d^5*g^3*i^2)*x^2 + (2*a*b^5*c^5*g^3*i^2 - 7*a^2*b^4*c^4*d*g^
3*i^2 + 8*a^3*b^3*c^3*d^2*g^3*i^2 - 2*a^4*b^2*c^2*d^3*g^3*i^2 - 2*a^5*b*c*d^4*g^3*i^2 + a^6*d^5*g^3*i^2)*x)

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mupad [B]  time = 9.11, size = 984, normalized size = 2.70 \[ \frac {3\,B\,b\,d^2\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{2\,g^3\,i^2\,{\left (a\,d-b\,c\right )}^4}-\frac {A\,a^2\,d^2}{g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}+\frac {A\,b^2\,c^2}{2\,g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}+\frac {B\,a^2\,d^2}{g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}+\frac {B\,b^2\,c^2}{4\,g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {B\,a\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i^2\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {B\,b\,c\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^3\,i^2\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {3\,A\,b^2\,d^2\,x^2}{g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {3\,B\,b^2\,d^2\,x^2}{2\,g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {5\,A\,a\,b\,c\,d}{2\,g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {11\,B\,a\,b\,c\,d}{4\,g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {3\,B\,b\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^3\,i^2\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {3\,B\,b^2\,d^2\,x^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {9\,A\,a\,b\,d^2\,x}{2\,g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {3\,B\,a\,b\,d^2\,x}{4\,g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {3\,A\,b^2\,c\,d\,x}{2\,g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {9\,B\,b^2\,c\,d\,x}{4\,g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {3\,B\,a\,b\,c\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {3\,B\,a\,b\,d^2\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {3\,B\,b^2\,c\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i^2\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}-\frac {A\,b\,d^2\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,6{}\mathrm {i}}{g^3\,i^2\,{\left (a\,d-b\,c\right )}^4}-\frac {B\,b\,d^2\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,3{}\mathrm {i}}{g^3\,i^2\,{\left (a\,d-b\,c\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 52.28, size = 1562, normalized size = 4.29 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*B*b*d**2*log(e*(a + b*x)/(c + d*x))**2/(2*a**4*d**4*g**3*i**2 - 8*a**3*b*c*d**3*g**3*i**2 + 12*a**2*b**2*c**
2*d**2*g**3*i**2 - 8*a*b**3*c**3*d*g**3*i**2 + 2*b**4*c**4*g**3*i**2) - 3*b*d**2*(2*A + B)*log(x + (6*A*a*b*d*
*3 + 6*A*b**2*c*d**2 + 3*B*a*b*d**3 + 3*B*b**2*c*d**2 - 3*a**5*b*d**7*(2*A + B)/(a*d - b*c)**4 + 15*a**4*b**2*
c*d**6*(2*A + B)/(a*d - b*c)**4 - 30*a**3*b**3*c**2*d**5*(2*A + B)/(a*d - b*c)**4 + 30*a**2*b**4*c**3*d**4*(2*
A + B)/(a*d - b*c)**4 - 15*a*b**5*c**4*d**3*(2*A + B)/(a*d - b*c)**4 + 3*b**6*c**5*d**2*(2*A + B)/(a*d - b*c)*
*4)/(12*A*b**2*d**3 + 6*B*b**2*d**3))/(2*g**3*i**2*(a*d - b*c)**4) + 3*b*d**2*(2*A + B)*log(x + (6*A*a*b*d**3
+ 6*A*b**2*c*d**2 + 3*B*a*b*d**3 + 3*B*b**2*c*d**2 + 3*a**5*b*d**7*(2*A + B)/(a*d - b*c)**4 - 15*a**4*b**2*c*d
**6*(2*A + B)/(a*d - b*c)**4 + 30*a**3*b**3*c**2*d**5*(2*A + B)/(a*d - b*c)**4 - 30*a**2*b**4*c**3*d**4*(2*A +
B)/(a*d - b*c)**4 + 15*a*b**5*c**4*d**3*(2*A + B)/(a*d - b*c)**4 - 3*b**6*c**5*d**2*(2*A + B)/(a*d - b*c)**4)
/(12*A*b**2*d**3 + 6*B*b**2*d**3))/(2*g**3*i**2*(a*d - b*c)**4) + (-2*B*a**2*d**2 - 5*B*a*b*c*d - 9*B*a*b*d**2
*x + B*b**2*c**2 - 3*B*b**2*c*d*x - 6*B*b**2*d**2*x**2)*log(e*(a + b*x)/(c + d*x))/(2*a**5*c*d**3*g**3*i**2 +
2*a**5*d**4*g**3*i**2*x - 6*a**4*b*c**2*d**2*g**3*i**2 - 2*a**4*b*c*d**3*g**3*i**2*x + 4*a**4*b*d**4*g**3*i**2
*x**2 + 6*a**3*b**2*c**3*d*g**3*i**2 - 6*a**3*b**2*c**2*d**2*g**3*i**2*x - 10*a**3*b**2*c*d**3*g**3*i**2*x**2
+ 2*a**3*b**2*d**4*g**3*i**2*x**3 - 2*a**2*b**3*c**4*g**3*i**2 + 10*a**2*b**3*c**3*d*g**3*i**2*x + 6*a**2*b**3
*c**2*d**2*g**3*i**2*x**2 - 6*a**2*b**3*c*d**3*g**3*i**2*x**3 - 4*a*b**4*c**4*g**3*i**2*x + 2*a*b**4*c**3*d*g*
*3*i**2*x**2 + 6*a*b**4*c**2*d**2*g**3*i**2*x**3 - 2*b**5*c**4*g**3*i**2*x**2 - 2*b**5*c**3*d*g**3*i**2*x**3)
- (4*A*a**2*d**2 + 10*A*a*b*c*d - 2*A*b**2*c**2 - 4*B*a**2*d**2 + 11*B*a*b*c*d - B*b**2*c**2 + x**2*(12*A*b**2
*d**2 + 6*B*b**2*d**2) + x*(18*A*a*b*d**2 + 6*A*b**2*c*d + 3*B*a*b*d**2 + 9*B*b**2*c*d))/(4*a**5*c*d**3*g**3*i
**2 - 12*a**4*b*c**2*d**2*g**3*i**2 + 12*a**3*b**2*c**3*d*g**3*i**2 - 4*a**2*b**3*c**4*g**3*i**2 + x**3*(4*a**
3*b**2*d**4*g**3*i**2 - 12*a**2*b**3*c*d**3*g**3*i**2 + 12*a*b**4*c**2*d**2*g**3*i**2 - 4*b**5*c**3*d*g**3*i**
2) + x**2*(8*a**4*b*d**4*g**3*i**2 - 20*a**3*b**2*c*d**3*g**3*i**2 + 12*a**2*b**3*c**2*d**2*g**3*i**2 + 4*a*b*
*4*c**3*d*g**3*i**2 - 4*b**5*c**4*g**3*i**2) + x*(4*a**5*d**4*g**3*i**2 - 4*a**4*b*c*d**3*g**3*i**2 - 12*a**3*
b**2*c**2*d**2*g**3*i**2 + 20*a**2*b**3*c**3*d*g**3*i**2 - 8*a*b**4*c**4*g**3*i**2))

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