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

Optimal. Leaf size=255 \[ -\frac {b^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 i (a+b x)^2 (b c-a d)^3}+\frac {d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i (b c-a d)^3}+\frac {2 b d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i (a+b x) (b c-a d)^3}-\frac {B d^2 \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^3 i (b c-a d)^3}-\frac {B (c+d x)^2 \left (b-\frac {4 d (a+b x)}{c+d x}\right )^2}{4 g^3 i (a+b x)^2 (b c-a d)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 0.88, antiderivative size = 535, normalized size of antiderivative = 2.10, number of steps used = 28, 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} \[ \frac {B d^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac {B d^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac {d^2 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i (b c-a d)^3}-\frac {d^2 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i (b c-a d)^3}+\frac {d \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i (a+b x) (b c-a d)^2}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 g^3 i (a+b x)^2 (b c-a d)}-\frac {B d^2 \log ^2(a+b x)}{2 g^3 i (b c-a d)^3}-\frac {B d^2 \log ^2(c+d x)}{2 g^3 i (b c-a d)^3}+\frac {3 B d^2 \log (a+b x)}{2 g^3 i (b c-a d)^3}+\frac {B d^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}-\frac {3 B d^2 \log (c+d x)}{2 g^3 i (b c-a d)^3}+\frac {B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac {3 B d}{2 g^3 i (a+b x) (b c-a d)^2}-\frac {B}{4 g^3 i (a+b x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-B/(4*(b*c - a*d)*g^3*i*(a + b*x)^2) + (3*B*d)/(2*(b*c - a*d)^2*g^3*i*(a + b*x)) + (3*B*d^2*Log[a + b*x])/(2*(
b*c - a*d)^3*g^3*i) - (B*d^2*Log[a + b*x]^2)/(2*(b*c - a*d)^3*g^3*i) - (A + B*Log[(e*(a + b*x))/(c + d*x)])/(2
*(b*c - a*d)*g^3*i*(a + b*x)^2) + (d*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((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)]))/((b*c - a*d)^3*g^3*i) - (3*B*d^2*Log[c + d*x])/(2*(b*c
- a*d)^3*g^3*i) + (B*d^2*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*L
og[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/((b*c - a*d)^3*g^3*i) - (B*d^2*Log[c + d*x]^2)/(2*(b*c - a*d)^3*g^3
*i) + (B*d^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^3*g^3*i) + (B*d^2*PolyLog[2, -((d*(a +
b*x))/(b*c - a*d))])/((b*c - a*d)^3*g^3*i) + (B*d^2*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 (\frac {e (a+b x)}{c+d x}\right )}{(37 c+37 d x) (a g+b g x)^3} \, dx &=\int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d) g^3 (a+b x)^3}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (b d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{37 (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} \, dx}{37 (b c-a d)^3 g^3}-\frac {(b d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{37 (b c-a d)^2 g^3}+\frac {b \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{37 (b c-a d) g^3}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{74 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}-\frac {\left (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}{37 (b c-a d)^3 g^3}+\frac {\left (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}{37 (b c-a d)^3 g^3}-\frac {(B d) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{37 (b c-a d)^2 g^3}+\frac {B \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{74 (b c-a d) g^3}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{74 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}+\frac {B \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{74 g^3}-\frac {(B d) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{37 (b c-a d) g^3}-\frac {\left (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}{37 (b c-a d)^3 e g^3}+\frac {\left (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}{37 (b c-a d)^3 e g^3}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{74 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}+\frac {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}{74 g^3}-\frac {(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}{37 (b c-a d) g^3}-\frac {\left (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}{37 (b c-a d)^3 e g^3}+\frac {\left (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}{37 (b c-a d)^3 e g^3}\\ &=-\frac {B}{148 (b c-a d) g^3 (a+b x)^2}+\frac {3 B d}{74 (b c-a d)^2 g^3 (a+b x)}+\frac {3 B d^2 \log (a+b x)}{74 (b c-a d)^3 g^3}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{74 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3}-\frac {3 B d^2 \log (c+d x)}{74 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}-\frac {\left (b B d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{37 (b c-a d)^3 g^3}+\frac {\left (b B d^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{37 (b c-a d)^3 g^3}+\frac {\left (B d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{37 (b c-a d)^3 g^3}-\frac {\left (B d^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{37 (b c-a d)^3 g^3}\\ &=-\frac {B}{148 (b c-a d) g^3 (a+b x)^2}+\frac {3 B d}{74 (b c-a d)^2 g^3 (a+b x)}+\frac {3 B d^2 \log (a+b x)}{74 (b c-a d)^3 g^3}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{74 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3}-\frac {3 B d^2 \log (c+d x)}{74 (b c-a d)^3 g^3}+\frac {B d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}+\frac {B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{37 (b c-a d)^3 g^3}-\frac {\left (B d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{37 (b c-a d)^3 g^3}-\frac {\left (B d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{37 (b c-a d)^3 g^3}-\frac {\left (b B d^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{37 (b c-a d)^3 g^3}-\frac {\left (B d^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{37 (b c-a d)^3 g^3}\\ &=-\frac {B}{148 (b c-a d) g^3 (a+b x)^2}+\frac {3 B d}{74 (b c-a d)^2 g^3 (a+b x)}+\frac {3 B d^2 \log (a+b x)}{74 (b c-a d)^3 g^3}-\frac {B d^2 \log ^2(a+b x)}{74 (b c-a d)^3 g^3}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{74 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3}-\frac {3 B d^2 \log (c+d x)}{74 (b c-a d)^3 g^3}+\frac {B d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}-\frac {B d^2 \log ^2(c+d x)}{74 (b c-a d)^3 g^3}+\frac {B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{37 (b c-a d)^3 g^3}-\frac {\left (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 )}{37 (b c-a d)^3 g^3}-\frac {\left (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 )}{37 (b c-a d)^3 g^3}\\ &=-\frac {B}{148 (b c-a d) g^3 (a+b x)^2}+\frac {3 B d}{74 (b c-a d)^2 g^3 (a+b x)}+\frac {3 B d^2 \log (a+b x)}{74 (b c-a d)^3 g^3}-\frac {B d^2 \log ^2(a+b x)}{74 (b c-a d)^3 g^3}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{74 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{37 (b c-a d)^3 g^3}-\frac {3 B d^2 \log (c+d x)}{74 (b c-a d)^3 g^3}+\frac {B d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{37 (b c-a d)^3 g^3}-\frac {B d^2 \log ^2(c+d x)}{74 (b c-a d)^3 g^3}+\frac {B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{37 (b c-a d)^3 g^3}+\frac {B d^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{37 (b c-a d)^3 g^3}+\frac {B d^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{37 (b c-a d)^3 g^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.43, size = 418, normalized size = 1.64 \[ \frac {4 d^2 (a+b x)^2 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-4 d^2 (a+b x)^2 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+4 d (a+b x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+2 B d^2 (a+b x)^2 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-B \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )+4 B d (a+b x) (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)}{4 g^3 i (a+b x)^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 349, normalized size = 1.37 \[ -\frac {{\left (2 \, A + B\right )} b^{2} c^{2} - 8 \, {\left (A + B\right )} a b c d + {\left (6 \, A + 7 \, B\right )} a^{2} d^{2} - 2 \, {\left (B b^{2} d^{2} x^{2} + 2 \, B a b d^{2} x + B a^{2} d^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 2 \, {\left ({\left (2 \, A + 3 \, B\right )} b^{2} c d - {\left (2 \, A + 3 \, B\right )} a b d^{2}\right )} x - 2 \, {\left ({\left (2 \, A + 3 \, B\right )} b^{2} d^{2} x^{2} - B b^{2} c^{2} + 4 \, B a b c d + 2 \, A a^{2} d^{2} + 2 \, {\left (B b^{2} c d + 2 \, {\left (A + B\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} g^{3} i x + {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} g^{3} i\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [B]  time = 0.05, size = 1040, normalized size = 4.08 \[ \frac {B a \,b^{2} d \,e^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{2 \left (a d -b c \right )^{4} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3} i}-\frac {B \,b^{3} c \,e^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{2 \left (a d -b c \right )^{4} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3} i}+\frac {A a \,b^{2} d \,e^{2}}{2 \left (a d -b c \right )^{4} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3} i}-\frac {A \,b^{3} c \,e^{2}}{2 \left (a d -b c \right )^{4} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3} i}+\frac {B a \,b^{2} d \,e^{2}}{4 \left (a d -b c \right )^{4} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3} i}-\frac {2 B a b \,d^{2} e \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{4} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{3} i}-\frac {B a \,d^{3} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{2 \left (a d -b c \right )^{4} g^{3} i}-\frac {B \,b^{3} c \,e^{2}}{4 \left (a d -b c \right )^{4} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3} i}+\frac {2 B \,b^{2} c d e \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{4} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{3} i}+\frac {B b c \,d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{2 \left (a d -b c \right )^{4} g^{3} i}-\frac {2 A a b \,d^{2} e}{\left (a d -b c \right )^{4} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{3} i}-\frac {A a \,d^{3} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{4} g^{3} i}+\frac {2 A \,b^{2} c d e}{\left (a d -b c \right )^{4} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{3} i}+\frac {A b c \,d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{4} g^{3} i}-\frac {2 B a b \,d^{2} e}{\left (a d -b c \right )^{4} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{3} i}+\frac {2 B \,b^{2} c d e}{\left (a d -b c \right )^{4} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{3} i} \]

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),x)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 1.72, size = 885, normalized size = 3.47 \[ \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 (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\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 )} - \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}{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 )}} \]

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),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(b*e*x/(d*x + c) + a*e/(d*x + c)) + 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)) - 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/(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)

________________________________________________________________________________________

mupad [B]  time = 6.93, size = 545, normalized size = 2.14 \[ \frac {3\,A\,a\,d}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}-\frac {B\,d^2\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^3}-\frac {A\,b\,c}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}+\frac {7\,B\,a\,d}{4\,g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}-\frac {B\,b\,c}{4\,g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}+\frac {3\,B\,a^2\,d^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2}+\frac {B\,b^2\,c^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2}+\frac {A\,b\,d\,x}{g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}+\frac {3\,B\,b\,d\,x}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}+\frac {B\,a\,b\,d^2\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2}-\frac {B\,b^2\,c\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2}-\frac {2\,B\,a\,b\,c\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2}+\frac {A\,d^2\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3}+\frac {B\,d^2\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,3{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 6.98, size = 889, normalized size = 3.49 \[ - \frac {B d^{2} \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{3} d^{3} g^{3} i - 6 a^{2} b c d^{2} g^{3} i + 6 a b^{2} c^{2} d g^{3} i - 2 b^{3} c^{3} g^{3} i} + \frac {d^{2} \left (2 A + 3 B\right ) \log {\left (x + \frac {2 A a d^{3} + 2 A b c d^{2} + 3 B a d^{3} + 3 B b c d^{2} - \frac {a^{4} d^{6} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b c d^{5} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{2} c^{2} d^{4} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {4 a b^{3} c^{3} d^{3} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {b^{4} c^{4} d^{2} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}}}{4 A b d^{3} + 6 B b d^{3}} \right )}}{2 g^{3} i \left (a d - b c\right )^{3}} - \frac {d^{2} \left (2 A + 3 B\right ) \log {\left (x + \frac {2 A a d^{3} + 2 A b c d^{2} + 3 B a d^{3} + 3 B b c d^{2} + \frac {a^{4} d^{6} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b c d^{5} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{2} c^{2} d^{4} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {4 a b^{3} c^{3} d^{3} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {b^{4} c^{4} d^{2} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}}}{4 A b d^{3} + 6 B b d^{3}} \right )}}{2 g^{3} i \left (a d - b c\right )^{3}} + \frac {\left (3 B a d - B b c + 2 B b d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 a^{4} d^{2} g^{3} i - 4 a^{3} b c d g^{3} i + 4 a^{3} b d^{2} g^{3} i x + 2 a^{2} b^{2} c^{2} g^{3} i - 8 a^{2} b^{2} c d g^{3} i x + 2 a^{2} b^{2} d^{2} g^{3} i x^{2} + 4 a b^{3} c^{2} g^{3} i x - 4 a b^{3} c d g^{3} i x^{2} + 2 b^{4} c^{2} g^{3} i x^{2}} + \frac {6 A a d - 2 A b c + 7 B a d - B b c + x \left (4 A b d + 6 B b d\right )}{4 a^{4} d^{2} g^{3} i - 8 a^{3} b c d g^{3} i + 4 a^{2} b^{2} c^{2} g^{3} i + x^{2} \left (4 a^{2} b^{2} d^{2} g^{3} i - 8 a b^{3} c d g^{3} i + 4 b^{4} c^{2} g^{3} i\right ) + x \left (8 a^{3} b d^{2} g^{3} i - 16 a^{2} b^{2} c d g^{3} i + 8 a b^{3} c^{2} g^{3} i\right )} \]

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),x)

[Out]

-B*d**2*log(e*(a + b*x)/(c + d*x))**2/(2*a**3*d**3*g**3*i - 6*a**2*b*c*d**2*g**3*i + 6*a*b**2*c**2*d*g**3*i -
2*b**3*c**3*g**3*i) + d**2*(2*A + 3*B)*log(x + (2*A*a*d**3 + 2*A*b*c*d**2 + 3*B*a*d**3 + 3*B*b*c*d**2 - a**4*d
**6*(2*A + 3*B)/(a*d - b*c)**3 + 4*a**3*b*c*d**5*(2*A + 3*B)/(a*d - b*c)**3 - 6*a**2*b**2*c**2*d**4*(2*A + 3*B
)/(a*d - b*c)**3 + 4*a*b**3*c**3*d**3*(2*A + 3*B)/(a*d - b*c)**3 - b**4*c**4*d**2*(2*A + 3*B)/(a*d - b*c)**3)/
(4*A*b*d**3 + 6*B*b*d**3))/(2*g**3*i*(a*d - b*c)**3) - d**2*(2*A + 3*B)*log(x + (2*A*a*d**3 + 2*A*b*c*d**2 + 3
*B*a*d**3 + 3*B*b*c*d**2 + a**4*d**6*(2*A + 3*B)/(a*d - b*c)**3 - 4*a**3*b*c*d**5*(2*A + 3*B)/(a*d - b*c)**3 +
6*a**2*b**2*c**2*d**4*(2*A + 3*B)/(a*d - b*c)**3 - 4*a*b**3*c**3*d**3*(2*A + 3*B)/(a*d - b*c)**3 + b**4*c**4*
d**2*(2*A + 3*B)/(a*d - b*c)**3)/(4*A*b*d**3 + 6*B*b*d**3))/(2*g**3*i*(a*d - b*c)**3) + (3*B*a*d - B*b*c + 2*B
*b*d*x)*log(e*(a + b*x)/(c + d*x))/(2*a**4*d**2*g**3*i - 4*a**3*b*c*d*g**3*i + 4*a**3*b*d**2*g**3*i*x + 2*a**2
*b**2*c**2*g**3*i - 8*a**2*b**2*c*d*g**3*i*x + 2*a**2*b**2*d**2*g**3*i*x**2 + 4*a*b**3*c**2*g**3*i*x - 4*a*b**
3*c*d*g**3*i*x**2 + 2*b**4*c**2*g**3*i*x**2) + (6*A*a*d - 2*A*b*c + 7*B*a*d - B*b*c + x*(4*A*b*d + 6*B*b*d))/(
4*a**4*d**2*g**3*i - 8*a**3*b*c*d*g**3*i + 4*a**2*b**2*c**2*g**3*i + x**2*(4*a**2*b**2*d**2*g**3*i - 8*a*b**3*
c*d*g**3*i + 4*b**4*c**2*g**3*i) + x*(8*a**3*b*d**2*g**3*i - 16*a**2*b**2*c*d*g**3*i + 8*a*b**3*c**2*g**3*i))

________________________________________________________________________________________