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

Optimal. Leaf size=141 \[ -\frac {i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^3 (a+b x)^2 (b c-a d)}-\frac {B i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 (a+b x)^2 (b c-a d)}-\frac {B^2 i (c+d x)^2}{4 g^3 (a+b x)^2 (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 1.94, antiderivative size = 639, normalized size of antiderivative = 4.53, number of steps used = 58, 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} \[ -\frac {B^2 d^2 i \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac {B^2 d^2 i \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac {B d^2 i \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (b c-a d)}+\frac {B d^2 i \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (b c-a d)}-\frac {B d i \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (a+b x)}-\frac {B i (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b^2 g^3 (a+b x)^2}-\frac {d i \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g^3 (a+b x)}-\frac {i (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 b^2 g^3 (a+b x)^2}+\frac {B^2 d^2 i \log ^2(a+b x)}{2 b^2 g^3 (b c-a d)}+\frac {B^2 d^2 i \log ^2(c+d x)}{2 b^2 g^3 (b c-a d)}-\frac {B^2 d^2 i \log (a+b x)}{2 b^2 g^3 (b c-a d)}+\frac {B^2 d^2 i \log (c+d x)}{2 b^2 g^3 (b c-a d)}-\frac {B^2 d^2 i \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac {B^2 d^2 i \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac {B^2 i (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {B^2 d i}{2 b^2 g^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 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 {(61 c+61 d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx &=\int \left (\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g^3 (a+b x)^3}+\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g^3 (a+b x)^2}\right ) \, dx\\ &=\frac {(61 d) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{b g^3}+\frac {(61 (b c-a d)) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^3} \, dx}{b g^3}\\ &=-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {(122 B d) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {(61 B (b c-a d)) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^3}\\ &=-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {(122 B d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (61 B (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^3}\\ &=-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {(122 B d (b c-a d)) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^3}+\frac {\left (61 B (b c-a d)^2\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^3}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^2 g^3}\\ &=-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}-\frac {(61 B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b g^3}+\frac {(122 B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b g^3}+\frac {\left (61 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (122 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (61 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {\left (122 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {(61 B (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b g^3}\\ &=-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (122 B^2 d\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}-\frac {\left (61 B^2 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}{b^2 (b c-a d) g^3}+\frac {\left (61 B^2 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}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 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}{b^2 (b c-a d) g^3}-\frac {\left (122 B^2 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}{b^2 (b c-a d) g^3}+\frac {\left (61 B^2 (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}\\ &=-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d (b c-a d)\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (122 B^2 d (b c-a d)\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (61 B^2 (b c-a d)^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}-\frac {\left (61 B^2 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}{b^2 (b c-a d) e g^3}+\frac {\left (61 B^2 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}{b^2 (b c-a d) e g^3}+\frac {\left (122 B^2 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}{b^2 (b c-a d) e g^3}-\frac {\left (122 B^2 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}{b^2 (b c-a d) e g^3}\\ &=-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d (b c-a d)\right ) \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}{b^2 g^3}+\frac {\left (122 B^2 d (b c-a d)\right ) \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}{b^2 g^3}+\frac {\left (61 B^2 (b c-a d)^2\right ) \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}{2 b^2 g^3}-\frac {\left (61 B^2 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}{b^2 (b c-a d) e g^3}+\frac {\left (61 B^2 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}{b^2 (b c-a d) e g^3}+\frac {\left (122 B^2 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}{b^2 (b c-a d) e g^3}-\frac {\left (122 B^2 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}{b^2 (b c-a d) e g^3}\\ &=-\frac {61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac {61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (61 B^2 d^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (122 B^2 d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (122 B^2 d^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (61 B^2 d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}-\frac {\left (122 B^2 d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 d^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}\\ &=-\frac {61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac {61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (122 B^2 d^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (61 B^2 d^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 d^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}\\ &=-\frac {61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac {61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}+\frac {61 B^2 d^2 \log ^2(a+b x)}{2 b^2 (b c-a d) g^3}-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {61 B^2 d^2 \log ^2(c+d x)}{2 b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 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 )}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 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 )}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 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 )}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 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 )}{b^2 (b c-a d) g^3}\\ &=-\frac {61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac {61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}+\frac {61 B^2 d^2 \log ^2(a+b x)}{2 b^2 (b c-a d) g^3}-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {61 B^2 d^2 \log ^2(c+d x)}{2 b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}\\ \end {align*}

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Mathematica [C]  time = 0.91, size = 765, normalized size = 5.43 \[ -\frac {i \left (B \left (-4 d^2 (a+b x)^2 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+4 d^2 (a+b x)^2 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+4 d (a+b x) (a d-b c) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )-2 B d^2 (a+b x)^2 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+B \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )-4 B d (a+b x) (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)\right )+4 B d (a+b x) \left (2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 d (a+b x) \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-2 d (a+b x) \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+B d (a+b x) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 B (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)\right )+2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2-4 d (a+b x) (a d-b c) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2\right )}{4 b^2 g^3 (a+b x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(i*(2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - 4*d*(-(b*c) + a*d)*(a + b*x)*(A + B*Log[(e*(
a + b*x))/(c + d*x)])^2 + 4*B*d*(a + b*x)*(2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2*d*(a + b*x)*
Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 2*d*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c +
d*x] + 2*B*(b*c - a*d + d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) - B*d*(a + b*x)*(Log[a + b*x]*(L
og[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) + B*d*(a + b*x)*
((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)]))
+ B*(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)]))))/(b^2*(b*c - a*d)*g^3*(a + b*x)^2)

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fricas [B]  time = 0.85, size = 289, normalized size = 2.05 \[ -\frac {2 \, {\left ({\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{2} c d - {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a b d^{2}\right )} i x + 2 \, {\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x + B^{2} b^{2} c^{2} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + {\left ({\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{2} c^{2} - {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a^{2} d^{2}\right )} i + 2 \, {\left ({\left (2 \, A B + B^{2}\right )} b^{2} d^{2} i x^{2} + 2 \, {\left (2 \, A B + B^{2}\right )} b^{2} c d i x + {\left (2 \, A B + B^{2}\right )} b^{2} c^{2} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.29, size = 185, normalized size = 1.31 \[ -\frac {{\left (2 \, B^{2} i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )^{2} + 4 \, A B i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right ) + 2 \, B^{2} i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right ) + 2 \, A^{2} i e^{3} + 2 \, A B i e^{3} + B^{2} i e^{3}\right )} {\left (d x + c\right )}^{2} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{4 \, {\left (b x e + a e\right )}^{2} g^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 865, normalized size = 6.13 \[ \frac {B^{2} a d \,e^{2} i \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 )^{2} \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}}-\frac {B^{2} b c \,e^{2} i \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 )^{2} \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}}+\frac {A B a d \,e^{2} i \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 )^{2} \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}}-\frac {A B b c \,e^{2} i \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 )^{2} \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}}+\frac {B^{2} a d \,e^{2} i \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 )^{2} \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}}-\frac {B^{2} b c \,e^{2} i \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 )^{2} \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}}+\frac {A^{2} a d \,e^{2} i}{2 \left (a d -b c \right )^{2} \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}}-\frac {A^{2} b c \,e^{2} i}{2 \left (a d -b c \right )^{2} \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}}+\frac {A B a d \,e^{2} i}{2 \left (a d -b c \right )^{2} \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}}-\frac {A B b c \,e^{2} i}{2 \left (a d -b c \right )^{2} \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}}+\frac {B^{2} a d \,e^{2} i}{4 \left (a d -b c \right )^{2} \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}}-\frac {B^{2} b c \,e^{2} i}{4 \left (a d -b c \right )^{2} \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}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 2.39, size = 1987, normalized size = 14.09 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 6.18, size = 469, normalized size = 3.33 \[ -\frac {x\,\left (2\,b\,d\,i\,A^2+2\,b\,d\,i\,A\,B+b\,d\,i\,B^2\right )+A^2\,a\,d\,i+A^2\,b\,c\,i+\frac {B^2\,a\,d\,i}{2}+\frac {B^2\,b\,c\,i}{2}+A\,B\,a\,d\,i+A\,B\,b\,c\,i}{2\,a^2\,b^2\,g^3+4\,a\,b^3\,g^3\,x+2\,b^4\,g^3\,x^2}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {\frac {B^2\,c\,i}{2\,b^2\,g^3}+\frac {B^2\,a\,d\,i}{2\,b^3\,g^3}+\frac {B^2\,d\,i\,x}{b^2\,g^3}}{2\,a\,x+b\,x^2+\frac {a^2}{b}}-\frac {B^2\,d^2\,i}{2\,b^2\,g^3\,\left (a\,d-b\,c\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (x\,\left (\frac {B^2\,i}{b^2\,g^3}+\frac {2\,A\,B\,i}{b^2\,g^3}\right )+\frac {A\,B\,a\,i}{b^3\,g^3}+\frac {B\,i\,\left (A\,b\,c-B\,a\,d+B\,b\,c\right )}{b^3\,d\,g^3}+\frac {B^2\,d^2\,i\,\left (\frac {2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2}{2\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,b\,d^2}\right )}{b^2\,g^3\,\left (a\,d-b\,c\right )}\right )}{\frac {b\,x^2}{d}+\frac {a^2}{b\,d}+\frac {2\,a\,x}{d}}-\frac {B\,d^2\,i\,\mathrm {atan}\left (\frac {\left (\frac {2\,c\,b^3\,g^3+2\,a\,d\,b^2\,g^3}{2\,b^2\,g^3}+2\,b\,d\,x\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (2\,A+B\right )\,1{}\mathrm {i}}{b^2\,g^3\,\left (a\,d-b\,c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 14.31, size = 714, normalized size = 5.06 \[ - \frac {B d^{2} i \left (2 A + B\right ) \log {\left (x + \frac {2 A B a d^{3} i + 2 A B b c d^{2} i + B^{2} a d^{3} i + B^{2} b c d^{2} i - \frac {B a^{2} d^{4} i \left (2 A + B\right )}{a d - b c} + \frac {2 B a b c d^{3} i \left (2 A + B\right )}{a d - b c} - \frac {B b^{2} c^{2} d^{2} i \left (2 A + B\right )}{a d - b c}}{4 A B b d^{3} i + 2 B^{2} b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {B d^{2} i \left (2 A + B\right ) \log {\left (x + \frac {2 A B a d^{3} i + 2 A B b c d^{2} i + B^{2} a d^{3} i + B^{2} b c d^{2} i + \frac {B a^{2} d^{4} i \left (2 A + B\right )}{a d - b c} - \frac {2 B a b c d^{3} i \left (2 A + B\right )}{a d - b c} + \frac {B b^{2} c^{2} d^{2} i \left (2 A + B\right )}{a d - b c}}{4 A B b d^{3} i + 2 B^{2} b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {\left (B^{2} c^{2} i + 2 B^{2} c d i x + B^{2} d^{2} i x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{3} d g^{3} - 2 a^{2} b c g^{3} + 4 a^{2} b d g^{3} x - 4 a b^{2} c g^{3} x + 2 a b^{2} d g^{3} x^{2} - 2 b^{3} c g^{3} x^{2}} + \frac {- 2 A^{2} a d i - 2 A^{2} b c i - 2 A B a d i - 2 A B b c i - B^{2} a d i - B^{2} b c i + x \left (- 4 A^{2} b d i - 4 A B b d i - 2 B^{2} b d i\right )}{4 a^{2} b^{2} g^{3} + 8 a b^{3} g^{3} x + 4 b^{4} g^{3} x^{2}} + \frac {\left (- 2 A B a d i - 2 A B b c i - 4 A B b d i x - B^{2} a d i - B^{2} b c i - 2 B^{2} b d i x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 a^{2} b^{2} g^{3} + 4 a b^{3} g^{3} x + 2 b^{4} g^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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