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

Optimal. Leaf size=130 \[ -\frac {4 B (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g^2 (a+b x) (b c-a d)}-\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{g^2 (a+b x) (b c-a d)}-\frac {8 B^2 (c+d x)}{g^2 (a+b x) (b c-a d)} \]

[Out]

-8*B^2*(d*x+c)/(-a*d+b*c)/g^2/(b*x+a)-4*B*(d*x+c)*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(-a*d+b*c)/g^2/(b*x+a)-(d*x+
c)*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(-a*d+b*c)/g^2/(b*x+a)

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Rubi [C]  time = 0.89, antiderivative size = 480, normalized size of antiderivative = 3.69, number of steps used = 26, number of rules used = 11, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {2525, 12, 2528, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac {8 B^2 d \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac {8 B^2 d \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac {4 B d \log (a+b x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g^2 (b c-a d)}-\frac {4 B \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g^2 (a+b x)}+\frac {4 B d \log (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g^2 (b c-a d)}-\frac {\left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{b g^2 (a+b x)}+\frac {4 B^2 d \log ^2(a+b x)}{b g^2 (b c-a d)}+\frac {4 B^2 d \log ^2(c+d x)}{b g^2 (b c-a d)}-\frac {8 B^2 d \log (a+b x)}{b g^2 (b c-a d)}-\frac {8 B^2 d \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b g^2 (b c-a d)}+\frac {8 B^2 d \log (c+d x)}{b g^2 (b c-a d)}-\frac {8 B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac {8 B^2}{b g^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*B^2)/(b*g^2*(a + b*x)) - (8*B^2*d*Log[a + b*x])/(b*(b*c - a*d)*g^2) + (4*B^2*d*Log[a + b*x]^2)/(b*(b*c - a
*d)*g^2) - (4*B*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(b*g^2*(a + b*x)) - (4*B*d*Log[a + b*x]*(A + B*Log[(
e*(a + b*x)^2)/(c + d*x)^2]))/(b*(b*c - a*d)*g^2) - (A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(b*g^2*(a + b*x
)) + (8*B^2*d*Log[c + d*x])/(b*(b*c - a*d)*g^2) - (8*B^2*d*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(b*
(b*c - a*d)*g^2) + (4*B*d*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[c + d*x])/(b*(b*c - a*d)*g^2) + (4*B^2*
d*Log[c + d*x]^2)/(b*(b*c - a*d)*g^2) - (8*B^2*d*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(b*(b*c - a*d)*g
^2) - (8*B^2*d*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b*(b*c - a*d)*g^2) - (8*B^2*d*PolyLog[2, (b*(c + d*x
))/(b*c - a*d)])/(b*(b*c - a*d)*g^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 {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac {(2 B) \int \frac {2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (a+b x)^2 (c+d x)} \, dx}{b g}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac {(4 B (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^2 (c+d x)} \, dx}{b g^2}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac {(4 B (b c-a d)) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) (a+b x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b g^2}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac {(4 B) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^2} \, dx}{g^2}-\frac {(4 B d) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{a+b x} \, dx}{(b c-a d) g^2}+\frac {\left (4 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{c+d x} \, dx}{b (b c-a d) g^2}\\ &=-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac {4 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac {4 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac {\left (4 B^2\right ) \int \frac {2 (b c-a d)}{(a+b x)^2 (c+d x)} \, dx}{b g^2}+\frac {\left (4 B^2 d\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{e (a+b x)^2} \, dx}{b (b c-a d) g^2}-\frac {\left (4 B^2 d\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{e (a+b x)^2} \, dx}{b (b c-a d) g^2}\\ &=-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac {4 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac {4 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac {\left (8 B^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b g^2}+\frac {\left (4 B^2 d\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{(a+b x)^2} \, dx}{b (b c-a d) e g^2}-\frac {\left (4 B^2 d\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{(a+b x)^2} \, dx}{b (b c-a d) e g^2}\\ &=-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac {4 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac {4 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac {\left (8 B^2 (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 g^2}+\frac {\left (4 B^2 d\right ) \int \left (\frac {2 b e \log (a+b x)}{a+b x}-\frac {2 d e \log (a+b x)}{c+d x}\right ) \, dx}{b (b c-a d) e g^2}-\frac {\left (4 B^2 d\right ) \int \left (\frac {2 b e \log (c+d x)}{a+b x}-\frac {2 d e \log (c+d x)}{c+d x}\right ) \, dx}{b (b c-a d) e g^2}\\ &=-\frac {8 B^2}{b g^2 (a+b x)}-\frac {8 B^2 d \log (a+b x)}{b (b c-a d) g^2}-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac {4 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac {8 B^2 d \log (c+d x)}{b (b c-a d) g^2}+\frac {4 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac {\left (8 B^2 d\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{(b c-a d) g^2}-\frac {\left (8 B^2 d\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{(b c-a d) g^2}-\frac {\left (8 B^2 d^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b (b c-a d) g^2}+\frac {\left (8 B^2 d^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b (b c-a d) g^2}\\ &=-\frac {8 B^2}{b g^2 (a+b x)}-\frac {8 B^2 d \log (a+b x)}{b (b c-a d) g^2}-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac {4 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac {8 B^2 d \log (c+d x)}{b (b c-a d) g^2}-\frac {8 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac {4 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}-\frac {8 B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}+\frac {\left (8 B^2 d\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{(b c-a d) g^2}+\frac {\left (8 B^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b (b c-a d) g^2}+\frac {\left (8 B^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b (b c-a d) g^2}+\frac {\left (8 B^2 d^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b (b c-a d) g^2}\\ &=-\frac {8 B^2}{b g^2 (a+b x)}-\frac {8 B^2 d \log (a+b x)}{b (b c-a d) g^2}+\frac {4 B^2 d \log ^2(a+b x)}{b (b c-a d) g^2}-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac {4 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac {8 B^2 d \log (c+d x)}{b (b c-a d) g^2}-\frac {8 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac {4 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac {4 B^2 d \log ^2(c+d x)}{b (b c-a d) g^2}-\frac {8 B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}+\frac {\left (8 B^2 d\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 (b c-a d) g^2}+\frac {\left (8 B^2 d\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 (b c-a d) g^2}\\ &=-\frac {8 B^2}{b g^2 (a+b x)}-\frac {8 B^2 d \log (a+b x)}{b (b c-a d) g^2}+\frac {4 B^2 d \log ^2(a+b x)}{b (b c-a d) g^2}-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac {4 B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac {8 B^2 d \log (c+d x)}{b (b c-a d) g^2}-\frac {8 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac {4 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac {4 B^2 d \log ^2(c+d x)}{b (b c-a d) g^2}-\frac {8 B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}-\frac {8 B^2 d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b (b c-a d) g^2}-\frac {8 B^2 d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}\\ \end {align*}

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Mathematica [C]  time = 0.43, size = 321, normalized size = 2.47 \[ -\frac {\frac {4 B \left ((b c-a d) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )+d (a+b x) \log (a+b x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )-d (a+b x) \log (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+B d (a+b x) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 B (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)\right )}{b c-a d}+\left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{b g^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.76, size = 200, normalized size = 1.54 \[ -\frac {{\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} b c - {\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} a d + {\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} + 2 \, {\left ({\left (A B + 2 \, B^{2}\right )} b d x + {\left (A B + 2 \, B^{2}\right )} b c\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{{\left (b^{3} c - a b^{2} d\right )} g^{2} x + {\left (a b^{2} c - a^{2} b d\right )} g^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.74, size = 378, normalized size = 2.91 \[ -{\left (\frac {B^{2} d}{b^{2} c g^{2} - a b d g^{2}} + \frac {B^{2}}{{\left (b g x + a g\right )} b g}\right )} \log \left (\frac {b^{2}}{\frac {b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d g}{b g x + a g} - \frac {2 \, a d^{2} g}{b g x + a g} + d^{2}}\right )^{2} + \frac {4 \, {\left (A B d + 3 \, B^{2} d\right )} \log \left (\frac {b c g}{b g x + a g} - \frac {a d g}{b g x + a g} + d\right )}{b^{2} c g^{2} - a b d g^{2}} - \frac {2 \, {\left (A B + 3 \, B^{2}\right )} \log \left (\frac {b^{2}}{\frac {b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d g}{b g x + a g} - \frac {2 \, a d^{2} g}{b g x + a g} + d^{2}}\right )}{{\left (b g x + a g\right )} b g} - \frac {A^{2} + 6 \, A B + 13 \, B^{2}}{{\left (b g x + a g\right )} b g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(B^2*d/(b^2*c*g^2 - a*b*d*g^2) + B^2/((b*g*x + a*g)*b*g))*log(b^2/(b^2*c^2*g^2/(b*g*x + a*g)^2 - 2*a*b*c*d*g^
2/(b*g*x + a*g)^2 + a^2*d^2*g^2/(b*g*x + a*g)^2 + 2*b*c*d*g/(b*g*x + a*g) - 2*a*d^2*g/(b*g*x + a*g) + d^2))^2
+ 4*(A*B*d + 3*B^2*d)*log(b*c*g/(b*g*x + a*g) - a*d*g/(b*g*x + a*g) + d)/(b^2*c*g^2 - a*b*d*g^2) - 2*(A*B + 3*
B^2)*log(b^2/(b^2*c^2*g^2/(b*g*x + a*g)^2 - 2*a*b*c*d*g^2/(b*g*x + a*g)^2 + a^2*d^2*g^2/(b*g*x + a*g)^2 + 2*b*
c*d*g/(b*g*x + a*g) - 2*a*d^2*g/(b*g*x + a*g) + d^2))/((b*g*x + a*g)*b*g) - (A^2 + 6*A*B + 13*B^2)/((b*g*x + a
*g)*b*g)

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maple [B]  time = 0.10, size = 357, normalized size = 2.75 \[ \frac {B^{2} d \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )^{2}}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right ) \left (a d -b c \right ) g^{2}}+\frac {2 A B d \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right ) \left (a d -b c \right ) g^{2}}+\frac {4 B^{2} d \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right ) \left (a d -b c \right ) g^{2}}+\frac {A^{2} d}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right ) \left (a d -b c \right ) g^{2}}-\frac {4 A B d}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right ) \left (d x +c \right ) b \,g^{2}}-\frac {8 B^{2} d}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right ) \left (d x +c \right ) b \,g^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d/g^2*A^2/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)/(a*d-b*c)-8*d/g^2/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*B^2/b/(d*x+c)+4*d/
g^2/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*B^2/(a*d-b*c)*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)+d/g^2/(1/(d*x+c)
*a*d-1/(d*x+c)*b*c+b)*B^2/(a*d-b*c)*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)^2-4*d/g^2/(1/(d*x+c)*a*d-1/(d*
x+c)*b*c+b)*A*B/b/(d*x+c)+2*d/g^2/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*A*B/(a*d-b*c)*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*
c+b)^2/d^2*e)

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maxima [B]  time = 1.59, size = 574, normalized size = 4.42 \[ -4 \, {\left ({\left (\frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac {{\left (b d x + a d\right )} \log \left (b x + a\right )^{2} + {\left (b d x + a d\right )} \log \left (d x + c\right )^{2} - 2 \, b c + 2 \, a d - 2 \, {\left (b d x + a d\right )} \log \left (b x + a\right ) + 2 \, {\left (b d x + a d - {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b^{2} c g^{2} - a^{2} b d g^{2} + {\left (b^{3} c g^{2} - a b^{2} d g^{2}\right )} x}\right )} B^{2} - 2 \, A B {\left (\frac {\log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b^{2} g^{2} x + a b g^{2}} + \frac {2}{b^{2} g^{2} x + a b g^{2}} + \frac {2 \, d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {2 \, d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac {B^{2} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2}}{b^{2} g^{2} x + a b g^{2}} - \frac {A^{2}}{b^{2} g^{2} x + a b g^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.97, size = 228, normalized size = 1.75 \[ -{\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}^2\,\left (\frac {B^2}{b^2\,g^2\,\left (x+\frac {a}{b}\right )}-\frac {B^2\,d}{b\,g^2\,\left (a\,d-b\,c\right )}\right )-\frac {A^2+4\,A\,B+8\,B^2}{x\,b^2\,g^2+a\,b\,g^2}-\frac {\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\,\left (\frac {4\,B^2}{b^2\,d\,g^2}+\frac {2\,A\,B}{b^2\,d\,g^2}\right )}{\frac {x}{d}+\frac {a}{b\,d}}-\frac {B\,d\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {c\,b^2\,g^2+a\,d\,b\,g^2}{b\,g^2}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A+2\,B\right )\,8{}\mathrm {i}}{b\,g^2\,\left (a\,d-b\,c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 3.66, size = 454, normalized size = 3.49 \[ - \frac {4 B d \left (A + 2 B\right ) \log {\left (x + \frac {4 A B a d^{2} + 4 A B b c d + 8 B^{2} a d^{2} + 8 B^{2} b c d - \frac {4 B a^{2} d^{3} \left (A + 2 B\right )}{a d - b c} + \frac {8 B a b c d^{2} \left (A + 2 B\right )}{a d - b c} - \frac {4 B b^{2} c^{2} d \left (A + 2 B\right )}{a d - b c}}{8 A B b d^{2} + 16 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {4 B d \left (A + 2 B\right ) \log {\left (x + \frac {4 A B a d^{2} + 4 A B b c d + 8 B^{2} a d^{2} + 8 B^{2} b c d + \frac {4 B a^{2} d^{3} \left (A + 2 B\right )}{a d - b c} - \frac {8 B a b c d^{2} \left (A + 2 B\right )}{a d - b c} + \frac {4 B b^{2} c^{2} d \left (A + 2 B\right )}{a d - b c}}{8 A B b d^{2} + 16 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {\left (- 2 A B - 4 B^{2}\right ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}}{a b g^{2} + b^{2} g^{2} x} + \frac {\left (B^{2} c + B^{2} d x\right ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}^{2}}{a^{2} d g^{2} - a b c g^{2} + a b d g^{2} x - b^{2} c g^{2} x} + \frac {- A^{2} - 4 A B - 8 B^{2}}{a b g^{2} + b^{2} g^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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