3.135 \(\int \frac {(A+B \log (\frac {e (a+b x)^2}{(c+d x)^2}))^2}{(a g+b g x)^4} \, dx\)

Optimal. Leaf size=429 \[ -\frac {b^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{3 g^4 (a+b x)^3 (b c-a d)^3}-\frac {4 b^2 B (c+d x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{9 g^4 (a+b x)^3 (b c-a d)^3}-\frac {d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{g^4 (a+b x) (b c-a d)^3}-\frac {4 B d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g^4 (a+b x) (b c-a d)^3}+\frac {b d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{g^4 (a+b x)^2 (b c-a d)^3}+\frac {2 b B d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g^4 (a+b x)^2 (b c-a d)^3}-\frac {8 b^2 B^2 (c+d x)^3}{27 g^4 (a+b x)^3 (b c-a d)^3}-\frac {8 B^2 d^2 (c+d x)}{g^4 (a+b x) (b c-a d)^3}+\frac {2 b B^2 d (c+d x)^2}{g^4 (a+b x)^2 (b c-a d)^3} \]

[Out]

-8*B^2*d^2*(d*x+c)/(-a*d+b*c)^3/g^4/(b*x+a)+2*b*B^2*d*(d*x+c)^2/(-a*d+b*c)^3/g^4/(b*x+a)^2-8/27*b^2*B^2*(d*x+c
)^3/(-a*d+b*c)^3/g^4/(b*x+a)^3-4*B*d^2*(d*x+c)*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(-a*d+b*c)^3/g^4/(b*x+a)+2*b*B*
d*(d*x+c)^2*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(-a*d+b*c)^3/g^4/(b*x+a)^2-4/9*b^2*B*(d*x+c)^3*(A+B*ln(e*(b*x+a)^2
/(d*x+c)^2))/(-a*d+b*c)^3/g^4/(b*x+a)^3-d^2*(d*x+c)*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(-a*d+b*c)^3/g^4/(b*x+a)
+b*d*(d*x+c)^2*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(-a*d+b*c)^3/g^4/(b*x+a)^2-1/3*b^2*(d*x+c)^3*(A+B*ln(e*(b*x+a
)^2/(d*x+c)^2))^2/(-a*d+b*c)^3/g^4/(b*x+a)^3

________________________________________________________________________________________

Rubi [C]  time = 1.23, antiderivative size = 692, normalized size of antiderivative = 1.61, number of steps used = 34, 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} \[ -\frac {8 B^2 d^3 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}-\frac {8 B^2 d^3 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}-\frac {4 B d^3 \log (a+b x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 b g^4 (b c-a d)^3}+\frac {4 B d^3 \log (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 b g^4 (b c-a d)^3}-\frac {4 B d^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 b g^4 (a+b x) (b c-a d)^2}+\frac {2 B d \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 b g^4 (a+b x)^2 (b c-a d)}-\frac {\left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{3 b g^4 (a+b x)^3}-\frac {4 B \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{9 b g^4 (a+b x)^3}-\frac {44 B^2 d^2}{9 b g^4 (a+b x) (b c-a d)^2}+\frac {4 B^2 d^3 \log ^2(a+b x)}{3 b g^4 (b c-a d)^3}+\frac {4 B^2 d^3 \log ^2(c+d x)}{3 b g^4 (b c-a d)^3}-\frac {44 B^2 d^3 \log (a+b x)}{9 b g^4 (b c-a d)^3}-\frac {8 B^2 d^3 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}+\frac {44 B^2 d^3 \log (c+d x)}{9 b g^4 (b c-a d)^3}-\frac {8 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}+\frac {10 B^2 d}{9 b g^4 (a+b x)^2 (b c-a d)}-\frac {8 B^2}{27 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*B^2)/(27*b*g^4*(a + b*x)^3) + (10*B^2*d)/(9*b*(b*c - a*d)*g^4*(a + b*x)^2) - (44*B^2*d^2)/(9*b*(b*c - a*d)
^2*g^4*(a + b*x)) - (44*B^2*d^3*Log[a + b*x])/(9*b*(b*c - a*d)^3*g^4) + (4*B^2*d^3*Log[a + b*x]^2)/(3*b*(b*c -
a*d)^3*g^4) - (4*B*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(9*b*g^4*(a + b*x)^3) + (2*B*d*(A + B*Log[(e*(a
+ b*x)^2)/(c + d*x)^2]))/(3*b*(b*c - a*d)*g^4*(a + b*x)^2) - (4*B*d^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])
)/(3*b*(b*c - a*d)^2*g^4*(a + b*x)) - (4*B*d^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*b*(b*
c - a*d)^3*g^4) - (A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(3*b*g^4*(a + b*x)^3) + (44*B^2*d^3*Log[c + d*x])
/(9*b*(b*c - a*d)^3*g^4) - (8*B^2*d^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(3*b*(b*c - a*d)^3*g^4)
+ (4*B*d^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[c + d*x])/(3*b*(b*c - a*d)^3*g^4) + (4*B^2*d^3*Log[c +
d*x]^2)/(3*b*(b*c - a*d)^3*g^4) - (8*B^2*d^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(3*b*(b*c - a*d)^3*
g^4) - (8*B^2*d^3*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(3*b*(b*c - a*d)^3*g^4) - (8*B^2*d^3*PolyLog[2, (b
*(c + d*x))/(b*c - a*d)])/(3*b*(b*c - a*d)^3*g^4)

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)^4} \, dx &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\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^3 (a+b x)^4 (c+d x)} \, dx}{3 b g}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\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)^4 (c+d x)} \, dx}{3 b g^4}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\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)^4}-\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)^3}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b g^4}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {(4 B) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^4} \, dx}{3 g^4}-\frac {\left (4 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{a+b x} \, dx}{3 (b c-a d)^3 g^4}+\frac {\left (4 B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}+\frac {\left (4 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^2} \, dx}{3 (b c-a d)^2 g^4}-\frac {(4 B d) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^3} \, dx}{3 (b c-a d) g^4}\\ &=-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 b g^4 (a+b x)^3}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac {4 B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {4 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {4 B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {\left (4 B^2\right ) \int \frac {2 (b c-a d)}{(a+b x)^4 (c+d x)} \, dx}{9 b g^4}+\frac {\left (4 B^2 d^3\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}{3 b (b c-a d)^3 g^4}-\frac {\left (4 B^2 d^3\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}{3 b (b c-a d)^3 g^4}+\frac {\left (4 B^2 d^2\right ) \int \frac {2 (b c-a d)}{(a+b x)^2 (c+d x)} \, dx}{3 b (b c-a d)^2 g^4}-\frac {\left (2 B^2 d\right ) \int \frac {2 (b c-a d)}{(a+b x)^3 (c+d x)} \, dx}{3 b (b c-a d) g^4}\\ &=-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 b g^4 (a+b x)^3}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac {4 B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {4 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {4 B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}-\frac {\left (4 B^2 d\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{3 b g^4}+\frac {\left (8 B^2 d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3 b (b c-a d) g^4}+\frac {\left (8 B^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{9 b g^4}+\frac {\left (4 B^2 d^3\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}{3 b (b c-a d)^3 e g^4}-\frac {\left (4 B^2 d^3\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}{3 b (b c-a d)^3 e g^4}\\ &=-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 b g^4 (a+b x)^3}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac {4 B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {4 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {4 B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}-\frac {\left (4 B^2 d\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}{3 b g^4}+\frac {\left (8 B^2 d^2\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}{3 b (b c-a d) g^4}+\frac {\left (8 B^2 (b c-a d)\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{9 b g^4}+\frac {\left (4 B^2 d^3\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}{3 b (b c-a d)^3 e g^4}-\frac {\left (4 B^2 d^3\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}{3 b (b c-a d)^3 e g^4}\\ &=-\frac {8 B^2}{27 b g^4 (a+b x)^3}+\frac {10 B^2 d}{9 b (b c-a d) g^4 (a+b x)^2}-\frac {44 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac {44 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 b g^4 (a+b x)^3}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac {4 B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {4 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {44 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}+\frac {4 B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {\left (8 B^2 d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 (b c-a d)^3 g^4}-\frac {\left (8 B^2 d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 (b c-a d)^3 g^4}-\frac {\left (8 B^2 d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}+\frac {\left (8 B^2 d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}\\ &=-\frac {8 B^2}{27 b g^4 (a+b x)^3}+\frac {10 B^2 d}{9 b (b c-a d) g^4 (a+b x)^2}-\frac {44 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac {44 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 b g^4 (a+b x)^3}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac {4 B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {4 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {44 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {4 B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}+\frac {\left (8 B^2 d^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 (b c-a d)^3 g^4}+\frac {\left (8 B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b (b c-a d)^3 g^4}+\frac {\left (8 B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b (b c-a d)^3 g^4}+\frac {\left (8 B^2 d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}\\ &=-\frac {8 B^2}{27 b g^4 (a+b x)^3}+\frac {10 B^2 d}{9 b (b c-a d) g^4 (a+b x)^2}-\frac {44 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac {44 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac {4 B^2 d^3 \log ^2(a+b x)}{3 b (b c-a d)^3 g^4}-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 b g^4 (a+b x)^3}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac {4 B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {4 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {44 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {4 B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {4 B^2 d^3 \log ^2(c+d x)}{3 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}+\frac {\left (8 B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3 b (b c-a d)^3 g^4}+\frac {\left (8 B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b (b c-a d)^3 g^4}\\ &=-\frac {8 B^2}{27 b g^4 (a+b x)^3}+\frac {10 B^2 d}{9 b (b c-a d) g^4 (a+b x)^2}-\frac {44 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac {44 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac {4 B^2 d^3 \log ^2(a+b x)}{3 b (b c-a d)^3 g^4}-\frac {4 B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 b g^4 (a+b x)^3}+\frac {2 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac {4 B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {4 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {44 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {4 B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {4 B^2 d^3 \log ^2(c+d x)}{3 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}-\frac {8 B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}\\ \end {align*}

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Mathematica [C]  time = 0.64, size = 598, normalized size = 1.39 \[ -\frac {\frac {2 B \left (18 d^3 (a+b x)^3 \log (a+b x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )-18 d^3 (a+b x)^3 \log (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )+18 d^2 (a+b x)^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )+6 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )-9 d (a+b x) (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )-18 B d^3 (a+b x)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+18 B d^3 (a+b x)^3 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+36 B d^2 (a+b x)^2 (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)-9 B d (a+b x) \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )+2 B \left (-6 d^3 (a+b x)^3 \log (c+d x)+6 d^2 (a+b x)^2 (b c-a d)-3 d (a+b x) (b c-a d)^2+2 (b c-a d)^3+6 d^3 (a+b x)^3 \log (a+b x)\right )\right )}{(b c-a d)^3}+9 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{27 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/27*(9*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2 + (2*B*(6*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*
x)^2]) - 9*d*(b*c - a*d)^2*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + 18*d^2*(b*c - a*d)*(a + b*x)^2
*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + 18*d^3*(a + b*x)^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x)^2)/(c + d*
x)^2]) - 18*d^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[c + d*x] + 36*B*d^2*(a + b*x)^2*(b*c
- a*d + d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) - 9*B*d*(a + b*x)*((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*(2*(b*c - a*d)^3 - 3*d
*(b*c - a*d)^2*(a + b*x) + 6*d^2*(b*c - a*d)*(a + b*x)^2 + 6*d^3*(a + b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*
Log[c + d*x]) - 18*B*d^3*(a + b*x)^3*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyL
og[2, (d*(a + b*x))/(-(b*c) + a*d)]) + 18*B*d^3*(a + b*x)^3*((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)^3)/(b*g^4*(a + b*x)^3)

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fricas [A]  time = 0.92, size = 719, normalized size = 1.68 \[ -\frac {{\left (9 \, A^{2} + 12 \, A B + 8 \, B^{2}\right )} b^{3} c^{3} - 27 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a b^{2} c^{2} d + 27 \, {\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} a^{2} b c d^{2} - {\left (9 \, A^{2} + 66 \, A B + 170 \, B^{2}\right )} a^{3} d^{3} + 12 \, {\left ({\left (3 \, A B + 11 \, B^{2}\right )} b^{3} c d^{2} - {\left (3 \, A B + 11 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 9 \, {\left (B^{2} b^{3} d^{3} x^{3} + 3 \, B^{2} a b^{2} d^{3} x^{2} + 3 \, B^{2} a^{2} b d^{3} x + B^{2} b^{3} c^{3} - 3 \, B^{2} a b^{2} c^{2} d + 3 \, B^{2} a^{2} b c d^{2}\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} - 6 \, {\left ({\left (3 \, A B + 5 \, B^{2}\right )} b^{3} c^{2} d - 18 \, {\left (A B + 3 \, B^{2}\right )} a b^{2} c d^{2} + {\left (15 \, A B + 49 \, B^{2}\right )} a^{2} b d^{3}\right )} x + 6 \, {\left ({\left (3 \, A B + 11 \, B^{2}\right )} b^{3} d^{3} x^{3} + {\left (3 \, A B + 2 \, B^{2}\right )} b^{3} c^{3} - 9 \, {\left (A B + B^{2}\right )} a b^{2} c^{2} d + 9 \, {\left (A B + 2 \, B^{2}\right )} a^{2} b c d^{2} + 3 \, {\left (2 \, B^{2} b^{3} c d^{2} + 3 \, {\left (A B + 3 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (B^{2} b^{3} c^{2} d - 6 \, B^{2} a b^{2} c d^{2} - 3 \, {\left (A B + 2 \, B^{2}\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{27 \, {\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} g^{4} x + {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} g^{4}\right )}} \]

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)^4,x, algorithm="fricas")

[Out]

-1/27*((9*A^2 + 12*A*B + 8*B^2)*b^3*c^3 - 27*(A^2 + 2*A*B + 2*B^2)*a*b^2*c^2*d + 27*(A^2 + 4*A*B + 8*B^2)*a^2*
b*c*d^2 - (9*A^2 + 66*A*B + 170*B^2)*a^3*d^3 + 12*((3*A*B + 11*B^2)*b^3*c*d^2 - (3*A*B + 11*B^2)*a*b^2*d^3)*x^
2 + 9*(B^2*b^3*d^3*x^3 + 3*B^2*a*b^2*d^3*x^2 + 3*B^2*a^2*b*d^3*x + B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a^2
*b*c*d^2)*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 - 6*((3*A*B + 5*B^2)*b^3*c^2*d - 18
*(A*B + 3*B^2)*a*b^2*c*d^2 + (15*A*B + 49*B^2)*a^2*b*d^3)*x + 6*((3*A*B + 11*B^2)*b^3*d^3*x^3 + (3*A*B + 2*B^2
)*b^3*c^3 - 9*(A*B + B^2)*a*b^2*c^2*d + 9*(A*B + 2*B^2)*a^2*b*c*d^2 + 3*(2*B^2*b^3*c*d^2 + 3*(A*B + 3*B^2)*a*b
^2*d^3)*x^2 - 3*(B^2*b^3*c^2*d - 6*B^2*a*b^2*c*d^2 - 3*(A*B + 2*B^2)*a^2*b*d^3)*x)*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^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*g^4*x^3 + 3*(a
*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*g^4*x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4
*b^3*c*d^2 - a^5*b^2*d^3)*g^4*x + (a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*g^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{4}}\,{d x} \]

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)^4,x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)^2*e/(d*x + c)^2) + A)^2/(b*g*x + a*g)^4, x)

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maple [B]  time = 0.23, size = 1343, normalized size = 3.13 \[ \text {result too large to display} \]

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)^4,x)

[Out]

1/3*d^3/g^4*A^2*b^2/(a*d-b*c)^3/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3+d^3/g^4*A^2/(a*d-b*c)^3/(1/(d*x+c)*a*d-1/(d*
x+c)*b*c+b)-d^3/g^4*A^2*b/(a*d-b*c)^3/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2-170/27*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c
)*b*c+b)^3*B^2/b/(d*x+c)^3+22/9*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3*b^2*B^2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b
^2*c^2*d-b^3*c^3)*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)-44/9*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3*B
^2*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)-98/9*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3*B^2/(a*d-b*c)/(d*x+c)^
2+4*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3*B^2/(a*d-b*c)/(d*x+c)^2*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2
*e)+1/3*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3*B^2*b^2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*ln((1/
(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)^2+d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3*B^2/(a*d-b*c)/(d*x+c)^2*ln((
1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)^2+6*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3*B^2*b/(a^2*d^2-2*a*b*c*d
+b^2*c^2)/(d*x+c)*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)+d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3*B^2*b/
(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)^2-22/9*d^3/g^4/(1/(d*x+c)*a*d-
1/(d*x+c)*b*c+b)^3*A*B/b/(d*x+c)^3+2/3*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3*b^2*A*B/(a^3*d^3-3*a^2*b*c*d^
2+3*a*b^2*c^2*d-b^3*c^3)*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)-4/3*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+
b)^3*A*B*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)-10/3*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3*A*B/(a*d-b*c)/(d
*x+c)^2+2*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3*A*B/(a*d-b*c)/(d*x+c)^2*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)
^2/d^2*e)+2*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3*A*B*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+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 = 2.67, size = 1575, normalized size = 3.67 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2/(b*g*x+a*g)^4,x, algorithm="maxima")

[Out]

-2/27*(3*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5
*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3
*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*
a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d
^2 - a^3*b*d^3)*g^4))*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)) + (4*b^3*c^3 - 27*a*b^2*c^2*d + 108*a^2*b*c*d^2 - 85*a^3*d^3 + 66*(b^3*c*d^2 - a*b^2*
d^3)*x^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a)^2 - 18*(b^3*d^3*x^3 + 3*a
*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(d*x + c)^2 - 3*(5*b^3*c^2*d - 54*a*b^2*c*d^2 + 49*a^2*b*d^3)*x + 6
6*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a) - 6*(11*b^3*d^3*x^3 + 33*a*b^2*d^3*x^
2 + 33*a^2*b*d^3*x + 11*a^3*d^3 - 6*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a))*lo
g(d*x + c))/(a^3*b^4*c^3*g^4 - 3*a^4*b^3*c^2*d*g^4 + 3*a^5*b^2*c*d^2*g^4 - a^6*b*d^3*g^4 + (b^7*c^3*g^4 - 3*a*
b^6*c^2*d*g^4 + 3*a^2*b^5*c*d^2*g^4 - a^3*b^4*d^3*g^4)*x^3 + 3*(a*b^6*c^3*g^4 - 3*a^2*b^5*c^2*d*g^4 + 3*a^3*b^
4*c*d^2*g^4 - a^4*b^3*d^3*g^4)*x^2 + 3*(a^2*b^5*c^3*g^4 - 3*a^3*b^4*c^2*d*g^4 + 3*a^4*b^3*c*d^2*g^4 - a^5*b^2*
d^3*g^4)*x))*B^2 - 2/9*A*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/(
(b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b
^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 3*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^4*g^4*x
^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2
*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4)) -
1/3*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^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) - 1/3*A^2/(b^4*g^4*x^3 + 3*a*b^3*g
^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4)

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mupad [B]  time = 7.67, size = 1069, normalized size = 2.49 \[ \frac {\frac {9\,A^2\,a^2\,d^2-18\,A^2\,a\,b\,c\,d+9\,A^2\,b^2\,c^2+66\,A\,B\,a^2\,d^2-42\,A\,B\,a\,b\,c\,d+12\,A\,B\,b^2\,c^2+170\,B^2\,a^2\,d^2-46\,B^2\,a\,b\,c\,d+8\,B^2\,b^2\,c^2}{3\,\left (a\,d-b\,c\right )}+\frac {2\,x\,\left (-5\,c\,B^2\,b^2\,d+49\,a\,B^2\,b\,d^2-3\,A\,c\,B\,b^2\,d+15\,A\,a\,B\,b\,d^2\right )}{a\,d-b\,c}+\frac {4\,d\,x^2\,\left (11\,d\,B^2\,b^2+3\,A\,d\,B\,b^2\right )}{a\,d-b\,c}}{x\,\left (27\,a^2\,b^3\,c\,g^4-27\,a^3\,b^2\,d\,g^4\right )-x^2\,\left (27\,a^2\,b^3\,d\,g^4-27\,a\,b^4\,c\,g^4\right )+x^3\,\left (9\,b^5\,c\,g^4-9\,a\,b^4\,d\,g^4\right )+9\,a^3\,b^2\,c\,g^4-9\,a^4\,b\,d\,g^4}-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}^2\,\left (\frac {B^2}{3\,b^2\,g^4\,\left (3\,a^2\,x+\frac {a^3}{b}+b^2\,x^3+3\,a\,b\,x^2\right )}-\frac {B^2\,d^3}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )-\frac {\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\,\left (\frac {2\,A\,B}{3\,b^2\,d\,g^4}+\frac {2\,B^2\,d^3\,\left (a\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{3\,b\,d^3}+\frac {2\,a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {2\,\left (3\,a^3\,d^3-6\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{3\,b\,d^4}\right )}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {2\,B^2\,d^3\,x^2\,\left (\frac {2\,\left (b^2\,c-a\,b\,d\right )}{3\,d^2}-\frac {4\,b\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {2\,B^2\,d^3\,x\,\left (b\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{3\,b\,d^3}+\frac {2\,a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {2\,\left (3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2\right )}{3\,d^3}+\frac {4\,a\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )}{\frac {3\,a^2\,x}{d}+\frac {a^3}{b\,d}+\frac {b^2\,x^3}{d}+\frac {3\,a\,b\,x^2}{d}}-\frac {B\,d^3\,\mathrm {atan}\left (\frac {B\,d^3\,\left (\frac {a^3\,b\,d^3\,g^4-a^2\,b^2\,c\,d^2\,g^4-a\,b^3\,c^2\,d\,g^4+b^4\,c^3\,g^4}{a^2\,b\,d^2\,g^4-2\,a\,b^2\,c\,d\,g^4+b^3\,c^2\,g^4}+2\,b\,d\,x\right )\,\left (3\,A+11\,B\right )\,\left (a^2\,b\,d^2\,g^4-2\,a\,b^2\,c\,d\,g^4+b^3\,c^2\,g^4\right )\,4{}\mathrm {i}}{b\,g^4\,{\left (a\,d-b\,c\right )}^3\,\left (44\,B^2\,d^3+12\,A\,B\,d^3\right )}\right )\,\left (3\,A+11\,B\right )\,8{}\mathrm {i}}{9\,b\,g^4\,{\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)^2)/(c + d*x)^2))^2/(a*g + b*g*x)^4,x)

[Out]

((9*A^2*a^2*d^2 + 9*A^2*b^2*c^2 + 170*B^2*a^2*d^2 + 8*B^2*b^2*c^2 + 66*A*B*a^2*d^2 + 12*A*B*b^2*c^2 - 18*A^2*a
*b*c*d - 46*B^2*a*b*c*d - 42*A*B*a*b*c*d)/(3*(a*d - b*c)) + (2*x*(49*B^2*a*b*d^2 - 5*B^2*b^2*c*d + 15*A*B*a*b*
d^2 - 3*A*B*b^2*c*d))/(a*d - b*c) + (4*d*x^2*(11*B^2*b^2*d + 3*A*B*b^2*d))/(a*d - b*c))/(x*(27*a^2*b^3*c*g^4 -
27*a^3*b^2*d*g^4) - x^2*(27*a^2*b^3*d*g^4 - 27*a*b^4*c*g^4) + x^3*(9*b^5*c*g^4 - 9*a*b^4*d*g^4) + 9*a^3*b^2*c
*g^4 - 9*a^4*b*d*g^4) - log((e*(a + b*x)^2)/(c + d*x)^2)^2*(B^2/(3*b^2*g^4*(3*a^2*x + a^3/b + b^2*x^3 + 3*a*b*
x^2)) - (B^2*d^3)/(3*b*g^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - (log((e*(a + b*x)^2)/(c + d
*x)^2)*((2*A*B)/(3*b^2*d*g^4) + (2*B^2*d^3*(a*((3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(3*b*d^3) + (2*a*(a*d - b*c))
/(3*b*d^2)) + (2*(3*a^3*d^3 - b^3*c^3 + 4*a*b^2*c^2*d - 6*a^2*b*c*d^2))/(3*b*d^4)))/(3*b*g^4*(a^3*d^3 - b^3*c^
3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (2*B^2*d^3*x^2*((2*(b^2*c - a*b*d))/(3*d^2) - (4*b*(a*d - b*c))/(3*d^2))
)/(3*b*g^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (2*B^2*d^3*x*(b*((3*a^2*d^2 + b^2*c^2 - 4*a*
b*c*d)/(3*b*d^3) + (2*a*(a*d - b*c))/(3*b*d^2)) + (2*(3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(3*d^3) + (4*a*(a*d -
b*c))/(3*d^2)))/(3*b*g^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))))/((3*a^2*x)/d + a^3/(b*d) + (b^
2*x^3)/d + (3*a*b*x^2)/d) - (B*d^3*atan((B*d^3*((b^4*c^3*g^4 + a^3*b*d^3*g^4 - a*b^3*c^2*d*g^4 - a^2*b^2*c*d^2
*g^4)/(b^3*c^2*g^4 + a^2*b*d^2*g^4 - 2*a*b^2*c*d*g^4) + 2*b*d*x)*(3*A + 11*B)*(b^3*c^2*g^4 + a^2*b*d^2*g^4 - 2
*a*b^2*c*d*g^4)*4i)/(b*g^4*(a*d - b*c)^3*(44*B^2*d^3 + 12*A*B*d^3)))*(3*A + 11*B)*8i)/(9*b*g^4*(a*d - b*c)^3)

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sympy [B]  time = 34.03, size = 1561, normalized size = 3.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*(b*x+a)**2/(d*x+c)**2))**2/(b*g*x+a*g)**4,x)

[Out]

-4*B*d**3*(3*A + 11*B)*log(x + (12*A*B*a*d**4 + 12*A*B*b*c*d**3 + 44*B**2*a*d**4 + 44*B**2*b*c*d**3 - 4*B*a**4
*d**7*(3*A + 11*B)/(a*d - b*c)**3 + 16*B*a**3*b*c*d**6*(3*A + 11*B)/(a*d - b*c)**3 - 24*B*a**2*b**2*c**2*d**5*
(3*A + 11*B)/(a*d - b*c)**3 + 16*B*a*b**3*c**3*d**4*(3*A + 11*B)/(a*d - b*c)**3 - 4*B*b**4*c**4*d**3*(3*A + 11
*B)/(a*d - b*c)**3)/(24*A*B*b*d**4 + 88*B**2*b*d**4))/(9*b*g**4*(a*d - b*c)**3) + 4*B*d**3*(3*A + 11*B)*log(x
+ (12*A*B*a*d**4 + 12*A*B*b*c*d**3 + 44*B**2*a*d**4 + 44*B**2*b*c*d**3 + 4*B*a**4*d**7*(3*A + 11*B)/(a*d - b*c
)**3 - 16*B*a**3*b*c*d**6*(3*A + 11*B)/(a*d - b*c)**3 + 24*B*a**2*b**2*c**2*d**5*(3*A + 11*B)/(a*d - b*c)**3 -
16*B*a*b**3*c**3*d**4*(3*A + 11*B)/(a*d - b*c)**3 + 4*B*b**4*c**4*d**3*(3*A + 11*B)/(a*d - b*c)**3)/(24*A*B*b
*d**4 + 88*B**2*b*d**4))/(9*b*g**4*(a*d - b*c)**3) + (3*B**2*a**2*c*d**2 + 3*B**2*a**2*d**3*x - 3*B**2*a*b*c**
2*d + 3*B**2*a*b*d**3*x**2 + B**2*b**2*c**3 + B**2*b**2*d**3*x**3)*log(e*(a + b*x)**2/(c + d*x)**2)**2/(3*a**6
*d**3*g**4 - 9*a**5*b*c*d**2*g**4 + 9*a**5*b*d**3*g**4*x + 9*a**4*b**2*c**2*d*g**4 - 27*a**4*b**2*c*d**2*g**4*
x + 9*a**4*b**2*d**3*g**4*x**2 - 3*a**3*b**3*c**3*g**4 + 27*a**3*b**3*c**2*d*g**4*x - 27*a**3*b**3*c*d**2*g**4
*x**2 + 3*a**3*b**3*d**3*g**4*x**3 - 9*a**2*b**4*c**3*g**4*x + 27*a**2*b**4*c**2*d*g**4*x**2 - 9*a**2*b**4*c*d
**2*g**4*x**3 - 9*a*b**5*c**3*g**4*x**2 + 9*a*b**5*c**2*d*g**4*x**3 - 3*b**6*c**3*g**4*x**3) + (-6*A*B*a**2*d*
*2 + 12*A*B*a*b*c*d - 6*A*B*b**2*c**2 - 22*B**2*a**2*d**2 + 14*B**2*a*b*c*d - 30*B**2*a*b*d**2*x - 4*B**2*b**2
*c**2 + 6*B**2*b**2*c*d*x - 12*B**2*b**2*d**2*x**2)*log(e*(a + b*x)**2/(c + d*x)**2)/(9*a**5*b*d**2*g**4 - 18*
a**4*b**2*c*d*g**4 + 27*a**4*b**2*d**2*g**4*x + 9*a**3*b**3*c**2*g**4 - 54*a**3*b**3*c*d*g**4*x + 27*a**3*b**3
*d**2*g**4*x**2 + 27*a**2*b**4*c**2*g**4*x - 54*a**2*b**4*c*d*g**4*x**2 + 9*a**2*b**4*d**2*g**4*x**3 + 27*a*b*
*5*c**2*g**4*x**2 - 18*a*b**5*c*d*g**4*x**3 + 9*b**6*c**2*g**4*x**3) - (9*A**2*a**2*d**2 - 18*A**2*a*b*c*d + 9
*A**2*b**2*c**2 + 66*A*B*a**2*d**2 - 42*A*B*a*b*c*d + 12*A*B*b**2*c**2 + 170*B**2*a**2*d**2 - 46*B**2*a*b*c*d
+ 8*B**2*b**2*c**2 + x**2*(36*A*B*b**2*d**2 + 132*B**2*b**2*d**2) + x*(90*A*B*a*b*d**2 - 18*A*B*b**2*c*d + 294
*B**2*a*b*d**2 - 30*B**2*b**2*c*d))/(27*a**5*b*d**2*g**4 - 54*a**4*b**2*c*d*g**4 + 27*a**3*b**3*c**2*g**4 + x*
*3*(27*a**2*b**4*d**2*g**4 - 54*a*b**5*c*d*g**4 + 27*b**6*c**2*g**4) + x**2*(81*a**3*b**3*d**2*g**4 - 162*a**2
*b**4*c*d*g**4 + 81*a*b**5*c**2*g**4) + x*(81*a**4*b**2*d**2*g**4 - 162*a**3*b**3*c*d*g**4 + 81*a**2*b**4*c**2
*g**4))

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