3.136 \(\int \frac{(A+B \log (\frac{e (a+b x)^2}{(c+d x)^2}))^2}{(a g+b g x)^5} \, dx\)

Optimal. Leaf size=587 \[ -\frac{b^3 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{4 g^5 (a+b x)^4 (b c-a d)^4}-\frac{b^3 B (c+d x)^4 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{4 g^5 (a+b x)^4 (b c-a d)^4}+\frac{b^2 d (c+d x)^3 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{g^5 (a+b x)^3 (b c-a d)^4}+\frac{4 b^2 B d (c+d x)^3 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^4}-\frac{3 b d^2 (c+d x)^2 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{2 g^5 (a+b x)^2 (b c-a d)^4}-\frac{3 b B d^2 (c+d x)^2 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g^5 (a+b x)^2 (b c-a d)^4}+\frac{d^3 (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{g^5 (a+b x) (b c-a d)^4}+\frac{4 B d^3 (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g^5 (a+b x) (b c-a d)^4}-\frac{b^3 B^2 (c+d x)^4}{8 g^5 (a+b x)^4 (b c-a d)^4}+\frac{8 b^2 B^2 d (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^4}-\frac{3 b B^2 d^2 (c+d x)^2}{g^5 (a+b x)^2 (b c-a d)^4}+\frac{8 B^2 d^3 (c+d x)}{g^5 (a+b x) (b c-a d)^4} \]

[Out]

(8*B^2*d^3*(c + d*x))/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*B^2*d^2*(c + d*x)^2)/((b*c - a*d)^4*g^5*(a + b*x)^2
) + (8*b^2*B^2*d*(c + d*x)^3)/(9*(b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*B^2*(c + d*x)^4)/(8*(b*c - a*d)^4*g^5*(
a + b*x)^4) + (4*B*d^3*(c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/((b*c - a*d)^4*g^5*(a + b*x)) - (3*
b*B*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/((b*c - a*d)^4*g^5*(a + b*x)^2) + (4*b^2*B*d*(c
+ d*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*(b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*B*(c + d*x)^4*(A +
B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(4*(b*c - a*d)^4*g^5*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[(e*(a + b*
x)^2)/(c + d*x)^2])^2)/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*
x)^2])^2)/(2*(b*c - a*d)^4*g^5*(a + b*x)^2) + (b^2*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(
(b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(4*(b*c - a*d)^4
*g^5*(a + b*x)^4)

________________________________________________________________________________________

Rubi [C]  time = 1.39387, antiderivative size = 757, normalized size of antiderivative = 1.29, number of steps used = 38, 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} \[ \frac{2 B^2 d^4 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b g^5 (b c-a d)^4}+\frac{2 B^2 d^4 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b g^5 (b c-a d)^4}+\frac{B d^4 \log (a+b x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g^5 (b c-a d)^4}-\frac{B d^4 \log (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g^5 (b c-a d)^4}+\frac{B d^3 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g^5 (a+b x) (b c-a d)^3}-\frac{B d^2 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{2 b g^5 (a+b x)^2 (b c-a d)^2}+\frac{B d \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 b g^5 (a+b x)^3 (b c-a d)}-\frac{\left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{4 b g^5 (a+b x)^4}-\frac{B \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{4 b g^5 (a+b x)^4}+\frac{25 B^2 d^3}{6 b g^5 (a+b x) (b c-a d)^3}-\frac{13 B^2 d^2}{12 b g^5 (a+b x)^2 (b c-a d)^2}-\frac{B^2 d^4 \log ^2(a+b x)}{b g^5 (b c-a d)^4}-\frac{B^2 d^4 \log ^2(c+d x)}{b g^5 (b c-a d)^4}+\frac{25 B^2 d^4 \log (a+b x)}{6 b g^5 (b c-a d)^4}+\frac{2 B^2 d^4 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b g^5 (b c-a d)^4}-\frac{25 B^2 d^4 \log (c+d x)}{6 b g^5 (b c-a d)^4}+\frac{2 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g^5 (b c-a d)^4}+\frac{7 B^2 d}{18 b g^5 (a+b x)^3 (b c-a d)}-\frac{B^2}{8 b g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(a*g + b*g*x)^5,x]

[Out]

-B^2/(8*b*g^5*(a + b*x)^4) + (7*B^2*d)/(18*b*(b*c - a*d)*g^5*(a + b*x)^3) - (13*B^2*d^2)/(12*b*(b*c - a*d)^2*g
^5*(a + b*x)^2) + (25*B^2*d^3)/(6*b*(b*c - a*d)^3*g^5*(a + b*x)) + (25*B^2*d^4*Log[a + b*x])/(6*b*(b*c - a*d)^
4*g^5) - (B^2*d^4*Log[a + b*x]^2)/(b*(b*c - a*d)^4*g^5) - (B*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(4*b*g^
5*(a + b*x)^4) + (B*d*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*b*(b*c - a*d)*g^5*(a + b*x)^3) - (B*d^2*(A
+ B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(2*b*(b*c - a*d)^2*g^5*(a + b*x)^2) + (B*d^3*(A + B*Log[(e*(a + b*x)^2)
/(c + d*x)^2]))/(b*(b*c - a*d)^3*g^5*(a + b*x)) + (B*d^4*Log[a + b*x]*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])
)/(b*(b*c - a*d)^4*g^5) - (A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(4*b*g^5*(a + b*x)^4) - (25*B^2*d^4*Log[c
+ d*x])/(6*b*(b*c - a*d)^4*g^5) + (2*B^2*d^4*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(b*(b*c - a*d)^4
*g^5) - (B*d^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[c + d*x])/(b*(b*c - a*d)^4*g^5) - (B^2*d^4*Log[c +
d*x]^2)/(b*(b*c - a*d)^4*g^5) + (2*B^2*d^4*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(b*(b*c - a*d)^4*g^5)
+ (2*B^2*d^4*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b*(b*c - a*d)^4*g^5) + (2*B^2*d^4*PolyLog[2, (b*(c +
d*x))/(b*c - a*d)])/(b*(b*c - a*d)^4*g^5)

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 12

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

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]

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 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 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 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 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 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 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 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]

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)^5} \, dx &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac{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^4 (a+b x)^5 (c+d x)} \, dx}{2 b g}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac{(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)^5 (c+d x)} \, dx}{b g^5}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac{(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)^5}-\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)^4}+\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)^3}-\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)^2}+\frac{b d^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^5 (a+b x)}-\frac{d^5 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{b g^5}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac{B \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^5} \, dx}{g^5}+\frac{\left (B d^4\right ) \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)^4 g^5}-\frac{\left (B d^5\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)^4 g^5}-\frac{\left (B d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^2} \, dx}{(b c-a d)^3 g^5}+\frac{\left (B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^3} \, dx}{(b c-a d)^2 g^5}-\frac{(B d) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^4} \, dx}{(b c-a d) g^5}\\ &=-\frac{B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 b g^5 (a+b x)^4}+\frac{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^5 (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 )}{2 b (b c-a d)^2 g^5 (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 (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 \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)^4 g^5}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{B d^4 \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)^4 g^5}+\frac{B^2 \int \frac{2 (b c-a d)}{(a+b x)^5 (c+d x)} \, dx}{4 b g^5}-\frac{\left (B^2 d^4\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)^4 g^5}+\frac{\left (B^2 d^4\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)^4 g^5}-\frac{\left (B^2 d^3\right ) \int \frac{2 (b c-a d)}{(a+b x)^2 (c+d x)} \, dx}{b (b c-a d)^3 g^5}+\frac{\left (B^2 d^2\right ) \int \frac{2 (b c-a d)}{(a+b x)^3 (c+d x)} \, dx}{2 b (b c-a d)^2 g^5}-\frac{\left (B^2 d\right ) \int \frac{2 (b c-a d)}{(a+b x)^4 (c+d x)} \, dx}{3 b (b c-a d) g^5}\\ &=-\frac{B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 b g^5 (a+b x)^4}+\frac{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^5 (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 )}{2 b (b c-a d)^2 g^5 (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 (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 \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)^4 g^5}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{B d^4 \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)^4 g^5}-\frac{\left (2 B^2 d\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{3 b g^5}-\frac{\left (2 B^2 d^3\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b (b c-a d)^2 g^5}+\frac{\left (B^2 d^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b (b c-a d) g^5}+\frac{\left (B^2 (b c-a d)\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{2 b g^5}-\frac{\left (B^2 d^4\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)^4 e g^5}+\frac{\left (B^2 d^4\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)^4 e g^5}\\ &=-\frac{B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 b g^5 (a+b x)^4}+\frac{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^5 (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 )}{2 b (b c-a d)^2 g^5 (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 (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 \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)^4 g^5}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{B d^4 \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)^4 g^5}-\frac{\left (2 B^2 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}{3 b g^5}-\frac{\left (2 B^2 d^3\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 (b c-a d)^2 g^5}+\frac{\left (B^2 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}{b (b c-a d) g^5}+\frac{\left (B^2 (b c-a d)\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{2 b g^5}-\frac{\left (B^2 d^4\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)^4 e g^5}+\frac{\left (B^2 d^4\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)^4 e g^5}\\ &=-\frac{B^2}{8 b g^5 (a+b x)^4}+\frac{7 B^2 d}{18 b (b c-a d) g^5 (a+b x)^3}-\frac{13 B^2 d^2}{12 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{25 B^2 d^3}{6 b (b c-a d)^3 g^5 (a+b x)}+\frac{25 B^2 d^4 \log (a+b x)}{6 b (b c-a d)^4 g^5}-\frac{B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 b g^5 (a+b x)^4}+\frac{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^5 (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 )}{2 b (b c-a d)^2 g^5 (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 (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 \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)^4 g^5}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{25 B^2 d^4 \log (c+d x)}{6 b (b c-a d)^4 g^5}-\frac{B d^4 \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)^4 g^5}-\frac{\left (2 B^2 d^4\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{(b c-a d)^4 g^5}+\frac{\left (2 B^2 d^4\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{(b c-a d)^4 g^5}+\frac{\left (2 B^2 d^5\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b (b c-a d)^4 g^5}-\frac{\left (2 B^2 d^5\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b (b c-a d)^4 g^5}\\ &=-\frac{B^2}{8 b g^5 (a+b x)^4}+\frac{7 B^2 d}{18 b (b c-a d) g^5 (a+b x)^3}-\frac{13 B^2 d^2}{12 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{25 B^2 d^3}{6 b (b c-a d)^3 g^5 (a+b x)}+\frac{25 B^2 d^4 \log (a+b x)}{6 b (b c-a d)^4 g^5}-\frac{B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 b g^5 (a+b x)^4}+\frac{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^5 (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 )}{2 b (b c-a d)^2 g^5 (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 (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 \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)^4 g^5}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{25 B^2 d^4 \log (c+d x)}{6 b (b c-a d)^4 g^5}+\frac{2 B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^4 g^5}-\frac{B d^4 \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)^4 g^5}+\frac{2 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)^4 g^5}-\frac{\left (2 B^2 d^4\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{(b c-a d)^4 g^5}-\frac{\left (2 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b (b c-a d)^4 g^5}-\frac{\left (2 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b (b c-a d)^4 g^5}-\frac{\left (2 B^2 d^5\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b (b c-a d)^4 g^5}\\ &=-\frac{B^2}{8 b g^5 (a+b x)^4}+\frac{7 B^2 d}{18 b (b c-a d) g^5 (a+b x)^3}-\frac{13 B^2 d^2}{12 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{25 B^2 d^3}{6 b (b c-a d)^3 g^5 (a+b x)}+\frac{25 B^2 d^4 \log (a+b x)}{6 b (b c-a d)^4 g^5}-\frac{B^2 d^4 \log ^2(a+b x)}{b (b c-a d)^4 g^5}-\frac{B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 b g^5 (a+b x)^4}+\frac{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^5 (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 )}{2 b (b c-a d)^2 g^5 (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 (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 \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)^4 g^5}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{25 B^2 d^4 \log (c+d x)}{6 b (b c-a d)^4 g^5}+\frac{2 B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^4 g^5}-\frac{B d^4 \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)^4 g^5}-\frac{B^2 d^4 \log ^2(c+d x)}{b (b c-a d)^4 g^5}+\frac{2 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)^4 g^5}-\frac{\left (2 B^2 d^4\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)^4 g^5}-\frac{\left (2 B^2 d^4\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)^4 g^5}\\ &=-\frac{B^2}{8 b g^5 (a+b x)^4}+\frac{7 B^2 d}{18 b (b c-a d) g^5 (a+b x)^3}-\frac{13 B^2 d^2}{12 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{25 B^2 d^3}{6 b (b c-a d)^3 g^5 (a+b x)}+\frac{25 B^2 d^4 \log (a+b x)}{6 b (b c-a d)^4 g^5}-\frac{B^2 d^4 \log ^2(a+b x)}{b (b c-a d)^4 g^5}-\frac{B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 b g^5 (a+b x)^4}+\frac{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^5 (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 )}{2 b (b c-a d)^2 g^5 (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 (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 \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)^4 g^5}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{25 B^2 d^4 \log (c+d x)}{6 b (b c-a d)^4 g^5}+\frac{2 B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^4 g^5}-\frac{B d^4 \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)^4 g^5}-\frac{B^2 d^4 \log ^2(c+d x)}{b (b c-a d)^4 g^5}+\frac{2 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)^4 g^5}+\frac{2 B^2 d^4 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b (b c-a d)^4 g^5}+\frac{2 B^2 d^4 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)^4 g^5}\\ \end{align*}

Mathematica [C]  time = 1.02916, size = 762, normalized size = 1.3 \[ -\frac{\frac{B \left (72 B d^4 (a+b x)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-72 B d^4 (a+b x)^4 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+36 d^2 (a+b x)^2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+72 d^3 (a+b x)^3 (a d-b c) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )-72 d^4 (a+b x)^4 \log (a+b x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+72 d^4 (a+b x)^4 \log (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+18 (b c-a d)^4 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+24 d (a+b x) (a d-b c)^3 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )-144 B d^3 (a+b x)^3 (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)+36 B d^2 (a+b x)^2 \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )-8 B d (a+b x) \left (6 d^2 (a+b x)^2 (b c-a d)-6 d^3 (a+b x)^3 \log (c+d x)-3 d (a+b x) (b c-a d)^2+2 (b c-a d)^3+6 d^3 (a+b x)^3 \log (a+b x)\right )+3 B \left (6 d^2 (a+b x)^2 (b c-a d)^2+12 d^3 (a+b x)^3 (a d-b c)+12 d^4 (a+b x)^4 \log (c+d x)+4 d (a+b x) (a d-b c)^3+3 (b c-a d)^4-12 d^4 (a+b x)^4 \log (a+b x)\right )\right )}{(b c-a d)^4}+18 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{72 b g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(a*g + b*g*x)^5,x]

[Out]

-(18*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2 + (B*(18*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]
) + 24*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + 36*d^2*(b*c - a*d)^2*(a + b*x)^
2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + 72*d^3*(-(b*c) + a*d)*(a + b*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c +
d*x)^2]) - 72*d^4*(a + b*x)^4*Log[a + b*x]*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + 72*d^4*(a + b*x)^4*(A +
B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[c + d*x] - 144*B*d^3*(a + b*x)^3*(b*c - a*d + d*(a + b*x)*Log[a + b*x
] - d*(a + b*x)*Log[c + d*x]) + 36*B*d^2*(a + b*x)^2*((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]) - 8*B*d*(a + b*x)*(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])
+ 3*B*(3*(b*c - a*d)^4 + 4*d*(-(b*c) + a*d)^3*(a + b*x) + 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 12*d^3*(-(b*c) + a
*d)*(a + b*x)^3 - 12*d^4*(a + b*x)^4*Log[a + b*x] + 12*d^4*(a + b*x)^4*Log[c + d*x]) + 72*B*d^4*(a + b*x)^4*(L
og[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)]) -
72*B*d^4*(a + b*x)^4*((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)^4)/(72*b*g^5*(a + b*x)^4)

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Maple [B]  time = 0.297, size = 1943, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(b*g*x+a*g)^5,x)

[Out]

-3/2*d^4/g^5*A^2*b/(a*d-b*c)^4/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2+d^4/g^5*A^2/(a*d-b*c)^4/(1/(d*x+c)*a*d-b*c/(d*x
+c)+b)+d^4/g^5*A^2*b^2/(a*d-b*c)^4/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^3-1/4*d^4/g^5*A^2*b^3/(a*d-b*c)^4/(1/(d*x+c)*
a*d-b*c/(d*x+c)+b)^4-415/72*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*B^2/b/(d*x+c)^4+25/12*d^4/g^5/(1/(d*x+c)*a
*d-b*c/(d*x+c)+b)^4*b^3*B^2/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*ln(e*(1/(d*x+c)*a*
d-b*c/(d*x+c)+b)^2/d^2)-25/6*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*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)/(d*x+c)-163/12*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*b*B^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)^
2-271/18*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*B^2/(a*d-b*c)/(d*x+c)^3+4*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+
b)^4*B^2/(a*d-b*c)/(d*x+c)^3*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)+1/4*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+
b)^4*B^2*b^3/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b
)^2/d^2)^2+d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*B^2/(a*d-b*c)/(d*x+c)^3*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^
2/d^2)^2+9*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*B^2*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)^2*ln(e*(1/(d*x+c)
*a*d-b*c/(d*x+c)+b)^2/d^2)+22/3*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*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)/(d*x+c)*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)+d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*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)/(d*x+c)*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)^2+3/2*d^4/
g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*b*B^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)^2*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c
)+b)^2/d^2)^2-d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*A*B*b^2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(d
*x+c)-25/12*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*A*B/b/(d*x+c)^4+1/2*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^
4*b^3*A*B/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2
/d^2)-7/2*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*A*B*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)^2-13/3*d^4/g^5/(1/
(d*x+c)*a*d-b*c/(d*x+c)+b)^4*A*B/(a*d-b*c)/(d*x+c)^3+2*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*A*B/(a*d-b*c)/(
d*x+c)^3*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)+3*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4*A*B*b/(a^2*d^2-2*
a*b*c*d+b^2*c^2)/(d*x+c)^2*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)+2*d^4/g^5/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^4
*A*B*b^2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(d*x+c)*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)

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Maxima [B]  time = 2.25964, size = 3077, normalized size = 5.24 \begin{align*} \text{result too large to display} \end{align*}

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)^5,x, algorithm="maxima")

[Out]

1/72*(6*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d
^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^
5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*
a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 -
a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) + 12*d^4*log(b*x + a)/
((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*
c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5))*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)) - (9*b^4*c^4 - 64*a*b^3*c^3*d
+ 216*a^2*b^2*c^2*d^2 - 576*a^3*b*c*d^3 + 415*a^4*d^4 - 300*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 6*(13*b^4*c^2*d^2 -
176*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x
+ a^4*d^4)*log(b*x + a)^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*
log(d*x + c)^2 - 4*(7*b^4*c^3*d - 60*a*b^3*c^2*d^2 + 324*a^2*b^2*c*d^3 - 271*a^3*b*d^4)*x - 300*(b^4*d^4*x^4 +
4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a) + 12*(25*b^4*d^4*x^4 + 100*a*b^3*
d^4*x^3 + 150*a^2*b^2*d^4*x^2 + 100*a^3*b*d^4*x + 25*a^4*d^4 - 12*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d
^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a))*log(d*x + c))/(a^4*b^5*c^4*g^5 - 4*a^5*b^4*c^3*d*g^5 + 6*a^6*b
^3*c^2*d^2*g^5 - 4*a^7*b^2*c*d^3*g^5 + a^8*b*d^4*g^5 + (b^9*c^4*g^5 - 4*a*b^8*c^3*d*g^5 + 6*a^2*b^7*c^2*d^2*g^
5 - 4*a^3*b^6*c*d^3*g^5 + a^4*b^5*d^4*g^5)*x^4 + 4*(a*b^8*c^4*g^5 - 4*a^2*b^7*c^3*d*g^5 + 6*a^3*b^6*c^2*d^2*g^
5 - 4*a^4*b^5*c*d^3*g^5 + a^5*b^4*d^4*g^5)*x^3 + 6*(a^2*b^7*c^4*g^5 - 4*a^3*b^6*c^3*d*g^5 + 6*a^4*b^5*c^2*d^2*
g^5 - 4*a^5*b^4*c*d^3*g^5 + a^6*b^3*d^4*g^5)*x^2 + 4*(a^3*b^6*c^4*g^5 - 4*a^4*b^5*c^3*d*g^5 + 6*a^5*b^4*c^2*d^
2*g^5 - 4*a^6*b^3*c*d^3*g^5 + a^7*b^2*d^4*g^5)*x))*B^2 + 1/12*A*B*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*
d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^
3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*
a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x
^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d
+ 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) - 6*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^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*
x + a^4*b*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d
^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^
5)) - 1/4*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^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) - 1/4*A^2
/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5)

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Fricas [A]  time = 1.20757, size = 2229, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

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)^5,x, algorithm="fricas")

[Out]

-1/72*(9*(2*A^2 + 2*A*B + B^2)*b^4*c^4 - 8*(9*A^2 + 12*A*B + 8*B^2)*a*b^3*c^3*d + 108*(A^2 + 2*A*B + 2*B^2)*a^
2*b^2*c^2*d^2 - 72*(A^2 + 4*A*B + 8*B^2)*a^3*b*c*d^3 + (18*A^2 + 150*A*B + 415*B^2)*a^4*d^4 - 12*((6*A*B + 25*
B^2)*b^4*c*d^3 - (6*A*B + 25*B^2)*a*b^3*d^4)*x^3 + 6*((6*A*B + 13*B^2)*b^4*c^2*d^2 - 16*(3*A*B + 11*B^2)*a*b^3
*c*d^3 + (42*A*B + 163*B^2)*a^2*b^2*d^4)*x^2 - 18*(B^2*b^4*d^4*x^4 + 4*B^2*a*b^3*d^4*x^3 + 6*B^2*a^2*b^2*d^4*x
^2 + 4*B^2*a^3*b*d^4*x - B^2*b^4*c^4 + 4*B^2*a*b^3*c^3*d - 6*B^2*a^2*b^2*c^2*d^2 + 4*B^2*a^3*b*c*d^3)*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 - 4*((6*A*B + 7*B^2)*b^4*c^3*d - 12*(3*A*B + 5*B^2)*a
*b^3*c^2*d^2 + 108*(A*B + 3*B^2)*a^2*b^2*c*d^3 - (78*A*B + 271*B^2)*a^3*b*d^4)*x - 6*((6*A*B + 25*B^2)*b^4*d^4
*x^4 - 3*(2*A*B + B^2)*b^4*c^4 + 8*(3*A*B + 2*B^2)*a*b^3*c^3*d - 36*(A*B + B^2)*a^2*b^2*c^2*d^2 + 24*(A*B + 2*
B^2)*a^3*b*c*d^3 + 4*(3*B^2*b^4*c*d^3 + 2*(3*A*B + 11*B^2)*a*b^3*d^4)*x^3 - 6*(B^2*b^4*c^2*d^2 - 8*B^2*a*b^3*c
*d^3 - 6*(A*B + 3*B^2)*a^2*b^2*d^4)*x^2 + 4*(B^2*b^4*c^3*d - 6*B^2*a*b^3*c^2*d^2 + 18*B^2*a^2*b^2*c*d^3 + 6*(A
*B + 2*B^2)*a^3*b*d^4)*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^9*c^4 - 4*a*b^8*
c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*g^5*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^
6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*g^5*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*
a^5*b^4*c*d^3 + a^6*b^3*d^4)*g^5*x^2 + 4*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3
+ a^7*b^2*d^4)*g^5*x + (a^4*b^5*c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4)*g^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)**5,x)

[Out]

Timed out

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Giac [A]  time = 1.64779, size = 1180, normalized size = 2.01 \begin{align*} \text{result too large to display} \end{align*}

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)^5,x, algorithm="giac")

[Out]

1/4*(B^2*d^4/(b^5*c^4*g^5 - 4*a*b^4*c^3*d*g^5 + 6*a^2*b^3*c^2*d^2*g^5 - 4*a^3*b^2*c*d^3*g^5 + a^4*b*d^4*g^5) -
B^2/((b*g*x + a*g)^4*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 + 1/12*(12*B^2*d^3/((b^3*c^3*g^
3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^3)*(b*g*x + a*g)*b*g) - 6*B^2*d^2/((b^2*c^2*g - 2*a*b*c*
d*g + a^2*d^2*g)*(b*g*x + a*g)^2*b*g^2) + 4*B^2*d/((b*g*x + a*g)^3*(b*c - a*d)*b*g^2) - 3*(2*A*B*b^3*g^3 + 3*B
^2*b^3*g^3)/((b*g*x + a*g)^4*b^4*g^4))*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)) - 1/6*(6*A*B*d^4 + 31*
B^2*d^4)*log(-b*c*g/(b*g*x + a*g) + a*d*g/(b*g*x + a*g) - d)/(b^5*c^4*g^5 - 4*a*b^4*c^3*d*g^5 + 6*a^2*b^3*c^2*
d^2*g^5 - 4*a^3*b^2*c*d^3*g^5 + a^4*b*d^4*g^5) + 1/6*(6*A*B*d^3 + 31*B^2*d^3)/((b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^
3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^3)*(b*g*x + a*g)*b*g) - 1/12*(6*A*B*b*d^2 + 19*B^2*b*d^2)/((b^2*c^2*g - 2*a*
b*c*d*g + a^2*d^2*g)*(b*g*x + a*g)^2*b^2*g^2) + 1/18*(6*A*B*b^2*d*g + 13*B^2*b^2*d*g)/((b*g*x + a*g)^3*(b*c -
a*d)*b^3*g^3) - 1/8*(2*A^2*b^3*g^3 + 6*A*B*b^3*g^3 + 5*B^2*b^3*g^3)/((b*g*x + a*g)^4*b^4*g^4)