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

Optimal. Leaf size=575 \[ -\frac{b^3 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{4 g^5 (a+b x)^4 (b c-a d)^4}-\frac{b^3 B (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{8 g^5 (a+b x)^4 (b c-a d)^4}+\frac{b^2 d (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{g^5 (a+b x)^3 (b c-a d)^4}+\frac{2 b^2 B d (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^4}-\frac{3 b d^2 (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^5 (a+b x)^2 (b c-a d)^4}-\frac{3 b B d^2 (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^5 (a+b x)^2 (b c-a d)^4}+\frac{d^3 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{g^5 (a+b x) (b c-a d)^4}+\frac{2 B d^3 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^5 (a+b x) (b c-a d)^4}-\frac{b^3 B^2 (c+d x)^4}{32 g^5 (a+b x)^4 (b c-a d)^4}+\frac{2 b^2 B^2 d (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^4}-\frac{3 b B^2 d^2 (c+d x)^2}{4 g^5 (a+b x)^2 (b c-a d)^4}+\frac{2 B^2 d^3 (c+d x)}{g^5 (a+b x) (b c-a d)^4} \]

[Out]

(2*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)/(4*(b*c - a*d)^4*g^5*(a + b*x)
^2) + (2*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)/(32*(b*c - a*d)^4*g^
5*(a + b*x)^4) + (2*B*d^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((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))/(c + d*x)]))/(2*(b*c - a*d)^4*g^5*(a + b*x)^2) + (2*b^2*B*d*(c + d
*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(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))/(c + d*x)]))/(8*(b*c - a*d)^4*g^5*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*
x)])^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))/(c + d*x)])^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))/(c + d*x)])^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))/(c + d*x)])^2)/(4*(b*c - a*d)^4*g^5*(a + b*x)^4)

________________________________________________________________________________________

Rubi [C]  time = 1.22722, antiderivative size = 763, normalized size of antiderivative = 1.33, number of steps used = 38, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {2525, 12, 2528, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{B^2 d^4 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{2 b g^5 (b c-a d)^4}+\frac{B^2 d^4 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{2 b g^5 (b c-a d)^4}+\frac{B d^4 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b g^5 (b c-a d)^4}-\frac{B d^4 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b g^5 (b c-a d)^4}+\frac{B d^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b g^5 (a+b x) (b c-a d)^3}-\frac{B d^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b g^5 (a+b x)^2 (b c-a d)^2}+\frac{B d \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{6 b g^5 (a+b x)^3 (b c-a d)}-\frac{\left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{4 b g^5 (a+b x)^4}-\frac{B \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{8 b g^5 (a+b x)^4}+\frac{25 B^2 d^3}{24 b g^5 (a+b x) (b c-a d)^3}-\frac{13 B^2 d^2}{48 b g^5 (a+b x)^2 (b c-a d)^2}-\frac{B^2 d^4 \log ^2(a+b x)}{4 b g^5 (b c-a d)^4}-\frac{B^2 d^4 \log ^2(c+d x)}{4 b g^5 (b c-a d)^4}+\frac{25 B^2 d^4 \log (a+b x)}{24 b g^5 (b c-a d)^4}+\frac{B^2 d^4 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{2 b g^5 (b c-a d)^4}-\frac{25 B^2 d^4 \log (c+d x)}{24 b g^5 (b c-a d)^4}+\frac{B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b g^5 (b c-a d)^4}+\frac{7 B^2 d}{72 b g^5 (a+b x)^3 (b c-a d)}-\frac{B^2}{32 b g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-B^2/(32*b*g^5*(a + b*x)^4) + (7*B^2*d)/(72*b*(b*c - a*d)*g^5*(a + b*x)^3) - (13*B^2*d^2)/(48*b*(b*c - a*d)^2*
g^5*(a + b*x)^2) + (25*B^2*d^3)/(24*b*(b*c - a*d)^3*g^5*(a + b*x)) + (25*B^2*d^4*Log[a + b*x])/(24*b*(b*c - a*
d)^4*g^5) - (B^2*d^4*Log[a + b*x]^2)/(4*b*(b*c - a*d)^4*g^5) - (B*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8*b*g
^5*(a + b*x)^4) + (B*d*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b*(b*c - a*d)*g^5*(a + b*x)^3) - (B*d^2*(A + B
*Log[(e*(a + b*x))/(c + d*x)]))/(4*b*(b*c - a*d)^2*g^5*(a + b*x)^2) + (B*d^3*(A + B*Log[(e*(a + b*x))/(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))/(c + d*x)]))/(2*b*(b*c -
a*d)^4*g^5) - (A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(4*b*g^5*(a + b*x)^4) - (25*B^2*d^4*Log[c + d*x])/(24*b*
(b*c - a*d)^4*g^5) + (B^2*d^4*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(2*b*(b*c - a*d)^4*g^5) - (B*d^4
*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(2*b*(b*c - a*d)^4*g^5) - (B^2*d^4*Log[c + d*x]^2)/(4*b*(b
*c - a*d)^4*g^5) + (B^2*d^4*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(2*b*(b*c - a*d)^4*g^5) + (B^2*d^4*Po
lyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(2*b*(b*c - a*d)^4*g^5) + (B^2*d^4*PolyLog[2, (b*(c + d*x))/(b*c - a*d
)])/(2*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)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac{B \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\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)}{c+d x}\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)}{c+d x}\right )}{(a+b x)^5 (c+d x)} \, dx}{2 b g^5}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\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)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^5}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\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)}{c+d x}\right )\right )}{(b c-a d)^5 (a+b x)}-\frac{d^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{2 b g^5}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac{B \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{2 g^5}+\frac{\left (B d^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{2 (b c-a d)^4 g^5}-\frac{\left (B d^5\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{2 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)}{c+d x}\right )}{(a+b x)^2} \, dx}{2 (b c-a d)^3 g^5}+\frac{\left (B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{2 (b c-a d)^2 g^5}-\frac{(B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{2 (b c-a d) g^5}\\ &=-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b g^5 (a+b x)^4}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 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)}{c+d x}\right )\right )}{4 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)}{c+d x}\right )\right )}{2 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)}{c+d x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\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)}{c+d x}\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}+\frac{B^2 \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{8 b g^5}-\frac{\left (B^2 d^4\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{2 b (b c-a d)^4 g^5}+\frac{\left (B^2 d^4\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d^3\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{2 b (b c-a d)^3 g^5}+\frac{\left (B^2 d^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{4 b (b c-a d)^2 g^5}-\frac{\left (B^2 d\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{6 b (b c-a d) g^5}\\ &=-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b g^5 (a+b x)^4}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 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)}{c+d x}\right )\right )}{4 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)}{c+d x}\right )\right )}{2 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)}{c+d x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\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)}{c+d x}\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{6 b g^5}-\frac{\left (B^2 d^3\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{2 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}{4 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}{8 b g^5}-\frac{\left (B^2 d^4\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{2 b (b c-a d)^4 e g^5}+\frac{\left (B^2 d^4\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{2 b (b c-a d)^4 e g^5}\\ &=-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b g^5 (a+b x)^4}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 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)}{c+d x}\right )\right )}{4 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)}{c+d x}\right )\right )}{2 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)}{c+d x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\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)}{c+d x}\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{\left (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}{6 b g^5}-\frac{\left (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}{2 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}{4 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}{8 b g^5}-\frac{\left (B^2 d^4\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{2 b (b c-a d)^4 e g^5}+\frac{\left (B^2 d^4\right ) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{2 b (b c-a d)^4 e g^5}\\ &=-\frac{B^2}{32 b g^5 (a+b x)^4}+\frac{7 B^2 d}{72 b (b c-a d) g^5 (a+b x)^3}-\frac{13 B^2 d^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{25 B^2 d^3}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac{25 B^2 d^4 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b g^5 (a+b x)^4}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 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)}{c+d x}\right )\right )}{4 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)}{c+d x}\right )\right )}{2 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)}{c+d x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{25 B^2 d^4 \log (c+d x)}{24 b (b c-a d)^4 g^5}-\frac{B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d^4\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{2 (b c-a d)^4 g^5}+\frac{\left (B^2 d^4\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{2 (b c-a d)^4 g^5}+\frac{\left (B^2 d^5\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d^5\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}\\ &=-\frac{B^2}{32 b g^5 (a+b x)^4}+\frac{7 B^2 d}{72 b (b c-a d) g^5 (a+b x)^3}-\frac{13 B^2 d^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{25 B^2 d^3}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac{25 B^2 d^4 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b g^5 (a+b x)^4}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 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)}{c+d x}\right )\right )}{4 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)}{c+d x}\right )\right )}{2 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)}{c+d x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{25 B^2 d^4 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac{B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}+\frac{B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d^4\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{2 (b c-a d)^4 g^5}-\frac{\left (B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d^5\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}\\ &=-\frac{B^2}{32 b g^5 (a+b x)^4}+\frac{7 B^2 d}{72 b (b c-a d) g^5 (a+b x)^3}-\frac{13 B^2 d^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{25 B^2 d^3}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac{25 B^2 d^4 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac{B^2 d^4 \log ^2(a+b x)}{4 b (b c-a d)^4 g^5}-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b g^5 (a+b x)^4}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 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)}{c+d x}\right )\right )}{4 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)}{c+d x}\right )\right )}{2 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)}{c+d x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{25 B^2 d^4 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac{B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{B^2 d^4 \log ^2(c+d x)}{4 b (b c-a d)^4 g^5}+\frac{B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (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 )}{2 b (b c-a d)^4 g^5}-\frac{\left (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 )}{2 b (b c-a d)^4 g^5}\\ &=-\frac{B^2}{32 b g^5 (a+b x)^4}+\frac{7 B^2 d}{72 b (b c-a d) g^5 (a+b x)^3}-\frac{13 B^2 d^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{25 B^2 d^3}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac{25 B^2 d^4 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac{B^2 d^4 \log ^2(a+b x)}{4 b (b c-a d)^4 g^5}-\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b g^5 (a+b x)^4}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6 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)}{c+d x}\right )\right )}{4 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)}{c+d x}\right )\right )}{2 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)}{c+d x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{25 B^2 d^4 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac{B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{B^2 d^4 \log ^2(c+d x)}{4 b (b c-a d)^4 g^5}+\frac{B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}+\frac{B^2 d^4 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}+\frac{B^2 d^4 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}\\ \end{align*}

Mathematica [C]  time = 1.11006, size = 748, normalized size = 1.3 \[ -\frac{\frac{B \left (72 B d^4 (a+b x)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-72 B d^4 (a+b x)^4 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+72 d^2 (a+b x)^2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+144 d^3 (a+b x)^3 (a d-b c) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-144 d^4 (a+b x)^4 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+144 d^4 (a+b x)^4 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+36 (b c-a d)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+48 d (a+b x) (a d-b c)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-144 B d^3 (a+b x)^3 (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)+36 B d^2 (a+b x)^2 \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )-8 B d (a+b x) \left (6 d^2 (a+b x)^2 (b c-a d)-6 d^3 (a+b x)^3 \log (c+d x)-3 d (a+b x) (b c-a d)^2+2 (b c-a d)^3+6 d^3 (a+b x)^3 \log (a+b x)\right )+3 B \left (6 d^2 (a+b x)^2 (b c-a d)^2+12 d^3 (a+b x)^3 (a d-b c)+12 d^4 (a+b x)^4 \log (c+d x)+4 d (a+b x) (a d-b c)^3+3 (b c-a d)^4-12 d^4 (a+b x)^4 \log (a+b x)\right )\right )}{(b c-a d)^4}+72 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{288 b g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(72*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + (B*(36*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 48*d
*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 72*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log
[(e*(a + b*x))/(c + d*x)]) + 144*d^3*(-(b*c) + a*d)*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 144*d^4
*(a + b*x)^4*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 144*d^4*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/
(c + d*x)])*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*(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)]) - 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)/(288*b*g^5*(a + b*x)^4)

________________________________________________________________________________________

Maple [B]  time = 0.049, size = 3538, normalized size = 6.2 \begin{align*} \text{output 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)/(d*x+c)))^2/(b*g*x+a*g)^5,x)

[Out]

e*d^4/(a*d-b*c)^5/g^5*A^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*a+1/4*e^4/(a*d-b*c)^5/g^5*A^2*b^4/(b*e/d+e/(d*x+
c)*a-e/d/(d*x+c)*b*c)^4*c+1/32*e^4/(a*d-b*c)^5/g^5*B^2*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*c+2*e*d^4/(a*
d-b*c)^5/g^5*B^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*a+2*e*d^4/(a*d-b*c)^5/g^5*A*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x
+c)*b*c)*a-1/2*e^4*d/(a*d-b*c)^5/g^5*A*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x
+c))*a-e*d^3/(a*d-b*c)^5/g^5*A^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*b*c+2/9*e^3*d^2/(a*d-b*c)^5/g^5*B^2*b^2/(
b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a-2/9*e^3*d/(a*d-b*c)^5/g^5*B^2*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3
*c-1/32*e^4*d/(a*d-b*c)^5/g^5*B^2*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a-3/2*e^2*d^3/(a*d-b*c)^5/g^5*A^2*
b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*a+2*e*d^4/(a*d-b*c)^5/g^5*B^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*ln(b
*e/d+(a*d-b*c)*e/d/(d*x+c))*a+3/2*e^2*d^2/(a*d-b*c)^5/g^5*A^2*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*c+e^3*
d^2/(a*d-b*c)^5/g^5*A^2*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a-e^3*d/(a*d-b*c)^5/g^5*A^2*b^3/(b*e/d+e/(d*
x+c)*a-e/d/(d*x+c)*b*c)^3*c+3/4*e^2*d^2/(a*d-b*c)^5/g^5*B^2*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*c+1/8*e^
4/(a*d-b*c)^5/g^5*A*B*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*c-1/4*e^4*d/(a*d-b*c)^5/g^5*A^2*b^3/(b*e/d+e/(
d*x+c)*a-e/d/(d*x+c)*b*c)^4*a+1/4*e^4/(a*d-b*c)^5/g^5*B^2*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e/d+(
a*d-b*c)*e/d/(d*x+c))^2*c+1/8*e^4/(a*d-b*c)^5/g^5*B^2*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e/d+(a*d-
b*c)*e/d/(d*x+c))*c+e*d^4/(a*d-b*c)^5/g^5*B^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*ln(b*e/d+(a*d-b*c)*e/d/(d*x+
c))^2*a+2*e^3*d^2/(a*d-b*c)^5/g^5*A*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c)
)*a-2*e*d^3/(a*d-b*c)^5/g^5*A*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b*c-2*e^3*
d/(a*d-b*c)^5/g^5*A*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c-3*e^2*d^3/(a
*d-b*c)^5/g^5*A*B*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+3*e^2*d^2/(a*d-b*c
)^5/g^5*A*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c-2*e*d^3/(a*d-b*c)^5/g^
5*B^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*b*c-3/4*e^2*d^3/(a*d-b*c)^5/g^5*B^2*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)
*b*c)^2*a-2/3*e^3*d/(a*d-b*c)^5/g^5*A*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*c-1/8*e^4*d/(a*d-b*c)^5/g^5*
A*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a-2*e*d^3/(a*d-b*c)^5/g^5*A*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c
)*b*c+1/2*e^4/(a*d-b*c)^5/g^5*A*B*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c-
e*d^3/(a*d-b*c)^5/g^5*B^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*b*c-2*e*d^3/(a
*d-b*c)^5/g^5*B^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b*c+2*e*d^4/(a*d-b*c)^5/
g^5*A*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+3/2*e^2*d^2/(a*d-b*c)^5/g^5*B^2*
b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*c-3/2*e^2*d^3/(a*d-b*c)^5/g^5*B^2*
b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+2/3*e^3*d^2/(a*d-b*c)^5/g^5*B^2*b^2/
(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-2/3*e^3*d/(a*d-b*c)^5/g^5*B^2*b^3/(b*e
/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c-1/4*e^4*d/(a*d-b*c)^5/g^5*B^2*b^3/(b*e/d+e
/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*a-1/8*e^4*d/(a*d-b*c)^5/g^5*B^2*b^3/(b*e/d+e/(
d*x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+e^3*d^2/(a*d-b*c)^5/g^5*B^2*b^2/(b*e/d+e/(d*x+c)
*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*a-e^3*d/(a*d-b*c)^5/g^5*B^2*b^3/(b*e/d+e/(d*x+c)*a-e/d
/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*c-3/2*e^2*d^3/(a*d-b*c)^5/g^5*B^2*b/(b*e/d+e/(d*x+c)*a-e/d/(
d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*a-3/2*e^2*d^3/(a*d-b*c)^5/g^5*A*B*b/(b*e/d+e/(d*x+c)*a-e/d/(d*
x+c)*b*c)^2*a+3/2*e^2*d^2/(a*d-b*c)^5/g^5*A*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*c+2/3*e^3*d^2/(a*d-b*c
)^5/g^5*A*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a+3/2*e^2*d^2/(a*d-b*c)^5/g^5*B^2*b^2/(b*e/d+e/(d*x+c)*a
-e/d/(d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c

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Maxima [B]  time = 2.18354, size = 2866, normalized size = 4.98 \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)/(d*x+c)))^2/(b*g*x+a*g)^5,x, algorithm="maxima")

[Out]

1/288*(12*((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*e*x/(d*x + c) + a*e/(d*x
+ c)) - (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 - 2
71*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 + 1/24*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) - 12*log(b*e*x/(d*x + c) + a*e/(d*x +
c))/(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*e*x/(d*x + c) + a*e
/(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) - 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 = 1.16413, size = 2152, normalized size = 3.74 \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)/(d*x+c)))^2/(b*g*x+a*g)^5,x, algorithm="fricas")

[Out]

-1/288*(9*(8*A^2 + 4*A*B + B^2)*b^4*c^4 - 32*(9*A^2 + 6*A*B + 2*B^2)*a*b^3*c^3*d + 216*(2*A^2 + 2*A*B + B^2)*a
^2*b^2*c^2*d^2 - 288*(A^2 + 2*A*B + 2*B^2)*a^3*b*c*d^3 + (72*A^2 + 300*A*B + 415*B^2)*a^4*d^4 - 12*((12*A*B +
25*B^2)*b^4*c*d^3 - (12*A*B + 25*B^2)*a*b^3*d^4)*x^3 + 6*((12*A*B + 13*B^2)*b^4*c^2*d^2 - 16*(6*A*B + 11*B^2)*
a*b^3*c*d^3 + (84*A*B + 163*B^2)*a^2*b^2*d^4)*x^2 - 72*(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*e*x + a*e)/(d*x + c))^2 - 4*((12*A*B + 7*B^2)*b^4*c^3*d - 12*(6*A*B + 5*B^2)*a*b^3*c^2*d^2 + 108*(2*A*B +
3*B^2)*a^2*b^2*c*d^3 - (156*A*B + 271*B^2)*a^3*b*d^4)*x - 12*((12*A*B + 25*B^2)*b^4*d^4*x^4 - 3*(4*A*B + B^2)*
b^4*c^4 + 16*(3*A*B + B^2)*a*b^3*c^3*d - 36*(2*A*B + B^2)*a^2*b^2*c^2*d^2 + 48*(A*B + B^2)*a^3*b*c*d^3 + 4*(3*
B^2*b^4*c*d^3 + 2*(6*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*(2*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 + 12*(A*B + B^2)*a^3*b*d^4)*
x)*log((b*e*x + a*e)/(d*x + c)))/((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)/(d*x+c)))**2/(b*g*x+a*g)**5,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x, algorithm="giac")

[Out]

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