3.1.79 \(\int \frac {(c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^2} \, dx\) [79]

Optimal. Leaf size=692 \[ -\frac {2 B^2 (b c-a d)^2 i^3 (c+d x)}{b^3 g^2 (a+b x)}-\frac {B d^2 (b c-a d) i^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}-\frac {2 B (b c-a d)^2 i^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {4 B d (b c-a d)^2 i^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {2 d^2 (b c-a d) i^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2}-\frac {(b c-a d)^2 i^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}+\frac {d i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^2}+\frac {B^2 d (b c-a d)^2 i^3 \log (c+d x)}{b^4 g^2}+\frac {B d (b c-a d)^2 i^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}-\frac {3 d (b c-a d)^2 i^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}+\frac {4 B^2 d (b c-a d)^2 i^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 g^2}-\frac {B^2 d (b c-a d)^2 i^3 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}+\frac {6 B d (b c-a d)^2 i^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}+\frac {6 B^2 d (b c-a d)^2 i^3 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2} \]

[Out]

-2*B^2*(-a*d+b*c)^2*i^3*(d*x+c)/b^3/g^2/(b*x+a)-B*d^2*(-a*d+b*c)*i^3*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^4/g
^2-2*B*(-a*d+b*c)^2*i^3*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^3/g^2/(b*x+a)+4*B*d*(-a*d+b*c)^2*i^3*ln((-a*d+b*
c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^4/g^2+2*d^2*(-a*d+b*c)*i^3*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b
^4/g^2-(-a*d+b*c)^2*i^3*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b^3/g^2/(b*x+a)+1/2*d*i^3*(d*x+c)^2*(A+B*ln(e*(b
*x+a)/(d*x+c)))^2/b^2/g^2+B^2*d*(-a*d+b*c)^2*i^3*ln(d*x+c)/b^4/g^2+B*d*(-a*d+b*c)^2*i^3*(A+B*ln(e*(b*x+a)/(d*x
+c)))*ln(1-b*(d*x+c)/d/(b*x+a))/b^4/g^2-3*d*(-a*d+b*c)^2*i^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2*ln(1-b*(d*x+c)/d/(b
*x+a))/b^4/g^2+4*B^2*d*(-a*d+b*c)^2*i^3*polylog(2,d*(b*x+a)/b/(d*x+c))/b^4/g^2-B^2*d*(-a*d+b*c)^2*i^3*polylog(
2,b*(d*x+c)/d/(b*x+a))/b^4/g^2+6*B*d*(-a*d+b*c)^2*i^3*(A+B*ln(e*(b*x+a)/(d*x+c)))*polylog(2,b*(d*x+c)/d/(b*x+a
))/b^4/g^2+6*B^2*d*(-a*d+b*c)^2*i^3*polylog(3,b*(d*x+c)/d/(b*x+a))/b^4/g^2

________________________________________________________________________________________

Rubi [A]
time = 0.52, antiderivative size = 692, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2562, 2395, 2342, 2341, 2356, 2389, 2379, 2438, 2351, 31, 2355, 2354, 2421, 6724} \begin {gather*} \frac {6 B d i^3 (b c-a d)^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^2}+\frac {4 B^2 d i^3 (b c-a d)^2 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 g^2}-\frac {B^2 d i^3 (b c-a d)^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}+\frac {6 B^2 d i^3 (b c-a d)^2 \text {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}+\frac {2 d^2 i^3 (a+b x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b^4 g^2}-\frac {B d^2 i^3 (a+b x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^2}+\frac {4 B d i^3 (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^2}-\frac {3 d i^3 (b c-a d)^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b^4 g^2}+\frac {B d i^3 (b c-a d)^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^2}-\frac {i^3 (c+d x) (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b^3 g^2 (a+b x)}-\frac {2 B i^3 (c+d x) (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^2 (a+b x)}+\frac {d i^3 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 b^2 g^2}+\frac {B^2 d i^3 (b c-a d)^2 \log (c+d x)}{b^4 g^2}-\frac {2 B^2 i^3 (c+d x) (b c-a d)^2}{b^3 g^2 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*B^2*(b*c - a*d)^2*i^3*(c + d*x))/(b^3*g^2*(a + b*x)) - (B*d^2*(b*c - a*d)*i^3*(a + b*x)*(A + B*Log[(e*(a +
b*x))/(c + d*x)]))/(b^4*g^2) - (2*B*(b*c - a*d)^2*i^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g^
2*(a + b*x)) + (4*B*d*(b*c - a*d)^2*i^3*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(
b^4*g^2) + (2*d^2*(b*c - a*d)*i^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^4*g^2) - ((b*c - a*d)^2
*i^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^3*g^2*(a + b*x)) + (d*i^3*(c + d*x)^2*(A + B*Log[(e*
(a + b*x))/(c + d*x)])^2)/(2*b^2*g^2) + (B^2*d*(b*c - a*d)^2*i^3*Log[c + d*x])/(b^4*g^2) + (B*d*(b*c - a*d)^2*
i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2) - (3*d*(b*c - a*d)^2*
i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2) + (4*B^2*d*(b*c - a
*d)^2*i^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^4*g^2) - (B^2*d*(b*c - a*d)^2*i^3*PolyLog[2, (b*(c + d*x
))/(d*(a + b*x))])/(b^4*g^2) + (6*B*d*(b*c - a*d)^2*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c
+ d*x))/(d*(a + b*x))])/(b^4*g^2) + (6*B^2*d*(b*c - a*d)^2*i^3*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g
^2)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[x*(d + e*x^r)^(q +
1)*((a + b*Log[c*x^n])/d), x] - Dist[b*(n/d), Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2355

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[x*((a + b*Log[c*x^n])
^p/(d*(d + e*x))), x] - Dist[b*n*(p/d), Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[p, 0]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +
d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^
(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2395

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2438

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 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*(
(A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h,
i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i
, 0] && IntegersQ[m, q]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {(79 c+79 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx &=\int \left (\frac {493039 d^2 (3 b c-2 a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {493039 d^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^2}+\frac {493039 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)^2}+\frac {1479117 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2 (a+b x)}\right ) \, dx\\ &=\frac {\left (493039 d^3\right ) \int x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{b^2 g^2}+\frac {\left (493039 d^2 (3 b c-2 a d)\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{b^3 g^2}+\frac {\left (1479117 d (b c-a d)^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{b^3 g^2}+\frac {\left (493039 (b c-a d)^3\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{b^3 g^2}\\ &=\frac {493039 d^2 (3 b c-2 a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {493039 d^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^2}-\frac {493039 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2 (a+b x)}+\frac {1479117 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2}-\frac {\left (493039 B d^3\right ) \int \frac {(b c-a d) x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b^2 g^2}-\frac {\left (986078 B d^2 (3 b c-2 a d)\right ) \int \frac {(b c-a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b^3 g^2}-\frac {\left (2958234 B d (b c-a d)^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{e (a+b x)} \, dx}{b^4 g^2}+\frac {\left (986078 B (b c-a d)^3\right ) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^2}\\ &=\frac {493039 d^2 (3 b c-2 a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {493039 d^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^2}-\frac {493039 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2 (a+b x)}+\frac {1479117 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2}-\frac {\left (493039 B d^3 (b c-a d)\right ) \int \frac {x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b^2 g^2}-\frac {\left (986078 B d^2 (3 b c-2 a d) (b c-a d)\right ) \int \frac {x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b^3 g^2}+\frac {\left (986078 B (b c-a d)^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^2}-\frac {\left (2958234 B d (b c-a d)^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{b^4 e g^2}\\ &=\frac {493039 d^2 (3 b c-2 a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {493039 d^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^2}-\frac {493039 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2 (a+b x)}+\frac {1479117 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2}-\frac {\left (493039 B d^3 (b c-a d)\right ) \int \left (\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{b d}+\frac {a^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b (b c-a d) (a+b x)}+\frac {c^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d (-b c+a d) (c+d x)}\right ) \, dx}{b^2 g^2}-\frac {\left (986078 B d^2 (3 b c-2 a d) (b c-a d)\right ) \int \left (-\frac {a \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)}+\frac {c \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)}\right ) \, dx}{b^3 g^2}+\frac {\left (986078 B (b c-a d)^4\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^4 g^2}-\frac {\left (2958234 B d (b c-a d)^2\right ) \int \frac {(b c-a d) e \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b^4 e g^2}\\ &=\frac {493039 d^2 (3 b c-2 a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {493039 d^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^2}-\frac {493039 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2 (a+b x)}+\frac {1479117 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2}+\frac {\left (493039 B c^2 d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^2 g^2}-\frac {\left (493039 a^2 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^3 g^2}+\frac {\left (986078 a B d^2 (3 b c-2 a d)\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^3 g^2}-\frac {\left (986078 B c d^2 (3 b c-2 a d)\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^3 g^2}-\frac {\left (493039 B d^2 (b c-a d)\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^3 g^2}-\frac {\left (986078 B d (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^3 g^2}+\frac {\left (986078 B d^2 (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^4 g^2}+\frac {\left (986078 B (b c-a d)^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^3 g^2}-\frac {\left (2958234 B d (b c-a d)^3\right ) \int \frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b^4 g^2}\\ &=-\frac {493039 A B d^2 (b c-a d) x}{b^3 g^2}-\frac {986078 B (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2 (a+b x)}-\frac {493039 a^2 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {986078 a B d^2 (3 b c-2 a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}-\frac {986078 B d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {493039 d^2 (3 b c-2 a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {493039 d^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^2}-\frac {493039 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2 (a+b x)}+\frac {1479117 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2}+\frac {493039 B c^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}-\frac {986078 B c d (3 b c-2 a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}+\frac {986078 B d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^4 g^2}-\frac {\left (493039 B^2 c^2 d\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^2 g^2}+\frac {\left (493039 a^2 B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^4 g^2}+\frac {\left (986078 B^2 c d (3 b c-2 a d)\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^3 g^2}-\frac {\left (986078 a B^2 d^2 (3 b c-2 a d)\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^4 g^2}-\frac {\left (493039 B^2 d^2 (b c-a d)\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{b^3 g^2}+\frac {\left (986078 B^2 d (b c-a d)^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^4 g^2}-\frac {\left (986078 B^2 d (b c-a d)^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^4 g^2}+\frac {\left (986078 B^2 (b c-a d)^3\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^2}-\frac {\left (2958234 B d (b c-a d)^3\right ) \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)}\right ) \, dx}{b^4 g^2}\\ &=-\frac {493039 A B d^2 (b c-a d) x}{b^3 g^2}-\frac {493039 B^2 d^2 (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac {986078 B (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2 (a+b x)}-\frac {493039 a^2 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {986078 a B d^2 (3 b c-2 a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}-\frac {986078 B d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {493039 d^2 (3 b c-2 a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {493039 d^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^2}-\frac {493039 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2 (a+b x)}+\frac {1479117 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2}+\frac {493039 B c^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}-\frac {986078 B c d (3 b c-2 a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}+\frac {986078 B d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^4 g^2}+\frac {\left (493039 B^2 d^2 (b c-a d)^2\right ) \int \frac {1}{c+d x} \, dx}{b^4 g^2}-\frac {\left (2958234 A B d (b c-a d)^3\right ) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{b^4 g^2}-\frac {\left (2958234 B^2 d (b c-a d)^3\right ) \int \frac {\log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{b^4 g^2}+\frac {\left (986078 B^2 (b c-a d)^4\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^2}-\frac {\left (493039 B^2 c^2 d\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^2 e g^2}+\frac {\left (493039 a^2 B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^4 e g^2}+\frac {\left (986078 B^2 c d (3 b c-2 a d)\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^3 e g^2}-\frac {\left (986078 a B^2 d^2 (3 b c-2 a d)\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^4 e g^2}+\frac {\left (986078 B^2 d (b c-a d)^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^4 e g^2}-\frac {\left (986078 B^2 d (b c-a d)^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^4 e g^2}\\ &=-\frac {493039 A B d^2 (b c-a d) x}{b^3 g^2}-\frac {493039 B^2 d^2 (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac {1479117 B^2 d (b c-a d)^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac {986078 B (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2 (a+b x)}-\frac {493039 a^2 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {986078 a B d^2 (3 b c-2 a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}-\frac {986078 B d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {493039 d^2 (3 b c-2 a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {493039 d^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^2}-\frac {493039 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2 (a+b x)}+\frac {1479117 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2}+\frac {493039 B^2 d (b c-a d)^2 \log (c+d x)}{b^4 g^2}+\frac {493039 B c^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}-\frac {986078 B c d (3 b c-2 a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}+\frac {986078 B d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^4 g^2}+\frac {\left (1479117 B^2 d (b c-a d)^2\right ) \int \frac {\log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^3 g^2}-\frac {\left (2958234 A B d (b c-a d)^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{b^5 g^2}+\frac {\left (986078 B^2 (b c-a d)^4\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^4 g^2}-\frac {\left (493039 B^2 c^2 d\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{b^2 e g^2}+\frac {\left (493039 a^2 B^2 d^3\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^4 e g^2}+\frac {\left (986078 B^2 c d (3 b c-2 a d)\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{b^3 e g^2}-\frac {\left (986078 a B^2 d^2 (3 b c-2 a d)\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^4 e g^2}+\frac {\left (986078 B^2 d (b c-a d)^2\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^4 e g^2}-\frac {\left (986078 B^2 d (b c-a d)^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{b^4 e g^2}\\ &=-\frac {493039 A B d^2 (b c-a d) x}{b^3 g^2}-\frac {986078 B^2 (b c-a d)^3}{b^4 g^2 (a+b x)}-\frac {986078 B^2 d (b c-a d)^2 \log (a+b x)}{b^4 g^2}-\frac {493039 B^2 d^2 (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac {1479117 B^2 d (b c-a d)^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac {1479117 B^2 d (b c-a d)^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac {986078 B (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2 (a+b x)}-\frac {493039 a^2 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {986078 a B d^2 (3 b c-2 a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}-\frac {986078 B d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {493039 d^2 (3 b c-2 a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {493039 d^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^2}-\frac {493039 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2 (a+b x)}+\frac {1479117 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2}+\frac {1479117 B^2 d (b c-a d)^2 \log (c+d x)}{b^4 g^2}+\frac {493039 B c^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}-\frac {986078 B c d (3 b c-2 a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}+\frac {986078 B d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^4 g^2}-\frac {\left (493039 B^2 c^2 d\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b g^2}+\frac {\left (493039 B^2 c^2 d^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 g^2}+\frac {\left (493039 a^2 B^2 d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 g^2}-\frac {\left (493039 a^2 B^2 d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 g^2}+\frac {\left (986078 B^2 c d (3 b c-2 a d)\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 g^2}-\frac {\left (986078 a B^2 d^2 (3 b c-2 a d)\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 g^2}-\frac {\left (986078 B^2 c d^2 (3 b c-2 a d)\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 g^2}+\frac {\left (986078 a B^2 d^3 (3 b c-2 a d)\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 g^2}-\frac {\left (2958234 A B d (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^2}+\frac {\left (986078 B^2 d (b c-a d)^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 g^2}-\frac {\left (986078 B^2 d (b c-a d)^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^3 g^2}+\frac {\left (2958234 A B d^2 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{b^5 g^2}-\frac {\left (986078 B^2 d^2 (b c-a d)^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 g^2}+\frac {\left (986078 B^2 d^2 (b c-a d)^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^4 g^2}+\frac {\left (2958234 B^2 d (b c-a d)^3\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{b^4 g^2}\\ &=-\frac {493039 A B d^2 (b c-a d) x}{b^3 g^2}-\frac {986078 B^2 (b c-a d)^3}{b^4 g^2 (a+b x)}-\frac {986078 B^2 d (b c-a d)^2 \log (a+b x)}{b^4 g^2}-\frac {1479117 A B d (b c-a d)^2 \log ^2(a+b x)}{b^4 g^2}-\frac {493039 B^2 d^2 (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac {1479117 B^2 d (b c-a d)^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac {1479117 B^2 d (b c-a d)^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac {986078 B (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2 (a+b x)}-\frac {493039 a^2 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {986078 a B d^2 (3 b c-2 a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}-\frac {986078 B d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {493039 d^2 (3 b c-2 a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {493039 d^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^2}-\frac {493039 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2 (a+b x)}+\frac {1479117 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2}+\frac {1479117 B^2 d (b c-a d)^2 \log (c+d x)}{b^4 g^2}-\frac {493039 B^2 c^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}+\frac {986078 B^2 c d (3 b c-2 a d) \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac {986078 B^2 d (b c-a d)^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^4 g^2}+\frac {493039 B c^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}-\frac {986078 B c d (3 b c-2 a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}+\frac {986078 B d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^4 g^2}-\frac {493039 a^2 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac {986078 a B^2 d^2 (3 b c-2 a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac {2958234 A B d (b c-a d)^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac {986078 B^2 d (b c-a d)^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac {2958234 B^2 d (b c-a d)^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^4 g^2}+\frac {\left (493039 B^2 c^2 d\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 g^2}+\frac {\left (493039 B^2 c^2 d^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 g^2}+\frac {\left (493039 a^2 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^2}+\frac {\left (493039 a^2 B^2 d^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^2}-\frac {\left (986078 B^2 c d (3 b c-2 a d)\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 g^2}-\frac {\left (986078 a B^2 d^2 (3 b c-2 a d)\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^2}-\frac {\left (986078 a B^2 d^2 (3 b c-2 a d)\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^2}-\frac {\left (986078 B^2 c d^2 (3 b c-2 a d)\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 g^2}-\frac {\left (2958234 A B d (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 g^2}+\frac {\left (986078 B^2 d (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^2}+\frac {\left (986078 B^2 d (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^4 g^2}+\frac {\left (986078 B^2 d (b c-a d)^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^2}+\frac {\left (986078 B^2 d^2 (b c-a d)^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^4 g^2}-\frac {\left (2958234 B^2 d (b c-a d)^3\right ) \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^4 g^2}\\ &=-\frac {493039 A B d^2 (b c-a d) x}{b^3 g^2}-\frac {986078 B^2 (b c-a d)^3}{b^4 g^2 (a+b x)}-\frac {986078 B^2 d (b c-a d)^2 \log (a+b x)}{b^4 g^2}+\frac {493039 a^2 B^2 d^3 \log ^2(a+b x)}{2 b^4 g^2}-\frac {493039 a B^2 d^2 (3 b c-2 a d) \log ^2(a+b x)}{b^4 g^2}-\frac {1479117 A B d (b c-a d)^2 \log ^2(a+b x)}{b^4 g^2}+\frac {493039 B^2 d (b c-a d)^2 \log ^2(a+b x)}{b^4 g^2}-\frac {493039 B^2 d^2 (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac {1479117 B^2 d (b c-a d)^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac {1479117 B^2 d (b c-a d)^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac {986078 B (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2 (a+b x)}-\frac {493039 a^2 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {986078 a B d^2 (3 b c-2 a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}-\frac {986078 B d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {493039 d^2 (3 b c-2 a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {493039 d^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^2}-\frac {493039 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2 (a+b x)}+\frac {1479117 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2}+\frac {1479117 B^2 d (b c-a d)^2 \log (c+d x)}{b^4 g^2}-\frac {493039 B^2 c^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}+\frac {986078 B^2 c d (3 b c-2 a d) \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac {986078 B^2 d (b c-a d)^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^4 g^2}+\frac {493039 B c^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}-\frac {986078 B c d (3 b c-2 a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}+\frac {986078 B d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^4 g^2}+\frac {493039 B^2 c^2 d \log ^2(c+d x)}{2 b^2 g^2}-\frac {493039 B^2 c d (3 b c-2 a d) \log ^2(c+d x)}{b^3 g^2}+\frac {493039 B^2 d (b c-a d)^2 \log ^2(c+d x)}{b^4 g^2}-\frac {493039 a^2 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac {986078 a B^2 d^2 (3 b c-2 a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac {2958234 A B d (b c-a d)^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac {986078 B^2 d (b c-a d)^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac {2958234 A B d (b c-a d)^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^4 g^2}+\frac {2958234 B^2 d (b c-a d)^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^4 g^2}+\frac {2958234 B^2 d (b c-a d)^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^4 g^2}+\frac {\left (493039 B^2 c^2 d\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2 g^2}+\frac {\left (493039 a^2 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 g^2}-\frac {\left (986078 B^2 c d (3 b c-2 a d)\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^3 g^2}-\frac {\left (986078 a B^2 d^2 (3 b c-2 a d)\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 g^2}+\frac {\left (986078 B^2 d (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 g^2}+\frac {\left (986078 B^2 d (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^4 g^2}\\ &=-\frac {493039 A B d^2 (b c-a d) x}{b^3 g^2}-\frac {986078 B^2 (b c-a d)^3}{b^4 g^2 (a+b x)}-\frac {986078 B^2 d (b c-a d)^2 \log (a+b x)}{b^4 g^2}+\frac {493039 a^2 B^2 d^3 \log ^2(a+b x)}{2 b^4 g^2}-\frac {493039 a B^2 d^2 (3 b c-2 a d) \log ^2(a+b x)}{b^4 g^2}-\frac {1479117 A B d (b c-a d)^2 \log ^2(a+b x)}{b^4 g^2}+\frac {493039 B^2 d (b c-a d)^2 \log ^2(a+b x)}{b^4 g^2}-\frac {493039 B^2 d^2 (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac {1479117 B^2 d (b c-a d)^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac {1479117 B^2 d (b c-a d)^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^2}-\frac {986078 B (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2 (a+b x)}-\frac {493039 a^2 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {986078 a B d^2 (3 b c-2 a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}-\frac {986078 B d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}+\frac {493039 d^2 (3 b c-2 a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^2}+\frac {493039 d^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^2}-\frac {493039 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2 (a+b x)}+\frac {1479117 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^2}+\frac {1479117 B^2 d (b c-a d)^2 \log (c+d x)}{b^4 g^2}-\frac {493039 B^2 c^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g^2}+\frac {986078 B^2 c d (3 b c-2 a d) \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 g^2}-\frac {986078 B^2 d (b c-a d)^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^4 g^2}+\frac {493039 B c^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 g^2}-\frac {986078 B c d (3 b c-2 a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 g^2}+\frac {986078 B d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^4 g^2}+\frac {493039 B^2 c^2 d \log ^2(c+d x)}{2 b^2 g^2}-\frac {493039 B^2 c d (3 b c-2 a d) \log ^2(c+d x)}{b^3 g^2}+\frac {493039 B^2 d (b c-a d)^2 \log ^2(c+d x)}{b^4 g^2}-\frac {493039 a^2 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac {986078 a B^2 d^2 (3 b c-2 a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac {2958234 A B d (b c-a d)^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac {986078 B^2 d (b c-a d)^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac {493039 a^2 B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^4 g^2}+\frac {986078 a B^2 d^2 (3 b c-2 a d) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^4 g^2}+\frac {2958234 A B d (b c-a d)^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^4 g^2}-\frac {986078 B^2 d (b c-a d)^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^4 g^2}-\frac {493039 B^2 c^2 d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac {986078 B^2 c d (3 b c-2 a d) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac {986078 B^2 d (b c-a d)^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac {2958234 B^2 d (b c-a d)^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^4 g^2}+\frac {2958234 B^2 d (b c-a d)^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^4 g^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(5108\) vs. \(2(692)=1384\).
time = 13.07, size = 5108, normalized size = 7.38 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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Maple [F]
time = 0.34, size = 0, normalized size = 0.00 \[\int \frac {\left (d i x +c i \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{\left (b g x +a g \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*I*A^2*(a^2/(b^4*g^2*x + a*b^3*g^2) - x/(b^2*g^2) + 2*a*log(b*x + a)/(b^3*g^2))*c*d^2 - 1/2*I*(2*a^3/(b^5*g^2
*x + a*b^4*g^2) + 6*a^2*log(b*x + a)/(b^4*g^2) + (b*x^2 - 4*a*x)/(b^3*g^2))*A^2*d^3 - 3*I*A^2*c^2*d*(a/(b^3*g^
2*x + a*b^2*g^2) + log(b*x + a)/(b^2*g^2)) + 2*I*A*B*c^3*(log(b*x*e/(d*x + c) + a*e/(d*x + c))/(b^2*g^2*x + a*
b*g^2) + 1/(b^2*g^2*x + a*b*g^2) + d*log(b*x + a)/((b^2*c - a*b*d)*g^2) - d*log(d*x + c)/((b^2*c - a*b*d)*g^2)
) + I*A^2*c^3/(b^2*g^2*x + a*b*g^2) - 1/2*(I*B^2*b^3*d^3*x^3 - 3*(-2*I*b^3*c*d^2 + I*a*b^2*d^3)*B^2*x^2 - 2*(-
3*I*a*b^2*c*d^2 + 2*I*a^2*b*d^3)*B^2*x - 2*(I*b^3*c^3 - 3*I*a*b^2*c^2*d + 3*I*a^2*b*c*d^2 - I*a^3*d^3)*B^2 - 6
*((-I*b^3*c^2*d + 2*I*a*b^2*c*d^2 - I*a^2*b*d^3)*B^2*x + (-I*a*b^2*c^2*d + 2*I*a^2*b*c*d^2 - I*a^3*d^3)*B^2)*l
og(b*x + a))*log(d*x + c)^2/(b^5*g^2*x + a*b^4*g^2) + integrate((-I*B^2*b^4*c^4 + (-2*I*A*B*b^4*d^4 - I*B^2*b^
4*d^4)*x^4 - 4*(2*I*A*B*b^4*c*d^3 + I*B^2*b^4*c*d^3)*x^3 - 6*(2*I*A*B*b^4*c^2*d^2 + I*B^2*b^4*c^2*d^2)*x^2 + (
-I*B^2*b^4*d^4*x^4 - 4*I*B^2*b^4*c*d^3*x^3 - 6*I*B^2*b^4*c^2*d^2*x^2 - 4*I*B^2*b^4*c^3*d*x - I*B^2*b^4*c^4)*lo
g(b*x + a)^2 - 2*(3*I*A*B*b^4*c^3*d + 2*I*B^2*b^4*c^3*d)*x - 2*(I*B^2*b^4*c^4 + (I*A*B*b^4*d^4 + I*B^2*b^4*d^4
)*x^4 + 4*(I*A*B*b^4*c*d^3 + I*B^2*b^4*c*d^3)*x^3 + 6*(I*A*B*b^4*c^2*d^2 + I*B^2*b^4*c^2*d^2)*x^2 + (3*I*A*B*b
^4*c^3*d + 4*I*B^2*b^4*c^3*d)*x)*log(b*x + a) + ((2*I*A*B*b^4*d^4 + 3*I*B^2*b^4*d^4)*x^4 - 2*(-4*I*A*B*b^4*c*d
^3 + (-7*I*b^4*c*d^3 + I*a*b^3*d^4)*B^2)*x^3 - 2*(-I*b^4*c^4 + I*a*b^3*c^3*d - 3*I*a^2*b^2*c^2*d^2 + 3*I*a^3*b
*c*d^3 - I*a^4*d^4)*B^2 + (12*I*A*B*b^4*c^2*d^2 + (12*I*b^4*c^2*d^2 + 12*I*a*b^3*c*d^3 - 7*I*a^2*b^2*d^4)*B^2)
*x^2 - 2*(-3*I*A*B*b^4*c^3*d + (-3*I*b^4*c^3*d - 3*I*a*b^3*c^2*d^2 + I*a^3*b*d^4)*B^2)*x - 2*(-I*B^2*b^4*d^4*x
^4 - 4*I*B^2*b^4*c*d^3*x^3 + 3*(-3*I*b^4*c^2*d^2 + 2*I*a*b^3*c*d^3 - I*a^2*b^2*d^4)*B^2*x^2 + 2*(-2*I*b^4*c^3*
d - 3*I*a*b^3*c^2*d^2 + 6*I*a^2*b^2*c*d^3 - 3*I*a^3*b*d^4)*B^2*x + (-I*b^4*c^4 - 3*I*a^2*b^2*c^2*d^2 + 6*I*a^3
*b*c*d^3 - 3*I*a^4*d^4)*B^2)*log(b*x + a))*log(d*x + c))/(b^6*d*g^2*x^3 + a^2*b^4*c*g^2 + (b^6*c*g^2 + 2*a*b^5
*d*g^2)*x^2 + (2*a*b^5*c*g^2 + a^2*b^4*d*g^2)*x), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-I*A^2*d^3*x^3 - 3*I*A^2*c*d^2*x^2 - 3*I*A^2*c^2*d*x - I*A^2*c^3 + (-I*B^2*d^3*x^3 - 3*I*B^2*c*d^2*x
^2 - 3*I*B^2*c^2*d*x - I*B^2*c^3)*log((b*x + a)*e/(d*x + c))^2 - 2*(I*A*B*d^3*x^3 + 3*I*A*B*c*d^2*x^2 + 3*I*A*
B*c^2*d*x + I*A*B*c^3)*log((b*x + a)*e/(d*x + c)))/(b^2*g^2*x^2 + 2*a*b*g^2*x + a^2*g^2), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,i+d\,i\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{{\left (a\,g+b\,g\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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