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

Optimal. Leaf size=761 \[ \frac {B g^2 i^2 (b c-a d)^5 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+3 B\right )}{30 b^3 d^3}+\frac {B g^2 i^2 (a+b x) (b c-a d)^4 \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+B\right )}{30 b^3 d^2}-\frac {B g^2 i^2 (a+b x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{30 b^3 d}+\frac {g^2 i^2 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{30 b^3}-\frac {B g^2 i^2 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{15 b^3}+\frac {g^2 i^2 (a+b x)^3 (c+d x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{10 b^2}-\frac {B g^2 i^2 (c+d x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 b d^3}+\frac {4 B g^2 i^2 (c+d x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{15 d^3}-\frac {b B g^2 i^2 (c+d x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{10 d^3}+\frac {g^2 i^2 (a+b x)^3 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 b}+\frac {B^2 g^2 i^2 (b c-a d)^5 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{15 b^3 d^3}+\frac {B^2 g^2 i^2 (b c-a d)^5 \log \left (\frac {a+b x}{c+d x}\right )}{30 b^3 d^3}+\frac {B^2 g^2 i^2 (b c-a d)^5 \log (c+d x)}{10 b^3 d^3}-\frac {B^2 g^2 i^2 x (b c-a d)^4}{10 b^2 d^2}-\frac {B^2 g^2 i^2 (c+d x)^2 (b c-a d)^3}{20 b d^3}+\frac {B^2 g^2 i^2 (c+d x)^3 (b c-a d)^2}{30 d^3} \]

[Out]

-1/10*B^2*(-a*d+b*c)^4*g^2*i^2*x/b^2/d^2-1/20*B^2*(-a*d+b*c)^3*g^2*i^2*(d*x+c)^2/b/d^3+1/30*B^2*(-a*d+b*c)^2*g
^2*i^2*(d*x+c)^3/d^3+1/30*B^2*(-a*d+b*c)^5*g^2*i^2*ln((b*x+a)/(d*x+c))/b^3/d^3-1/30*B*(-a*d+b*c)^3*g^2*i^2*(b*
x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^3/d-1/15*B*(-a*d+b*c)^2*g^2*i^2*(b*x+a)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^3
-1/5*B*(-a*d+b*c)^3*g^2*i^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b/d^3+4/15*B*(-a*d+b*c)^2*g^2*i^2*(d*x+c)^3*
(A+B*ln(e*(b*x+a)/(d*x+c)))/d^3-1/10*b*B*(-a*d+b*c)*g^2*i^2*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^3+1/30*(-a
*d+b*c)^2*g^2*i^2*(b*x+a)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b^3+1/10*(-a*d+b*c)*g^2*i^2*(b*x+a)^3*(d*x+c)*(A+B*l
n(e*(b*x+a)/(d*x+c)))^2/b^2+1/5*g^2*i^2*(b*x+a)^3*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b+1/30*B*(-a*d+b*c)^
4*g^2*i^2*(b*x+a)*(2*A+B+2*B*ln(e*(b*x+a)/(d*x+c)))/b^3/d^2+1/30*B*(-a*d+b*c)^5*g^2*i^2*ln((-a*d+b*c)/b/(d*x+c
))*(2*A+3*B+2*B*ln(e*(b*x+a)/(d*x+c)))/b^3/d^3+1/10*B^2*(-a*d+b*c)^5*g^2*i^2*ln(d*x+c)/b^3/d^3+1/15*B^2*(-a*d+
b*c)^5*g^2*i^2*polylog(2,d*(b*x+a)/b/(d*x+c))/b^3/d^3

________________________________________________________________________________________

Rubi [A]  time = 2.40, antiderivative size = 666, normalized size of antiderivative = 0.88, number of steps used = 74, number of rules used = 13, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2528, 2525, 12, 2486, 31, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac {B^2 g^2 i^2 (b c-a d)^5 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{15 b^3 d^3}-\frac {B g^2 i^2 (b c-a d)^5 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{15 b^3 d^3}+\frac {d^2 g^2 i^2 (a+b x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 b^3}+\frac {A B g^2 i^2 x (b c-a d)^4}{15 b^2 d^2}-\frac {B g^2 i^2 (a+b x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{30 b^3 d}+\frac {g^2 i^2 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 b^3}-\frac {B g^2 i^2 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 b^3}+\frac {d g^2 i^2 (a+b x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 b^3}-\frac {B d g^2 i^2 (a+b x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{10 b^3}+\frac {B^2 g^2 i^2 (a+b x) (b c-a d)^4 \log \left (\frac {e (a+b x)}{c+d x}\right )}{15 b^3 d^2}-\frac {B^2 g^2 i^2 x (b c-a d)^4}{15 b^2 d^2}-\frac {B^2 g^2 i^2 (b c-a d)^5 \log ^2(c+d x)}{30 b^3 d^3}+\frac {B^2 g^2 i^2 (b c-a d)^5 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{15 b^3 d^3}+\frac {B^2 g^2 i^2 (a+b x)^2 (b c-a d)^3}{20 b^3 d}+\frac {B^2 g^2 i^2 (a+b x)^3 (b c-a d)^2}{30 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*B*(b*c - a*d)^4*g^2*i^2*x)/(15*b^2*d^2) - (B^2*(b*c - a*d)^4*g^2*i^2*x)/(15*b^2*d^2) + (B^2*(b*c - a*d)^3*g
^2*i^2*(a + b*x)^2)/(20*b^3*d) + (B^2*(b*c - a*d)^2*g^2*i^2*(a + b*x)^3)/(30*b^3) + (B^2*(b*c - a*d)^4*g^2*i^2
*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(15*b^3*d^2) - (B*(b*c - a*d)^3*g^2*i^2*(a + b*x)^2*(A + B*Log[(e*(a
+ b*x))/(c + d*x)]))/(30*b^3*d) - (B*(b*c - a*d)^2*g^2*i^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(
5*b^3) - (B*d*(b*c - a*d)*g^2*i^2*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*b^3) + ((b*c - a*d)^2*
g^2*i^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*b^3) + (d*(b*c - a*d)*g^2*i^2*(a + b*x)^4*(A +
B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*b^3) + (d^2*g^2*i^2*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/
(5*b^3) + (B^2*(b*c - a*d)^5*g^2*i^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(15*b^3*d^3) - (B*(b*c -
a*d)^5*g^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(15*b^3*d^3) - (B^2*(b*c - a*d)^5*g^2*i^2*Lo
g[c + d*x]^2)/(30*b^3*d^3) + (B^2*(b*c - a*d)^5*g^2*i^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(15*b^3*d^3)

Rule 12

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

Rule 31

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

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2390

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 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 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int (65 c+65 d x)^2 (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx &=\int \left (\frac {(-b c+a d)^2 g^2 (65 c+65 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2}-\frac {2 b (b c-a d) g^2 (65 c+65 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{65 d^2}+\frac {b^2 g^2 (65 c+65 d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4225 d^2}\right ) \, dx\\ &=\frac {\left (b^2 g^2\right ) \int (65 c+65 d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{4225 d^2}-\frac {\left (2 b (b c-a d) g^2\right ) \int (65 c+65 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{65 d^2}+\frac {\left ((b c-a d)^2 g^2\right ) \int (65 c+65 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{d^2}\\ &=\frac {4225 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 d^3}-\frac {4225 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {845 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}-\frac {\left (2 b^2 B g^2\right ) \int \frac {1160290625 (b c-a d) (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{1373125 d^3}+\frac {\left (b B (b c-a d) g^2\right ) \int \frac {17850625 (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{4225 d^3}-\frac {\left (2 B (b c-a d)^2 g^2\right ) \int \frac {274625 (b c-a d) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{195 d^3}\\ &=\frac {4225 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 d^3}-\frac {4225 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {845 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}-\frac {\left (1690 b^2 B (b c-a d) g^2\right ) \int \frac {(c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{d^3}+\frac {\left (4225 b B (b c-a d)^2 g^2\right ) \int \frac {(c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{d^3}-\frac {\left (8450 B (b c-a d)^3 g^2\right ) \int \frac {(c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{3 d^3}\\ &=\frac {4225 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 d^3}-\frac {4225 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {845 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}-\frac {\left (1690 b^2 B (b c-a d) g^2\right ) \int \left (\frac {d (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4}+\frac {(b c-a d)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 (a+b x)}+\frac {d (b c-a d)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {d (b c-a d) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac {d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b}\right ) \, dx}{d^3}+\frac {\left (4225 b B (b c-a d)^2 g^2\right ) \int \left (\frac {d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac {(b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac {d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b}\right ) \, dx}{d^3}-\frac {\left (8450 B (b c-a d)^3 g^2\right ) \int \left (\frac {d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac {(b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (a+b x)}+\frac {d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b}\right ) \, dx}{3 d^3}\\ &=\frac {4225 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 d^3}-\frac {4225 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {845 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}-\frac {\left (1690 b B (b c-a d) g^2\right ) \int (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{d^2}-\frac {\left (1690 B (b c-a d)^2 g^2\right ) \int (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{d^2}+\frac {\left (4225 B (b c-a d)^2 g^2\right ) \int (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{d^2}-\frac {\left (1690 B (b c-a d)^3 g^2\right ) \int (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b d^2}-\frac {\left (8450 B (b c-a d)^3 g^2\right ) \int (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{3 b d^2}+\frac {\left (4225 B (b c-a d)^3 g^2\right ) \int (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b d^2}-\frac {\left (1690 B (b c-a d)^4 g^2\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d^2}-\frac {\left (8450 B (b c-a d)^4 g^2\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{3 b^2 d^2}+\frac {\left (4225 B (b c-a d)^4 g^2\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 d^2}-\frac {\left (1690 B (b c-a d)^5 g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 d^3}-\frac {\left (8450 B (b c-a d)^5 g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{3 b^2 d^3}+\frac {\left (4225 B (b c-a d)^5 g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 d^3}\\ &=-\frac {845 A B (b c-a d)^4 g^2 x}{3 b^2 d^2}-\frac {845 B (b c-a d)^3 g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d^3}+\frac {845 B (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {845 b B (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^3}-\frac {845 B (b c-a d)^5 g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 d^3}+\frac {4225 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 d^3}-\frac {4225 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {845 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {\left (845 b B^2 (b c-a d) g^2\right ) \int \frac {(b c-a d) (c+d x)^3}{a+b x} \, dx}{2 d^3}+\frac {\left (1690 B^2 (b c-a d)^2 g^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{3 d^3}-\frac {\left (4225 B^2 (b c-a d)^2 g^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{3 d^3}+\frac {\left (845 B^2 (b c-a d)^3 g^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{b d^3}+\frac {\left (4225 B^2 (b c-a d)^3 g^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{3 b d^3}-\frac {\left (4225 B^2 (b c-a d)^3 g^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{2 b d^3}-\frac {\left (1690 B^2 (b c-a d)^4 g^2\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{b^2 d^2}-\frac {\left (8450 B^2 (b c-a d)^4 g^2\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{3 b^2 d^2}+\frac {\left (4225 B^2 (b c-a d)^4 g^2\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{b^2 d^2}+\frac {\left (1690 B^2 (b c-a d)^5 g^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^3 d^3}+\frac {\left (8450 B^2 (b c-a d)^5 g^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}{3 b^3 d^3}-\frac {\left (4225 B^2 (b c-a d)^5 g^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^3 d^3}\\ &=-\frac {845 A B (b c-a d)^4 g^2 x}{3 b^2 d^2}-\frac {845 B^2 (b c-a d)^4 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 b^3 d^2}-\frac {845 B (b c-a d)^3 g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d^3}+\frac {845 B (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {845 b B (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^3}-\frac {845 B (b c-a d)^5 g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 d^3}+\frac {4225 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 d^3}-\frac {4225 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {845 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {\left (845 b B^2 (b c-a d)^2 g^2\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{2 d^3}+\frac {\left (1690 B^2 (b c-a d)^3 g^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{3 d^3}-\frac {\left (4225 B^2 (b c-a d)^3 g^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{3 d^3}+\frac {\left (845 B^2 (b c-a d)^4 g^2\right ) \int \frac {c+d x}{a+b x} \, dx}{b d^3}+\frac {\left (4225 B^2 (b c-a d)^4 g^2\right ) \int \frac {c+d x}{a+b x} \, dx}{3 b d^3}-\frac {\left (4225 B^2 (b c-a d)^4 g^2\right ) \int \frac {c+d x}{a+b x} \, dx}{2 b d^3}+\frac {\left (1690 B^2 (b c-a d)^5 g^2\right ) \int \frac {1}{c+d x} \, dx}{b^3 d^2}+\frac {\left (8450 B^2 (b c-a d)^5 g^2\right ) \int \frac {1}{c+d x} \, dx}{3 b^3 d^2}-\frac {\left (4225 B^2 (b c-a d)^5 g^2\right ) \int \frac {1}{c+d x} \, dx}{b^3 d^2}+\frac {\left (1690 B^2 (b c-a d)^5 g^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^3 d^3 e}+\frac {\left (8450 B^2 (b c-a d)^5 g^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}{3 b^3 d^3 e}-\frac {\left (4225 B^2 (b c-a d)^5 g^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^3 d^3 e}\\ &=-\frac {845 A B (b c-a d)^4 g^2 x}{3 b^2 d^2}-\frac {845 B^2 (b c-a d)^4 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 b^3 d^2}-\frac {845 B (b c-a d)^3 g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d^3}+\frac {845 B (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {845 b B (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^3}-\frac {845 B (b c-a d)^5 g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 d^3}+\frac {4225 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 d^3}-\frac {4225 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {845 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {845 B^2 (b c-a d)^5 g^2 \log (c+d x)}{3 b^3 d^3}+\frac {\left (845 b B^2 (b c-a d)^2 g^2\right ) \int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx}{2 d^3}+\frac {\left (1690 B^2 (b c-a d)^3 g^2\right ) \int \left (\frac {d (b c-a d)}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)}+\frac {d (c+d x)}{b}\right ) \, dx}{3 d^3}-\frac {\left (4225 B^2 (b c-a d)^3 g^2\right ) \int \left (\frac {d (b c-a d)}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)}+\frac {d (c+d x)}{b}\right ) \, dx}{3 d^3}+\frac {\left (845 B^2 (b c-a d)^4 g^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{b d^3}+\frac {\left (4225 B^2 (b c-a d)^4 g^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{3 b d^3}-\frac {\left (4225 B^2 (b c-a d)^4 g^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{2 b d^3}+\frac {\left (1690 B^2 (b c-a d)^5 g^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^3 d^3 e}+\frac {\left (8450 B^2 (b c-a d)^5 g^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}{3 b^3 d^3 e}-\frac {\left (4225 B^2 (b c-a d)^5 g^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^3 d^3 e}\\ &=-\frac {845 A B (b c-a d)^4 g^2 x}{3 b^2 d^2}-\frac {845 B^2 (b c-a d)^4 g^2 x}{3 b^2 d^2}-\frac {845 B^2 (b c-a d)^3 g^2 (c+d x)^2}{4 b d^3}+\frac {845 B^2 (b c-a d)^2 g^2 (c+d x)^3}{6 d^3}-\frac {845 B^2 (b c-a d)^5 g^2 \log (a+b x)}{3 b^3 d^3}-\frac {845 B^2 (b c-a d)^4 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 b^3 d^2}-\frac {845 B (b c-a d)^3 g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d^3}+\frac {845 B (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {845 b B (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^3}-\frac {845 B (b c-a d)^5 g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 d^3}+\frac {4225 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 d^3}-\frac {4225 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {845 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {845 B^2 (b c-a d)^5 g^2 \log (c+d x)}{3 b^3 d^3}+\frac {\left (1690 B^2 (b c-a d)^5 g^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 d^3}+\frac {\left (8450 B^2 (b c-a d)^5 g^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 b^2 d^3}-\frac {\left (4225 B^2 (b c-a d)^5 g^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 d^3}-\frac {\left (1690 B^2 (b c-a d)^5 g^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 d^2}-\frac {\left (8450 B^2 (b c-a d)^5 g^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 b^3 d^2}+\frac {\left (4225 B^2 (b c-a d)^5 g^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 d^2}\\ &=-\frac {845 A B (b c-a d)^4 g^2 x}{3 b^2 d^2}-\frac {845 B^2 (b c-a d)^4 g^2 x}{3 b^2 d^2}-\frac {845 B^2 (b c-a d)^3 g^2 (c+d x)^2}{4 b d^3}+\frac {845 B^2 (b c-a d)^2 g^2 (c+d x)^3}{6 d^3}-\frac {845 B^2 (b c-a d)^5 g^2 \log (a+b x)}{3 b^3 d^3}-\frac {845 B^2 (b c-a d)^4 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 b^3 d^2}-\frac {845 B (b c-a d)^3 g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d^3}+\frac {845 B (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {845 b B (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^3}-\frac {845 B (b c-a d)^5 g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 d^3}+\frac {4225 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 d^3}-\frac {4225 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {845 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {845 B^2 (b c-a d)^5 g^2 \log (c+d x)}{3 b^3 d^3}-\frac {845 B^2 (b c-a d)^5 g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 d^3}+\frac {\left (1690 B^2 (b c-a d)^5 g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 d^3}+\frac {\left (8450 B^2 (b c-a d)^5 g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b^3 d^3}-\frac {\left (4225 B^2 (b c-a d)^5 g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 d^3}+\frac {\left (1690 B^2 (b c-a d)^5 g^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 d^3}+\frac {\left (8450 B^2 (b c-a d)^5 g^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 b^2 d^3}-\frac {\left (4225 B^2 (b c-a d)^5 g^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 d^3}\\ &=-\frac {845 A B (b c-a d)^4 g^2 x}{3 b^2 d^2}-\frac {845 B^2 (b c-a d)^4 g^2 x}{3 b^2 d^2}-\frac {845 B^2 (b c-a d)^3 g^2 (c+d x)^2}{4 b d^3}+\frac {845 B^2 (b c-a d)^2 g^2 (c+d x)^3}{6 d^3}-\frac {845 B^2 (b c-a d)^5 g^2 \log (a+b x)}{3 b^3 d^3}+\frac {845 B^2 (b c-a d)^5 g^2 \log ^2(a+b x)}{6 b^3 d^3}-\frac {845 B^2 (b c-a d)^4 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 b^3 d^2}-\frac {845 B (b c-a d)^3 g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d^3}+\frac {845 B (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {845 b B (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^3}-\frac {845 B (b c-a d)^5 g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 d^3}+\frac {4225 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 d^3}-\frac {4225 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {845 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {845 B^2 (b c-a d)^5 g^2 \log (c+d x)}{3 b^3 d^3}-\frac {845 B^2 (b c-a d)^5 g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 d^3}+\frac {\left (1690 B^2 (b c-a d)^5 g^2\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^3 d^3}+\frac {\left (8450 B^2 (b c-a d)^5 g^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3 b^3 d^3}-\frac {\left (4225 B^2 (b c-a d)^5 g^2\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^3 d^3}\\ &=-\frac {845 A B (b c-a d)^4 g^2 x}{3 b^2 d^2}-\frac {845 B^2 (b c-a d)^4 g^2 x}{3 b^2 d^2}-\frac {845 B^2 (b c-a d)^3 g^2 (c+d x)^2}{4 b d^3}+\frac {845 B^2 (b c-a d)^2 g^2 (c+d x)^3}{6 d^3}-\frac {845 B^2 (b c-a d)^5 g^2 \log (a+b x)}{3 b^3 d^3}+\frac {845 B^2 (b c-a d)^5 g^2 \log ^2(a+b x)}{6 b^3 d^3}-\frac {845 B^2 (b c-a d)^4 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 b^3 d^2}-\frac {845 B (b c-a d)^3 g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d^3}+\frac {845 B (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3}-\frac {845 b B (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^3}-\frac {845 B (b c-a d)^5 g^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 d^3}+\frac {4225 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 d^3}-\frac {4225 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3}+\frac {845 b^2 g^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3}+\frac {845 B^2 (b c-a d)^5 g^2 \log (c+d x)}{3 b^3 d^3}-\frac {845 B^2 (b c-a d)^5 g^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 d^3}-\frac {845 B^2 (b c-a d)^5 g^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b^3 d^3}\\ \end {align*}

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Mathematica [A]  time = 0.95, size = 1194, normalized size = 1.57 \[ \frac {g^2 i^2 \left (12 d^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (a+b x)^5+30 d^4 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (a+b x)^4+20 d^3 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (a+b x)^3+20 B (b c-a d)^3 \left (-2 B \log (c+d x) (b c-a d)^2-2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x) (b c-a d)^2+B \left (\left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right ) (b c-a d)^2+2 A b d x (b c-a d)+2 B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right ) (b c-a d)+B (b d x+(a d-b c) \log (c+d x)) (b c-a d)-d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right )-10 B (b c-a d)^2 \left (-6 B \log (c+d x) (b c-a d)^3-6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x) (b c-a d)^3+3 B \left (\left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right ) (b c-a d)^3+6 A b d x (b c-a d)^2+6 B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right ) (b c-a d)^2+3 B (b d x+(a d-b c) \log (c+d x)) (b c-a d)^2+B \left (-2 \log (c+d x) (b c-a d)^2+2 b d x (b c-a d)-d^2 (a+b x)^2\right ) (b c-a d)+2 d^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+3 d^2 (a d-b c) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right )+B (b c-a d) \left (-24 B \log (c+d x) (b c-a d)^4-24 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x) (b c-a d)^4+12 B \left (\left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right ) (b c-a d)^4+24 A b d x (b c-a d)^3+24 B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right ) (b c-a d)^3+12 B (b d x+(a d-b c) \log (c+d x)) (b c-a d)^3-12 d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^2+4 B \left (-2 \log (c+d x) (b c-a d)^2+2 b d x (b c-a d)-d^2 (a+b x)^2\right ) (b c-a d)^2+8 d^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)+B \left (-6 \log (c+d x) (b c-a d)^3+6 b d x (b c-a d)^2+2 d^3 (a+b x)^3+3 d^2 (a d-b c) (a+b x)^2\right ) (b c-a d)-6 d^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right )\right )}{60 b^3 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^2*i^2*(20*d^3*(b*c - a*d)^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 30*d^4*(b*c - a*d)*(a + b*
x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 12*d^5*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 20*B
*(b*c - a*d)^3*(2*A*b*d*(b*c - a*d)*x + 2*B*d*(b*c - a*d)*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] - d^2*(a + b*
x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 2*B*(b*c - a*d)^2*Log[c + d*x] - 2*(b*c - a*d)^2*(A + B*Log[(e*(a
+ b*x))/(c + d*x)])*Log[c + d*x] + B*(b*c - a*d)*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + B*(b*c - a*d)^2*((2*L
og[(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)])) - 10
*B*(b*c - a*d)^2*(6*A*b*d*(b*c - a*d)^2*x + 6*B*d*(b*c - a*d)^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + 3*d^2
*(-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2*d^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/
(c + d*x)]) - 6*B*(b*c - a*d)^3*Log[c + d*x] - 6*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*
x] + B*(b*c - a*d)*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]) + 3*B*(b*c - a*d)^2*
(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 3*B*(b*c - a*d)^3*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x]
)*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) + B*(b*c - a*d)*(24*A*b*d*(b*c - a*d)^3*x + 24*B*d*
(b*c - a*d)^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] - 12*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x
))/(c + d*x)]) + 8*d^3*(b*c - a*d)*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 6*d^4*(a + b*x)^4*(A + B
*Log[(e*(a + b*x))/(c + d*x)]) - 24*B*(b*c - a*d)^4*Log[c + d*x] - 24*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x))/(
c + d*x)])*Log[c + d*x] + 4*B*(b*c - a*d)^2*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d
*x]) + B*(b*c - a*d)*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c -
a*d)^3*Log[c + d*x]) + 12*B*(b*c - a*d)^3*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 12*B*(b*c - a*d)^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)]))))/(60*b^
3*d^3)

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fricas [F]  time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} b^{2} d^{2} g^{2} i^{2} x^{4} + A^{2} a^{2} c^{2} g^{2} i^{2} + 2 \, {\left (A^{2} b^{2} c d + A^{2} a b d^{2}\right )} g^{2} i^{2} x^{3} + {\left (A^{2} b^{2} c^{2} + 4 \, A^{2} a b c d + A^{2} a^{2} d^{2}\right )} g^{2} i^{2} x^{2} + 2 \, {\left (A^{2} a b c^{2} + A^{2} a^{2} c d\right )} g^{2} i^{2} x + {\left (B^{2} b^{2} d^{2} g^{2} i^{2} x^{4} + B^{2} a^{2} c^{2} g^{2} i^{2} + 2 \, {\left (B^{2} b^{2} c d + B^{2} a b d^{2}\right )} g^{2} i^{2} x^{3} + {\left (B^{2} b^{2} c^{2} + 4 \, B^{2} a b c d + B^{2} a^{2} d^{2}\right )} g^{2} i^{2} x^{2} + 2 \, {\left (B^{2} a b c^{2} + B^{2} a^{2} c d\right )} g^{2} i^{2} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left (A B b^{2} d^{2} g^{2} i^{2} x^{4} + A B a^{2} c^{2} g^{2} i^{2} + 2 \, {\left (A B b^{2} c d + A B a b d^{2}\right )} g^{2} i^{2} x^{3} + {\left (A B b^{2} c^{2} + 4 \, A B a b c d + A B a^{2} d^{2}\right )} g^{2} i^{2} x^{2} + 2 \, {\left (A B a b c^{2} + A B a^{2} c d\right )} g^{2} i^{2} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(A^2*b^2*d^2*g^2*i^2*x^4 + A^2*a^2*c^2*g^2*i^2 + 2*(A^2*b^2*c*d + A^2*a*b*d^2)*g^2*i^2*x^3 + (A^2*b^2*
c^2 + 4*A^2*a*b*c*d + A^2*a^2*d^2)*g^2*i^2*x^2 + 2*(A^2*a*b*c^2 + A^2*a^2*c*d)*g^2*i^2*x + (B^2*b^2*d^2*g^2*i^
2*x^4 + B^2*a^2*c^2*g^2*i^2 + 2*(B^2*b^2*c*d + B^2*a*b*d^2)*g^2*i^2*x^3 + (B^2*b^2*c^2 + 4*B^2*a*b*c*d + B^2*a
^2*d^2)*g^2*i^2*x^2 + 2*(B^2*a*b*c^2 + B^2*a^2*c*d)*g^2*i^2*x)*log((b*e*x + a*e)/(d*x + c))^2 + 2*(A*B*b^2*d^2
*g^2*i^2*x^4 + A*B*a^2*c^2*g^2*i^2 + 2*(A*B*b^2*c*d + A*B*a*b*d^2)*g^2*i^2*x^3 + (A*B*b^2*c^2 + 4*A*B*a*b*c*d
+ A*B*a^2*d^2)*g^2*i^2*x^2 + 2*(A*B*a*b*c^2 + A*B*a^2*c*d)*g^2*i^2*x)*log((b*e*x + a*e)/(d*x + c)), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 2.56, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right )^{2} \left (d i x +c i \right )^{2} \left (B \ln \left (\frac {\left (b x +a \right ) e}{d x +c}\right )+A \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 2.63, size = 3656, normalized size = 4.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*A^2*b^2*d^2*g^2*i^2*x^5 + 1/2*A^2*b^2*c*d*g^2*i^2*x^4 + 1/2*A^2*a*b*d^2*g^2*i^2*x^4 + 1/3*A^2*b^2*c^2*g^2*
i^2*x^3 + 4/3*A^2*a*b*c*d*g^2*i^2*x^3 + 1/3*A^2*a^2*d^2*g^2*i^2*x^3 + A^2*a*b*c^2*g^2*i^2*x^2 + A^2*a^2*c*d*g^
2*i^2*x^2 + 2*(x*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + a*log(b*x + a)/b - c*log(d*x + c)/d)*A*B*a^2*c^2*g^2*i
^2 + 2*(x^2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x
/(b*d))*A*B*a*b*c^2*g^2*i^2 + 1/3*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3
*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*A*B*b^2*c^2*g^2*i^2 + 2*(x^
2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*A*
B*a^2*c*d*g^2*i^2 + 4/3*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x +
c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*A*B*a*b*c*d*g^2*i^2 + 1/6*(6*x^4*log(
b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3
)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*A*B*b^2*c*d*g^2*i^2 + 1/3*(2*x^3*l
og(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x
^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*A*B*a^2*d^2*g^2*i^2 + 1/6*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c))
- 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3
)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*A*B*a*b*d^2*g^2*i^2 + 1/30*(12*x^5*log(b*e*x/(d*x + c) + a*e/(d*x
+ c)) + 12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 -
a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))*A*B*b^2*d^2*g^2*i^2 +
A^2*a^2*c^2*g^2*i^2*x - 1/30*(2*b^4*c^5*g^2*i^2*log(e) + 9*a^3*b*c^2*d^3*g^2*i^2 - 2*a^4*c*d^4*g^2*i^2 - 2*(5*
g^2*i^2*log(e) - g^2*i^2)*a*b^3*c^4*d + (20*g^2*i^2*log(e) - 9*g^2*i^2)*a^2*b^2*c^3*d^2)*B^2*log(d*x + c)/(b^2
*d^3) - 1/15*(b^5*c^5*g^2*i^2 - 5*a*b^4*c^4*d*g^2*i^2 + 10*a^2*b^3*c^3*d^2*g^2*i^2 - 10*a^3*b^2*c^2*d^3*g^2*i^
2 + 5*a^4*b*c*d^4*g^2*i^2 - a^5*d^5*g^2*i^2)*(log(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x
+ a*d)/(b*c - a*d)))*B^2/(b^3*d^3) + 1/60*(12*B^2*b^5*d^5*g^2*i^2*x^5*log(e)^2 + 6*((5*g^2*i^2*log(e)^2 - g^2*
i^2*log(e))*b^5*c*d^4 + (5*g^2*i^2*log(e)^2 + g^2*i^2*log(e))*a*b^4*d^5)*B^2*x^4 + 2*((10*g^2*i^2*log(e)^2 - 6
*g^2*i^2*log(e) + g^2*i^2)*b^5*c^2*d^3 + 2*(20*g^2*i^2*log(e)^2 - g^2*i^2)*a*b^4*c*d^4 + (10*g^2*i^2*log(e)^2
+ 6*g^2*i^2*log(e) + g^2*i^2)*a^2*b^3*d^5)*B^2*x^3 - ((2*g^2*i^2*log(e) - 3*g^2*i^2)*b^5*c^3*d^2 - 3*(20*g^2*i
^2*log(e)^2 - 10*g^2*i^2*log(e) - g^2*i^2)*a*b^4*c^2*d^3 - 3*(20*g^2*i^2*log(e)^2 + 10*g^2*i^2*log(e) - g^2*i^
2)*a^2*b^3*c*d^4 - (2*g^2*i^2*log(e) + 3*g^2*i^2)*a^3*b^2*d^5)*B^2*x^2 + 2*(2*(g^2*i^2*log(e) - g^2*i^2)*b^5*c
^4*d - (10*g^2*i^2*log(e) - 11*g^2*i^2)*a*b^4*c^3*d^2 + 6*(5*g^2*i^2*log(e)^2 - 3*g^2*i^2)*a^2*b^3*c^2*d^3 + (
10*g^2*i^2*log(e) + 11*g^2*i^2)*a^3*b^2*c*d^4 - 2*(g^2*i^2*log(e) + g^2*i^2)*a^4*b*d^5)*B^2*x + 2*(6*B^2*b^5*d
^5*g^2*i^2*x^5 + 30*B^2*a^2*b^3*c^2*d^3*g^2*i^2*x + 15*(b^5*c*d^4*g^2*i^2 + a*b^4*d^5*g^2*i^2)*B^2*x^4 + 10*(b
^5*c^2*d^3*g^2*i^2 + 4*a*b^4*c*d^4*g^2*i^2 + a^2*b^3*d^5*g^2*i^2)*B^2*x^3 + 30*(a*b^4*c^2*d^3*g^2*i^2 + a^2*b^
3*c*d^4*g^2*i^2)*B^2*x^2 + (10*a^3*b^2*c^2*d^3*g^2*i^2 - 5*a^4*b*c*d^4*g^2*i^2 + a^5*d^5*g^2*i^2)*B^2)*log(b*x
+ a)^2 + 2*(6*B^2*b^5*d^5*g^2*i^2*x^5 + 30*B^2*a^2*b^3*c^2*d^3*g^2*i^2*x + 15*(b^5*c*d^4*g^2*i^2 + a*b^4*d^5*
g^2*i^2)*B^2*x^4 + 10*(b^5*c^2*d^3*g^2*i^2 + 4*a*b^4*c*d^4*g^2*i^2 + a^2*b^3*d^5*g^2*i^2)*B^2*x^3 + 30*(a*b^4*
c^2*d^3*g^2*i^2 + a^2*b^3*c*d^4*g^2*i^2)*B^2*x^2 + (b^5*c^5*g^2*i^2 - 5*a*b^4*c^4*d*g^2*i^2 + 10*a^2*b^3*c^3*d
^2*g^2*i^2)*B^2)*log(d*x + c)^2 + 2*(12*B^2*b^5*d^5*g^2*i^2*x^5*log(e) + 3*((10*g^2*i^2*log(e) - g^2*i^2)*b^5*
c*d^4 + (10*g^2*i^2*log(e) + g^2*i^2)*a*b^4*d^5)*B^2*x^4 + 2*(40*a*b^4*c*d^4*g^2*i^2*log(e) + (10*g^2*i^2*log(
e) - 3*g^2*i^2)*b^5*c^2*d^3 + (10*g^2*i^2*log(e) + 3*g^2*i^2)*a^2*b^3*d^5)*B^2*x^3 - (b^5*c^3*d^2*g^2*i^2 - a^
3*b^2*d^5*g^2*i^2 - 15*(4*g^2*i^2*log(e) - g^2*i^2)*a*b^4*c^2*d^3 - 15*(4*g^2*i^2*log(e) + g^2*i^2)*a^2*b^3*c*
d^4)*B^2*x^2 + 2*(30*a^2*b^3*c^2*d^3*g^2*i^2*log(e) + b^5*c^4*d*g^2*i^2 - 5*a*b^4*c^3*d^2*g^2*i^2 + 5*a^3*b^2*
c*d^4*g^2*i^2 - a^4*b*d^5*g^2*i^2)*B^2*x + (2*a^5*d^5*g^2*i^2*log(e) + 2*a*b^4*c^4*d*g^2*i^2 - 9*a^2*b^3*c^3*d
^2*g^2*i^2 + (20*g^2*i^2*log(e) + 9*g^2*i^2)*a^3*b^2*c^2*d^3 - 2*(5*g^2*i^2*log(e) + g^2*i^2)*a^4*b*c*d^4)*B^2
)*log(b*x + a) - 2*(12*B^2*b^5*d^5*g^2*i^2*x^5*log(e) + 3*((10*g^2*i^2*log(e) - g^2*i^2)*b^5*c*d^4 + (10*g^2*i
^2*log(e) + g^2*i^2)*a*b^4*d^5)*B^2*x^4 + 2*(40*a*b^4*c*d^4*g^2*i^2*log(e) + (10*g^2*i^2*log(e) - 3*g^2*i^2)*b
^5*c^2*d^3 + (10*g^2*i^2*log(e) + 3*g^2*i^2)*a^2*b^3*d^5)*B^2*x^3 - (b^5*c^3*d^2*g^2*i^2 - a^3*b^2*d^5*g^2*i^2
- 15*(4*g^2*i^2*log(e) - g^2*i^2)*a*b^4*c^2*d^3 - 15*(4*g^2*i^2*log(e) + g^2*i^2)*a^2*b^3*c*d^4)*B^2*x^2 + 2*
(30*a^2*b^3*c^2*d^3*g^2*i^2*log(e) + b^5*c^4*d*g^2*i^2 - 5*a*b^4*c^3*d^2*g^2*i^2 + 5*a^3*b^2*c*d^4*g^2*i^2 - a
^4*b*d^5*g^2*i^2)*B^2*x + 2*(6*B^2*b^5*d^5*g^2*i^2*x^5 + 30*B^2*a^2*b^3*c^2*d^3*g^2*i^2*x + 15*(b^5*c*d^4*g^2*
i^2 + a*b^4*d^5*g^2*i^2)*B^2*x^4 + 10*(b^5*c^2*d^3*g^2*i^2 + 4*a*b^4*c*d^4*g^2*i^2 + a^2*b^3*d^5*g^2*i^2)*B^2*
x^3 + 30*(a*b^4*c^2*d^3*g^2*i^2 + a^2*b^3*c*d^4*g^2*i^2)*B^2*x^2 + (10*a^3*b^2*c^2*d^3*g^2*i^2 - 5*a^4*b*c*d^4
*g^2*i^2 + a^5*d^5*g^2*i^2)*B^2)*log(b*x + a))*log(d*x + c))/(b^3*d^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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