\(\int \frac {(c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^2} \, dx\) [60]

Optimal result

Integrand size = 40, antiderivative size = 241 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {2 B^2 i (c+d x)}{b g^2 (a+b x)}-\frac {2 B i (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^2 (a+b x)}-\frac {i (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g^2 (a+b x)}-\frac {d i \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 g^2}+\frac {2 B d i \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 g^2}+\frac {2 B^2 d i \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 g^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {2562, 2380, 2342, 2341, 2379, 2421, 6724} \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=\frac {2 B d i \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^2}-\frac {d i \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g^2}-\frac {2 B i (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g^2 (a+b x)}-\frac {i (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b g^2 (a+b x)}+\frac {2 B^2 d i \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 g^2}-\frac {2 B^2 i (c+d x)}{b g^2 (a+b x)} \]

[In]

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

[Out]

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

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 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 2380

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

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 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*} \text {integral}& = \frac {i \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{x^2 (b-d x)} \, dx,x,\frac {a+b x}{c+d x}\right )}{g^2} \\ & = \frac {i \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{b g^2}+\frac {(d i) \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{x (b-d x)} \, dx,x,\frac {a+b x}{c+d x}\right )}{b g^2} \\ & = -\frac {i (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g^2 (a+b x)}-\frac {d i \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^2 g^2}+\frac {(2 B i) \text {Subst}\left (\int \frac {A+B \log (e x)}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{b g^2}+\frac {(2 B d i) \text {Subst}\left (\int \frac {\log \left (1-\frac {b}{d x}\right ) (A+B \log (e x))}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{b^2 g^2} \\ & = -\frac {2 B^2 i (c+d x)}{b g^2 (a+b x)}-\frac {2 B i (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^2 (a+b x)}-\frac {i (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g^2 (a+b x)}-\frac {d i \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^2 g^2}+\frac {2 B d i \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^2 g^2}-\frac {\left (2 B^2 d i\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {b}{d x}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{b^2 g^2} \\ & = -\frac {2 B^2 i (c+d x)}{b g^2 (a+b x)}-\frac {2 B i (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^2 (a+b x)}-\frac {i (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g^2 (a+b x)}-\frac {d i \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^2 g^2}+\frac {2 B d i \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^2 g^2}+\frac {2 B^2 d i \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 g^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1407\) vs. \(2(241)=482\).

Time = 1.21 (sec) , antiderivative size = 1407, normalized size of antiderivative = 5.84 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=\frac {i \left (\frac {3 A^2 (-b c+a d)}{a+b x}+3 A^2 d \log (a+b x)-\frac {6 A b B c \left (-d (a+b x) \log \left (\frac {c}{d}+x\right )+d (a+b x) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+(b c-a d) \left (1+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right )}{(b c-a d) (a+b x)}+\frac {3 b B^2 c \left (-2 b c+2 a d-2 d (a+b x) \log (a+b x)-2 (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 d (a+b x) \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-(b c-a d) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+2 d (a+b x) \log (c+d x)-2 d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+d (a+b x) \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 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+d (a+b x) \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{(b c-a d) (a+b x)}+3 A B d \left (\log ^2\left (\frac {a}{b}+x\right )-2 \log \left (\frac {a}{b}+x\right ) \log (a+b x)-2 \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+2 \log (a+b x) \left (\frac {a d}{b c-a d}+\log \left (\frac {c}{d}+x\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 a \left (\frac {1}{a+b x}+\frac {\log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x}+\frac {d \log (c+d x)}{-b c+a d}\right )-2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )+\frac {B^2 d \left ((b c-a d) (a+b x) \log ^3\left (\frac {a}{b}+x\right )+3 a (b c-a d) \left (2+2 \log \left (\frac {a}{b}+x\right )+\log ^2\left (\frac {a}{b}+x\right )\right )+3 (b c-a d) (a+(a+b x) \log (a+b x)) \left (-\log \left (\frac {a}{b}+x\right )+\log \left (\frac {c}{d}+x\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+3 a \left (d (a+b x) \log ^2\left (\frac {a}{b}+x\right )+2 \left ((-b c+a d) \log \left (\frac {c}{d}+x\right )+d (a+b x) (\log (a+b x)-\log (c+d x))\right )-2 \log \left (\frac {a}{b}+x\right ) \left ((b c-a d) \log \left (\frac {c}{d}+x\right )+d (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 d (a+b x) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+3 a \left (\log \left (\frac {c}{d}+x\right ) \left (b (c+d x) \log \left (\frac {c}{d}+x\right )-2 d (a+b x) \log \left (\frac {d (a+b x)}{-b c+a d}\right )\right )-2 d (a+b x) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )-3 \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \left ((b c-a d) (a+b x) \log ^2\left (\frac {a}{b}+x\right )+2 a (b c-a d) \left (1+\log \left (\frac {a}{b}+x\right )\right )+2 a (-b c+a d) \log \left (\frac {c}{d}+x\right )+2 a d (a+b x) (\log (a+b x)-\log (c+d x))-2 (b c-a d) (a+b x) \left (\log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )-3 (b c-a d) (a+b x) \left (\log ^2\left (\frac {a}{b}+x\right ) \left (\log \left (\frac {c}{d}+x\right )-\log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \log \left (\frac {a}{b}+x\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{-b c+a d}\right )\right )+3 (b c-a d) (a+b x) \left (\log ^2\left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+2 \log \left (\frac {c}{d}+x\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )-2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{(b c-a d) (a+b x)}\right )}{3 b^2 g^2} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (d i x +c i \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{\left (b g x +a g \right )^{2}}d x\]

[In]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=\int \frac {\left (c\,i+d\,i\,x\right )\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{{\left (a\,g+b\,g\,x\right )}^2} \,d x \]

[In]

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