\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^3} \, dx\) [103]

Optimal result

Integrand size = 32, antiderivative size = 268 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=\frac {2 B^2 d (c+d x)}{(b c-a d)^2 g^3 (a+b x)}-\frac {b B^2 (c+d x)^2}{4 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 B d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 g^3 (a+b x)}-\frac {b B (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^2 g^3 (a+b x)^2}+\frac {d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^2 g^3 (a+b x)}-\frac {b (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^2 g^3 (a+b x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2550, 2395, 2342, 2341} \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {b B (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 (a+b x)^2 (b c-a d)^2}+\frac {2 B d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 (a+b x) (b c-a d)^2}-\frac {b (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^3 (a+b x)^2 (b c-a d)^2}+\frac {d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g^3 (a+b x) (b c-a d)^2}-\frac {b B^2 (c+d x)^2}{4 g^3 (a+b x)^2 (b c-a d)^2}+\frac {2 B^2 d (c+d x)}{g^3 (a+b x) (b c-a d)^2} \]

[In]

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

[Out]

(2*B^2*d*(c + d*x))/((b*c - a*d)^2*g^3*(a + b*x)) - (b*B^2*(c + d*x)^2)/(4*(b*c - a*d)^2*g^3*(a + b*x)^2) + (2
*B*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^2*g^3*(a + b*x)) - (b*B*(c + d*x)^2*(A + B*L
og[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^2*g^3*(a + b*x)^2) + (d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d
*x)])^2)/((b*c - a*d)^2*g^3*(a + b*x)) - (b*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)
^2*g^3*(a + b*x)^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 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 2550

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(b-d x) (A+B \log (e x))^2}{x^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^2 g^3} \\ & = \frac {\text {Subst}\left (\int \left (\frac {b (A+B \log (e x))^2}{x^3}-\frac {d (A+B \log (e x))^2}{x^2}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^2 g^3} \\ & = \frac {b \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{x^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^2 g^3}-\frac {d \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^2 g^3} \\ & = \frac {d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^2 g^3 (a+b x)}-\frac {b (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^2 g^3 (a+b x)^2}+\frac {(b B) \text {Subst}\left (\int \frac {A+B \log (e x)}{x^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^2 g^3}-\frac {(2 B d) \text {Subst}\left (\int \frac {A+B \log (e x)}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^2 g^3} \\ & = \frac {2 B^2 d (c+d x)}{(b c-a d)^2 g^3 (a+b x)}-\frac {b B^2 (c+d x)^2}{4 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 B d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 g^3 (a+b x)}-\frac {b B (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^2 g^3 (a+b x)^2}+\frac {d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^2 g^3 (a+b x)}-\frac {b (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^2 g^3 (a+b x)^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.29 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.65 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B \left (2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d (-b c+a d) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-4 B d (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+B \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-2 B d^2 (a+b x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{(b c-a d)^2}}{4 b g^3 (a+b x)^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.81

method	result	size
norman	\(\frac {\frac {B d \left (2 A a d +2 B a d +B b c \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {B^{2} a \,d^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (2 A^{2} a d -2 A^{2} b c +4 A B a d -2 A B b c +4 B^{2} a d -B^{2} b c \right ) x}{2 a g \left (a d -c b \right )}+\frac {B c \left (4 A a d -2 A b c +4 B a d -B b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {B^{2} c \left (2 a d -c b \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (2 A^{2} a d -2 A^{2} b c +6 A B a d -2 A B b c +7 B^{2} a d -B^{2} b c \right ) b \,x^{2}}{4 g \,a^{2} \left (a d -c b \right )}+\frac {b \,d^{2} B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {B b \,d^{2} \left (2 A +3 B \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right )^{2} g^{2}}\)	\(485\)
parallelrisch	\(-\frac {-4 A^{2} a \,b^{4} c \,d^{2}+6 A B \,a^{2} b^{3} d^{3}+2 A B \,b^{5} c^{2} d -8 B^{2} a \,b^{4} c \,d^{2}-8 A B a \,b^{4} c \,d^{2}-4 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c \,d^{2}-4 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{4} c \,d^{2}+4 A B x a \,b^{4} d^{3}-4 A B x \,b^{5} c \,d^{2}+4 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c^{2} d -4 A B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} d^{3}-4 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{4} d^{3}-8 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{4} d^{3}-8 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{4} c \,d^{2}-2 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{5} d^{3}-6 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} d^{3}+2 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{5} c^{2} d +6 B^{2} x a \,b^{4} d^{3}-6 B^{2} x \,b^{5} c \,d^{2}+2 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c^{2} d -8 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{4} d^{3}-8 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{4} c \,d^{2}+2 A^{2} a^{2} b^{3} d^{3}+2 A^{2} b^{5} c^{2} d +7 B^{2} a^{2} b^{3} d^{3}+B^{2} b^{5} c^{2} d}{4 g^{3} \left (b x +a \right )^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{4} d}\)	\(574\)
parts	\(-\frac {A^{2}}{2 g^{3} \left (b x +a \right )^{2} b}-\frac {B^{2} \left (a d -c b \right ) e \left (\frac {d^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{3}}-\frac {d^{2} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{3}}\right )}{g^{3} d^{2}}-\frac {2 B A \left (a d -c b \right ) e \left (\frac {d^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{3}}-\frac {d^{2} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{3}}\right )}{g^{3} d^{2}}\)	\(609\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A^{2} b e}{2 \left (a d -c b \right )^{3} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {d^{3} A^{2}}{\left (a d -c b \right )^{3} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}-\frac {2 d^{2} A B b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{3} g^{3}}+\frac {2 d^{3} A B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{3} g^{3}}-\frac {d^{2} B^{2} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{3} g^{3}}+\frac {d^{3} B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{3} g^{3}}\right )}{d^{2}}\)	\(689\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A^{2} b e}{2 \left (a d -c b \right )^{3} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {d^{3} A^{2}}{\left (a d -c b \right )^{3} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}-\frac {2 d^{2} A B b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{3} g^{3}}+\frac {2 d^{3} A B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{3} g^{3}}-\frac {d^{2} B^{2} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{3} g^{3}}+\frac {d^{3} B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{3} g^{3}}\right )}{d^{2}}\)	\(689\)
risch	\(\text {Expression too large to display}\)	\(1504\)

[In]

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

[Out]

(B/g*d*(2*A*a*d+2*B*a*d+B*b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x*ln(e*(b*x+a)/(d*x+c))+B^2*a*d^2/g/(a^2*d^2-2*a*b*
c*d+b^2*c^2)*x*ln(e*(b*x+a)/(d*x+c))^2+1/2*(2*A^2*a*d-2*A^2*b*c+4*A*B*a*d-2*A*B*b*c+4*B^2*a*d-B^2*b*c)/a/g/(a*
d-b*c)*x+1/2*B*c*(4*A*a*d-2*A*b*c+4*B*a*d-B*b*c)/g/(a^2*d^2-2*a*b*c*d+b^2*c^2)*ln(e*(b*x+a)/(d*x+c))+1/2*B^2*c
*(2*a*d-b*c)/g/(a^2*d^2-2*a*b*c*d+b^2*c^2)*ln(e*(b*x+a)/(d*x+c))^2+1/4*(2*A^2*a*d-2*A^2*b*c+6*A*B*a*d-2*A*B*b*
c+7*B^2*a*d-B^2*b*c)/g/a^2*b/(a*d-b*c)*x^2+1/2*b*d^2*B^2/g/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x^2*ln(e*(b*x+a)/(d*x+c
))^2+1/2*B*b/g*d^2*(2*A+3*B)/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x^2*ln(e*(b*x+a)/(d*x+c)))/(b*x+a)^2/g^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.37 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {{\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{2} c^{2} - 4 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a b c d + {\left (2 \, A^{2} + 6 \, A B + 7 \, B^{2}\right )} a^{2} d^{2} - 2 \, {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} a b d^{2} x - B^{2} b^{2} c^{2} + 2 \, B^{2} a b c d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 2 \, {\left ({\left (2 \, A B + 3 \, B^{2}\right )} b^{2} c d - {\left (2 \, A B + 3 \, B^{2}\right )} a b d^{2}\right )} x - 2 \, {\left ({\left (2 \, A B + 3 \, B^{2}\right )} b^{2} d^{2} x^{2} - {\left (2 \, A B + B^{2}\right )} b^{2} c^{2} + 4 \, {\left (A B + B^{2}\right )} a b c d + 2 \, {\left (B^{2} b^{2} c d + 2 \, {\left (A B + B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} g^{3} x + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} g^{3}\right )}} \]

[In]

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

[Out]

-1/4*((2*A^2 + 2*A*B + B^2)*b^2*c^2 - 4*(A^2 + 2*A*B + 2*B^2)*a*b*c*d + (2*A^2 + 6*A*B + 7*B^2)*a^2*d^2 - 2*(B
^2*b^2*d^2*x^2 + 2*B^2*a*b*d^2*x - B^2*b^2*c^2 + 2*B^2*a*b*c*d)*log((b*e*x + a*e)/(d*x + c))^2 - 2*((2*A*B + 3
*B^2)*b^2*c*d - (2*A*B + 3*B^2)*a*b*d^2)*x - 2*((2*A*B + 3*B^2)*b^2*d^2*x^2 - (2*A*B + B^2)*b^2*c^2 + 4*(A*B +
B^2)*a*b*c*d + 2*(B^2*b^2*c*d + 2*(A*B + B^2)*a*b*d^2)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^5*c^2 - 2*a*b^4*c
*d + a^2*b^3*d^2)*g^3*x^2 + 2*(a*b^4*c^2 - 2*a^2*b^3*c*d + a^3*b^2*d^2)*g^3*x + (a^2*b^3*c^2 - 2*a^3*b^2*c*d +
a^4*b*d^2)*g^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 894 vs. \(2 (241) = 482\).

Time = 2.11 (sec) , antiderivative size = 894, normalized size of antiderivative = 3.34 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=- \frac {B d^{2} \cdot \left (2 A + 3 B\right ) \log {\left (x + \frac {2 A B a d^{3} + 2 A B b c d^{2} + 3 B^{2} a d^{3} + 3 B^{2} b c d^{2} - \frac {B a^{3} d^{5} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {3 B a^{2} b c d^{4} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {3 B a b^{2} c^{2} d^{3} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {B b^{3} c^{3} d^{2} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}}}{4 A B b d^{3} + 6 B^{2} b d^{3}} \right )}}{2 b g^{3} \left (a d - b c\right )^{2}} + \frac {B d^{2} \cdot \left (2 A + 3 B\right ) \log {\left (x + \frac {2 A B a d^{3} + 2 A B b c d^{2} + 3 B^{2} a d^{3} + 3 B^{2} b c d^{2} + \frac {B a^{3} d^{5} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {3 B a^{2} b c d^{4} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {3 B a b^{2} c^{2} d^{3} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {B b^{3} c^{3} d^{2} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}}}{4 A B b d^{3} + 6 B^{2} b d^{3}} \right )}}{2 b g^{3} \left (a d - b c\right )^{2}} + \frac {\left (2 B^{2} a c d + 2 B^{2} a d^{2} x - B^{2} b c^{2} + B^{2} b d^{2} x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{4} d^{2} g^{3} - 4 a^{3} b c d g^{3} + 4 a^{3} b d^{2} g^{3} x + 2 a^{2} b^{2} c^{2} g^{3} - 8 a^{2} b^{2} c d g^{3} x + 2 a^{2} b^{2} d^{2} g^{3} x^{2} + 4 a b^{3} c^{2} g^{3} x - 4 a b^{3} c d g^{3} x^{2} + 2 b^{4} c^{2} g^{3} x^{2}} + \frac {\left (- 2 A B a d + 2 A B b c - 3 B^{2} a d + B^{2} b c - 2 B^{2} b d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 a^{3} b d g^{3} - 2 a^{2} b^{2} c g^{3} + 4 a^{2} b^{2} d g^{3} x - 4 a b^{3} c g^{3} x + 2 a b^{3} d g^{3} x^{2} - 2 b^{4} c g^{3} x^{2}} + \frac {- 2 A^{2} a d + 2 A^{2} b c - 6 A B a d + 2 A B b c - 7 B^{2} a d + B^{2} b c + x \left (- 4 A B b d - 6 B^{2} b d\right )}{4 a^{3} b d g^{3} - 4 a^{2} b^{2} c g^{3} + x^{2} \cdot \left (4 a b^{3} d g^{3} - 4 b^{4} c g^{3}\right ) + x \left (8 a^{2} b^{2} d g^{3} - 8 a b^{3} c g^{3}\right )} \]

[In]

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

[Out]

-B*d**2*(2*A + 3*B)*log(x + (2*A*B*a*d**3 + 2*A*B*b*c*d**2 + 3*B**2*a*d**3 + 3*B**2*b*c*d**2 - B*a**3*d**5*(2*
A + 3*B)/(a*d - b*c)**2 + 3*B*a**2*b*c*d**4*(2*A + 3*B)/(a*d - b*c)**2 - 3*B*a*b**2*c**2*d**3*(2*A + 3*B)/(a*d
- b*c)**2 + B*b**3*c**3*d**2*(2*A + 3*B)/(a*d - b*c)**2)/(4*A*B*b*d**3 + 6*B**2*b*d**3))/(2*b*g**3*(a*d - b*c
)**2) + B*d**2*(2*A + 3*B)*log(x + (2*A*B*a*d**3 + 2*A*B*b*c*d**2 + 3*B**2*a*d**3 + 3*B**2*b*c*d**2 + B*a**3*d
**5*(2*A + 3*B)/(a*d - b*c)**2 - 3*B*a**2*b*c*d**4*(2*A + 3*B)/(a*d - b*c)**2 + 3*B*a*b**2*c**2*d**3*(2*A + 3*
B)/(a*d - b*c)**2 - B*b**3*c**3*d**2*(2*A + 3*B)/(a*d - b*c)**2)/(4*A*B*b*d**3 + 6*B**2*b*d**3))/(2*b*g**3*(a*
d - b*c)**2) + (2*B**2*a*c*d + 2*B**2*a*d**2*x - B**2*b*c**2 + B**2*b*d**2*x**2)*log(e*(a + b*x)/(c + d*x))**2
/(2*a**4*d**2*g**3 - 4*a**3*b*c*d*g**3 + 4*a**3*b*d**2*g**3*x + 2*a**2*b**2*c**2*g**3 - 8*a**2*b**2*c*d*g**3*x
+ 2*a**2*b**2*d**2*g**3*x**2 + 4*a*b**3*c**2*g**3*x - 4*a*b**3*c*d*g**3*x**2 + 2*b**4*c**2*g**3*x**2) + (-2*A
*B*a*d + 2*A*B*b*c - 3*B**2*a*d + B**2*b*c - 2*B**2*b*d*x)*log(e*(a + b*x)/(c + d*x))/(2*a**3*b*d*g**3 - 2*a**
2*b**2*c*g**3 + 4*a**2*b**2*d*g**3*x - 4*a*b**3*c*g**3*x + 2*a*b**3*d*g**3*x**2 - 2*b**4*c*g**3*x**2) + (-2*A*
*2*a*d + 2*A**2*b*c - 6*A*B*a*d + 2*A*B*b*c - 7*B**2*a*d + B**2*b*c + x*(-4*A*B*b*d - 6*B**2*b*d))/(4*a**3*b*d
*g**3 - 4*a**2*b**2*c*g**3 + x**2*(4*a*b**3*d*g**3 - 4*b**4*c*g**3) + x*(8*a**2*b**2*d*g**3 - 8*a*b**3*c*g**3)
)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 848 vs. \(2 (262) = 524\).

Time = 0.24 (sec) , antiderivative size = 848, normalized size of antiderivative = 3.16 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=\frac {1}{4} \, {\left (2 \, {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {b^{2} c^{2} - 8 \, a b c d + 7 \, a^{2} d^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (d x + c\right )^{2} - 6 \, {\left (b^{2} c d - a b d^{2}\right )} x - 6 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 6 \, a b d^{2} x + 3 \, a^{2} d^{2} - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a^{2} b^{3} c^{2} g^{3} - 2 \, a^{3} b^{2} c d g^{3} + a^{4} b d^{2} g^{3} + {\left (b^{5} c^{2} g^{3} - 2 \, a b^{4} c d g^{3} + a^{2} b^{3} d^{2} g^{3}\right )} x^{2} + 2 \, {\left (a b^{4} c^{2} g^{3} - 2 \, a^{2} b^{3} c d g^{3} + a^{3} b^{2} d^{2} g^{3}\right )} x}\right )} B^{2} + \frac {1}{2} \, A B {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} - \frac {2 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac {B^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )^{2}}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} - \frac {A^{2}}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \]

[In]

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

[Out]

1/4*(2*((2*b*d*x - b*c + 3*a*d)/((b^4*c - a*b^3*d)*g^3*x^2 + 2*(a*b^3*c - a^2*b^2*d)*g^3*x + (a^2*b^2*c - a^3*
b*d)*g^3) + 2*d^2*log(b*x + a)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3) - 2*d^2*log(d*x + c)/((b^3*c^2 - 2*a*
b^2*c*d + a^2*b*d^2)*g^3))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - (b^2*c^2 - 8*a*b*c*d + 7*a^2*d^2 + 2*(b^2*d^
2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a)^2 + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(d*x + c)^2 - 6*(b^
2*c*d - a*b*d^2)*x - 6*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a) + 2*(3*b^2*d^2*x^2 + 6*a*b*d^2*x + 3
*a^2*d^2 - 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a))*log(d*x + c))/(a^2*b^3*c^2*g^3 - 2*a^3*b^2*c*
d*g^3 + a^4*b*d^2*g^3 + (b^5*c^2*g^3 - 2*a*b^4*c*d*g^3 + a^2*b^3*d^2*g^3)*x^2 + 2*(a*b^4*c^2*g^3 - 2*a^2*b^3*c
*d*g^3 + a^3*b^2*d^2*g^3)*x))*B^2 + 1/2*A*B*((2*b*d*x - b*c + 3*a*d)/((b^4*c - a*b^3*d)*g^3*x^2 + 2*(a*b^3*c -
a^2*b^2*d)*g^3*x + (a^2*b^2*c - a^3*b*d)*g^3) - 2*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^3*g^3*x^2 + 2*a*b^2
*g^3*x + a^2*b*g^3) + 2*d^2*log(b*x + a)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3) - 2*d^2*log(d*x + c)/((b^3*
c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3)) - 1/2*B^2*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^3*g^3*x^2 + 2*a*b^2*g
^3*x + a^2*b*g^3) - 1/2*A^2/(b^3*g^3*x^2 + 2*a*b^2*g^3*x + a^2*b*g^3)

Giac [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.64 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, {\left (B^{2} b e^{3} - \frac {2 \, {\left (b e x + a e\right )} B^{2} d e^{2}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{\frac {{\left (b e x + a e\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b e x + a e\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}} + \frac {2 \, {\left (2 \, A B b e^{3} + B^{2} b e^{3} - \frac {4 \, {\left (b e x + a e\right )} A B d e^{2}}{d x + c} - \frac {4 \, {\left (b e x + a e\right )} B^{2} d e^{2}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b e x + a e\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}} + \frac {2 \, A^{2} b e^{3} + 2 \, A B b e^{3} + B^{2} b e^{3} - \frac {4 \, {\left (b e x + a e\right )} A^{2} d e^{2}}{d x + c} - \frac {8 \, {\left (b e x + a e\right )} A B d e^{2}}{d x + c} - \frac {8 \, {\left (b e x + a e\right )} B^{2} d e^{2}}{d x + c}}{\frac {{\left (b e x + a e\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b e x + a e\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.60 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.89 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {\frac {2\,A^2\,a\,d-2\,A^2\,b\,c+7\,B^2\,a\,d-B^2\,b\,c+6\,A\,B\,a\,d-2\,A\,B\,b\,c}{2\,\left (a\,d-b\,c\right )}+\frac {x\,\left (3\,b\,d\,B^2+2\,A\,b\,d\,B\right )}{a\,d-b\,c}}{2\,a^2\,b\,g^3+4\,a\,b^2\,g^3\,x+2\,b^3\,g^3\,x^2}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B^2}{2\,b^2\,g^3\,\left (2\,a\,x+b\,x^2+\frac {a^2}{b}\right )}-\frac {B^2\,d^2}{2\,b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {A\,B}{b^2\,d\,g^3}+\frac {B^2\,x\,\left (a\,d-b\,c\right )}{b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B^2\,d^2\,\left (\frac {2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2}{2\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,b\,d^2}\right )}{b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )}{\frac {b\,x^2}{d}+\frac {a^2}{b\,d}+\frac {2\,a\,x}{d}}-\frac {B\,d^2\,\mathrm {atan}\left (\frac {B\,d^2\,\left (2\,b\,d\,x-\frac {b^3\,c^2\,g^3-a^2\,b\,d^2\,g^3}{b\,g^3\,\left (a\,d-b\,c\right )}\right )\,\left (2\,A+3\,B\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (3\,B^2\,d^2+2\,A\,B\,d^2\right )}\right )\,\left (2\,A+3\,B\right )\,1{}\mathrm {i}}{b\,g^3\,{\left (a\,d-b\,c\right )}^2} \]

[In]

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

[Out]

- ((2*A^2*a*d - 2*A^2*b*c + 7*B^2*a*d - B^2*b*c + 6*A*B*a*d - 2*A*B*b*c)/(2*(a*d - b*c)) + (x*(3*B^2*b*d + 2*A
*B*b*d))/(a*d - b*c))/(2*a^2*b*g^3 + 2*b^3*g^3*x^2 + 4*a*b^2*g^3*x) - log((e*(a + b*x))/(c + d*x))^2*(B^2/(2*b
^2*g^3*(2*a*x + b*x^2 + a^2/b)) - (B^2*d^2)/(2*b*g^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) - (log((e*(a + b*x))/(c
+ d*x))*((A*B)/(b^2*d*g^3) + (B^2*x*(a*d - b*c))/(b*g^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (B^2*d^2*((2*a^2*d
^2 + b^2*c^2 - 3*a*b*c*d)/(2*b*d^3) + (a*(a*d - b*c))/(2*b*d^2)))/(b*g^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))))/((
b*x^2)/d + a^2/(b*d) + (2*a*x)/d) - (B*d^2*atan((B*d^2*(2*b*d*x - (b^3*c^2*g^3 - a^2*b*d^2*g^3)/(b*g^3*(a*d -
b*c)))*(2*A + 3*B)*1i)/((a*d - b*c)*(3*B^2*d^2 + 2*A*B*d^2)))*(2*A + 3*B)*1i)/(b*g^3*(a*d - b*c)^2)