\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^4} \, dx\) [104]

Optimal result

Integrand size = 32, antiderivative size = 418 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {2 B^2 d^2 (c+d x)}{(b c-a d)^3 g^4 (a+b x)}+\frac {b B^2 d (c+d x)^2}{2 (b c-a d)^3 g^4 (a+b x)^2}-\frac {2 b^2 B^2 (c+d x)^3}{27 (b c-a d)^3 g^4 (a+b x)^3}-\frac {2 B d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^4 (a+b x)}+\frac {b B d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^4 (a+b x)^2}-\frac {2 b^2 B (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d)^3 g^4 (a+b x)^3}-\frac {d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g^4 (a+b x)}+\frac {b d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g^4 (a+b x)^2}-\frac {b^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d)^3 g^4 (a+b x)^3} \]

[Out]

-2*B^2*d^2*(d*x+c)/(-a*d+b*c)^3/g^4/(b*x+a)+1/2*b*B^2*d*(d*x+c)^2/(-a*d+b*c)^3/g^4/(b*x+a)^2-2/27*b^2*B^2*(d*x
+c)^3/(-a*d+b*c)^3/g^4/(b*x+a)^3-2*B*d^2*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^4/(b*x+a)+b*B*d*(d
*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^4/(b*x+a)^2-2/9*b^2*B*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))
/(-a*d+b*c)^3/g^4/(b*x+a)^3-d^2*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^3/g^4/(b*x+a)+b*d*(d*x+c)^2*(
A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^3/g^4/(b*x+a)^2-1/3*b^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+
b*c)^3/g^4/(b*x+a)^3

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2550, 2395, 2342, 2341} \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {b^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 g^4 (a+b x)^3 (b c-a d)^3}-\frac {2 b^2 B (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{9 g^4 (a+b x)^3 (b c-a d)^3}-\frac {d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g^4 (a+b x) (b c-a d)^3}-\frac {2 B d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 (a+b x) (b c-a d)^3}+\frac {b d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g^4 (a+b x)^2 (b c-a d)^3}+\frac {b B d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 (a+b x)^2 (b c-a d)^3}-\frac {2 b^2 B^2 (c+d x)^3}{27 g^4 (a+b x)^3 (b c-a d)^3}-\frac {2 B^2 d^2 (c+d x)}{g^4 (a+b x) (b c-a d)^3}+\frac {b B^2 d (c+d x)^2}{2 g^4 (a+b x)^2 (b c-a d)^3} \]

[In]

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

[Out]

(-2*B^2*d^2*(c + d*x))/((b*c - a*d)^3*g^4*(a + b*x)) + (b*B^2*d*(c + d*x)^2)/(2*(b*c - a*d)^3*g^4*(a + b*x)^2)
- (2*b^2*B^2*(c + d*x)^3)/(27*(b*c - a*d)^3*g^4*(a + b*x)^3) - (2*B*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/((b*c - a*d)^3*g^4*(a + b*x)) + (b*B*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*
d)^3*g^4*(a + b*x)^2) - (2*b^2*B*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(b*c - a*d)^3*g^4*(a + b
*x)^3) - (d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^3*g^4*(a + b*x)) + (b*d*(c + d*x)
^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^3*g^4*(a + b*x)^2) - (b^2*(c + d*x)^3*(A + B*Log[(e*(a
+ b*x))/(c + d*x)])^2)/(3*(b*c - a*d)^3*g^4*(a + b*x)^3)

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)^2 (A+B \log (e x))^2}{x^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^4} \\ & = \frac {\text {Subst}\left (\int \left (\frac {b^2 (A+B \log (e x))^2}{x^4}-\frac {2 b d (A+B \log (e x))^2}{x^3}+\frac {d^2 (A+B \log (e x))^2}{x^2}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^4} \\ & = \frac {b^2 \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{x^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^4}-\frac {(2 b d) \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)^3 g^4}+\frac {d^2 \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)^3 g^4} \\ & = -\frac {d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g^4 (a+b x)}+\frac {b d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g^4 (a+b x)^2}-\frac {b^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d)^3 g^4 (a+b x)^3}+\frac {\left (2 b^2 B\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{x^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 (b c-a d)^3 g^4}-\frac {(2 b B d) \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)^3 g^4}+\frac {\left (2 B d^2\right ) \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)^3 g^4} \\ & = -\frac {2 B^2 d^2 (c+d x)}{(b c-a d)^3 g^4 (a+b x)}+\frac {b B^2 d (c+d x)^2}{2 (b c-a d)^3 g^4 (a+b x)^2}-\frac {2 b^2 B^2 (c+d x)^3}{27 (b c-a d)^3 g^4 (a+b x)^3}-\frac {2 B d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^4 (a+b x)}+\frac {b B d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^4 (a+b x)^2}-\frac {2 b^2 B (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d)^3 g^4 (a+b x)^3}-\frac {d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g^4 (a+b x)}+\frac {b d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g^4 (a+b x)^2}-\frac {b^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d)^3 g^4 (a+b x)^3} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.40 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.39 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {18 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B \left (12 A (b c-a d)^3+4 B (b c-a d)^3-18 A d (b c-a d)^2 (a+b x)-15 B d (b c-a d)^2 (a+b x)+36 A d^2 (b c-a d) (a+b x)^2+66 B d^2 (b c-a d) (a+b x)^2+36 A d^3 (a+b x)^3 \log (a+b x)+66 B d^3 (a+b x)^3 \log (a+b x)-18 B d^3 (a+b x)^3 \log ^2(a+b x)+12 B (b c-a d)^3 \log \left (\frac {e (a+b x)}{c+d x}\right )-18 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+36 B d^2 (b c-a d) (a+b x)^2 \log \left (\frac {e (a+b x)}{c+d x}\right )+36 B d^3 (a+b x)^3 \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-36 A d^3 (a+b x)^3 \log (c+d x)-66 B d^3 (a+b x)^3 \log (c+d x)+36 B d^3 (a+b x)^3 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)-36 B d^3 (a+b x)^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)-18 B d^3 (a+b x)^3 \log ^2(c+d x)+36 B d^3 (a+b x)^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+36 B d^3 (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+36 B d^3 (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^3}}{54 b g^4 (a+b x)^3} \]

[In]

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

[Out]

-1/54*(18*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + (B*(12*A*(b*c - a*d)^3 + 4*B*(b*c - a*d)^3 - 18*A*d*(b*c -
a*d)^2*(a + b*x) - 15*B*d*(b*c - a*d)^2*(a + b*x) + 36*A*d^2*(b*c - a*d)*(a + b*x)^2 + 66*B*d^2*(b*c - a*d)*(a
+ b*x)^2 + 36*A*d^3*(a + b*x)^3*Log[a + b*x] + 66*B*d^3*(a + b*x)^3*Log[a + b*x] - 18*B*d^3*(a + b*x)^3*Log[a
+ b*x]^2 + 12*B*(b*c - a*d)^3*Log[(e*(a + b*x))/(c + d*x)] - 18*B*d*(b*c - a*d)^2*(a + b*x)*Log[(e*(a + b*x))
/(c + d*x)] + 36*B*d^2*(b*c - a*d)*(a + b*x)^2*Log[(e*(a + b*x))/(c + d*x)] + 36*B*d^3*(a + b*x)^3*Log[a + b*x
]*Log[(e*(a + b*x))/(c + d*x)] - 36*A*d^3*(a + b*x)^3*Log[c + d*x] - 66*B*d^3*(a + b*x)^3*Log[c + d*x] + 36*B*
d^3*(a + b*x)^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] - 36*B*d^3*(a + b*x)^3*Log[(e*(a + b*x))/(c + d
*x)]*Log[c + d*x] - 18*B*d^3*(a + b*x)^3*Log[c + d*x]^2 + 36*B*d^3*(a + b*x)^3*Log[a + b*x]*Log[(b*(c + d*x))/
(b*c - a*d)] + 36*B*d^3*(a + b*x)^3*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 36*B*d^3*(a + b*x)^3*PolyLog[2,
(b*(c + d*x))/(b*c - a*d)]))/(b*c - a*d)^3)/(b*g^4*(a + b*x)^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(891\) vs. \(2(410)=820\).

Time = 1.34 (sec) , antiderivative size = 892, normalized size of antiderivative = 2.13

method	result	size
parts	\(-\frac {A^{2}}{3 g^{4} \left (b x +a \right )^{3} b}-\frac {B^{2} \left (a d -c b \right ) e \left (\frac {d^{4} \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 )^{4}}-\frac {2 d^{3} 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 )^{4}}+\frac {d^{2} e^{2} 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}}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2}{27 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{4}}\right )}{g^{4} d^{2}}-\frac {2 B A \left (a d -c b \right ) e \left (\frac {d^{4} \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 )^{4}}-\frac {2 d^{3} 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 )^{4}}+\frac {d^{2} e^{2} 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 )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{4}}\right )}{g^{4} d^{2}}\)	\(892\)
norman	\(\frac {\frac {B^{2} a^{2} d^{3} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) g}+\frac {B^{2} a b \,d^{3} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {18 A^{2} a^{2} b^{2} d^{2}-36 A^{2} a \,b^{3} c d +18 A^{2} b^{4} c^{2}+66 A B \,a^{2} b^{2} d^{2}-42 A B a \,b^{3} c d +12 A B \,b^{4} c^{2}+85 B^{2} a^{2} b^{2} d^{2}-23 B^{2} a \,b^{3} c d +4 B^{2} b^{4} c^{2}}{54 g \left (a d -c b \right )^{2} b^{3}}-\frac {\left (30 A B a \,b^{2} d^{2}-6 A B \,b^{3} c d +49 B^{2} a \,b^{2} d^{2}-5 B^{2} b^{3} c d \right ) x}{18 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2}}-\frac {\left (6 A B \,b^{2} d^{2}+11 B^{2} b^{2} d^{2}\right ) x^{2}}{9 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) g}+\frac {B^{2} c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{3 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) g}+\frac {c B \left (18 A \,a^{2} d^{2}-18 A a b c d +6 A \,b^{2} c^{2}+18 B \,a^{2} d^{2}-9 B a b c d +2 B \,b^{2} c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{9 g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {B d \left (6 A \,a^{2} d^{2}+6 B \,a^{2} d^{2}+6 B a b c d -B \,b^{2} c^{2}\right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {b^{2} d^{3} B^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{3 g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {b^{2} d^{3} B \left (6 A +11 B \right ) x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{9 g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (6 A a d +9 B a d +2 B b c \right ) B b \,d^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{g^{3} \left (b x +a \right )^{3}}\)	\(929\)
parallelrisch	\(-\frac {66 A B \,a^{3} b^{4} d^{4}-12 A B \,b^{7} c^{3} d -108 B^{2} a^{2} b^{5} c \,d^{3}+27 B^{2} a \,b^{6} c^{2} d^{2}-108 A B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} d^{4}-108 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{5} d^{4}-108 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} c \,d^{3}-108 A B x a \,b^{6} c \,d^{3}-108 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{5} c \,d^{3}-54 A^{2} a^{2} b^{5} c \,d^{3}+54 A^{2} a \,b^{6} c^{2} d^{2}-36 A B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} d^{4}-54 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{6} d^{4}-18 B^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{7} d^{4}-66 B^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} d^{4}+66 B^{2} x^{2} a \,b^{6} d^{4}-66 B^{2} x^{2} b^{7} c \,d^{3}-18 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{7} c^{3} d +147 B^{2} x \,a^{2} b^{5} d^{4}+15 B^{2} x \,b^{7} c^{2} d^{2}-12 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} c^{3} d -108 A B \,a^{2} b^{5} c \,d^{3}+54 A B a \,b^{6} c^{2} d^{2}-162 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} d^{4}-36 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} c \,d^{3}-54 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{2} b^{5} d^{4}+36 A B \,x^{2} a \,b^{6} d^{4}-36 A B \,x^{2} b^{7} c \,d^{3}-108 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{5} d^{4}+18 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} c^{2} d^{2}-54 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{2} b^{5} c \,d^{3}+54 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{6} c^{2} d^{2}+90 A B x \,a^{2} b^{5} d^{4}+18 A B x \,b^{7} c^{2} d^{2}-36 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} c^{3} d -162 B^{2} x a \,b^{6} c \,d^{3}-108 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{5} c \,d^{3}+54 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} c^{2} d^{2}+108 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} c^{2} d^{2}+18 A^{2} a^{3} b^{4} d^{4}-18 A^{2} b^{7} c^{3} d +85 B^{2} a^{3} b^{4} d^{4}-4 B^{2} b^{7} c^{3} d}{54 g^{4} \left (b x +a \right )^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a d -c b \right ) b^{5} d}\)	\(983\)
derivativedivides	\(\text {Expression too large to display}\)	\(1039\)
default	\(\text {Expression too large to display}\)	\(1039\)
risch	\(\text {Expression too large to display}\)	\(2290\)

[In]

int((A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*A^2/g^4/(b*x+a)^3/b-B^2/g^4/d^2*(a*d-b*c)*e*(d^4/(a*d-b*c)^4*(-1/(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(
a*d-b*c)*e/d/(d*x+c))^2-2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-2/(b*e/d+(a*d-b*c)*e/d
/(d*x+c)))-2*d^3/(a*d-b*c)^4*b*e*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-1/2/(
b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2)+d^2/(a*d-b
*c)^4*e^2*b^2*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-2/9/(b*e/d+(a*d-b*c)*e/d
/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-2/27/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3))-2*B*A/g^4/d^2*(a*d-b*c)*e*(
d^4/(a*d-b*c)^4*(-1/(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/(b*e/d+(a*d-b*c)*e/d/(d*x+
c)))-2*d^3/(a*d-b*c)^4*b*e*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a
*d-b*c)*e/d/(d*x+c))^2)+d^2/(a*d-b*c)^4*e^2*b^2*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(
d*x+c))-1/9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.61 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {2 \, {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c^{3} - 27 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a b^{2} c^{2} d + 54 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a^{2} b c d^{2} - {\left (18 \, A^{2} + 66 \, A B + 85 \, B^{2}\right )} a^{3} d^{3} + 6 \, {\left ({\left (6 \, A B + 11 \, B^{2}\right )} b^{3} c d^{2} - {\left (6 \, A B + 11 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 18 \, {\left (B^{2} b^{3} d^{3} x^{3} + 3 \, B^{2} a b^{2} d^{3} x^{2} + 3 \, B^{2} a^{2} b d^{3} x + B^{2} b^{3} c^{3} - 3 \, B^{2} a b^{2} c^{2} d + 3 \, B^{2} a^{2} b c d^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 3 \, {\left ({\left (6 \, A B + 5 \, B^{2}\right )} b^{3} c^{2} d - 18 \, {\left (2 \, A B + 3 \, B^{2}\right )} a b^{2} c d^{2} + {\left (30 \, A B + 49 \, B^{2}\right )} a^{2} b d^{3}\right )} x + 6 \, {\left ({\left (6 \, A B + 11 \, B^{2}\right )} b^{3} d^{3} x^{3} + 2 \, {\left (3 \, A B + B^{2}\right )} b^{3} c^{3} - 9 \, {\left (2 \, A B + B^{2}\right )} a b^{2} c^{2} d + 18 \, {\left (A B + B^{2}\right )} a^{2} b c d^{2} + 3 \, {\left (2 \, B^{2} b^{3} c d^{2} + 3 \, {\left (2 \, A B + 3 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (B^{2} b^{3} c^{2} d - 6 \, B^{2} a b^{2} c d^{2} - 6 \, {\left (A B + B^{2}\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{54 \, {\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} g^{4} x + {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} g^{4}\right )}} \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algorithm="fricas")

[Out]

-1/54*(2*(9*A^2 + 6*A*B + 2*B^2)*b^3*c^3 - 27*(2*A^2 + 2*A*B + B^2)*a*b^2*c^2*d + 54*(A^2 + 2*A*B + 2*B^2)*a^2
*b*c*d^2 - (18*A^2 + 66*A*B + 85*B^2)*a^3*d^3 + 6*((6*A*B + 11*B^2)*b^3*c*d^2 - (6*A*B + 11*B^2)*a*b^2*d^3)*x^
2 + 18*(B^2*b^3*d^3*x^3 + 3*B^2*a*b^2*d^3*x^2 + 3*B^2*a^2*b*d^3*x + B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a^
2*b*c*d^2)*log((b*e*x + a*e)/(d*x + c))^2 - 3*((6*A*B + 5*B^2)*b^3*c^2*d - 18*(2*A*B + 3*B^2)*a*b^2*c*d^2 + (3
0*A*B + 49*B^2)*a^2*b*d^3)*x + 6*((6*A*B + 11*B^2)*b^3*d^3*x^3 + 2*(3*A*B + B^2)*b^3*c^3 - 9*(2*A*B + B^2)*a*b
^2*c^2*d + 18*(A*B + B^2)*a^2*b*c*d^2 + 3*(2*B^2*b^3*c*d^2 + 3*(2*A*B + 3*B^2)*a*b^2*d^3)*x^2 - 3*(B^2*b^3*c^2
*d - 6*B^2*a*b^2*c*d^2 - 6*(A*B + B^2)*a^2*b*d^3)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^7*c^3 - 3*a*b^6*c^2*d +
3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*g^4*x^3 + 3*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*g^4*
x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*g^4*x + (a^3*b^4*c^3 - 3*a^4*b^3*c^2*d
+ 3*a^5*b^2*c*d^2 - a^6*b*d^3)*g^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1544 vs. \(2 (384) = 768\).

Time = 11.68 (sec) , antiderivative size = 1544, normalized size of antiderivative = 3.69 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**4,x)

[Out]

-B*d**3*(6*A + 11*B)*log(x + (6*A*B*a*d**4 + 6*A*B*b*c*d**3 + 11*B**2*a*d**4 + 11*B**2*b*c*d**3 - B*a**4*d**7*
(6*A + 11*B)/(a*d - b*c)**3 + 4*B*a**3*b*c*d**6*(6*A + 11*B)/(a*d - b*c)**3 - 6*B*a**2*b**2*c**2*d**5*(6*A + 1
1*B)/(a*d - b*c)**3 + 4*B*a*b**3*c**3*d**4*(6*A + 11*B)/(a*d - b*c)**3 - B*b**4*c**4*d**3*(6*A + 11*B)/(a*d -
b*c)**3)/(12*A*B*b*d**4 + 22*B**2*b*d**4))/(9*b*g**4*(a*d - b*c)**3) + B*d**3*(6*A + 11*B)*log(x + (6*A*B*a*d*
*4 + 6*A*B*b*c*d**3 + 11*B**2*a*d**4 + 11*B**2*b*c*d**3 + B*a**4*d**7*(6*A + 11*B)/(a*d - b*c)**3 - 4*B*a**3*b
*c*d**6*(6*A + 11*B)/(a*d - b*c)**3 + 6*B*a**2*b**2*c**2*d**5*(6*A + 11*B)/(a*d - b*c)**3 - 4*B*a*b**3*c**3*d*
*4*(6*A + 11*B)/(a*d - b*c)**3 + B*b**4*c**4*d**3*(6*A + 11*B)/(a*d - b*c)**3)/(12*A*B*b*d**4 + 22*B**2*b*d**4
))/(9*b*g**4*(a*d - b*c)**3) + (3*B**2*a**2*c*d**2 + 3*B**2*a**2*d**3*x - 3*B**2*a*b*c**2*d + 3*B**2*a*b*d**3*
x**2 + B**2*b**2*c**3 + B**2*b**2*d**3*x**3)*log(e*(a + b*x)/(c + d*x))**2/(3*a**6*d**3*g**4 - 9*a**5*b*c*d**2
*g**4 + 9*a**5*b*d**3*g**4*x + 9*a**4*b**2*c**2*d*g**4 - 27*a**4*b**2*c*d**2*g**4*x + 9*a**4*b**2*d**3*g**4*x*
*2 - 3*a**3*b**3*c**3*g**4 + 27*a**3*b**3*c**2*d*g**4*x - 27*a**3*b**3*c*d**2*g**4*x**2 + 3*a**3*b**3*d**3*g**
4*x**3 - 9*a**2*b**4*c**3*g**4*x + 27*a**2*b**4*c**2*d*g**4*x**2 - 9*a**2*b**4*c*d**2*g**4*x**3 - 9*a*b**5*c**
3*g**4*x**2 + 9*a*b**5*c**2*d*g**4*x**3 - 3*b**6*c**3*g**4*x**3) + (-6*A*B*a**2*d**2 + 12*A*B*a*b*c*d - 6*A*B*
b**2*c**2 - 11*B**2*a**2*d**2 + 7*B**2*a*b*c*d - 15*B**2*a*b*d**2*x - 2*B**2*b**2*c**2 + 3*B**2*b**2*c*d*x - 6
*B**2*b**2*d**2*x**2)*log(e*(a + b*x)/(c + d*x))/(9*a**5*b*d**2*g**4 - 18*a**4*b**2*c*d*g**4 + 27*a**4*b**2*d*
*2*g**4*x + 9*a**3*b**3*c**2*g**4 - 54*a**3*b**3*c*d*g**4*x + 27*a**3*b**3*d**2*g**4*x**2 + 27*a**2*b**4*c**2*
g**4*x - 54*a**2*b**4*c*d*g**4*x**2 + 9*a**2*b**4*d**2*g**4*x**3 + 27*a*b**5*c**2*g**4*x**2 - 18*a*b**5*c*d*g*
*4*x**3 + 9*b**6*c**2*g**4*x**3) - (18*A**2*a**2*d**2 - 36*A**2*a*b*c*d + 18*A**2*b**2*c**2 + 66*A*B*a**2*d**2
- 42*A*B*a*b*c*d + 12*A*B*b**2*c**2 + 85*B**2*a**2*d**2 - 23*B**2*a*b*c*d + 4*B**2*b**2*c**2 + x**2*(36*A*B*b
**2*d**2 + 66*B**2*b**2*d**2) + x*(90*A*B*a*b*d**2 - 18*A*B*b**2*c*d + 147*B**2*a*b*d**2 - 15*B**2*b**2*c*d))/
(54*a**5*b*d**2*g**4 - 108*a**4*b**2*c*d*g**4 + 54*a**3*b**3*c**2*g**4 + x**3*(54*a**2*b**4*d**2*g**4 - 108*a*
b**5*c*d*g**4 + 54*b**6*c**2*g**4) + x**2*(162*a**3*b**3*d**2*g**4 - 324*a**2*b**4*c*d*g**4 + 162*a*b**5*c**2*
g**4) + x*(162*a**4*b**2*d**2*g**4 - 324*a**3*b**3*c*d*g**4 + 162*a**2*b**4*c**2*g**4))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1419 vs. \(2 (410) = 820\).

Time = 0.29 (sec) , antiderivative size = 1419, normalized size of antiderivative = 3.39 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algorithm="maxima")

[Out]

-1/54*(6*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5
*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3
*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*
a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d
^2 - a^3*b*d^3)*g^4))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (4*b^3*c^3 - 27*a*b^2*c^2*d + 108*a^2*b*c*d^2 - 8
5*a^3*d^3 + 66*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(
b*x + a)^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(d*x + c)^2 - 3*(5*b^3*c^2*d - 54
*a*b^2*c*d^2 + 49*a^2*b*d^3)*x + 66*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a) - 6
*(11*b^3*d^3*x^3 + 33*a*b^2*d^3*x^2 + 33*a^2*b*d^3*x + 11*a^3*d^3 - 6*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b
*d^3*x + a^3*d^3)*log(b*x + a))*log(d*x + c))/(a^3*b^4*c^3*g^4 - 3*a^4*b^3*c^2*d*g^4 + 3*a^5*b^2*c*d^2*g^4 - a
^6*b*d^3*g^4 + (b^7*c^3*g^4 - 3*a*b^6*c^2*d*g^4 + 3*a^2*b^5*c*d^2*g^4 - a^3*b^4*d^3*g^4)*x^3 + 3*(a*b^6*c^3*g^
4 - 3*a^2*b^5*c^2*d*g^4 + 3*a^3*b^4*c*d^2*g^4 - a^4*b^3*d^3*g^4)*x^2 + 3*(a^2*b^5*c^3*g^4 - 3*a^3*b^4*c^2*d*g^
4 + 3*a^4*b^3*c*d^2*g^4 - a^5*b^2*d^3*g^4)*x))*B^2 - 1/9*A*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*
d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d
+ a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d +
a^5*b*d^2)*g^4) + 6*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a
^3*b*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log(d*x +
c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4)) - 1/3*B^2*log(b*e*x/(d*x + c) + a*e/(d*x +
c))^2/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) - 1/3*A^2/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 +
3*a^2*b^2*g^4*x + a^3*b*g^4)

Giac [A] (verification not implemented)

none

Time = 0.51 (sec) , antiderivative size = 725, normalized size of antiderivative = 1.73 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {1}{54} \, {\left (\frac {18 \, {\left (B^{2} b^{2} e^{4} - \frac {3 \, {\left (b e x + a e\right )} B^{2} b d e^{3}}{d x + c} + \frac {3 \, {\left (b e x + a e\right )}^{2} B^{2} d^{2} e^{2}}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{\frac {{\left (b e x + a e\right )}^{3} b^{2} c^{2} g^{4}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b e x + a e\right )}^{3} a b c d g^{4}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b e x + a e\right )}^{3} a^{2} d^{2} g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {6 \, {\left (6 \, A B b^{2} e^{4} + 2 \, B^{2} b^{2} e^{4} - \frac {18 \, {\left (b e x + a e\right )} A B b d e^{3}}{d x + c} - \frac {9 \, {\left (b e x + a e\right )} B^{2} b d e^{3}}{d x + c} + \frac {18 \, {\left (b e x + a e\right )}^{2} A B d^{2} e^{2}}{{\left (d x + c\right )}^{2}} + \frac {18 \, {\left (b e x + a e\right )}^{2} B^{2} d^{2} e^{2}}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{3} b^{2} c^{2} g^{4}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b e x + a e\right )}^{3} a b c d g^{4}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b e x + a e\right )}^{3} a^{2} d^{2} g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {18 \, A^{2} b^{2} e^{4} + 12 \, A B b^{2} e^{4} + 4 \, B^{2} b^{2} e^{4} - \frac {54 \, {\left (b e x + a e\right )} A^{2} b d e^{3}}{d x + c} - \frac {54 \, {\left (b e x + a e\right )} A B b d e^{3}}{d x + c} - \frac {27 \, {\left (b e x + a e\right )} B^{2} b d e^{3}}{d x + c} + \frac {54 \, {\left (b e x + a e\right )}^{2} A^{2} d^{2} e^{2}}{{\left (d x + c\right )}^{2}} + \frac {108 \, {\left (b e x + a e\right )}^{2} A B d^{2} e^{2}}{{\left (d x + c\right )}^{2}} + \frac {108 \, {\left (b e x + a e\right )}^{2} B^{2} d^{2} e^{2}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b e x + a e\right )}^{3} b^{2} c^{2} g^{4}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b e x + a e\right )}^{3} a b c d g^{4}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b e x + a e\right )}^{3} a^{2} d^{2} g^{4}}{{\left (d x + c\right )}^{3}}}\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)^4,x, algorithm="giac")

[Out]

-1/54*(18*(B^2*b^2*e^4 - 3*(b*e*x + a*e)*B^2*b*d*e^3/(d*x + c) + 3*(b*e*x + a*e)^2*B^2*d^2*e^2/(d*x + c)^2)*lo
g((b*e*x + a*e)/(d*x + c))^2/((b*e*x + a*e)^3*b^2*c^2*g^4/(d*x + c)^3 - 2*(b*e*x + a*e)^3*a*b*c*d*g^4/(d*x + c
)^3 + (b*e*x + a*e)^3*a^2*d^2*g^4/(d*x + c)^3) + 6*(6*A*B*b^2*e^4 + 2*B^2*b^2*e^4 - 18*(b*e*x + a*e)*A*B*b*d*e
^3/(d*x + c) - 9*(b*e*x + a*e)*B^2*b*d*e^3/(d*x + c) + 18*(b*e*x + a*e)^2*A*B*d^2*e^2/(d*x + c)^2 + 18*(b*e*x
+ a*e)^2*B^2*d^2*e^2/(d*x + c)^2)*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^3*b^2*c^2*g^4/(d*x + c)^3 - 2*(b
*e*x + a*e)^3*a*b*c*d*g^4/(d*x + c)^3 + (b*e*x + a*e)^3*a^2*d^2*g^4/(d*x + c)^3) + (18*A^2*b^2*e^4 + 12*A*B*b^
2*e^4 + 4*B^2*b^2*e^4 - 54*(b*e*x + a*e)*A^2*b*d*e^3/(d*x + c) - 54*(b*e*x + a*e)*A*B*b*d*e^3/(d*x + c) - 27*(
b*e*x + a*e)*B^2*b*d*e^3/(d*x + c) + 54*(b*e*x + a*e)^2*A^2*d^2*e^2/(d*x + c)^2 + 108*(b*e*x + a*e)^2*A*B*d^2*
e^2/(d*x + c)^2 + 108*(b*e*x + a*e)^2*B^2*d^2*e^2/(d*x + c)^2)/((b*e*x + a*e)^3*b^2*c^2*g^4/(d*x + c)^3 - 2*(b
*e*x + a*e)^3*a*b*c*d*g^4/(d*x + c)^3 + (b*e*x + a*e)^3*a^2*d^2*g^4/(d*x + c)^3))*(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 = 4.43 (sec) , antiderivative size = 1064, normalized size of antiderivative = 2.55 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=\frac {\frac {18\,A^2\,a^2\,d^2-36\,A^2\,a\,b\,c\,d+18\,A^2\,b^2\,c^2+66\,A\,B\,a^2\,d^2-42\,A\,B\,a\,b\,c\,d+12\,A\,B\,b^2\,c^2+85\,B^2\,a^2\,d^2-23\,B^2\,a\,b\,c\,d+4\,B^2\,b^2\,c^2}{6\,\left (a\,d-b\,c\right )}+\frac {x\,\left (-5\,c\,B^2\,b^2\,d+49\,a\,B^2\,b\,d^2-6\,A\,c\,B\,b^2\,d+30\,A\,a\,B\,b\,d^2\right )}{2\,\left (a\,d-b\,c\right )}+\frac {d\,x^2\,\left (11\,d\,B^2\,b^2+6\,A\,d\,B\,b^2\right )}{a\,d-b\,c}}{x\,\left (27\,a^2\,b^3\,c\,g^4-27\,a^3\,b^2\,d\,g^4\right )-x^2\,\left (27\,a^2\,b^3\,d\,g^4-27\,a\,b^4\,c\,g^4\right )+x^3\,\left (9\,b^5\,c\,g^4-9\,a\,b^4\,d\,g^4\right )+9\,a^3\,b^2\,c\,g^4-9\,a^4\,b\,d\,g^4}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B^2}{3\,b^2\,g^4\,\left (3\,a^2\,x+\frac {a^3}{b}+b^2\,x^3+3\,a\,b\,x^2\right )}-\frac {B^2\,d^3}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {2\,A\,B}{3\,b^2\,d\,g^4}+\frac {2\,B^2\,d^3\,\left (a\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^3\,d^3-6\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d-b^3\,c^3}{3\,b\,d^4}\right )}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {2\,B^2\,d^3\,x^2\,\left (\frac {b^2\,c-a\,b\,d}{3\,d^2}-\frac {2\,b\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {2\,B^2\,d^3\,x\,\left (b\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{3\,d^3}+\frac {2\,a\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )}{\frac {3\,a^2\,x}{d}+\frac {a^3}{b\,d}+\frac {b^2\,x^3}{d}+\frac {3\,a\,b\,x^2}{d}}-\frac {B\,d^3\,\mathrm {atan}\left (\frac {B\,d^3\,\left (\frac {a^3\,b\,d^3\,g^4-a^2\,b^2\,c\,d^2\,g^4-a\,b^3\,c^2\,d\,g^4+b^4\,c^3\,g^4}{a^2\,b\,d^2\,g^4-2\,a\,b^2\,c\,d\,g^4+b^3\,c^2\,g^4}+2\,b\,d\,x\right )\,\left (6\,A+11\,B\right )\,\left (a^2\,b\,d^2\,g^4-2\,a\,b^2\,c\,d\,g^4+b^3\,c^2\,g^4\right )\,1{}\mathrm {i}}{b\,g^4\,{\left (a\,d-b\,c\right )}^3\,\left (11\,B^2\,d^3+6\,A\,B\,d^3\right )}\right )\,\left (6\,A+11\,B\right )\,2{}\mathrm {i}}{9\,b\,g^4\,{\left (a\,d-b\,c\right )}^3} \]

[In]

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

[Out]

((18*A^2*a^2*d^2 + 18*A^2*b^2*c^2 + 85*B^2*a^2*d^2 + 4*B^2*b^2*c^2 + 66*A*B*a^2*d^2 + 12*A*B*b^2*c^2 - 36*A^2*
a*b*c*d - 23*B^2*a*b*c*d - 42*A*B*a*b*c*d)/(6*(a*d - b*c)) + (x*(49*B^2*a*b*d^2 - 5*B^2*b^2*c*d + 30*A*B*a*b*d
^2 - 6*A*B*b^2*c*d))/(2*(a*d - b*c)) + (d*x^2*(11*B^2*b^2*d + 6*A*B*b^2*d))/(a*d - b*c))/(x*(27*a^2*b^3*c*g^4
- 27*a^3*b^2*d*g^4) - x^2*(27*a^2*b^3*d*g^4 - 27*a*b^4*c*g^4) + x^3*(9*b^5*c*g^4 - 9*a*b^4*d*g^4) + 9*a^3*b^2*
c*g^4 - 9*a^4*b*d*g^4) - log((e*(a + b*x))/(c + d*x))^2*(B^2/(3*b^2*g^4*(3*a^2*x + a^3/b + b^2*x^3 + 3*a*b*x^2
)) - (B^2*d^3)/(3*b*g^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - (log((e*(a + b*x))/(c + d*x))*
((2*A*B)/(3*b^2*d*g^4) + (2*B^2*d^3*(a*((3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(3*b*d^2
)) + (3*a^3*d^3 - b^3*c^3 + 4*a*b^2*c^2*d - 6*a^2*b*c*d^2)/(3*b*d^4)))/(3*b*g^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c
^2*d - 3*a^2*b*c*d^2)) - (2*B^2*d^3*x^2*((b^2*c - a*b*d)/(3*d^2) - (2*b*(a*d - b*c))/(3*d^2)))/(3*b*g^4*(a^3*d
^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (2*B^2*d^3*x*(b*((3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(6*b*d^3)
+ (a*(a*d - b*c))/(3*b*d^2)) + (3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(3*d^3) + (2*a*(a*d - b*c))/(3*d^2)))/(3*b*g^
4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))))/((3*a^2*x)/d + a^3/(b*d) + (b^2*x^3)/d + (3*a*b*x^2)/
d) - (B*d^3*atan((B*d^3*((b^4*c^3*g^4 + a^3*b*d^3*g^4 - a*b^3*c^2*d*g^4 - a^2*b^2*c*d^2*g^4)/(b^3*c^2*g^4 + a^
2*b*d^2*g^4 - 2*a*b^2*c*d*g^4) + 2*b*d*x)*(6*A + 11*B)*(b^3*c^2*g^4 + a^2*b*d^2*g^4 - 2*a*b^2*c*d*g^4)*1i)/(b*
g^4*(a*d - b*c)^3*(11*B^2*d^3 + 6*A*B*d^3)))*(6*A + 11*B)*2i)/(9*b*g^4*(a*d - b*c)^3)