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

Optimal result

Integrand size = 32, antiderivative size = 296 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=-\frac {B^2 d (a+b x)^2}{4 (b c-a d)^2 i^3 (c+d x)^2}-\frac {2 A b B (a+b x)}{(b c-a d)^2 i^3 (c+d x)}+\frac {2 b B^2 (a+b x)}{(b c-a d)^2 i^3 (c+d x)}-\frac {2 b B^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d)^2 i^3 (c+d x)}+\frac {B d (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^2 i^3 (c+d x)^2}-\frac {d (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^2 i^3 (c+d x)^2}+\frac {b (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^2 i^3 (c+d x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2552, 2367, 2333, 2332, 2342, 2341} \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=\frac {B d (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 i^3 (c+d x)^2 (b c-a d)^2}+\frac {b (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{i^3 (c+d x) (b c-a d)^2}-\frac {d (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 i^3 (c+d x)^2 (b c-a d)^2}-\frac {2 A b B (a+b x)}{i^3 (c+d x) (b c-a d)^2}-\frac {2 b B^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{i^3 (c+d x) (b c-a d)^2}+\frac {2 b B^2 (a+b x)}{i^3 (c+d x) (b c-a d)^2}-\frac {B^2 d (a+b x)^2}{4 i^3 (c+d x)^2 (b c-a d)^2} \]

[In]

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

[Out]

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

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

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

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 2367

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

Rule 2552

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/d)^m, Subst[Int[(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[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])

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

Mathematica [C] (verified)

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

Time = 0.23 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.50 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=\frac {-2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B \left (2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 b (b c-a d) (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 b^2 (c+d x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-4 b^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-4 b B (c+d x) (b c-a d+b (c+d x) \log (a+b x)-b (c+d x) \log (c+d x))-B \left ((b c-a d)^2+2 b (b c-a d) (c+d x)+2 b^2 (c+d x)^2 \log (a+b x)-2 b^2 (c+d x)^2 \log (c+d x)\right )-2 b^2 B (c+d x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+2 b^2 B (c+d x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{(b c-a d)^2}}{4 d i^3 (c+d x)^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.63

method	result	size
norman	\(\frac {\frac {B b \left (2 A b c -B a d -2 B b c \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {B^{2} b^{2} c x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (2 A^{2} a d -2 A^{2} b c -2 A B a d +4 A B b c +B^{2} a d -4 B^{2} b c \right ) x}{2 c i \left (a d -c b \right )}-\frac {B a \left (2 A a d -4 A b c -B a d +4 B b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {B^{2} a \left (a d -2 c b \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (2 A^{2} a d -2 A^{2} b c -2 A B a d +6 A B b c +B^{2} a d -7 B^{2} b c \right ) d \,x^{2}}{4 c^{2} i \left (a d -c b \right )}+\frac {b^{2} d \,B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {d B \,b^{2} \left (2 A -3 B \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{i^{2} \left (d x +c \right )^{2}}\)	\(483\)
parts	\(-\frac {A^{2}}{2 i^{3} \left (d x +c \right )^{2} d}-\frac {B^{2} d \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}+\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}-\frac {b e \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}-2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (a d -c b \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{d}\right )}{i^{3} \left (a d -c b \right )^{2} e^{2}}-\frac {2 B A d \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}-\frac {b e \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{d}\right )}{i^{3} \left (a d -c b \right )^{2} e^{2}}\)	\(549\)
parallelrisch	\(-\frac {-8 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} c \,d^{4}-8 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{2} c \,d^{4}-4 A^{2} a \,b^{2} c \,d^{4}-2 A B \,a^{2} b \,d^{5}-6 A B \,b^{3} c^{2} d^{3}-8 B^{2} a \,b^{2} c \,d^{4}+8 A B a \,b^{2} c \,d^{4}-4 A B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{5}-4 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{3} c \,d^{4}+4 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{2} d^{5}+8 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} c \,d^{4}-4 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{2} c \,d^{4}+4 A B x a \,b^{2} d^{5}-4 A B x \,b^{3} c \,d^{4}+4 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b \,d^{5}+8 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{2} c \,d^{4}-2 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{3} d^{5}+6 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{5}+2 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{2} b \,d^{5}-6 B^{2} x a \,b^{2} d^{5}+6 B^{2} x \,b^{3} c \,d^{4}-2 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b \,d^{5}+2 A^{2} a^{2} b \,d^{5}+2 A^{2} b^{3} c^{2} d^{3}+B^{2} a^{2} b \,d^{5}+7 B^{2} b^{3} c^{2} d^{3}}{4 i^{3} \left (d x +c \right )^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b \,d^{4}}\)	\(574\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} A^{2} b \left (\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} e^{2} i^{3}}+\frac {d^{3} A^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{3} e^{3} i^{3}}-\frac {2 d^{2} A B b \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (a d -c b \right )^{3} e^{2} i^{3}}+\frac {2 d^{3} A B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{3} e^{3} i^{3}}-\frac {d^{2} B^{2} b \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \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 ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (a d -c b \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{\left (a d -c b \right )^{3} e^{2} i^{3}}+\frac {d^{3} B^{2} \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}+\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{3} e^{3} i^{3}}\right )}{d^{2}}\)	\(688\)
default	\(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} A^{2} b \left (\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} e^{2} i^{3}}+\frac {d^{3} A^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{3} e^{3} i^{3}}-\frac {2 d^{2} A B b \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (a d -c b \right )^{3} e^{2} i^{3}}+\frac {2 d^{3} A B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{3} e^{3} i^{3}}-\frac {d^{2} B^{2} b \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \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 ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (a d -c b \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{\left (a d -c b \right )^{3} e^{2} i^{3}}+\frac {d^{3} B^{2} \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}+\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{3} e^{3} i^{3}}\right )}{d^{2}}\)	\(688\)
risch	\(\text {Expression too large to display}\)	\(1158\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.26 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=-\frac {{\left (2 \, A^{2} - 6 \, A B + 7 \, B^{2}\right )} b^{2} c^{2} - 4 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a b c d + {\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} a^{2} d^{2} - 2 \, {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} b^{2} c d x + 2 \, B^{2} a b c d - B^{2} a^{2} d^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 2 \, {\left ({\left (2 \, A B - 3 \, B^{2}\right )} b^{2} c d - {\left (2 \, A B - 3 \, B^{2}\right )} a b d^{2}\right )} x - 2 \, {\left ({\left (2 \, A B - 3 \, B^{2}\right )} b^{2} d^{2} x^{2} + 4 \, {\left (A B - B^{2}\right )} a b c d - {\left (2 \, A B - B^{2}\right )} a^{2} d^{2} - 2 \, {\left (B^{2} a b d^{2} - 2 \, {\left (A B - B^{2}\right )} b^{2} c d\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} i^{3} x^{2} + 2 \, {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} i^{3} x + {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} i^{3}\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 892 vs. \(2 (269) = 538\).

Time = 2.21 (sec) , antiderivative size = 892, normalized size of antiderivative = 3.01 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=- \frac {B b^{2} \cdot \left (2 A - 3 B\right ) \log {\left (x + \frac {2 A B a b^{2} d + 2 A B b^{3} c - 3 B^{2} a b^{2} d - 3 B^{2} b^{3} c - \frac {B a^{3} b^{2} d^{3} \cdot \left (2 A - 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {3 B a^{2} b^{3} c d^{2} \cdot \left (2 A - 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {3 B a b^{4} c^{2} d \left (2 A - 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {B b^{5} c^{3} \cdot \left (2 A - 3 B\right )}{\left (a d - b c\right )^{2}}}{4 A B b^{3} d - 6 B^{2} b^{3} d} \right )}}{2 d i^{3} \left (a d - b c\right )^{2}} + \frac {B b^{2} \cdot \left (2 A - 3 B\right ) \log {\left (x + \frac {2 A B a b^{2} d + 2 A B b^{3} c - 3 B^{2} a b^{2} d - 3 B^{2} b^{3} c + \frac {B a^{3} b^{2} d^{3} \cdot \left (2 A - 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {3 B a^{2} b^{3} c d^{2} \cdot \left (2 A - 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {3 B a b^{4} c^{2} d \left (2 A - 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {B b^{5} c^{3} \cdot \left (2 A - 3 B\right )}{\left (a d - b c\right )^{2}}}{4 A B b^{3} d - 6 B^{2} b^{3} d} \right )}}{2 d i^{3} \left (a d - b c\right )^{2}} + \frac {\left (- B^{2} a^{2} d + 2 B^{2} a b c + 2 B^{2} b^{2} c x + B^{2} b^{2} d x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{2} c^{2} d^{2} i^{3} + 4 a^{2} c d^{3} i^{3} x + 2 a^{2} d^{4} i^{3} x^{2} - 4 a b c^{3} d i^{3} - 8 a b c^{2} d^{2} i^{3} x - 4 a b c d^{3} i^{3} x^{2} + 2 b^{2} c^{4} i^{3} + 4 b^{2} c^{3} d i^{3} x + 2 b^{2} c^{2} d^{2} i^{3} x^{2}} + \frac {\left (- 2 A B a d + 2 A B b c + B^{2} a d - 3 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 c^{2} d^{2} i^{3} + 4 a c d^{3} i^{3} x + 2 a d^{4} i^{3} x^{2} - 2 b c^{3} d i^{3} - 4 b c^{2} d^{2} i^{3} x - 2 b c d^{3} i^{3} x^{2}} + \frac {- 2 A^{2} a d + 2 A^{2} b c + 2 A B a d - 6 A B b c - B^{2} a d + 7 B^{2} b c + x \left (- 4 A B b d + 6 B^{2} b d\right )}{4 a c^{2} d^{2} i^{3} - 4 b c^{3} d i^{3} + x^{2} \cdot \left (4 a d^{4} i^{3} - 4 b c d^{3} i^{3}\right ) + x \left (8 a c d^{3} i^{3} - 8 b c^{2} d^{2} i^{3}\right )} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 848 vs. \(2 (290) = 580\).

Time = 0.23 (sec) , antiderivative size = 848, normalized size of antiderivative = 2.86 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=\frac {1}{4} \, {\left (2 \, {\left (\frac {2 \, b d x + 3 \, b c - a d}{{\left (b c d^{3} - a d^{4}\right )} i^{3} x^{2} + 2 \, {\left (b c^{2} d^{2} - a c d^{3}\right )} i^{3} x + {\left (b c^{3} d - a c^{2} d^{2}\right )} i^{3}} + \frac {2 \, b^{2} \log \left (b x + a\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}} - \frac {2 \, b^{2} \log \left (d x + c\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {7 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{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 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c d x + 3 \, b^{2} c^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{b^{2} c^{4} d i^{3} - 2 \, a b c^{3} d^{2} i^{3} + a^{2} c^{2} d^{3} i^{3} + {\left (b^{2} c^{2} d^{3} i^{3} - 2 \, a b c d^{4} i^{3} + a^{2} d^{5} i^{3}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d^{2} i^{3} - 2 \, a b c^{2} d^{3} i^{3} + a^{2} c d^{4} i^{3}\right )} x}\right )} B^{2} + \frac {1}{2} \, A B {\left (\frac {2 \, b d x + 3 \, b c - a d}{{\left (b c d^{3} - a d^{4}\right )} i^{3} x^{2} + 2 \, {\left (b c^{2} d^{2} - a c d^{3}\right )} i^{3} x + {\left (b c^{3} d - a c^{2} d^{2}\right )} i^{3}} - \frac {2 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{d^{3} i^{3} x^{2} + 2 \, c d^{2} i^{3} x + c^{2} d i^{3}} + \frac {2 \, b^{2} \log \left (b x + a\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}} - \frac {2 \, b^{2} \log \left (d x + c\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}}\right )} - \frac {B^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )^{2}}{2 \, {\left (d^{3} i^{3} x^{2} + 2 \, c d^{2} i^{3} x + c^{2} d i^{3}\right )}} - \frac {A^{2}}{2 \, {\left (d^{3} i^{3} x^{2} + 2 \, c d^{2} i^{3} x + c^{2} d i^{3}\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.25 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=\frac {1}{4} \, {\left (2 \, {\left (\frac {2 \, {\left (b e x + a e\right )} B^{2} b}{{\left (b c i^{3} - a d i^{3}\right )} {\left (d x + c\right )}} - \frac {{\left (b e x + a e\right )}^{2} B^{2} d}{{\left (b c e i^{3} - a d e i^{3}\right )} {\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 2 \, {\left (\frac {{\left (2 \, A B d - B^{2} d\right )} {\left (b e x + a e\right )}^{2}}{{\left (b c e i^{3} - a d e i^{3}\right )} {\left (d x + c\right )}^{2}} - \frac {4 \, {\left (A B b - B^{2} b\right )} {\left (b e x + a e\right )}}{{\left (b c i^{3} - a d i^{3}\right )} {\left (d x + c\right )}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right ) - \frac {{\left (2 \, A^{2} d - 2 \, A B d + B^{2} d\right )} {\left (b e x + a e\right )}^{2}}{{\left (b c e i^{3} - a d e i^{3}\right )} {\left (d x + c\right )}^{2}} + \frac {4 \, {\left (A^{2} b - 2 \, A B b + 2 \, B^{2} b\right )} {\left (b e x + a e\right )}}{{\left (b c i^{3} - a d i^{3}\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]

[In]

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

[Out]

1/4*(2*(2*(b*e*x + a*e)*B^2*b/((b*c*i^3 - a*d*i^3)*(d*x + c)) - (b*e*x + a*e)^2*B^2*d/((b*c*e*i^3 - a*d*e*i^3)
*(d*x + c)^2))*log((b*e*x + a*e)/(d*x + c))^2 - 2*((2*A*B*d - B^2*d)*(b*e*x + a*e)^2/((b*c*e*i^3 - a*d*e*i^3)*
(d*x + c)^2) - 4*(A*B*b - B^2*b)*(b*e*x + a*e)/((b*c*i^3 - a*d*i^3)*(d*x + c)))*log((b*e*x + a*e)/(d*x + c)) -
(2*A^2*d - 2*A*B*d + B^2*d)*(b*e*x + a*e)^2/((b*c*e*i^3 - a*d*e*i^3)*(d*x + c)^2) + 4*(A^2*b - 2*A*B*b + 2*B^
2*b)*(b*e*x + a*e)/((b*c*i^3 - a*d*i^3)*(d*x + c)))*(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 = 2.88 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.71 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=-\frac {\frac {2\,A^2\,a\,d-2\,A^2\,b\,c+B^2\,a\,d-7\,B^2\,b\,c-2\,A\,B\,a\,d+6\,A\,B\,b\,c}{2\,\left (a\,d-b\,c\right )}-\frac {x\,\left (3\,B^2\,b\,d-2\,A\,B\,b\,d\right )}{a\,d-b\,c}}{2\,c^2\,d\,i^3+4\,c\,d^2\,i^3\,x+2\,d^3\,i^3\,x^2}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B^2}{2\,d^2\,i^3\,\left (2\,c\,x+d\,x^2+\frac {c^2}{d}\right )}-\frac {B^2\,b^2}{2\,d\,i^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {A\,B}{b\,d^2\,i^3}+\frac {B^2\,x\,\left (a\,d-b\,c\right )}{d\,i^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {B^2\,b^2\,\left (\frac {a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2}{2\,b^3\,d}-\frac {c\,\left (a\,d-b\,c\right )}{2\,b^2\,d}\right )}{d\,i^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )}{\frac {d\,x^2}{b}+\frac {c^2}{b\,d}+\frac {2\,c\,x}{b}}+\frac {B\,b^2\,\mathrm {atan}\left (\frac {B\,b^2\,\left (2\,b\,d\,x+\frac {a^2\,d^3\,i^3-b^2\,c^2\,d\,i^3}{d\,i^3\,\left (a\,d-b\,c\right )}\right )\,\left (2\,A-3\,B\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (3\,B^2\,b^2-2\,A\,B\,b^2\right )}\right )\,\left (2\,A-3\,B\right )\,1{}\mathrm {i}}{d\,i^3\,{\left (a\,d-b\,c\right )}^2} \]

[In]

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

[Out]

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