\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^3 (c i+d i x)} \, dx\) [90]

Optimal result

Integrand size = 42, antiderivative size = 343 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx=\frac {4 b B^2 d (c+d x)}{(b c-a d)^3 g^3 i (a+b x)}-\frac {b^2 B^2 (c+d x)^2}{4 (b c-a d)^3 g^3 i (a+b x)^2}+\frac {4 b B d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^3 i (a+b x)}-\frac {b^2 B (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^3 i (a+b x)^2}+\frac {2 b d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g^3 i (a+b x)}-\frac {b^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^3 g^3 i (a+b x)^2}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^3 g^3 i} \]

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2562, 2395, 2342, 2341, 2339, 30} \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx=-\frac {b^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^3 i (a+b x)^2 (b c-a d)^3}-\frac {b^2 B (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 i (a+b x)^2 (b c-a d)^3}+\frac {d^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g^3 i (b c-a d)^3}+\frac {2 b d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g^3 i (a+b x) (b c-a d)^3}+\frac {4 b B d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i (a+b x) (b c-a d)^3}-\frac {b^2 B^2 (c+d x)^2}{4 g^3 i (a+b x)^2 (b c-a d)^3}+\frac {4 b B^2 d (c+d x)}{g^3 i (a+b x) (b c-a d)^3} \]

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

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 2562

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

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

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.93 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx=\frac {-3 \left (2 A^2+2 A B+B^2\right ) (b c-a d)^2+6 \left (2 A^2+6 A B+7 B^2\right ) d (b c-a d) (a+b x)+6 \left (2 A^2+6 A B+7 B^2\right ) d^2 (a+b x)^2 \log (a+b x)-6 B (b c-a d) (-6 a A d-7 a B d+b B (c-6 d x)+2 A b (c-2 d x)) \log \left (\frac {e (a+b x)}{c+d x}\right )-6 B \left (-2 a^2 A d^2-4 a b d (A d x+B (c+d x))+b^2 \left (-2 A d^2 x^2+B \left (c^2-2 c d x-3 d^2 x^2\right )\right )\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+4 B^2 d^2 (a+b x)^2 \log ^3\left (\frac {e (a+b x)}{c+d x}\right )-6 \left (2 A^2+6 A B+7 B^2\right ) d^2 (a+b x)^2 \log (c+d x)}{12 (b c-a d)^3 g^3 i (a+b x)^2} \]

[In]

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

[Out]

(-3*(2*A^2 + 2*A*B + B^2)*(b*c - a*d)^2 + 6*(2*A^2 + 6*A*B + 7*B^2)*d*(b*c - a*d)*(a + b*x) + 6*(2*A^2 + 6*A*B
+ 7*B^2)*d^2*(a + b*x)^2*Log[a + b*x] - 6*B*(b*c - a*d)*(-6*a*A*d - 7*a*B*d + b*B*(c - 6*d*x) + 2*A*b*(c - 2*
d*x))*Log[(e*(a + b*x))/(c + d*x)] - 6*B*(-2*a^2*A*d^2 - 4*a*b*d*(A*d*x + B*(c + d*x)) + b^2*(-2*A*d^2*x^2 + B
*(c^2 - 2*c*d*x - 3*d^2*x^2)))*Log[(e*(a + b*x))/(c + d*x)]^2 + 4*B^2*d^2*(a + b*x)^2*Log[(e*(a + b*x))/(c + d
*x)]^3 - 6*(2*A^2 + 6*A*B + 7*B^2)*d^2*(a + b*x)^2*Log[c + d*x])/(12*(b*c - a*d)^3*g^3*i*(a + b*x)^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(766\) vs. \(2(335)=670\).

Time = 1.31 (sec) , antiderivative size = 767, normalized size of antiderivative = 2.24

method	result	size
parts	\(\frac {A^{2} \left (\frac {d^{2} \ln \left (d x +c \right )}{\left (a d -c b \right )^{3}}+\frac {1}{2 \left (a d -c b \right ) \left (b x +a \right )^{2}}+\frac {d}{\left (a d -c b \right )^{2} \left (b x +a \right )}-\frac {d^{2} \ln \left (b x +a \right )}{\left (a d -c b \right )^{3}}\right )}{g^{3} i}-\frac {B^{2} \left (\frac {d^{3} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 \left (a d -c b \right )^{3}}-\frac {2 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}}{\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 \,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}}{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} i d}-\frac {2 B A \left (\frac {d^{3} \ln \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}}-\frac {2 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 )}{\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 \,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 \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} i d}\)	\(767\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} e \,A^{2} b^{2}}{2 i \left (a d -c b \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {2 d^{3} A^{2} b}{i \left (a d -c b \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{4} A^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{4} g^{3}}+\frac {2 d^{2} e A B \,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 \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 )}{i \left (a d -c b \right )^{4} g^{3}}-\frac {4 d^{3} A B 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 )}{i \left (a d -c b \right )^{4} g^{3}}+\frac {d^{4} A B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{e i \left (a d -c b \right )^{4} g^{3}}+\frac {d^{2} e \,B^{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}}{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 )}{i \left (a d -c b \right )^{4} g^{3}}-\frac {2 d^{3} B^{2} b \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 )}{i \left (a d -c b \right )^{4} g^{3}}+\frac {d^{4} B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e i \left (a d -c b \right )^{4} g^{3}}\right )}{d^{2}}\)	\(882\)
default	\(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} e \,A^{2} b^{2}}{2 i \left (a d -c b \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {2 d^{3} A^{2} b}{i \left (a d -c b \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{4} A^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{4} g^{3}}+\frac {2 d^{2} e A B \,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 \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 )}{i \left (a d -c b \right )^{4} g^{3}}-\frac {4 d^{3} A B 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 )}{i \left (a d -c b \right )^{4} g^{3}}+\frac {d^{4} A B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{e i \left (a d -c b \right )^{4} g^{3}}+\frac {d^{2} e \,B^{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}}{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 )}{i \left (a d -c b \right )^{4} g^{3}}-\frac {2 d^{3} B^{2} b \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 )}{i \left (a d -c b \right )^{4} g^{3}}+\frac {d^{4} B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e i \left (a d -c b \right )^{4} g^{3}}\right )}{d^{2}}\)	\(882\)
norman	\(\frac {\frac {6 A^{2} a \,b^{2} d -2 A^{2} b^{3} c +14 A B a \,b^{2} d -2 A B \,b^{3} c +15 B^{2} a \,b^{2} d -B^{2} b^{3} c}{4 g i \,b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (2 A^{2} a^{2} d^{2}+8 A B a b c d -2 A B \,b^{2} c^{2}+8 B^{2} a b c d -B^{2} b^{2} c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 i 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 \left (2 A \,a^{2} d^{2}+4 B a b c d -B \,b^{2} c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 i 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 (2 A^{2} b^{2} d +6 A B \,b^{2} d +7 B^{2} b^{2} d \right ) x}{2 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) i g}-\frac {B^{2} a^{2} d^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 i 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 \left (2 A^{2} a \,d^{2}+4 A B a \,d^{2}+2 A B b c d +4 B^{2} a \,d^{2}+3 B^{2} b c d \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{i 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} B^{2} d^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 i 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 (2 A^{2} d^{2}+6 A B \,d^{2}+7 B^{2} d^{2}\right ) b^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g i \left (a d -c b \right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {2 b \,B^{2} a \,d^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 i 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 B \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 )^{2}}{i 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} 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}}{2 i 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 )}}{\left (b x +a \right )^{2} g^{2}}\)	\(938\)
risch	\(\frac {A^{2} d^{2} \ln \left (d x +c \right )}{g^{3} i \left (a d -c b \right )^{3}}+\frac {A^{2}}{2 g^{3} i \left (a d -c b \right ) \left (b x +a \right )^{2}}+\frac {A^{2} d}{g^{3} i \left (a d -c b \right )^{2} \left (b x +a \right )}-\frac {A^{2} d^{2} \ln \left (b x +a \right )}{g^{3} i \left (a d -c b \right )^{3}}-\frac {B^{2} d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 g^{3} i \left (a d -c b \right )^{3}}-\frac {2 B^{2} d b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{g^{3} i \left (a d -c b \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {4 B^{2} d b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{3} i \left (a d -c b \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {4 B^{2} d b e}{g^{3} i \left (a d -c b \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}+\frac {B^{2} e^{2} b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g^{3} i \left (a d -c b \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{2}}+\frac {B^{2} e^{2} b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 g^{3} i \left (a d -c b \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{2}}+\frac {B^{2} e^{2} b^{2}}{4 g^{3} i \left (a d -c b \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{2}}-\frac {B A \,d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{g^{3} i \left (a d -c b \right )^{3}}-\frac {4 B A d b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{3} i \left (a d -c b \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {4 B A d b e}{g^{3} i \left (a d -c b \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}+\frac {B A \,e^{2} b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{3} i \left (a d -c b \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{2}}+\frac {B A \,e^{2} b^{2}}{2 g^{3} i \left (a d -c b \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{2}}\)	\(986\)
parallelrisch	\(\text {Expression too large to display}\)	\(1122\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.57 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx=-\frac {3 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{2} c^{2} - 24 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a b c d + 3 \, {\left (6 \, A^{2} + 14 \, A B + 15 \, B^{2}\right )} a^{2} d^{2} - 4 \, {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} a b d^{2} x + B^{2} a^{2} d^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} - 6 \, {\left ({\left (2 \, A B + 3 \, B^{2}\right )} b^{2} d^{2} x^{2} - B^{2} b^{2} c^{2} + 4 \, B^{2} a b c d + 2 \, A B a^{2} d^{2} + 2 \, {\left (B^{2} b^{2} c d + 2 \, {\left (A B + B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 6 \, {\left ({\left (2 \, A^{2} + 6 \, A B + 7 \, B^{2}\right )} b^{2} c d - {\left (2 \, A^{2} + 6 \, A B + 7 \, B^{2}\right )} a b d^{2}\right )} x - 6 \, {\left ({\left (2 \, A^{2} + 6 \, A B + 7 \, B^{2}\right )} b^{2} d^{2} x^{2} + 2 \, A^{2} a^{2} d^{2} - {\left (2 \, A B + B^{2}\right )} b^{2} c^{2} + 8 \, {\left (A B + B^{2}\right )} a b c d + 2 \, {\left ({\left (2 \, A B + 3 \, B^{2}\right )} b^{2} c d + 2 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{12 \, {\left ({\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} g^{3} i x + {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} g^{3} i\right )}} \]

[In]

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

[Out]

-1/12*(3*(2*A^2 + 2*A*B + B^2)*b^2*c^2 - 24*(A^2 + 2*A*B + 2*B^2)*a*b*c*d + 3*(6*A^2 + 14*A*B + 15*B^2)*a^2*d^
2 - 4*(B^2*b^2*d^2*x^2 + 2*B^2*a*b*d^2*x + B^2*a^2*d^2)*log((b*e*x + a*e)/(d*x + c))^3 - 6*((2*A*B + 3*B^2)*b^
2*d^2*x^2 - B^2*b^2*c^2 + 4*B^2*a*b*c*d + 2*A*B*a^2*d^2 + 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))^2 - 6*((2*A^2 + 6*A*B + 7*B^2)*b^2*c*d - (2*A^2 + 6*A*B + 7*B^2)*a*b*d^2)*x - 6*((2*A^2 +
6*A*B + 7*B^2)*b^2*d^2*x^2 + 2*A^2*a^2*d^2 - (2*A*B + B^2)*b^2*c^2 + 8*(A*B + B^2)*a*b*c*d + 2*((2*A*B + 3*B^2
)*b^2*c*d + 2*(A^2 + 2*A*B + 2*B^2)*a*b*d^2)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^
2*b^3*c*d^2 - a^3*b^2*d^3)*g^3*i*x^2 + 2*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 - a^4*b*d^3)*g^3*i*x +
(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*g^3*i)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1488 vs. \(2 (303) = 606\).

Time = 4.24 (sec) , antiderivative size = 1488, normalized size of antiderivative = 4.34 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-B**2*d**2*log(e*(a + b*x)/(c + d*x))**3/(3*a**3*d**3*g**3*i - 9*a**2*b*c*d**2*g**3*i + 9*a*b**2*c**2*d*g**3*i
- 3*b**3*c**3*g**3*i) + d**2*(2*A**2 + 6*A*B + 7*B**2)*log(x + (2*A**2*a*d**3 + 2*A**2*b*c*d**2 + 6*A*B*a*d**
3 + 6*A*B*b*c*d**2 + 7*B**2*a*d**3 + 7*B**2*b*c*d**2 - a**4*d**6*(2*A**2 + 6*A*B + 7*B**2)/(a*d - b*c)**3 + 4*
a**3*b*c*d**5*(2*A**2 + 6*A*B + 7*B**2)/(a*d - b*c)**3 - 6*a**2*b**2*c**2*d**4*(2*A**2 + 6*A*B + 7*B**2)/(a*d
- b*c)**3 + 4*a*b**3*c**3*d**3*(2*A**2 + 6*A*B + 7*B**2)/(a*d - b*c)**3 - b**4*c**4*d**2*(2*A**2 + 6*A*B + 7*B
**2)/(a*d - b*c)**3)/(4*A**2*b*d**3 + 12*A*B*b*d**3 + 14*B**2*b*d**3))/(2*g**3*i*(a*d - b*c)**3) - d**2*(2*A**
2 + 6*A*B + 7*B**2)*log(x + (2*A**2*a*d**3 + 2*A**2*b*c*d**2 + 6*A*B*a*d**3 + 6*A*B*b*c*d**2 + 7*B**2*a*d**3 +
7*B**2*b*c*d**2 + a**4*d**6*(2*A**2 + 6*A*B + 7*B**2)/(a*d - b*c)**3 - 4*a**3*b*c*d**5*(2*A**2 + 6*A*B + 7*B*
*2)/(a*d - b*c)**3 + 6*a**2*b**2*c**2*d**4*(2*A**2 + 6*A*B + 7*B**2)/(a*d - b*c)**3 - 4*a*b**3*c**3*d**3*(2*A*
*2 + 6*A*B + 7*B**2)/(a*d - b*c)**3 + b**4*c**4*d**2*(2*A**2 + 6*A*B + 7*B**2)/(a*d - b*c)**3)/(4*A**2*b*d**3
+ 12*A*B*b*d**3 + 14*B**2*b*d**3))/(2*g**3*i*(a*d - b*c)**3) + (6*A*B*a*d - 2*A*B*b*c + 4*A*B*b*d*x + 7*B**2*a
*d - B**2*b*c + 6*B**2*b*d*x)*log(e*(a + b*x)/(c + d*x))/(2*a**4*d**2*g**3*i - 4*a**3*b*c*d*g**3*i + 4*a**3*b*
d**2*g**3*i*x + 2*a**2*b**2*c**2*g**3*i - 8*a**2*b**2*c*d*g**3*i*x + 2*a**2*b**2*d**2*g**3*i*x**2 + 4*a*b**3*c
**2*g**3*i*x - 4*a*b**3*c*d*g**3*i*x**2 + 2*b**4*c**2*g**3*i*x**2) + (-2*A*B*a**2*d**2 - 4*A*B*a*b*d**2*x - 2*
A*B*b**2*d**2*x**2 - 4*B**2*a*b*c*d - 4*B**2*a*b*d**2*x + B**2*b**2*c**2 - 2*B**2*b**2*c*d*x - 3*B**2*b**2*d**
2*x**2)*log(e*(a + b*x)/(c + d*x))**2/(2*a**5*d**3*g**3*i - 6*a**4*b*c*d**2*g**3*i + 4*a**4*b*d**3*g**3*i*x +
6*a**3*b**2*c**2*d*g**3*i - 12*a**3*b**2*c*d**2*g**3*i*x + 2*a**3*b**2*d**3*g**3*i*x**2 - 2*a**2*b**3*c**3*g**
3*i + 12*a**2*b**3*c**2*d*g**3*i*x - 6*a**2*b**3*c*d**2*g**3*i*x**2 - 4*a*b**4*c**3*g**3*i*x + 6*a*b**4*c**2*d
*g**3*i*x**2 - 2*b**5*c**3*g**3*i*x**2) + (6*A**2*a*d - 2*A**2*b*c + 14*A*B*a*d - 2*A*B*b*c + 15*B**2*a*d - B*
*2*b*c + x*(4*A**2*b*d + 12*A*B*b*d + 14*B**2*b*d))/(4*a**4*d**2*g**3*i - 8*a**3*b*c*d*g**3*i + 4*a**2*b**2*c*
*2*g**3*i + x**2*(4*a**2*b**2*d**2*g**3*i - 8*a*b**3*c*d*g**3*i + 4*b**4*c**2*g**3*i) + x*(8*a**3*b*d**2*g**3*
i - 16*a**2*b**2*c*d*g**3*i + 8*a*b**3*c**2*g**3*i))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2115 vs. \(2 (335) = 670\).

Time = 0.35 (sec) , antiderivative size = 2115, normalized size of antiderivative = 6.17 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*B^2*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*g^3*i*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c
*d + a^3*b*d^2)*g^3*i*x + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*g^3*i) + 2*d^2*log(b*x + a)/((b^3*c^3 - 3*a*b^
2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^2*log(d*x + c)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3
*d^3)*g^3*i))*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2 + A*B*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a*b^3*c*d +
a^2*b^2*d^2)*g^3*i*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*g^3*i*x + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*
d^2)*g^3*i) + 2*d^2*log(b*x + a)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^2*log(d*x +
c)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 1/12*B
^2*(6*(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))*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(a^2*b^3*c^3*g^3*i - 3*a^3*b^2*c^2*d*g^3*i + 3*a^4*b*c
*d^2*g^3*i - a^5*d^3*g^3*i + (b^5*c^3*g^3*i - 3*a*b^4*c^2*d*g^3*i + 3*a^2*b^3*c*d^2*g^3*i - a^3*b^2*d^3*g^3*i)
*x^2 + 2*(a*b^4*c^3*g^3*i - 3*a^2*b^3*c^2*d*g^3*i + 3*a^3*b^2*c*d^2*g^3*i - a^4*b*d^3*g^3*i)*x) + (3*b^2*c^2 -
48*a*b*c*d + 45*a^2*d^2 - 4*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a)^3 + 4*(b^2*d^2*x^2 + 2*a*b*d^2
*x + a^2*d^2)*log(d*x + c)^3 + 18*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a)^2 + 6*(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)^2 - 42*(b^2*c*d - a
*b*d^2)*x - 42*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a) + 6*(7*b^2*d^2*x^2 + 14*a*b*d^2*x + 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 - 6*(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^3*g^3*i - 3*a^3*b^2*c^2*d*g^3*i + 3*a^4*b*c*d^2*g^3*i - a^5*d^3*g^3*i + (b^5*c^3
*g^3*i - 3*a*b^4*c^2*d*g^3*i + 3*a^2*b^3*c*d^2*g^3*i - a^3*b^2*d^3*g^3*i)*x^2 + 2*(a*b^4*c^3*g^3*i - 3*a^2*b^3
*c^2*d*g^3*i + 3*a^3*b^2*c*d^2*g^3*i - a^4*b*d^3*g^3*i)*x)) + 1/2*A^2*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a
*b^3*c*d + a^2*b^2*d^2)*g^3*i*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*g^3*i*x + (a^2*b^2*c^2 - 2*a^3*b
*c*d + a^4*d^2)*g^3*i) + 2*d^2*log(b*x + a)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^
2*log(d*x + c)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i)) - 1/2*(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*B/(a^2*b^3*c^
3*g^3*i - 3*a^3*b^2*c^2*d*g^3*i + 3*a^4*b*c*d^2*g^3*i - a^5*d^3*g^3*i + (b^5*c^3*g^3*i - 3*a*b^4*c^2*d*g^3*i +
3*a^2*b^3*c*d^2*g^3*i - a^3*b^2*d^3*g^3*i)*x^2 + 2*(a*b^4*c^3*g^3*i - 3*a^2*b^3*c^2*d*g^3*i + 3*a^3*b^2*c*d^2
*g^3*i - a^4*b*d^3*g^3*i)*x)

Giac [A] (verification not implemented)

none

Time = 58.16 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.62 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, {\left (d x + c\right )}^{2} B^{2} e^{3} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{{\left (b e x + a e\right )}^{2} g^{3} i} + \frac {2 \, {\left (2 \, A B e^{3} + B^{2} e^{3}\right )} {\left (d x + c\right )}^{2} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b e x + a e\right )}^{2} g^{3} i} + \frac {{\left (2 \, A^{2} e^{3} + 2 \, A B e^{3} + B^{2} e^{3}\right )} {\left (d x + c\right )}^{2}}{{\left (b e x + a e\right )}^{2} g^{3} i}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.84 (sec) , antiderivative size = 981, normalized size of antiderivative = 2.86 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx={\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {\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 )}{g^3\,i\,\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\,x\,\left (a\,d-b\,c\right )}{g^3\,i\,\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\,x^2}{d}+\frac {a^2}{b\,d}+\frac {2\,a\,x}{d}}-\frac {B\,d^2\,\left (2\,A+3\,B\right )}{2\,g^3\,i\,\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 {\frac {6\,A^2\,a\,d-2\,A^2\,b\,c+15\,B^2\,a\,d-B^2\,b\,c+14\,A\,B\,a\,d-2\,A\,B\,b\,c}{2\,\left (a\,d-b\,c\right )}+\frac {x\,\left (2\,b\,d\,A^2+6\,b\,d\,A\,B+7\,b\,d\,B^2\right )}{a\,d-b\,c}}{x^2\,\left (2\,b^3\,c\,g^3\,i-2\,a\,b^2\,d\,g^3\,i\right )+x\,\left (4\,a\,b^2\,c\,g^3\,i-4\,a^2\,b\,d\,g^3\,i\right )-2\,a^3\,d\,g^3\,i+2\,a^2\,b\,c\,g^3\,i}+\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,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 )\,\left (2\,A+3\,B\right )}{g^3\,i\,\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}{b\,d\,g^3\,i\,\left (a\,d-b\,c\right )}+\frac {B\,x\,\left (2\,A+3\,B\right )\,\left (a\,d-b\,c\right )}{g^3\,i\,\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 {b\,x^2}{d}+\frac {a^2}{b\,d}+\frac {2\,a\,x}{d}}-\frac {B^2\,d^2\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g^3\,i\,\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 {d^2\,\mathrm {atan}\left (\frac {d^2\,\left (A^2+3\,A\,B+\frac {7\,B^2}{2}\right )\,\left (2\,i\,a^3\,d^3\,g^3-2\,i\,a^2\,b\,c\,d^2\,g^3-2\,i\,a\,b^2\,c^2\,d\,g^3+2\,i\,b^3\,c^3\,g^3\right )\,1{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3\,\left (2\,A^2\,d^2+6\,A\,B\,d^2+7\,B^2\,d^2\right )}+\frac {b\,d^3\,x\,\left (i\,a^2\,d^2\,g^3-2\,i\,a\,b\,c\,d\,g^3+i\,b^2\,c^2\,g^3\right )\,\left (A^2+3\,A\,B+\frac {7\,B^2}{2}\right )\,4{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3\,\left (2\,A^2\,d^2+6\,A\,B\,d^2+7\,B^2\,d^2\right )}\right )\,\left (A^2+3\,A\,B+\frac {7\,B^2}{2}\right )\,2{}\mathrm {i}}{g^3\,i\,{\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)^3*(c*i + d*i*x)),x)

[Out]

log((e*(a + b*x))/(c + d*x))^2*(((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)))/(g^3*i*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (B^2*x*(a*d - b*c))/(g^3*i*(a^3*d^3 - b^3
*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))/((b*x^2)/d + a^2/(b*d) + (2*a*x)/d) - (B*d^2*(2*A + 3*B))/(2*g^3*i*(a^
3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - ((6*A^2*a*d - 2*A^2*b*c + 15*B^2*a*d - B^2*b*c + 14*A*B*a
*d - 2*A*B*b*c)/(2*(a*d - b*c)) + (x*(2*A^2*b*d + 7*B^2*b*d + 6*A*B*b*d))/(a*d - b*c))/(x^2*(2*b^3*c*g^3*i - 2
*a*b^2*d*g^3*i) + x*(4*a*b^2*c*g^3*i - 4*a^2*b*d*g^3*i) - 2*a^3*d*g^3*i + 2*a^2*b*c*g^3*i) + (log((e*(a + b*x)
)/(c + d*x))*((B*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))*(2*A + 3*B))/(g
^3*i*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - B^2/(b*d*g^3*i*(a*d - b*c)) + (B*x*(2*A + 3*B)*(a*
d - b*c))/(g^3*i*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))))/((b*x^2)/d + a^2/(b*d) + (2*a*x)/d) -
(B^2*d^2*log((e*(a + b*x))/(c + d*x))^3)/(3*g^3*i*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (d^2*
atan((d^2*(A^2 + (7*B^2)/2 + 3*A*B)*(2*a^3*d^3*g^3*i + 2*b^3*c^3*g^3*i - 2*a*b^2*c^2*d*g^3*i - 2*a^2*b*c*d^2*g
^3*i)*1i)/(g^3*i*(a*d - b*c)^3*(2*A^2*d^2 + 7*B^2*d^2 + 6*A*B*d^2)) + (b*d^3*x*(a^2*d^2*g^3*i + b^2*c^2*g^3*i
- 2*a*b*c*d*g^3*i)*(A^2 + (7*B^2)/2 + 3*A*B)*4i)/(g^3*i*(a*d - b*c)^3*(2*A^2*d^2 + 7*B^2*d^2 + 6*A*B*d^2)))*(A
^2 + (7*B^2)/2 + 3*A*B)*2i)/(g^3*i*(a*d - b*c)^3)