\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^2 (c i+d i x)} \, dx\) [89]

Optimal result

Integrand size = 42, antiderivative size = 183 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=-\frac {2 b B^2 (c+d x)}{(b c-a d)^2 g^2 i (a+b x)}-\frac {2 b B (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 g^2 i (a+b x)}-\frac {b (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^2 g^2 i (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^2 g^2 i} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2562, 2395, 2342, 2341, 2339, 30} \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=-\frac {d \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g^2 i (b c-a d)^2}-\frac {b (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g^2 i (a+b x) (b c-a d)^2}-\frac {2 b B (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i (a+b x) (b c-a d)^2}-\frac {2 b B^2 (c+d x)}{g^2 i (a+b x) (b c-a d)^2} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.02 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=-\frac {3 \left (A^2+2 A B+2 B^2\right ) d (a+b x) \log (a+b x)+6 B (A+B) (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 B (a A d+A b d x+b B (c+d x)) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+B^2 d (a+b x) \log ^3\left (\frac {e (a+b x)}{c+d x}\right )+3 \left (A^2+2 A B+2 B^2\right ) (b c-a d-d (a+b x) \log (c+d x))}{3 (b c-a d)^2 g^2 i (a+b x)} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(399\) vs. \(2(181)=362\).

Time = 0.98 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.19

method	result	size
norman	\(\frac {-\frac {\left (A^{2} a d +2 A B b c +2 B^{2} b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {B \left (A a d +B b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {B^{2} a d \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {b \left (A^{2} d +2 A B d +2 B^{2} d \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (A^{2}+2 B A +2 B^{2}\right ) b x}{g i a \left (a d -c b \right )}-\frac {b \,B^{2} d x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {B b d \left (A +B \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{g \left (b x +a \right )}\)	\(400\)
parts	\(\frac {A^{2} \left (\frac {d \ln \left (d x +c \right )}{\left (a d -c b \right )^{2}}+\frac {1}{\left (b x +a \right ) \left (a d -c b \right )}-\frac {d \ln \left (b x +a \right )}{\left (a d -c b \right )^{2}}\right )}{g^{2} i}-\frac {B^{2} \left (\frac {d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 \left (a d -c b \right )^{2}}-\frac {d b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{2}}\right )}{g^{2} i d}-\frac {2 B A \left (\frac {d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{2}}-\frac {d b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{2}}\right )}{g^{2} i d}\)	\(460\)
parallelrisch	\(-\frac {3 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{3} b \,c^{2} d +6 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{2} d +B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3} a^{4} c^{2} d +3 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{3} b \,c^{3}-3 A^{2} x \,a^{2} b^{2} c^{3}+3 A^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} c^{2} d -6 B^{2} x \,a^{2} b^{2} c^{3}+6 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{3}+3 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{4} c^{2} d +3 A^{2} x \,a^{3} b \,c^{2} d -6 A B x \,a^{2} b^{2} c^{3}+6 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{3}+6 B^{2} x \,a^{3} b \,c^{2} d +B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3} a^{3} b \,c^{2} d +3 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{3} b \,c^{2} d +3 A^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{2} d +6 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{2} d +6 A B x \,a^{3} b \,c^{2} d}{3 i \,g^{2} \left (b x +a \right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) c^{2} a^{3}}\)	\(473\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A^{2} b}{i \left (a d -c b \right )^{3} g^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{3} A^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{3} g^{2}}-\frac {2 d^{2} A B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{3} g^{2}}+\frac {d^{3} A B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{e i \left (a d -c b \right )^{3} g^{2}}-\frac {d^{2} B^{2} b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{3} g^{2}}+\frac {d^{3} B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e i \left (a d -c b \right )^{3} g^{2}}\right )}{d^{2}}\)	\(527\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A^{2} b}{i \left (a d -c b \right )^{3} g^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{3} A^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{3} g^{2}}-\frac {2 d^{2} A B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{3} g^{2}}+\frac {d^{3} A B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{e i \left (a d -c b \right )^{3} g^{2}}-\frac {d^{2} B^{2} b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{3} g^{2}}+\frac {d^{3} B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e i \left (a d -c b \right )^{3} g^{2}}\right )}{d^{2}}\)	\(527\)
risch	\(\frac {A^{2} d \ln \left (d x +c \right )}{g^{2} i \left (a d -c b \right )^{2}}+\frac {A^{2}}{g^{2} i \left (b x +a \right ) \left (a d -c b \right )}-\frac {A^{2} d \ln \left (b x +a \right )}{g^{2} i \left (a d -c b \right )^{2}}-\frac {B^{2} d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 g^{2} i \left (a d -c b \right )^{2}}-\frac {B^{2} b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{g^{2} i \left (a d -c b \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {2 B^{2} b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{2} i \left (a d -c b \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {2 B^{2} b e}{g^{2} i \left (a d -c b \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {B A d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{g^{2} i \left (a d -c b \right )^{2}}-\frac {2 B A b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{2} i \left (a d -c b \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {2 B A b e}{g^{2} i \left (a d -c b \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}\)	\(556\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.26 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=-\frac {{\left (B^{2} b d x + B^{2} a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 3 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b c - 3 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a d + 3 \, {\left (B^{2} b c + A B a d + {\left (A B + B^{2}\right )} b d x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, {\left (A^{2} a d + {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b d x + 2 \, {\left (A B + B^{2}\right )} b c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{3 \, {\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{2} i x + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} g^{2} i\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (158) = 316\).

Time = 0.74 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.96 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=- \frac {B^{2} d \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{3}}{3 a^{2} d^{2} g^{2} i - 6 a b c d g^{2} i + 3 b^{2} c^{2} g^{2} i} + \frac {\left (2 A B + 2 B^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a^{2} d g^{2} i - a b c g^{2} i + a b d g^{2} i x - b^{2} c g^{2} i x} + \left (A^{2} + 2 A B + 2 B^{2}\right ) \left (\frac {d \log {\left (x + \frac {- \frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} + \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} - \frac {d \log {\left (x + \frac {\frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} - \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} + \frac {1}{a^{2} d g^{2} i - a b c g^{2} i + x \left (a b d g^{2} i - b^{2} c g^{2} i\right )}\right ) + \frac {\left (- A B a d - A B b d x - B^{2} b c - B^{2} b d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a^{3} d^{2} g^{2} i - 2 a^{2} b c d g^{2} i + a^{2} b d^{2} g^{2} i x + a b^{2} c^{2} g^{2} i - 2 a b^{2} c d g^{2} i x + b^{3} c^{2} g^{2} i x} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1008 vs. \(2 (181) = 362\).

Time = 0.26 (sec) , antiderivative size = 1008, normalized size of antiderivative = 5.51 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=-B^{2} {\left (\frac {1}{{\left (b^{2} c - a b d\right )} g^{2} i x + {\left (a b c - a^{2} d\right )} g^{2} i} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )^{2} - 2 \, A B {\left (\frac {1}{{\left (b^{2} c - a b d\right )} g^{2} i x + {\left (a b c - a^{2} d\right )} g^{2} i} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {1}{3} \, B^{2} {\left (\frac {3 \, {\left ({\left (b d x + a d\right )} \log \left (b x + a\right )^{2} + {\left (b d x + a d\right )} \log \left (d x + c\right )^{2} - 2 \, b c + 2 \, a d - 2 \, {\left (b d x + a d\right )} \log \left (b x + a\right ) + 2 \, {\left (b d x + a d - {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{a b^{2} c^{2} g^{2} i - 2 \, a^{2} b c d g^{2} i + a^{3} d^{2} g^{2} i + {\left (b^{3} c^{2} g^{2} i - 2 \, a b^{2} c d g^{2} i + a^{2} b d^{2} g^{2} i\right )} x} - \frac {{\left (b d x + a d\right )} \log \left (b x + a\right )^{3} - {\left (b d x + a d\right )} \log \left (d x + c\right )^{3} - 3 \, {\left (b d x + a d\right )} \log \left (b x + a\right )^{2} - 3 \, {\left (b d x + a d - {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )^{2} + 6 \, b c - 6 \, a d + 6 \, {\left (b d x + a d\right )} \log \left (b x + a\right ) - 3 \, {\left (2 \, b d x + {\left (b d x + a d\right )} \log \left (b x + a\right )^{2} + 2 \, a d - 2 \, {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b^{2} c^{2} g^{2} i - 2 \, a^{2} b c d g^{2} i + a^{3} d^{2} g^{2} i + {\left (b^{3} c^{2} g^{2} i - 2 \, a b^{2} c d g^{2} i + a^{2} b d^{2} g^{2} i\right )} x}\right )} - A^{2} {\left (\frac {1}{{\left (b^{2} c - a b d\right )} g^{2} i x + {\left (a b c - a^{2} d\right )} g^{2} i} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i}\right )} + \frac {{\left ({\left (b d x + a d\right )} \log \left (b x + a\right )^{2} + {\left (b d x + a d\right )} \log \left (d x + c\right )^{2} - 2 \, b c + 2 \, a d - 2 \, {\left (b d x + a d\right )} \log \left (b x + a\right ) + 2 \, {\left (b d x + a d - {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} A B}{a b^{2} c^{2} g^{2} i - 2 \, a^{2} b c d g^{2} i + a^{3} d^{2} g^{2} i + {\left (b^{3} c^{2} g^{2} i - 2 \, a b^{2} c d g^{2} i + a^{2} b d^{2} g^{2} i\right )} x} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.97 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.29 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=\frac {A^2+2\,A\,B+2\,B^2}{\left (a\,d-b\,c\right )\,\left (a\,g^2\,i+b\,g^2\,i\,x\right )}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B\,d\,\left (A+B\right )}{g^2\,i\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {B^2\,\left (a\,d-b\,c\right )}{b\,d\,g^2\,i\,\left (\frac {x}{d}+\frac {a}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {B^2\,d\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g^2\,i\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {2\,B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (a\,d-b\,c\right )\,\left (A+B\right )}{b\,d\,g^2\,i\,\left (\frac {x}{d}+\frac {a}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d\,\mathrm {atan}\left (\frac {d\,\left (2\,b\,d\,x+\frac {a^2\,d^2\,g^2\,i-b^2\,c^2\,g^2\,i}{g^2\,i\,\left (a\,d-b\,c\right )}\right )\,\left (A^2+2\,A\,B+2\,B^2\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (d\,A^2+2\,d\,A\,B+2\,d\,B^2\right )}\right )\,\left (A^2+2\,A\,B+2\,B^2\right )\,2{}\mathrm {i}}{g^2\,i\,{\left (a\,d-b\,c\right )}^2} \]

[In]

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

[Out]

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