\(\int \frac {(A+B \log (\frac {e (c+d x)^2}{(a+b x)^2}))^2}{(a g+b g x)^2} \, dx\) [215]

Optimal result

Integrand size = 34, antiderivative size = 157 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=\frac {4 A B (c+d x)}{(b c-a d) g^2 (a+b x)}-\frac {8 B^2 (c+d x)}{(b c-a d) g^2 (a+b x)}+\frac {4 B^2 (c+d x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{(b c-a d) g^2 (a+b x)}-\frac {(c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(b c-a d) g^2 (a+b x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2552, 2333, 2332} \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {(c+d x) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{g^2 (a+b x) (b c-a d)}+\frac {4 A B (c+d x)}{g^2 (a+b x) (b c-a d)}+\frac {4 B^2 (c+d x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{g^2 (a+b x) (b c-a d)}-\frac {8 B^2 (c+d x)}{g^2 (a+b x) (b c-a d)} \]

[In]

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

[Out]

(4*A*B*(c + d*x))/((b*c - a*d)*g^2*(a + b*x)) - (8*B^2*(c + d*x))/((b*c - a*d)*g^2*(a + b*x)) + (4*B^2*(c + d*
x)*Log[(e*(c + d*x)^2)/(a + b*x)^2])/((b*c - a*d)*g^2*(a + b*x)) - ((c + d*x)*(A + B*Log[(e*(c + d*x)^2)/(a +
b*x)^2])^2)/((b*c - a*d)*g^2*(a + b*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 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 \left (A+B \log \left (e x^2\right )\right )^2 \, dx,x,\frac {c+d x}{a+b x}\right )}{(b c-a d) g^2} \\ & = -\frac {(c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(b c-a d) g^2 (a+b x)}+\frac {(4 B) \text {Subst}\left (\int \left (A+B \log \left (e x^2\right )\right ) \, dx,x,\frac {c+d x}{a+b x}\right )}{(b c-a d) g^2} \\ & = \frac {4 A B (c+d x)}{(b c-a d) g^2 (a+b x)}-\frac {(c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(b c-a d) g^2 (a+b x)}+\frac {\left (4 B^2\right ) \text {Subst}\left (\int \log \left (e x^2\right ) \, dx,x,\frac {c+d x}{a+b x}\right )}{(b c-a d) g^2} \\ & = \frac {4 A B (c+d x)}{(b c-a d) g^2 (a+b x)}-\frac {8 B^2 (c+d x)}{(b c-a d) g^2 (a+b x)}+\frac {4 B^2 (c+d x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{(b c-a d) g^2 (a+b x)}-\frac {(c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(b c-a d) g^2 (a+b x)} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.27 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.05 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2+\frac {4 B \left (2 B (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-(b c-a d) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )-d (a+b x) \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )+d (a+b x) \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+B d (a+b x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{b c-a d}}{b g^2 (a+b x)} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.96 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.20

method	result	size
norman	\(\frac {\frac {\left (A^{2}-4 B A +8 B^{2}\right ) x}{g a}+\frac {B^{2} c \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2}}{g \left (a d -c b \right )}+\frac {B^{2} d x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2}}{g \left (a d -c b \right )}+\frac {2 \left (A -2 B \right ) c B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{g \left (a d -c b \right )}+\frac {2 d \left (A -2 B \right ) B x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{g \left (a d -c b \right )}}{g \left (b x +a \right )}\)	\(188\)
parallelrisch	\(-\frac {2 A^{2} a \,b^{2} d^{2}-2 A^{2} b^{3} c d +16 B^{2} a \,b^{2} d^{2}-16 B^{2} b^{3} c d -2 B^{2} x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2} b^{3} d^{2}+8 B^{2} x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) b^{3} d^{2}-2 B^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2} b^{3} c d +8 B^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) b^{3} c d -4 A B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) b^{3} c d -4 A B x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) b^{3} d^{2}-8 A B a \,b^{2} d^{2}+8 A B \,b^{3} c d}{2 g^{2} \left (b x +a \right ) b^{3} d \left (a d -c b \right )}\)	\(264\)
parts	\(-\frac {A^{2}}{g^{2} \left (b x +a \right ) b}+\frac {\frac {8 B^{2} x}{a g}+\frac {B^{2} c \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2}}{g \left (a d -c b \right )}+\frac {B^{2} d x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2}}{g \left (a d -c b \right )}-\frac {4 B^{2} c \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{g \left (a d -c b \right )}-\frac {4 B^{2} d x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{g \left (a d -c b \right )}}{g \left (b x +a \right )}-\frac {2 A B \left (\frac {\ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{b x +a}-\left (2 a d -2 c b \right ) \left (\frac {1}{\left (b x +a \right ) \left (a d -c b \right )}+\frac {d \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{\left (a d -c b \right )^{2}}\right )\right )}{g^{2} b}\)	\(312\)
derivativedivides	\(-\frac {\frac {A^{2}}{g^{2} \left (b x +a \right )}+\frac {B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )^{2}}{g^{2} \left (b x +a \right )}+\frac {8 B^{2}}{g^{2} \left (b x +a \right )}-\frac {4 B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{g^{2} \left (b x +a \right )}+\frac {4 B^{2} d \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{g^{2} \left (a d -c b \right )}-\frac {B^{2} d \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )^{2}}{g^{2} \left (a d -c b \right )}+\frac {2 A B \left (\frac {\ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{b x +a}-\left (2 a d -2 c b \right ) \left (\frac {1}{\left (b x +a \right ) \left (a d -c b \right )}+\frac {d \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{\left (a d -c b \right )^{2}}\right )\right )}{g^{2}}}{b}\)	\(356\)
default	\(-\frac {\frac {A^{2}}{g^{2} \left (b x +a \right )}+\frac {B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )^{2}}{g^{2} \left (b x +a \right )}+\frac {8 B^{2}}{g^{2} \left (b x +a \right )}-\frac {4 B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{g^{2} \left (b x +a \right )}+\frac {4 B^{2} d \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{g^{2} \left (a d -c b \right )}-\frac {B^{2} d \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )^{2}}{g^{2} \left (a d -c b \right )}+\frac {2 A B \left (\frac {\ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{b x +a}-\left (2 a d -2 c b \right ) \left (\frac {1}{\left (b x +a \right ) \left (a d -c b \right )}+\frac {d \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{\left (a d -c b \right )^{2}}\right )\right )}{g^{2}}}{b}\)	\(356\)
risch	\(-\frac {A^{2}}{g^{2} \left (b x +a \right ) b}+\frac {\frac {8 B^{2} x}{a g}+\frac {B^{2} c \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2}}{g \left (a d -c b \right )}+\frac {B^{2} d x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )^{2}}{g \left (a d -c b \right )}-\frac {4 B^{2} c \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{g \left (a d -c b \right )}-\frac {4 B^{2} d x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{g \left (a d -c b \right )}}{g \left (b x +a \right )}-\frac {2 A B \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{g^{2} b \left (b x +a \right )}+\frac {4 A B a d}{g^{2} b \left (b x +a \right ) \left (a d -c b \right )}-\frac {4 A B c}{g^{2} \left (b x +a \right ) \left (a d -c b \right )}+\frac {4 A B \,d^{2} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right ) a}{g^{2} b \left (a d -c b \right )^{2}}-\frac {4 A B d \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right ) c}{g^{2} \left (a d -c b \right )^{2}}\)	\(391\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.27 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {{\left (A^{2} - 4 \, A B + 8 \, B^{2}\right )} b c - {\left (A^{2} - 4 \, A B + 8 \, B^{2}\right )} a d + {\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )^{2} + 2 \, {\left ({\left (A B - 2 \, B^{2}\right )} b d x + {\left (A B - 2 \, B^{2}\right )} b c\right )} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{{\left (b^{3} c - a b^{2} d\right )} g^{2} x + {\left (a b^{2} c - a^{2} b d\right )} g^{2}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (134) = 268\).

Time = 1.20 (sec) , antiderivative size = 450, normalized size of antiderivative = 2.87 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=\frac {4 B d \left (A - 2 B\right ) \log {\left (x + \frac {4 A B a d^{2} + 4 A B b c d - 8 B^{2} a d^{2} - 8 B^{2} b c d - \frac {4 B a^{2} d^{3} \left (A - 2 B\right )}{a d - b c} + \frac {8 B a b c d^{2} \left (A - 2 B\right )}{a d - b c} - \frac {4 B b^{2} c^{2} d \left (A - 2 B\right )}{a d - b c}}{8 A B b d^{2} - 16 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} - \frac {4 B d \left (A - 2 B\right ) \log {\left (x + \frac {4 A B a d^{2} + 4 A B b c d - 8 B^{2} a d^{2} - 8 B^{2} b c d + \frac {4 B a^{2} d^{3} \left (A - 2 B\right )}{a d - b c} - \frac {8 B a b c d^{2} \left (A - 2 B\right )}{a d - b c} + \frac {4 B b^{2} c^{2} d \left (A - 2 B\right )}{a d - b c}}{8 A B b d^{2} - 16 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {\left (- 2 A B + 4 B^{2}\right ) \log {\left (\frac {e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )}}{a b g^{2} + b^{2} g^{2} x} + \frac {\left (B^{2} c + B^{2} d x\right ) \log {\left (\frac {e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )}^{2}}{a^{2} d g^{2} - a b c g^{2} + a b d g^{2} x - b^{2} c g^{2} x} + \frac {- A^{2} + 4 A B - 8 B^{2}}{a b g^{2} + b^{2} g^{2} x} \]

[In]

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

[Out]

4*B*d*(A - 2*B)*log(x + (4*A*B*a*d**2 + 4*A*B*b*c*d - 8*B**2*a*d**2 - 8*B**2*b*c*d - 4*B*a**2*d**3*(A - 2*B)/(
a*d - b*c) + 8*B*a*b*c*d**2*(A - 2*B)/(a*d - b*c) - 4*B*b**2*c**2*d*(A - 2*B)/(a*d - b*c))/(8*A*B*b*d**2 - 16*
B**2*b*d**2))/(b*g**2*(a*d - b*c)) - 4*B*d*(A - 2*B)*log(x + (4*A*B*a*d**2 + 4*A*B*b*c*d - 8*B**2*a*d**2 - 8*B
**2*b*c*d + 4*B*a**2*d**3*(A - 2*B)/(a*d - b*c) - 8*B*a*b*c*d**2*(A - 2*B)/(a*d - b*c) + 4*B*b**2*c**2*d*(A -
2*B)/(a*d - b*c))/(8*A*B*b*d**2 - 16*B**2*b*d**2))/(b*g**2*(a*d - b*c)) + (-2*A*B + 4*B**2)*log(e*(c + d*x)**2
/(a + b*x)**2)/(a*b*g**2 + b**2*g**2*x) + (B**2*c + B**2*d*x)*log(e*(c + d*x)**2/(a + b*x)**2)**2/(a**2*d*g**2
- a*b*c*g**2 + a*b*d*g**2*x - b**2*c*g**2*x) + (-A**2 + 4*A*B - 8*B**2)/(a*b*g**2 + b**2*g**2*x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (157) = 314\).

Time = 0.24 (sec) , antiderivative size = 573, normalized size of antiderivative = 3.65 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=4 \, {\left ({\left (\frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + \frac {{\left (b d x + a d\right )} \log \left (b x + a\right )^{2} + {\left (b d x + a d\right )} \log \left (d x + c\right )^{2} - 2 \, b c + 2 \, a d - 2 \, {\left (b d x + a d\right )} \log \left (b x + a\right ) + 2 \, {\left (b d x + a d - {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b^{2} c g^{2} - a^{2} b d g^{2} + {\left (b^{3} c g^{2} - a b^{2} d g^{2}\right )} x}\right )} B^{2} - 2 \, A B {\left (\frac {\log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{b^{2} g^{2} x + a b g^{2}} - \frac {2}{b^{2} g^{2} x + a b g^{2}} - \frac {2 \, d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} + \frac {2 \, d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac {B^{2} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )^{2}}{b^{2} g^{2} x + a b g^{2}} - \frac {A^{2}}{b^{2} g^{2} x + a b g^{2}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (157) = 314\).

Time = 0.65 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.47 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-{\left (\frac {B^{2} d}{b^{2} c g^{2} - a b d g^{2}} + \frac {B^{2}}{{\left (b g x + a g\right )} b g}\right )} \log \left (\frac {\frac {b^{2} c^{2} e g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d e g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} e g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d e g}{b g x + a g} - \frac {2 \, a d^{2} e g}{b g x + a g} + d^{2} e}{b^{2}}\right )^{2} - \frac {4 \, {\left (A B d - 2 \, B^{2} d\right )} \log \left (\frac {b c g}{b g x + a g} - \frac {a d g}{b g x + a g} + d\right )}{b^{2} c g^{2} - a b d g^{2}} - \frac {2 \, {\left (A B - 2 \, B^{2}\right )} \log \left (\frac {\frac {b^{2} c^{2} e g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d e g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} e g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d e g}{b g x + a g} - \frac {2 \, a d^{2} e g}{b g x + a g} + d^{2} e}{b^{2}}\right )}{{\left (b g x + a g\right )} b g} - \frac {A^{2} - 4 \, A B + 8 \, B^{2}}{{\left (b g x + a g\right )} b g} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 3.28 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.45 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=\frac {\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\,\left (\frac {4\,B^2}{b^2\,d\,g^2}-\frac {2\,A\,B}{b^2\,d\,g^2}\right )}{\frac {x}{d}+\frac {a}{b\,d}}-\frac {A^2-4\,A\,B+8\,B^2}{x\,b^2\,g^2+a\,b\,g^2}-{\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )}^2\,\left (\frac {B^2}{b^2\,g^2\,\left (x+\frac {a}{b}\right )}-\frac {B^2\,d}{b\,g^2\,\left (a\,d-b\,c\right )}\right )+\frac {B\,d\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {c\,b^2\,g^2+a\,d\,b\,g^2}{b\,g^2}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A-2\,B\right )\,8{}\mathrm {i}}{b\,g^2\,\left (a\,d-b\,c\right )} \]

[In]

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

[Out]

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