3.1.61 \(\int \frac {(c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^3} \, dx\) [61]

Optimal. Leaf size=141 \[ -\frac {B^2 i (c+d x)^2}{4 (b c-a d) g^3 (a+b x)^2}-\frac {B i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d) g^3 (a+b x)^2}-\frac {i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d) g^3 (a+b x)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2562, 2342, 2341} \begin {gather*} -\frac {i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^3 (a+b x)^2 (b c-a d)}-\frac {B i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 (a+b x)^2 (b c-a d)}-\frac {B^2 i (c+d x)^2}{4 g^3 (a+b x)^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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*} \int \frac {(61 c+61 d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx &=\int \left (\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g^3 (a+b x)^3}+\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g^3 (a+b x)^2}\right ) \, dx\\ &=\frac {(61 d) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{b g^3}+\frac {(61 (b c-a d)) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^3} \, dx}{b g^3}\\ &=-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {(122 B d) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {(61 B (b c-a d)) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^3}\\ &=-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {(122 B d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (61 B (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^3}\\ &=-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {(122 B d (b c-a d)) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^3}+\frac {\left (61 B (b c-a d)^2\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^3}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^2 g^3}\\ &=-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}-\frac {(61 B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b g^3}+\frac {(122 B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b g^3}+\frac {\left (61 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (122 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (61 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {\left (122 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {(61 B (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b g^3}\\ &=-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (122 B^2 d\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}-\frac {\left (61 B^2 d^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^2 (b c-a d) g^3}+\frac {\left (61 B^2 d^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 d^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^2 (b c-a d) g^3}-\frac {\left (122 B^2 d^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^2 (b c-a d) g^3}+\frac {\left (61 B^2 (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}\\ &=-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d (b c-a d)\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (122 B^2 d (b c-a d)\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (61 B^2 (b c-a d)^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}-\frac {\left (61 B^2 d^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d) e g^3}+\frac {\left (61 B^2 d^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d) e g^3}+\frac {\left (122 B^2 d^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d) e g^3}-\frac {\left (122 B^2 d^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d) e g^3}\\ &=-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d (b c-a d)\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^3}+\frac {\left (122 B^2 d (b c-a d)\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^3}+\frac {\left (61 B^2 (b c-a d)^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b^2 g^3}-\frac {\left (61 B^2 d^2\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^2 (b c-a d) e g^3}+\frac {\left (61 B^2 d^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{b^2 (b c-a d) e g^3}+\frac {\left (122 B^2 d^2\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^2 (b c-a d) e g^3}-\frac {\left (122 B^2 d^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{b^2 (b c-a d) e g^3}\\ &=-\frac {61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac {61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (61 B^2 d^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (122 B^2 d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (122 B^2 d^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (61 B^2 d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}-\frac {\left (122 B^2 d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 d^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}\\ &=-\frac {61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac {61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 d^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 d^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b (b c-a d) g^3}+\frac {\left (122 B^2 d^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac {\left (61 B^2 d^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 d^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}\\ &=-\frac {61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac {61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}+\frac {61 B^2 d^2 \log ^2(a+b x)}{2 b^2 (b c-a d) g^3}-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {61 B^2 d^2 \log ^2(c+d x)}{2 b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}-\frac {\left (61 B^2 d^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 d^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}+\frac {\left (122 B^2 d^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}\\ &=-\frac {61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac {61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}+\frac {61 B^2 d^2 \log ^2(a+b x)}{2 b^2 (b c-a d) g^3}-\frac {61 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {61 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {61 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac {61 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac {61 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac {61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {61 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac {61 B^2 d^2 \log ^2(c+d x)}{2 b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac {61 B^2 d^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.59, size = 765, normalized size = 5.43 \begin {gather*} -\frac {i \left (2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2-4 d (-b c+a d) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+4 B d (a+b x) \left (2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 d (a+b x) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+2 B (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-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 \text {Li}_2\left (\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 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )+B \left (2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d (-b c+a d) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-4 B d (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+B \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B d^2 (a+b 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 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )-2 B d^2 (a+b 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 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )\right )}{4 b^2 (b c-a d) g^3 (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(354\) vs. \(2(135)=270\).
time = 0.59, size = 355, normalized size = 2.52

method	result	size
norman	\(\frac {\frac {B^{2} c d i x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g \left (a d -c b \right )}+\frac {c i B d \left (2 A +B \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g \left (a d -c b \right )}-\frac {2 A^{2} a i d +2 A^{2} b c i +2 a d i B A +2 b c i B A +a d i \,B^{2}+b c i \,B^{2}}{4 g \,b^{2}}-\frac {\left (2 A^{2} i d +2 d i B A +d i \,B^{2}\right ) x}{2 g b}+\frac {B^{2} i \,c^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g \left (a d -c b \right )}+\frac {B^{2} d^{2} i \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 \left (a d -c b \right ) g}+\frac {\left (2 A +B \right ) c^{2} i B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (a d -c b \right )}+\frac {d^{2} i B \left (2 A +B \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (a d -c b \right )}}{g^{2} \left (b x +a \right )^{2}}\)	\(336\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (-\frac {i \,d^{2} e \,A^{2}}{2 \left (a d -c b \right )^{2} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {2 i \,d^{2} e A B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{2} g^{3}}+\frac {i \,d^{2} e \,B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{2} g^{3}}\right )}{d^{2}}\)	\(355\)
default	\(-\frac {e \left (a d -c b \right ) \left (-\frac {i \,d^{2} e \,A^{2}}{2 \left (a d -c b \right )^{2} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {2 i \,d^{2} e A B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{2} g^{3}}+\frac {i \,d^{2} e \,B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{2} g^{3}}\right )}{d^{2}}\)	\(355\)
risch	\(\frac {i \,A^{2} a d}{2 g^{3} b^{2} \left (b x +a \right )^{2}}-\frac {i \,A^{2} c}{2 g^{3} b \left (b x +a \right )^{2}}-\frac {i \,A^{2} d}{g^{3} b^{2} \left (b x +a \right )}+\frac {i \,B^{2} e^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g^{3} \left (a d -c b \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{2}}+\frac {i \,B^{2} e^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 g^{3} \left (a d -c b \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{2}}+\frac {i \,B^{2} e^{2}}{4 g^{3} \left (a d -c b \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{2}}+\frac {i A B \,e^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{3} \left (a d -c b \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{2}}+\frac {i A B \,e^{2}}{2 g^{3} \left (a d -c b \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{2}}\)	\(426\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1991 vs. \(2 (134) = 268\).
time = 0.47, size = 1991, normalized size = 14.12 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (134) = 268\).
time = 0.38, size = 291, normalized size = 2.06 \begin {gather*} -\frac {{\left (2 i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{2} c^{2} + {\left (-2 i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} a^{2} d^{2} - 2 \, {\left (-i \, B^{2} b^{2} d^{2} x^{2} - 2 i \, B^{2} b^{2} c d x - i \, B^{2} b^{2} c^{2}\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )^{2} - 2 \, {\left ({\left (-2 i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} b^{2} c d + {\left (2 i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a b d^{2}\right )} x - 2 \, {\left ({\left (-2 i \, A B - i \, B^{2}\right )} b^{2} d^{2} x^{2} + 2 \, {\left (-2 i \, A B - i \, B^{2}\right )} b^{2} c d x + {\left (-2 i \, A B - i \, B^{2}\right )} b^{2} c^{2}\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 714 vs. \(2 (122) = 244\).
time = 6.53, size = 714, normalized size = 5.06 \begin {gather*} - \frac {B d^{2} i \left (2 A + B\right ) \log {\left (x + \frac {2 A B a d^{3} i + 2 A B b c d^{2} i + B^{2} a d^{3} i + B^{2} b c d^{2} i - \frac {B a^{2} d^{4} i \left (2 A + B\right )}{a d - b c} + \frac {2 B a b c d^{3} i \left (2 A + B\right )}{a d - b c} - \frac {B b^{2} c^{2} d^{2} i \left (2 A + B\right )}{a d - b c}}{4 A B b d^{3} i + 2 B^{2} b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {B d^{2} i \left (2 A + B\right ) \log {\left (x + \frac {2 A B a d^{3} i + 2 A B b c d^{2} i + B^{2} a d^{3} i + B^{2} b c d^{2} i + \frac {B a^{2} d^{4} i \left (2 A + B\right )}{a d - b c} - \frac {2 B a b c d^{3} i \left (2 A + B\right )}{a d - b c} + \frac {B b^{2} c^{2} d^{2} i \left (2 A + B\right )}{a d - b c}}{4 A B b d^{3} i + 2 B^{2} b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {\left (B^{2} c^{2} i + 2 B^{2} c d i x + B^{2} d^{2} i x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{3} d g^{3} - 2 a^{2} b c g^{3} + 4 a^{2} b d g^{3} x - 4 a b^{2} c g^{3} x + 2 a b^{2} d g^{3} x^{2} - 2 b^{3} c g^{3} x^{2}} + \frac {- 2 A^{2} a d i - 2 A^{2} b c i - 2 A B a d i - 2 A B b c i - B^{2} a d i - B^{2} b c i + x \left (- 4 A^{2} b d i - 4 A B b d i - 2 B^{2} b d i\right )}{4 a^{2} b^{2} g^{3} + 8 a b^{3} g^{3} x + 4 b^{4} g^{3} x^{2}} + \frac {\left (- 2 A B a d i - 2 A B b c i - 4 A B b d i x - B^{2} a d i - B^{2} b c i - 2 B^{2} b d i x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 a^{2} b^{2} g^{3} + 4 a b^{3} g^{3} x + 2 b^{4} g^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 3.05, size = 180, normalized size = 1.28 \begin {gather*} \frac {{\left (-2 i \, B^{2} e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )^{2} - 4 i \, A B e^{3} \log \left (\frac {b x e + a e}{d x + c}\right ) - 2 i \, B^{2} e^{3} \log \left (\frac {b x e + a e}{d x + c}\right ) - 2 i \, A^{2} e^{3} - 2 i \, A B e^{3} - i \, B^{2} e^{3}\right )} {\left (d x + c\right )}^{2} {\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 )}}{4 \, {\left (b x e + a e\right )}^{2} g^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 6.18, size = 469, normalized size = 3.33 \begin {gather*} -\frac {x\,\left (2\,b\,d\,i\,A^2+2\,b\,d\,i\,A\,B+b\,d\,i\,B^2\right )+A^2\,a\,d\,i+A^2\,b\,c\,i+\frac {B^2\,a\,d\,i}{2}+\frac {B^2\,b\,c\,i}{2}+A\,B\,a\,d\,i+A\,B\,b\,c\,i}{2\,a^2\,b^2\,g^3+4\,a\,b^3\,g^3\,x+2\,b^4\,g^3\,x^2}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {\frac {B^2\,c\,i}{2\,b^2\,g^3}+\frac {B^2\,a\,d\,i}{2\,b^3\,g^3}+\frac {B^2\,d\,i\,x}{b^2\,g^3}}{2\,a\,x+b\,x^2+\frac {a^2}{b}}-\frac {B^2\,d^2\,i}{2\,b^2\,g^3\,\left (a\,d-b\,c\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (x\,\left (\frac {B^2\,i}{b^2\,g^3}+\frac {2\,A\,B\,i}{b^2\,g^3}\right )+\frac {A\,B\,a\,i}{b^3\,g^3}+\frac {B\,i\,\left (A\,b\,c-B\,a\,d+B\,b\,c\right )}{b^3\,d\,g^3}+\frac {B^2\,d^2\,i\,\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 )}{b^2\,g^3\,\left (a\,d-b\,c\right )}\right )}{\frac {b\,x^2}{d}+\frac {a^2}{b\,d}+\frac {2\,a\,x}{d}}-\frac {B\,d^2\,i\,\mathrm {atan}\left (\frac {\left (\frac {2\,c\,b^3\,g^3+2\,a\,d\,b^2\,g^3}{2\,b^2\,g^3}+2\,b\,d\,x\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (2\,A+B\right )\,1{}\mathrm {i}}{b^2\,g^3\,\left (a\,d-b\,c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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