3.2.2 \(\int \frac {(a g+b g x) (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(c i+d i x)^3} \, dx\) [102]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.08, 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 {g (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 i^3 (c+d x)^2 (b c-a d)}-\frac {B g (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 i^3 (c+d x)^2 (b c-a d)}+\frac {B^2 g (a+b x)^2}{4 i^3 (c+d x)^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B^2*g*(a + b*x)^2)/(4*(b*c - a*d)*i^3*(c + d*x)^2) - (B*g*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(
2*(b*c - a*d)*i^3*(c + d*x)^2) + (g*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)*i^3*(c
+ d*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 {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(102 c+102 d x)^3} \, dx &=\int \left (\frac {(-b c+a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d (c+d x)^3}+\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d (c+d x)^2}\right ) \, dx\\ &=\frac {(b g) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^2} \, dx}{1061208 d}-\frac {((b c-a d) g) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^3} \, dx}{1061208 d}\\ &=\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {(b B g) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)^2} \, dx}{530604 d^2}-\frac {(B (b c-a d) g) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)^3} \, dx}{1061208 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {(b B (b c-a d) g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)^2} \, dx}{530604 d^2}-\frac {\left (B (b c-a d)^2 g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)^3} \, dx}{1061208 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {(b B (b c-a d) g) \int \left (\frac {b^2 \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 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d 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 (c+d x)}\right ) \, dx}{530604 d^2}-\frac {\left (B (b c-a d)^2 g\right ) \int \left (\frac {b^3 \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 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d 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 (c+d x)^2}-\frac {b^2 d \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}{1061208 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {(b B g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{1061208 d}-\frac {(b B g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{530604 d}-\frac {\left (b^3 B g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{1061208 d^2 (b c-a d)}+\frac {\left (b^3 B g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{530604 d^2 (b c-a d)}+\frac {\left (b^2 B g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{1061208 d (b c-a d)}-\frac {\left (b^2 B g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{530604 d (b c-a d)}+\frac {(B (b c-a d) g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{1061208 d}\\ &=-\frac {B (b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac {b B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac {b^2 B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}-\frac {b^2 B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}+\frac {\left (b B^2 g\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{1061208 d^2}-\frac {\left (b B^2 g\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{530604 d^2}+\frac {\left (b^2 B^2 g\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}{1061208 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g\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}{1061208 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g\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}{530604 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g\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}{530604 d^2 (b c-a d)}+\frac {\left (B^2 (b c-a d) g\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{2122416 d^2}\\ &=-\frac {B (b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac {b B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac {b^2 B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}-\frac {b^2 B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}+\frac {\left (b B^2 (b c-a d) g\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{1061208 d^2}-\frac {\left (b B^2 (b c-a d) g\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{530604 d^2}+\frac {\left (B^2 (b c-a d)^2 g\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{2122416 d^2}+\frac {\left (b^2 B^2 g\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}{1061208 d^2 (b c-a d) e}-\frac {\left (b^2 B^2 g\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}{1061208 d^2 (b c-a d) e}-\frac {\left (b^2 B^2 g\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}{530604 d^2 (b c-a d) e}+\frac {\left (b^2 B^2 g\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}{530604 d^2 (b c-a d) e}\\ &=-\frac {B (b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac {b B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac {b^2 B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}-\frac {b^2 B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}+\frac {\left (b B^2 (b c-a d) g\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{1061208 d^2}-\frac {\left (b B^2 (b c-a d) g\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{530604 d^2}+\frac {\left (B^2 (b c-a d)^2 g\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2122416 d^2}+\frac {\left (b^2 B^2 g\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}{1061208 d^2 (b c-a d) e}-\frac {\left (b^2 B^2 g\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}{1061208 d^2 (b c-a d) e}-\frac {\left (b^2 B^2 g\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}{530604 d^2 (b c-a d) e}+\frac {\left (b^2 B^2 g\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}{530604 d^2 (b c-a d) e}\\ &=\frac {B^2 (b c-a d) g}{4244832 d^2 (c+d x)^2}-\frac {b B^2 g}{2122416 d^2 (c+d x)}-\frac {b^2 B^2 g \log (a+b x)}{2122416 d^2 (b c-a d)}-\frac {B (b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac {b B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac {b^2 B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {b^2 B^2 g \log (c+d x)}{2122416 d^2 (b c-a d)}-\frac {b^2 B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}+\frac {\left (b^3 B^2 g\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{1061208 d^2 (b c-a d)}-\frac {\left (b^3 B^2 g\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{1061208 d^2 (b c-a d)}-\frac {\left (b^3 B^2 g\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{530604 d^2 (b c-a d)}+\frac {\left (b^3 B^2 g\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{530604 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{1061208 d (b c-a d)}+\frac {\left (b^2 B^2 g\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{1061208 d (b c-a d)}+\frac {\left (b^2 B^2 g\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{530604 d (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{530604 d (b c-a d)}\\ &=\frac {B^2 (b c-a d) g}{4244832 d^2 (c+d x)^2}-\frac {b B^2 g}{2122416 d^2 (c+d x)}-\frac {b^2 B^2 g \log (a+b x)}{2122416 d^2 (b c-a d)}-\frac {B (b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac {b B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac {b^2 B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {b^2 B^2 g \log (c+d x)}{2122416 d^2 (b c-a d)}+\frac {b^2 B^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}-\frac {b^2 B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}+\frac {b^2 B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1061208 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{1061208 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{1061208 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{530604 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{530604 d^2 (b c-a d)}+\frac {\left (b^3 B^2 g\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{1061208 d^2 (b c-a d)}-\frac {\left (b^3 B^2 g\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{530604 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{1061208 d (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{530604 d (b c-a d)}\\ &=\frac {B^2 (b c-a d) g}{4244832 d^2 (c+d x)^2}-\frac {b B^2 g}{2122416 d^2 (c+d x)}-\frac {b^2 B^2 g \log (a+b x)}{2122416 d^2 (b c-a d)}-\frac {b^2 B^2 g \log ^2(a+b x)}{2122416 d^2 (b c-a d)}-\frac {B (b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac {b B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac {b^2 B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {b^2 B^2 g \log (c+d x)}{2122416 d^2 (b c-a d)}+\frac {b^2 B^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}-\frac {b^2 B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}-\frac {b^2 B^2 g \log ^2(c+d x)}{2122416 d^2 (b c-a d)}+\frac {b^2 B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1061208 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{1061208 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{1061208 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{530604 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{530604 d^2 (b c-a d)}\\ &=\frac {B^2 (b c-a d) g}{4244832 d^2 (c+d x)^2}-\frac {b B^2 g}{2122416 d^2 (c+d x)}-\frac {b^2 B^2 g \log (a+b x)}{2122416 d^2 (b c-a d)}-\frac {b^2 B^2 g \log ^2(a+b x)}{2122416 d^2 (b c-a d)}-\frac {B (b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2122416 d^2 (c+d x)^2}+\frac {b B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (c+d x)}+\frac {b^2 B g \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1061208 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2122416 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1061208 d^2 (c+d x)}+\frac {b^2 B^2 g \log (c+d x)}{2122416 d^2 (b c-a d)}+\frac {b^2 B^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}-\frac {b^2 B g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1061208 d^2 (b c-a d)}-\frac {b^2 B^2 g \log ^2(c+d x)}{2122416 d^2 (b c-a d)}+\frac {b^2 B^2 g \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1061208 d^2 (b c-a d)}+\frac {b^2 B^2 g \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{1061208 d^2 (b c-a d)}+\frac {b^2 B^2 g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1061208 d^2 (b c-a d)}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.58, size = 767, normalized size = 5.44 \begin {gather*} \frac {g \left (2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2-4 b (b c-a d) (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+4 b B (c+d x) \left (2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 b (c+d x) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 b (c+d 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+b (c+d x) \log (a+b x)-b (c+d x) \log (c+d x))-b B (c+d 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 B (c+d 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 b (b c-a d) (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 b^2 (c+d x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-4 b^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-4 b B (c+d x) (b c-a d+b (c+d x) \log (a+b x)-b (c+d x) \log (c+d x))-B \left ((b c-a d)^2+2 b (b c-a d) (c+d x)+2 b^2 (c+d x)^2 \log (a+b x)-2 b^2 (c+d x)^2 \log (c+d x)\right )-2 b^2 B (c+d x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+2 b^2 B (c+d x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )\right )}{4 d^2 (b c-a d) i^3 (c+d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(360\) vs. \(2(135)=270\).
time = 0.55, size = 361, normalized size = 2.56

method	result	size
norman	\(\frac {-\frac {2 A^{2} a d g +2 A^{2} b c g -2 d g a B A -2 c b g B A +d g a \,B^{2}+c b g \,B^{2}}{4 i \,d^{2}}-\frac {\left (2 A^{2} b g -2 b g B A +b g \,B^{2}\right ) x}{2 i d}-\frac {B^{2} a^{2} g \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 i \left (a d -c b \right )}-\frac {B^{2} b^{2} g \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 \left (a d -c b \right ) i}-\frac {\left (-B +2 A \right ) g \,a^{2} B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 i \left (a d -c b \right )}-\frac {B^{2} a b g x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{i \left (a d -c b \right )}-\frac {b^{2} g B \left (-B +2 A \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 i \left (a d -c b \right )}-\frac {g a B b \left (-B +2 A \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{i \left (a d -c b \right )}}{i^{2} \left (d x +c \right )^{2}}\)	\(344\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {g \,d^{2} A^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{2} e^{3} i^{3}}+\frac {2 g \,d^{2} A B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{2} e^{3} i^{3}}+\frac {g \,d^{2} B^{2} \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}+\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{2} e^{3} i^{3}}\right )}{d^{2}}\)	\(361\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {g \,d^{2} A^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{2} e^{3} i^{3}}+\frac {2 g \,d^{2} A B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{2} e^{3} i^{3}}+\frac {g \,d^{2} B^{2} \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}+\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{2} e^{3} i^{3}}\right )}{d^{2}}\)	\(361\)
risch	\(\text {Expression too large to display}\)	\(2227\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d^2*e*(a*d-b*c)*(1/2*g*d^2/(a*d-b*c)^2/e^3/i^3*A^2*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2+2*g*d^2/(a*d-b*c)^2/e^3/
i^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)+g*d^2/(a*d-b*c)^2/e^3/i^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. 1867 vs. \(2 (128) = 256\).
time = 0.49, size = 1867, normalized size = 13.24 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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. \(2 (128) = 256\).
time = 0.38, size = 291, normalized size = 2.06 \begin {gather*} \frac {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 )} g x + 2 \, {\left (i \, B^{2} b^{2} d^{2} g x^{2} + 2 i \, B^{2} a b d^{2} g x + i \, B^{2} a^{2} d^{2} g\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )^{2} - {\left ({\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}\right )} g + 2 \, {\left ({\left (2 i \, A B - i \, B^{2}\right )} b^{2} d^{2} g x^{2} + 2 \, {\left (2 i \, A B - i \, B^{2}\right )} a b d^{2} g x + {\left (2 i \, A B - i \, B^{2}\right )} a^{2} d^{2} g\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{4 \, {\left (b c^{3} d^{2} - a c^{2} d^{3} + {\left (b c d^{4} - a d^{5}\right )} x^{2} + 2 \, {\left (b c^{2} d^{3} - a c d^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 712 vs. \(2 (121) = 242\).
time = 7.39, size = 712, normalized size = 5.05 \begin {gather*} \frac {B b^{2} g \left (2 A - B\right ) \log {\left (x + \frac {2 A B a b^{2} d g + 2 A B b^{3} c g - B^{2} a b^{2} d g - B^{2} b^{3} c g - \frac {B a^{2} b^{2} d^{2} g \left (2 A - B\right )}{a d - b c} + \frac {2 B a b^{3} c d g \left (2 A - B\right )}{a d - b c} - \frac {B b^{4} c^{2} g \left (2 A - B\right )}{a d - b c}}{4 A B b^{3} d g - 2 B^{2} b^{3} d g} \right )}}{2 d^{2} i^{3} \left (a d - b c\right )} - \frac {B b^{2} g \left (2 A - B\right ) \log {\left (x + \frac {2 A B a b^{2} d g + 2 A B b^{3} c g - B^{2} a b^{2} d g - B^{2} b^{3} c g + \frac {B a^{2} b^{2} d^{2} g \left (2 A - B\right )}{a d - b c} - \frac {2 B a b^{3} c d g \left (2 A - B\right )}{a d - b c} + \frac {B b^{4} c^{2} g \left (2 A - B\right )}{a d - b c}}{4 A B b^{3} d g - 2 B^{2} b^{3} d g} \right )}}{2 d^{2} i^{3} \left (a d - b c\right )} + \frac {\left (- B^{2} a^{2} g - 2 B^{2} a b g x - B^{2} b^{2} g x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a c^{2} d i^{3} + 4 a c d^{2} i^{3} x + 2 a d^{3} i^{3} x^{2} - 2 b c^{3} i^{3} - 4 b c^{2} d i^{3} x - 2 b c d^{2} i^{3} x^{2}} + \frac {- 2 A^{2} a d g - 2 A^{2} b c g + 2 A B a d g + 2 A B b c g - B^{2} a d g - B^{2} b c g + x \left (- 4 A^{2} b d g + 4 A B b d g - 2 B^{2} b d g\right )}{4 c^{2} d^{2} i^{3} + 8 c d^{3} i^{3} x + 4 d^{4} i^{3} x^{2}} + \frac {\left (- 2 A B a d g - 2 A B b c g - 4 A B b d g x + B^{2} a d g + B^{2} b c g + 2 B^{2} b d g x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 c^{2} d^{2} i^{3} + 4 c d^{3} i^{3} x + 2 d^{4} i^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (128) = 256\).
time = 2.33, size = 268, normalized size = 1.90 \begin {gather*} \frac {1}{4} \, {\left (\frac {2 i \, {\left (b x e + a e\right )}^{2} B^{2} g \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{{\left (d x + c\right )}^{2}} + \frac {4 i \, {\left (b x e + a e\right )}^{2} A B g \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} - \frac {2 i \, {\left (b x e + a e\right )}^{2} B^{2} g \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + \frac {2 i \, {\left (b x e + a e\right )}^{2} A^{2} g}{{\left (d x + c\right )}^{2}} - \frac {2 i \, {\left (b x e + a e\right )}^{2} A B g}{{\left (d x + c\right )}^{2}} + \frac {i \, {\left (b x e + a e\right )}^{2} B^{2} g}{{\left (d x + c\right )}^{2}}\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 )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 6.40, size = 474, normalized size = 3.36 \begin {gather*} -\frac {x\,\left (2\,b\,d\,g\,A^2-2\,b\,d\,g\,A\,B+b\,d\,g\,B^2\right )+A^2\,a\,d\,g+A^2\,b\,c\,g+\frac {B^2\,a\,d\,g}{2}+\frac {B^2\,b\,c\,g}{2}-A\,B\,a\,d\,g-A\,B\,b\,c\,g}{2\,c^2\,d^2\,i^3+4\,c\,d^3\,i^3\,x+2\,d^4\,i^3\,x^2}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {\frac {B^2\,a\,g}{2\,d^2\,i^3}+\frac {B^2\,b\,c\,g}{2\,d^3\,i^3}+\frac {B^2\,b\,g\,x}{d^2\,i^3}}{2\,c\,x+d\,x^2+\frac {c^2}{d}}+\frac {B^2\,b^2\,g}{2\,d^2\,i^3\,\left (a\,d-b\,c\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {A\,B\,c\,g}{d^3\,i^3}-x\,\left (\frac {B^2\,g}{d^2\,i^3}-\frac {2\,A\,B\,g}{d^2\,i^3}\right )+\frac {B\,g\,\left (A\,a\,d-B\,a\,d+B\,b\,c\right )}{b\,d^3\,i^3}+\frac {B^2\,b^2\,g\,\left (\frac {a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2}{2\,b^3\,d}-\frac {c\,\left (a\,d-b\,c\right )}{2\,b^2\,d}\right )}{d^2\,i^3\,\left (a\,d-b\,c\right )}\right )}{\frac {d\,x^2}{b}+\frac {c^2}{b\,d}+\frac {2\,c\,x}{b}}+\frac {B\,b^2\,g\,\mathrm {atan}\left (\frac {\left (\frac {2\,a\,d^3\,i^3+2\,b\,c\,d^2\,i^3}{2\,d^2\,i^3}+2\,b\,d\,x\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (2\,A-B\right )\,1{}\mathrm {i}}{d^2\,i^3\,\left (a\,d-b\,c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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