3.1.71 \(\int \frac {(c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^4} \, dx\) [71]

Optimal. Leaf size=147 \[ -\frac {2 B^2 i^2 (c+d x)^3}{27 (b c-a d) g^4 (a+b x)^3}-\frac {2 B i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d) g^4 (a+b x)^3}-\frac {i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d) g^4 (a+b x)^3} \]

[Out]

-2/27*B^2*i^2*(d*x+c)^3/(-a*d+b*c)/g^4/(b*x+a)^3-2/9*B*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)/g^
4/(b*x+a)^3-1/3*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)/g^4/(b*x+a)^3

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Rubi [A]
time = 0.12, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2562, 2342, 2341} \begin {gather*} -\frac {i^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 g^4 (a+b x)^3 (b c-a d)}-\frac {2 B i^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{9 g^4 (a+b x)^3 (b c-a d)}-\frac {2 B^2 i^2 (c+d x)^3}{27 g^4 (a+b x)^3 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*B^2*i^2*(c + d*x)^3)/(27*(b*c - a*d)*g^4*(a + b*x)^3) - (2*B*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c +
d*x)]))/(9*(b*c - a*d)*g^4*(a + b*x)^3) - (i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(b*c -
a*d)*g^4*(a + b*x)^3)

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 {(71 c+71 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx &=\int \left (\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^4}+\frac {10082 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^3}+\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}\right ) \, dx\\ &=\frac {\left (5041 d^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{b^2 g^4}+\frac {(10082 d (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^2 g^4}+\frac {\left (5041 (b c-a d)^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^4} \, dx}{b^2 g^4}\\ &=-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {\left (10082 B d^2\right ) \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^3 g^4}+\frac {(10082 B d (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^3 g^4}+\frac {\left (10082 B (b c-a d)^2\right ) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {\left (10082 B d^2 (b c-a d)\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (10082 B d (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^3 g^4}+\frac {\left (10082 B (b c-a d)^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {\left (10082 B d^2 (b c-a d)\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)^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^3 g^4}+\frac {\left (10082 B d (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^3 g^4}+\frac {\left (10082 B (b c-a d)^3\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)^4}-\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)^3}+\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)^2}-\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b^3 g^4}\\ &=-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {\left (10082 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{3 b^2 g^4}-\frac {\left (10082 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{3 b^2 (b c-a d) g^4}+\frac {\left (10082 B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{3 b^3 (b c-a d) g^4}-\frac {(10082 B d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{3 b^2 g^4}+\frac {(10082 B d (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^2 g^4}+\frac {\left (10082 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)^4} \, dx}{3 b^2 g^4}\\ &=-\frac {10082 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac {10082 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac {10082 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac {10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {10082 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 g^4}+\frac {\left (10082 B^2 d^3\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}{3 b^3 (b c-a d) g^4}-\frac {\left (10082 B^2 d^3\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}{3 b^3 (b c-a d) g^4}-\frac {\left (5041 B^2 d (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 g^4}+\frac {\left (5041 B^2 d (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (10082 B^2 (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^4}\\ &=-\frac {10082 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac {10082 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac {10082 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac {10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {10082 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 g^4}-\frac {\left (5041 B^2 d (b c-a d)^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 g^4}+\frac {\left (5041 B^2 d (b c-a d)^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (10082 B^2 (b c-a d)^3\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^4}+\frac {\left (10082 B^2 d^3\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}{3 b^3 (b c-a d) e g^4}-\frac {\left (10082 B^2 d^3\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}{3 b^3 (b c-a d) e g^4}\\ &=-\frac {10082 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac {10082 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac {10082 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac {10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {10082 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^2 (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}{3 b^3 g^4}-\frac {\left (5041 B^2 d (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}{3 b^3 g^4}+\frac {\left (5041 B^2 d (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}{b^3 g^4}+\frac {\left (10082 B^2 (b c-a d)^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{9 b^3 g^4}+\frac {\left (10082 B^2 d^3\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}{3 b^3 (b c-a d) e g^4}-\frac {\left (10082 B^2 d^3\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}{3 b^3 (b c-a d) e g^4}\\ &=-\frac {10082 B^2 (b c-a d)^2}{27 b^3 g^4 (a+b x)^3}-\frac {10082 B^2 d (b c-a d)}{9 b^3 g^4 (a+b x)^2}-\frac {10082 B^2 d^2}{9 b^3 g^4 (a+b x)}-\frac {10082 B^2 d^3 \log (a+b x)}{9 b^3 (b c-a d) g^4}-\frac {10082 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac {10082 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac {10082 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac {10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {10082 B^2 d^3 \log (c+d x)}{9 b^3 (b c-a d) g^4}+\frac {10082 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 b^2 (b c-a d) g^4}-\frac {\left (10082 B^2 d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 b^2 (b c-a d) g^4}-\frac {\left (10082 B^2 d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b^3 (b c-a d) g^4}\\ &=-\frac {10082 B^2 (b c-a d)^2}{27 b^3 g^4 (a+b x)^3}-\frac {10082 B^2 d (b c-a d)}{9 b^3 g^4 (a+b x)^2}-\frac {10082 B^2 d^2}{9 b^3 g^4 (a+b x)}-\frac {10082 B^2 d^3 \log (a+b x)}{9 b^3 (b c-a d) g^4}-\frac {10082 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac {10082 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac {10082 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac {10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {10082 B^2 d^3 \log (c+d x)}{9 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {10082 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 b^2 (b c-a d) g^4}+\frac {\left (10082 B^2 d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b^3 (b c-a d) g^4}\\ &=-\frac {10082 B^2 (b c-a d)^2}{27 b^3 g^4 (a+b x)^3}-\frac {10082 B^2 d (b c-a d)}{9 b^3 g^4 (a+b x)^2}-\frac {10082 B^2 d^2}{9 b^3 g^4 (a+b x)}-\frac {10082 B^2 d^3 \log (a+b x)}{9 b^3 (b c-a d) g^4}+\frac {5041 B^2 d^3 \log ^2(a+b x)}{3 b^3 (b c-a d) g^4}-\frac {10082 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac {10082 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac {10082 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac {10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {10082 B^2 d^3 \log (c+d x)}{9 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {10082 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {5041 B^2 d^3 \log ^2(c+d x)}{3 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3 b^3 (b c-a d) g^4}+\frac {\left (10082 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b^3 (b c-a d) g^4}\\ &=-\frac {10082 B^2 (b c-a d)^2}{27 b^3 g^4 (a+b x)^3}-\frac {10082 B^2 d (b c-a d)}{9 b^3 g^4 (a+b x)^2}-\frac {10082 B^2 d^2}{9 b^3 g^4 (a+b x)}-\frac {10082 B^2 d^3 \log (a+b x)}{9 b^3 (b c-a d) g^4}+\frac {5041 B^2 d^3 \log ^2(a+b x)}{3 b^3 (b c-a d) g^4}-\frac {10082 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac {10082 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac {10082 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac {10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac {5041 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {5041 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac {5041 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {10082 B^2 d^3 \log (c+d x)}{9 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {10082 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac {5041 B^2 d^3 \log ^2(c+d x)}{3 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}-\frac {10082 B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 1.23, size = 1355, normalized size = 9.22 \begin {gather*} -\frac {i^2 \left (18 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+54 d (b c-a d)^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2-54 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+54 B d^2 (a+b x)^2 \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 )+27 B d (a+b x) \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 )+B \left (12 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-18 d (b c-a d)^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+36 d^2 (b c-a d) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+36 d^3 (a+b x)^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-36 d^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+36 B d^2 (a+b x)^2 (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-9 B d (a+b x) \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 \left (2 (b c-a d)^3-3 d (b c-a d)^2 (a+b x)+6 d^2 (b c-a d) (a+b x)^2+6 d^3 (a+b x)^3 \log (a+b x)-6 d^3 (a+b x)^3 \log (c+d x)\right )-18 B d^3 (a+b x)^3 \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 )+18 B d^3 (a+b x)^3 \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 )}{54 b^3 (b c-a d) g^4 (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/54*(i^2*(18*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 54*d*(b*c - a*d)^2*(a + b*x)*(A + B*Log[
(e*(a + b*x))/(c + d*x)])^2 - 54*d^2*(-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 54*B*
d^2*(a + b*x)^2*(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]*(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)])) + 27*B*d*(a + b*x)*(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*Pol
yLog[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*(12*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(
c + d*x)]) - 18*d*(b*c - a*d)^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 36*d^2*(b*c - a*d)*(a + b*x)^
2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 36*d^3*(a + b*x)^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)])
- 36*d^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] + 36*B*d^2*(a + b*x)^2*(b*c - a*d + d*(
a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) - 9*B*d*(a + b*x)*((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*(2*(b*c - a*d)^3 - 3*d*(b*c - a*d
)^2*(a + b*x) + 6*d^2*(b*c - a*d)*(a + b*x)^2 + 6*d^3*(a + b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c + d*x
]) - 18*B*d^3*(a + b*x)^3*(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)]) + 18*B*d^3*(a + b*x)^3*((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^3*(b*c - a*d)*g^4*(a + b*x)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(366\) vs. \(2(141)=282\).
time = 0.70, size = 367, normalized size = 2.50 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x,method=_RETURNVERBOSE)

[Out]

-1/d^2*e*(a*d-b*c)*(-1/3*i^2*d^2*e^2/(a*d-b*c)^2/g^4*A^2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3+2*i^2*d^2*e^2/(a*d-b*
c)^2/g^4*A*B*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/9/(b*e/d+(a*d-b*c)*e/d/(d
*x+c))^3)+i^2*d^2*e^2/(a*d-b*c)^2/g^4*B^2*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c)
)^2-2/9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-2/27/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 5514 vs. \(2 (134) = 268\).
time = 0.83, size = 5514, normalized size = 37.51 \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)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algorithm="maxima")

[Out]

1/3*(3*b*x + a)*B^2*c*d*log(b*x*e/(d*x + c) + a*e/(d*x + c))^2/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*
x + a^3*b^2*g^4) + 1/3*(3*b^2*x^2 + 3*a*b*x + a^2)*B^2*d^2*log(b*x*e/(d*x + c) + a*e/(d*x + c))^2/(b^6*g^4*x^3
+ 3*a*b^5*g^4*x^2 + 3*a^2*b^4*g^4*x + a^3*b^3*g^4) + 1/54*(6*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2
*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*
d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d
+ a^5*b*d^2)*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*l
og(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4))*log(b*x*e/(d*x + c) + a*e/(d*x + c)
) + (4*b^3*c^3 - 27*a*b^2*c^2*d + 108*a^2*b*c*d^2 - 85*a^3*d^3 + 66*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 18*(b^3*d^3*
x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a)^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*
d^3*x + a^3*d^3)*log(d*x + c)^2 - 3*(5*b^3*c^2*d - 54*a*b^2*c*d^2 + 49*a^2*b*d^3)*x + 66*(b^3*d^3*x^3 + 3*a*b^
2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a) - 6*(11*b^3*d^3*x^3 + 33*a*b^2*d^3*x^2 + 33*a^2*b*d^3*x + 11
*a^3*d^3 - 6*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a))*log(d*x + c))/(a^3*b^4*c^
3*g^4 - 3*a^4*b^3*c^2*d*g^4 + 3*a^5*b^2*c*d^2*g^4 - a^6*b*d^3*g^4 + (b^7*c^3*g^4 - 3*a*b^6*c^2*d*g^4 + 3*a^2*b
^5*c*d^2*g^4 - a^3*b^4*d^3*g^4)*x^3 + 3*(a*b^6*c^3*g^4 - 3*a^2*b^5*c^2*d*g^4 + 3*a^3*b^4*c*d^2*g^4 - a^4*b^3*d
^3*g^4)*x^2 + 3*(a^2*b^5*c^3*g^4 - 3*a^3*b^4*c^2*d*g^4 + 3*a^4*b^3*c*d^2*g^4 - a^5*b^2*d^3*g^4)*x))*B^2*c^2 +
1/54*(6*((5*a*b^2*c^2 - 22*a^2*b*c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 16*a*b^2*c*d
+ 5*a^2*b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2
)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*
g^4) - 6*(3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4) + 6*
(3*b*c*d^2 - a*d^3)*log(d*x + c)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4))*log(b*x*e/(d
*x + c) + a*e/(d*x + c)) + (19*a*b^3*c^3 - 189*a^2*b^2*c^2*d + 189*a^3*b*c*d^2 - 19*a^4*d^3 - 6*(27*b^4*c^2*d
- 32*a*b^3*c*d^2 + 5*a^2*b^2*d^3)*x^2 + 18*(3*a^3*b*c*d^2 - a^4*d^3 + (3*b^4*c*d^2 - a*b^3*d^3)*x^3 + 3*(3*a*b
^3*c*d^2 - a^2*b^2*d^3)*x^2 + 3*(3*a^2*b^2*c*d^2 - a^3*b*d^3)*x)*log(b*x + a)^2 + 18*(3*a^3*b*c*d^2 - a^4*d^3
+ (3*b^4*c*d^2 - a*b^3*d^3)*x^3 + 3*(3*a*b^3*c*d^2 - a^2*b^2*d^3)*x^2 + 3*(3*a^2*b^2*c*d^2 - a^3*b*d^3)*x)*log
(d*x + c)^2 + 3*(9*b^4*c^3 - 125*a*b^3*c^2*d + 135*a^2*b^2*c*d^2 - 19*a^3*b*d^3)*x - 6*(27*a^3*b*c*d^2 - 5*a^4
*d^3 + (27*b^4*c*d^2 - 5*a*b^3*d^3)*x^3 + 3*(27*a*b^3*c*d^2 - 5*a^2*b^2*d^3)*x^2 + 3*(27*a^2*b^2*c*d^2 - 5*a^3
*b*d^3)*x)*log(b*x + a) + 6*(27*a^3*b*c*d^2 - 5*a^4*d^3 + (27*b^4*c*d^2 - 5*a*b^3*d^3)*x^3 + 3*(27*a*b^3*c*d^2
- 5*a^2*b^2*d^3)*x^2 + 3*(27*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x - 6*(3*a^3*b*c*d^2 - a^4*d^3 + (3*b^4*c*d^2 - a*b
^3*d^3)*x^3 + 3*(3*a*b^3*c*d^2 - a^2*b^2*d^3)*x^2 + 3*(3*a^2*b^2*c*d^2 - a^3*b*d^3)*x)*log(b*x + a))*log(d*x +
c))/(a^3*b^5*c^3*g^4 - 3*a^4*b^4*c^2*d*g^4 + 3*a^5*b^3*c*d^2*g^4 - a^6*b^2*d^3*g^4 + (b^8*c^3*g^4 - 3*a*b^7*c
^2*d*g^4 + 3*a^2*b^6*c*d^2*g^4 - a^3*b^5*d^3*g^4)*x^3 + 3*(a*b^7*c^3*g^4 - 3*a^2*b^6*c^2*d*g^4 + 3*a^3*b^5*c*d
^2*g^4 - a^4*b^4*d^3*g^4)*x^2 + 3*(a^2*b^6*c^3*g^4 - 3*a^3*b^5*c^2*d*g^4 + 3*a^4*b^4*c*d^2*g^4 - a^5*b^3*d^3*g
^4)*x))*B^2*c*d + 1/54*(6*((11*a^2*b^2*c^2 - 7*a^3*b*c*d + 2*a^4*d^2 + 6*(3*b^4*c^2 - 3*a*b^3*c*d + a^2*b^2*d^
2)*x^2 + 3*(9*a*b^3*c^2 - 7*a^2*b^2*c*d + 2*a^3*b*d^2)*x)/((b^8*c^2 - 2*a*b^7*c*d + a^2*b^6*d^2)*g^4*x^3 + 3*(
a*b^7*c^2 - 2*a^2*b^6*c*d + a^3*b^5*d^2)*g^4*x^2 + 3*(a^2*b^6*c^2 - 2*a^3*b^5*c*d + a^4*b^4*d^2)*g^4*x + (a^3*
b^5*c^2 - 2*a^4*b^4*c*d + a^5*b^3*d^2)*g^4) + 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(b*x + a)/((b^6*c^3 -
3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4) - 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(d*x + c)/((
b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4))*log(b*x*e/(d*x + c) + a*e/(d*x + c)) + (85*a^2*
b^3*c^3 - 108*a^3*b^2*c^2*d + 27*a^4*b*c*d^2 - 4*a^5*d^3 + 6*(18*b^5*c^3 - 27*a*b^4*c^2*d + 11*a^2*b^3*c*d^2 -
2*a^3*b^2*d^3)*x^2 - 18*(3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2 + a^5*d^3 + (3*b^5*c^2*d - 3*a*b^4*c*d^2 + a^2*b^3*d
^3)*x^3 + 3*(3*a*b^4*c^2*d - 3*a^2*b^3*c*d^2 + a^3*b^2*d^3)*x^2 + 3*(3*a^2*b^3*c^2*d - 3*a^3*b^2*c*d^2 + a^4*b
*d^3)*x)*log(b*x + a)^2 - 18*(3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2 + a^5*d^3 + (3*b^5*c^2*d - 3*a*b^4*c*d^2 + a^2*b
^3*d^3)*x^3 + 3*(3*a*b^4*c^2*d - 3*a^2*b^3*c*d^2 + a^3*b^2*d^3)*x^2 + 3*(3*a^2*b^3*c^2*d - 3*a^3*b^2*c*d^2 + a
^4*b*d^3)*x)*log(d*x + c)^2 + 3*(63*a*b^4*c^3 - 86*a^2*b^3*c^2*d + 27*a^3*b^2*c*d^2 - 4*a^4*b*d^3)*x + 6*(18*a
^3*b^2*c^2*d - 9*a^4*b*c*d^2 + 2*a^5*d^3 + (18*b^5*c^2*d - 9*a*b^4*c*d^2 + 2*a^2*b^3*d^3)*x^3 + 3*(18*a*b^4*c^
2*d - 9*a^2*b^3*c*d^2 + 2*a^3*b^2*d^3)*x^2 + 3*...

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (134) = 268\).
time = 0.37, size = 407, normalized size = 2.77 \begin {gather*} \frac {{\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c^{3} - {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a^{3} d^{3} + 3 \, {\left ({\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c d^{2} - {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 9 \, {\left (B^{2} b^{3} d^{3} x^{3} + 3 \, B^{2} b^{3} c d^{2} x^{2} + 3 \, B^{2} b^{3} c^{2} d x + B^{2} b^{3} c^{3}\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )^{2} + 3 \, {\left ({\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c^{2} d - {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a^{2} b d^{3}\right )} x + 6 \, {\left ({\left (3 \, A B + B^{2}\right )} b^{3} d^{3} x^{3} + 3 \, {\left (3 \, A B + B^{2}\right )} b^{3} c d^{2} x^{2} + 3 \, {\left (3 \, A B + B^{2}\right )} b^{3} c^{2} d x + {\left (3 \, A B + B^{2}\right )} b^{3} c^{3}\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{27 \, {\left ({\left (b^{7} c - a b^{6} d\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c - a^{2} b^{5} d\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c - a^{3} b^{4} d\right )} g^{4} x + {\left (a^{3} b^{4} c - a^{4} b^{3} d\right )} g^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algorithm="fricas")

[Out]

1/27*((9*A^2 + 6*A*B + 2*B^2)*b^3*c^3 - (9*A^2 + 6*A*B + 2*B^2)*a^3*d^3 + 3*((9*A^2 + 6*A*B + 2*B^2)*b^3*c*d^2
- (9*A^2 + 6*A*B + 2*B^2)*a*b^2*d^3)*x^2 + 9*(B^2*b^3*d^3*x^3 + 3*B^2*b^3*c*d^2*x^2 + 3*B^2*b^3*c^2*d*x + B^2
*b^3*c^3)*log((b*x + a)*e/(d*x + c))^2 + 3*((9*A^2 + 6*A*B + 2*B^2)*b^3*c^2*d - (9*A^2 + 6*A*B + 2*B^2)*a^2*b*
d^3)*x + 6*((3*A*B + B^2)*b^3*d^3*x^3 + 3*(3*A*B + B^2)*b^3*c*d^2*x^2 + 3*(3*A*B + B^2)*b^3*c^2*d*x + (3*A*B +
B^2)*b^3*c^3)*log((b*x + a)*e/(d*x + c)))/((b^7*c - a*b^6*d)*g^4*x^3 + 3*(a*b^6*c - a^2*b^5*d)*g^4*x^2 + 3*(a
^2*b^5*c - a^3*b^4*d)*g^4*x + (a^3*b^4*c - a^4*b^3*d)*g^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1182 vs. \(2 (131) = 262\).
time = 35.43, size = 1182, normalized size = 8.04 \begin {gather*} - \frac {2 B d^{3} i^{2} \cdot \left (3 A + B\right ) \log {\left (x + \frac {6 A B a d^{4} i^{2} + 6 A B b c d^{3} i^{2} + 2 B^{2} a d^{4} i^{2} + 2 B^{2} b c d^{3} i^{2} - \frac {2 B a^{2} d^{5} i^{2} \cdot \left (3 A + B\right )}{a d - b c} + \frac {4 B a b c d^{4} i^{2} \cdot \left (3 A + B\right )}{a d - b c} - \frac {2 B b^{2} c^{2} d^{3} i^{2} \cdot \left (3 A + B\right )}{a d - b c}}{12 A B b d^{4} i^{2} + 4 B^{2} b d^{4} i^{2}} \right )}}{9 b^{3} g^{4} \left (a d - b c\right )} + \frac {2 B d^{3} i^{2} \cdot \left (3 A + B\right ) \log {\left (x + \frac {6 A B a d^{4} i^{2} + 6 A B b c d^{3} i^{2} + 2 B^{2} a d^{4} i^{2} + 2 B^{2} b c d^{3} i^{2} + \frac {2 B a^{2} d^{5} i^{2} \cdot \left (3 A + B\right )}{a d - b c} - \frac {4 B a b c d^{4} i^{2} \cdot \left (3 A + B\right )}{a d - b c} + \frac {2 B b^{2} c^{2} d^{3} i^{2} \cdot \left (3 A + B\right )}{a d - b c}}{12 A B b d^{4} i^{2} + 4 B^{2} b d^{4} i^{2}} \right )}}{9 b^{3} g^{4} \left (a d - b c\right )} + \frac {\left (B^{2} c^{3} i^{2} + 3 B^{2} c^{2} d i^{2} x + 3 B^{2} c d^{2} i^{2} x^{2} + B^{2} d^{3} i^{2} x^{3}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{3 a^{4} d g^{4} - 3 a^{3} b c g^{4} + 9 a^{3} b d g^{4} x - 9 a^{2} b^{2} c g^{4} x + 9 a^{2} b^{2} d g^{4} x^{2} - 9 a b^{3} c g^{4} x^{2} + 3 a b^{3} d g^{4} x^{3} - 3 b^{4} c g^{4} x^{3}} + \frac {- 9 A^{2} a^{2} d^{2} i^{2} - 9 A^{2} a b c d i^{2} - 9 A^{2} b^{2} c^{2} i^{2} - 6 A B a^{2} d^{2} i^{2} - 6 A B a b c d i^{2} - 6 A B b^{2} c^{2} i^{2} - 2 B^{2} a^{2} d^{2} i^{2} - 2 B^{2} a b c d i^{2} - 2 B^{2} b^{2} c^{2} i^{2} + x^{2} \left (- 27 A^{2} b^{2} d^{2} i^{2} - 18 A B b^{2} d^{2} i^{2} - 6 B^{2} b^{2} d^{2} i^{2}\right ) + x \left (- 27 A^{2} a b d^{2} i^{2} - 27 A^{2} b^{2} c d i^{2} - 18 A B a b d^{2} i^{2} - 18 A B b^{2} c d i^{2} - 6 B^{2} a b d^{2} i^{2} - 6 B^{2} b^{2} c d i^{2}\right )}{27 a^{3} b^{3} g^{4} + 81 a^{2} b^{4} g^{4} x + 81 a b^{5} g^{4} x^{2} + 27 b^{6} g^{4} x^{3}} + \frac {\left (- 6 A B a^{2} d^{2} i^{2} - 6 A B a b c d i^{2} - 18 A B a b d^{2} i^{2} x - 6 A B b^{2} c^{2} i^{2} - 18 A B b^{2} c d i^{2} x - 18 A B b^{2} d^{2} i^{2} x^{2} - 2 B^{2} a^{2} d^{2} i^{2} - 2 B^{2} a b c d i^{2} - 6 B^{2} a b d^{2} i^{2} x - 2 B^{2} b^{2} c^{2} i^{2} - 6 B^{2} b^{2} c d i^{2} x - 6 B^{2} b^{2} d^{2} i^{2} x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{9 a^{3} b^{3} g^{4} + 27 a^{2} b^{4} g^{4} x + 27 a b^{5} g^{4} x^{2} + 9 b^{6} g^{4} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)**2*(A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**4,x)

[Out]

-2*B*d**3*i**2*(3*A + B)*log(x + (6*A*B*a*d**4*i**2 + 6*A*B*b*c*d**3*i**2 + 2*B**2*a*d**4*i**2 + 2*B**2*b*c*d*
*3*i**2 - 2*B*a**2*d**5*i**2*(3*A + B)/(a*d - b*c) + 4*B*a*b*c*d**4*i**2*(3*A + B)/(a*d - b*c) - 2*B*b**2*c**2
*d**3*i**2*(3*A + B)/(a*d - b*c))/(12*A*B*b*d**4*i**2 + 4*B**2*b*d**4*i**2))/(9*b**3*g**4*(a*d - b*c)) + 2*B*d
**3*i**2*(3*A + B)*log(x + (6*A*B*a*d**4*i**2 + 6*A*B*b*c*d**3*i**2 + 2*B**2*a*d**4*i**2 + 2*B**2*b*c*d**3*i**
2 + 2*B*a**2*d**5*i**2*(3*A + B)/(a*d - b*c) - 4*B*a*b*c*d**4*i**2*(3*A + B)/(a*d - b*c) + 2*B*b**2*c**2*d**3*
i**2*(3*A + B)/(a*d - b*c))/(12*A*B*b*d**4*i**2 + 4*B**2*b*d**4*i**2))/(9*b**3*g**4*(a*d - b*c)) + (B**2*c**3*
i**2 + 3*B**2*c**2*d*i**2*x + 3*B**2*c*d**2*i**2*x**2 + B**2*d**3*i**2*x**3)*log(e*(a + b*x)/(c + d*x))**2/(3*
a**4*d*g**4 - 3*a**3*b*c*g**4 + 9*a**3*b*d*g**4*x - 9*a**2*b**2*c*g**4*x + 9*a**2*b**2*d*g**4*x**2 - 9*a*b**3*
c*g**4*x**2 + 3*a*b**3*d*g**4*x**3 - 3*b**4*c*g**4*x**3) + (-9*A**2*a**2*d**2*i**2 - 9*A**2*a*b*c*d*i**2 - 9*A
**2*b**2*c**2*i**2 - 6*A*B*a**2*d**2*i**2 - 6*A*B*a*b*c*d*i**2 - 6*A*B*b**2*c**2*i**2 - 2*B**2*a**2*d**2*i**2
- 2*B**2*a*b*c*d*i**2 - 2*B**2*b**2*c**2*i**2 + x**2*(-27*A**2*b**2*d**2*i**2 - 18*A*B*b**2*d**2*i**2 - 6*B**2
*b**2*d**2*i**2) + x*(-27*A**2*a*b*d**2*i**2 - 27*A**2*b**2*c*d*i**2 - 18*A*B*a*b*d**2*i**2 - 18*A*B*b**2*c*d*
i**2 - 6*B**2*a*b*d**2*i**2 - 6*B**2*b**2*c*d*i**2))/(27*a**3*b**3*g**4 + 81*a**2*b**4*g**4*x + 81*a*b**5*g**4
*x**2 + 27*b**6*g**4*x**3) + (-6*A*B*a**2*d**2*i**2 - 6*A*B*a*b*c*d*i**2 - 18*A*B*a*b*d**2*i**2*x - 6*A*B*b**2
*c**2*i**2 - 18*A*B*b**2*c*d*i**2*x - 18*A*B*b**2*d**2*i**2*x**2 - 2*B**2*a**2*d**2*i**2 - 2*B**2*a*b*c*d*i**2
- 6*B**2*a*b*d**2*i**2*x - 2*B**2*b**2*c**2*i**2 - 6*B**2*b**2*c*d*i**2*x - 6*B**2*b**2*d**2*i**2*x**2)*log(e
*(a + b*x)/(c + d*x))/(9*a**3*b**3*g**4 + 27*a**2*b**4*g**4*x + 27*a*b**5*g**4*x**2 + 9*b**6*g**4*x**3)

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Giac [A]
time = 2.90, size = 180, normalized size = 1.22 \begin {gather*} \frac {{\left (9 \, B^{2} e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )^{2} + 18 \, A B e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) + 6 \, B^{2} e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) + 9 \, A^{2} e^{4} + 6 \, A B e^{4} + 2 \, B^{2} e^{4}\right )} {\left (d x + c\right )}^{3} {\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 )}}{27 \, {\left (b x e + a e\right )}^{3} g^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algorithm="giac")

[Out]

1/27*(9*B^2*e^4*log((b*x*e + a*e)/(d*x + c))^2 + 18*A*B*e^4*log((b*x*e + a*e)/(d*x + c)) + 6*B^2*e^4*log((b*x*
e + a*e)/(d*x + c)) + 9*A^2*e^4 + 6*A*B*e^4 + 2*B^2*e^4)*(d*x + c)^3*(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)^3*g^4)

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Mupad [B]
time = 7.41, size = 1153, normalized size = 7.84 \begin {gather*} -\frac {x^2\,\left (9\,A^2\,b^2\,d^2\,i^2+6\,A\,B\,b^2\,d^2\,i^2+2\,B^2\,b^2\,d^2\,i^2\right )+x\,\left (9\,c\,A^2\,b^2\,d\,i^2+9\,a\,A^2\,b\,d^2\,i^2+6\,c\,A\,B\,b^2\,d\,i^2+6\,a\,A\,B\,b\,d^2\,i^2+2\,c\,B^2\,b^2\,d\,i^2+2\,a\,B^2\,b\,d^2\,i^2\right )+3\,A^2\,a^2\,d^2\,i^2+3\,A^2\,b^2\,c^2\,i^2+\frac {2\,B^2\,a^2\,d^2\,i^2}{3}+\frac {2\,B^2\,b^2\,c^2\,i^2}{3}+2\,A\,B\,a^2\,d^2\,i^2+2\,A\,B\,b^2\,c^2\,i^2+3\,A^2\,a\,b\,c\,d\,i^2+\frac {2\,B^2\,a\,b\,c\,d\,i^2}{3}+2\,A\,B\,a\,b\,c\,d\,i^2}{9\,a^3\,b^3\,g^4+27\,a^2\,b^4\,g^4\,x+27\,a\,b^5\,g^4\,x^2+9\,b^6\,g^4\,x^3}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {x\,\left (b\,\left (\frac {B^2\,c\,d\,i^2}{3\,b^3\,g^4}+\frac {B^2\,a\,d^2\,i^2}{3\,b^4\,g^4}\right )+\frac {2\,B^2\,c\,d\,i^2}{3\,b^2\,g^4}+\frac {2\,B^2\,a\,d^2\,i^2}{3\,b^3\,g^4}\right )+a\,\left (\frac {B^2\,c\,d\,i^2}{3\,b^3\,g^4}+\frac {B^2\,a\,d^2\,i^2}{3\,b^4\,g^4}\right )+\frac {B^2\,c^2\,i^2}{3\,b^2\,g^4}+\frac {B^2\,d^2\,i^2\,x^2}{b^2\,g^4}}{3\,a^2\,x+\frac {a^3}{b}+b^2\,x^3+3\,a\,b\,x^2}-\frac {B^2\,d^3\,i^2}{3\,b^3\,g^4\,\left (a\,d-b\,c\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (x\,\left (b\,\left (\frac {B\,i^2\,\left (2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{3\,b^4\,g^4}+\frac {2\,A\,B\,a\,d\,i^2}{3\,b^4\,g^4}\right )+\frac {2\,B\,i^2\,\left (2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{3\,b^3\,g^4}+\frac {2\,B^2\,d^3\,i^2\,\left (b\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{3\,d^3}+\frac {2\,a\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b^3\,g^4\,\left (a\,d-b\,c\right )}+\frac {4\,A\,B\,a\,d\,i^2}{3\,b^3\,g^4}\right )+x^2\,\left (\frac {2\,A\,B\,d\,i^2}{b^2\,g^4}-\frac {2\,B^2\,d^3\,i^2\,\left (\frac {b^2\,c-a\,b\,d}{3\,d^2}-\frac {2\,b\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b^3\,g^4\,\left (a\,d-b\,c\right )}\right )+a\,\left (\frac {B\,i^2\,\left (2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{3\,b^4\,g^4}+\frac {2\,A\,B\,a\,d\,i^2}{3\,b^4\,g^4}\right )+\frac {2\,B\,i^2\,\left (-B\,a^2\,d^2+B\,a\,b\,c\,d+A\,b^2\,c^2\right )}{3\,b^4\,d\,g^4}+\frac {2\,B^2\,d^3\,i^2\,\left (a\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^3\,d^3-6\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d-b^3\,c^3}{3\,b\,d^4}\right )}{3\,b^3\,g^4\,\left (a\,d-b\,c\right )}\right )}{\frac {3\,a^2\,x}{d}+\frac {a^3}{b\,d}+\frac {b^2\,x^3}{d}+\frac {3\,a\,b\,x^2}{d}}-\frac {B\,d^3\,i^2\,\mathrm {atan}\left (\frac {\left (\frac {9\,c\,b^4\,g^4+9\,a\,d\,b^3\,g^4}{9\,b^3\,g^4}+2\,b\,d\,x\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (3\,A+B\right )\,4{}\mathrm {i}}{9\,b^3\,g^4\,\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)^2*(A + B*log((e*(a + b*x))/(c + d*x)))^2)/(a*g + b*g*x)^4,x)

[Out]

- (x^2*(9*A^2*b^2*d^2*i^2 + 2*B^2*b^2*d^2*i^2 + 6*A*B*b^2*d^2*i^2) + x*(9*A^2*a*b*d^2*i^2 + 2*B^2*a*b*d^2*i^2
+ 9*A^2*b^2*c*d*i^2 + 2*B^2*b^2*c*d*i^2 + 6*A*B*a*b*d^2*i^2 + 6*A*B*b^2*c*d*i^2) + 3*A^2*a^2*d^2*i^2 + 3*A^2*b
^2*c^2*i^2 + (2*B^2*a^2*d^2*i^2)/3 + (2*B^2*b^2*c^2*i^2)/3 + 2*A*B*a^2*d^2*i^2 + 2*A*B*b^2*c^2*i^2 + 3*A^2*a*b
*c*d*i^2 + (2*B^2*a*b*c*d*i^2)/3 + 2*A*B*a*b*c*d*i^2)/(9*a^3*b^3*g^4 + 9*b^6*g^4*x^3 + 27*a^2*b^4*g^4*x + 27*a
*b^5*g^4*x^2) - log((e*(a + b*x))/(c + d*x))^2*((x*(b*((B^2*c*d*i^2)/(3*b^3*g^4) + (B^2*a*d^2*i^2)/(3*b^4*g^4)
) + (2*B^2*c*d*i^2)/(3*b^2*g^4) + (2*B^2*a*d^2*i^2)/(3*b^3*g^4)) + a*((B^2*c*d*i^2)/(3*b^3*g^4) + (B^2*a*d^2*i
^2)/(3*b^4*g^4)) + (B^2*c^2*i^2)/(3*b^2*g^4) + (B^2*d^2*i^2*x^2)/(b^2*g^4))/(3*a^2*x + a^3/b + b^2*x^3 + 3*a*b
*x^2) - (B^2*d^3*i^2)/(3*b^3*g^4*(a*d - b*c))) - (log((e*(a + b*x))/(c + d*x))*(x*(b*((B*i^2*(2*A*b*c - B*a*d
+ B*b*c))/(3*b^4*g^4) + (2*A*B*a*d*i^2)/(3*b^4*g^4)) + (2*B*i^2*(2*A*b*c - B*a*d + B*b*c))/(3*b^3*g^4) + (2*B^
2*d^3*i^2*(b*((3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(3*b*d^2)) + (3*a^2*d^2 + b^2*c^2
- 4*a*b*c*d)/(3*d^3) + (2*a*(a*d - b*c))/(3*d^2)))/(3*b^3*g^4*(a*d - b*c)) + (4*A*B*a*d*i^2)/(3*b^3*g^4)) + x^
2*((2*A*B*d*i^2)/(b^2*g^4) - (2*B^2*d^3*i^2*((b^2*c - a*b*d)/(3*d^2) - (2*b*(a*d - b*c))/(3*d^2)))/(3*b^3*g^4*
(a*d - b*c))) + a*((B*i^2*(2*A*b*c - B*a*d + B*b*c))/(3*b^4*g^4) + (2*A*B*a*d*i^2)/(3*b^4*g^4)) + (2*B*i^2*(A*
b^2*c^2 - B*a^2*d^2 + B*a*b*c*d))/(3*b^4*d*g^4) + (2*B^2*d^3*i^2*(a*((3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(6*b*d^
3) + (a*(a*d - b*c))/(3*b*d^2)) + (3*a^3*d^3 - b^3*c^3 + 4*a*b^2*c^2*d - 6*a^2*b*c*d^2)/(3*b*d^4)))/(3*b^3*g^4
*(a*d - b*c))))/((3*a^2*x)/d + a^3/(b*d) + (b^2*x^3)/d + (3*a*b*x^2)/d) - (B*d^3*i^2*atan((((9*b^4*c*g^4 + 9*a
*b^3*d*g^4)/(9*b^3*g^4) + 2*b*d*x)*1i)/(a*d - b*c))*(3*A + B)*4i)/(9*b^3*g^4*(a*d - b*c))

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