3.3.16 \(\int \frac {(A+B \log (\frac {e (c+d x)^2}{(a+b x)^2}))^2}{(a g+b g x)^3} \, dx\) [216]

Optimal. Leaf size=299 \[ -\frac {4 A B d (c+d x)}{(b c-a d)^2 g^3 (a+b x)}+\frac {8 B^2 d (c+d x)}{(b c-a d)^2 g^3 (a+b x)}-\frac {b B^2 (c+d x)^2}{(b c-a d)^2 g^3 (a+b x)^2}-\frac {4 B^2 d (c+d x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{(b c-a d)^2 g^3 (a+b x)}+\frac {b B (c+d x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^2 g^3 (a+b x)^2}+\frac {d (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(b c-a d)^2 g^3 (a+b x)}-\frac {b (c+d x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 (b c-a d)^2 g^3 (a+b x)^2} \]

[Out]

-4*A*B*d*(d*x+c)/(-a*d+b*c)^2/g^3/(b*x+a)+8*B^2*d*(d*x+c)/(-a*d+b*c)^2/g^3/(b*x+a)-b*B^2*(d*x+c)^2/(-a*d+b*c)^
2/g^3/(b*x+a)^2-4*B^2*d*(d*x+c)*ln(e*(d*x+c)^2/(b*x+a)^2)/(-a*d+b*c)^2/g^3/(b*x+a)+b*B*(d*x+c)^2*(A+B*ln(e*(d*
x+c)^2/(b*x+a)^2))/(-a*d+b*c)^2/g^3/(b*x+a)^2+d*(d*x+c)*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))^2/(-a*d+b*c)^2/g^3/(b*
x+a)-1/2*b*(d*x+c)^2*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))^2/(-a*d+b*c)^2/g^3/(b*x+a)^2

________________________________________________________________________________________

Rubi [A]
time = 0.11, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2552, 2367, 2333, 2332, 2342, 2341} \begin {gather*} \frac {b B (c+d x)^2 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{g^3 (a+b x)^2 (b c-a d)^2}-\frac {b (c+d x)^2 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{2 g^3 (a+b x)^2 (b c-a d)^2}+\frac {d (c+d x) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{g^3 (a+b x) (b c-a d)^2}-\frac {4 A B d (c+d x)}{g^3 (a+b x) (b c-a d)^2}-\frac {4 B^2 d (c+d x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{g^3 (a+b x) (b c-a d)^2}-\frac {b B^2 (c+d x)^2}{g^3 (a+b x)^2 (b c-a d)^2}+\frac {8 B^2 d (c+d x)}{g^3 (a+b x) (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 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 2367

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rule 2552

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.), x_Symbol] :> Dist[(b*c - a*d)^(m + 1)*(g/d)^m, Subst[Int[(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x],
x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ
[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])

Rubi steps

\begin {align*} \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx &=-\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac {B \int \frac {2 (b c-a d) \left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{g^2 (a+b x)^3 (c+d x)} \, dx}{b g}\\ &=-\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac {(2 B (b c-a d)) \int \frac {-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{(a+b x)^3 (c+d x)} \, dx}{b g^3}\\ &=-\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac {(2 B (b c-a d)) \int \left (\frac {b \left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d) (a+b x)^3}-\frac {b d \left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2 \left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d^3 \left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b g^3}\\ &=-\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac {(2 B) \int \frac {-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{(a+b x)^3} \, dx}{g^3}+\frac {\left (2 B d^2\right ) \int \frac {-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{a+b x} \, dx}{(b c-a d)^2 g^3}-\frac {\left (2 B d^3\right ) \int \frac {-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{c+d x} \, dx}{b (b c-a d)^2 g^3}-\frac {(2 B d) \int \frac {-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{(a+b x)^2} \, dx}{(b c-a d) g^3}\\ &=\frac {B \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b g^3 (a+b x)^2}-\frac {2 B d \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d) g^3 (a+b x)}-\frac {2 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}+\frac {2 B d^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {B^2 \int \frac {-2 b c+2 a d}{(a+b x)^3 (c+d x)} \, dx}{b g^3}+\frac {\left (2 B^2 d^2\right ) \int \frac {(a+b x)^2 \left (\frac {2 d e (c+d x)}{(a+b x)^2}-\frac {2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (a+b x)}{e (c+d x)^2} \, dx}{b (b c-a d)^2 g^3}-\frac {\left (2 B^2 d^2\right ) \int \frac {(a+b x)^2 \left (\frac {2 d e (c+d x)}{(a+b x)^2}-\frac {2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (c+d x)}{e (c+d x)^2} \, dx}{b (b c-a d)^2 g^3}+\frac {\left (2 B^2 d\right ) \int \frac {2 (-b c+a d)}{(a+b x)^2 (c+d x)} \, dx}{b (b c-a d) g^3}\\ &=\frac {B \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b g^3 (a+b x)^2}-\frac {2 B d \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d) g^3 (a+b x)}-\frac {2 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}+\frac {2 B d^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {\left (4 B^2 d\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b g^3}+\frac {\left (2 B^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b g^3}+\frac {\left (2 B^2 d^2\right ) \int \frac {(a+b x)^2 \left (\frac {2 d e (c+d x)}{(a+b x)^2}-\frac {2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (a+b x)}{(c+d x)^2} \, dx}{b (b c-a d)^2 e g^3}-\frac {\left (2 B^2 d^2\right ) \int \frac {(a+b x)^2 \left (\frac {2 d e (c+d x)}{(a+b x)^2}-\frac {2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (c+d x)}{(c+d x)^2} \, dx}{b (b c-a d)^2 e g^3}\\ &=\frac {B \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b g^3 (a+b x)^2}-\frac {2 B d \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d) g^3 (a+b x)}-\frac {2 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}+\frac {2 B d^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {\left (4 B^2 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 g^3}+\frac {\left (2 B^2 (b c-a d)\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 g^3}+\frac {\left (2 B^2 d^2\right ) \int \left (-\frac {2 b e \log (a+b x)}{a+b x}+\frac {2 d e \log (a+b x)}{c+d x}\right ) \, dx}{b (b c-a d)^2 e g^3}-\frac {\left (2 B^2 d^2\right ) \int \left (-\frac {2 b e \log (c+d x)}{a+b x}+\frac {2 d e \log (c+d x)}{c+d x}\right ) \, dx}{b (b c-a d)^2 e g^3}\\ &=-\frac {B^2}{b g^3 (a+b x)^2}+\frac {6 B^2 d}{b (b c-a d) g^3 (a+b x)}+\frac {6 B^2 d^2 \log (a+b x)}{b (b c-a d)^2 g^3}-\frac {6 B^2 d^2 \log (c+d x)}{b (b c-a d)^2 g^3}+\frac {B \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b g^3 (a+b x)^2}-\frac {2 B d \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d) g^3 (a+b x)}-\frac {2 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}+\frac {2 B d^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {\left (4 B^2 d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{(b c-a d)^2 g^3}+\frac {\left (4 B^2 d^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{(b c-a d)^2 g^3}+\frac {\left (4 B^2 d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b (b c-a d)^2 g^3}-\frac {\left (4 B^2 d^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b (b c-a d)^2 g^3}\\ &=-\frac {B^2}{b g^3 (a+b x)^2}+\frac {6 B^2 d}{b (b c-a d) g^3 (a+b x)}+\frac {6 B^2 d^2 \log (a+b x)}{b (b c-a d)^2 g^3}-\frac {6 B^2 d^2 \log (c+d x)}{b (b c-a d)^2 g^3}+\frac {4 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^2 g^3}+\frac {4 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}+\frac {B \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b g^3 (a+b x)^2}-\frac {2 B d \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d) g^3 (a+b x)}-\frac {2 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}+\frac {2 B d^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {\left (4 B^2 d^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{(b c-a d)^2 g^3}-\frac {\left (4 B^2 d^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b (b c-a d)^2 g^3}-\frac {\left (4 B^2 d^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b (b c-a d)^2 g^3}-\frac {\left (4 B^2 d^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b (b c-a d)^2 g^3}\\ &=-\frac {B^2}{b g^3 (a+b x)^2}+\frac {6 B^2 d}{b (b c-a d) g^3 (a+b x)}+\frac {6 B^2 d^2 \log (a+b x)}{b (b c-a d)^2 g^3}-\frac {2 B^2 d^2 \log ^2(a+b x)}{b (b c-a d)^2 g^3}-\frac {6 B^2 d^2 \log (c+d x)}{b (b c-a d)^2 g^3}+\frac {4 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {2 B^2 d^2 \log ^2(c+d x)}{b (b c-a d)^2 g^3}+\frac {4 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}+\frac {B \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b g^3 (a+b x)^2}-\frac {2 B d \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d) g^3 (a+b x)}-\frac {2 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}+\frac {2 B d^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {\left (4 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 (b c-a d)^2 g^3}-\frac {\left (4 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 (b c-a d)^2 g^3}\\ &=-\frac {B^2}{b g^3 (a+b x)^2}+\frac {6 B^2 d}{b (b c-a d) g^3 (a+b x)}+\frac {6 B^2 d^2 \log (a+b x)}{b (b c-a d)^2 g^3}-\frac {2 B^2 d^2 \log ^2(a+b x)}{b (b c-a d)^2 g^3}-\frac {6 B^2 d^2 \log (c+d x)}{b (b c-a d)^2 g^3}+\frac {4 B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {2 B^2 d^2 \log ^2(c+d x)}{b (b c-a d)^2 g^3}+\frac {4 B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}+\frac {B \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b g^3 (a+b x)^2}-\frac {2 B d \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d) g^3 (a+b x)}-\frac {2 B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}+\frac {2 B d^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac {4 B^2 d^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}+\frac {4 B^2 d^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 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 = 0.31, size = 452, normalized size = 1.51 \begin {gather*} -\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2-\frac {2 B \left (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 )+(b c-a d)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )+2 d (-b c+a d) (a+b x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )-2 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )+2 d^2 (a+b x)^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\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 c-a d)^2}}{2 b g^3 (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2 - (2*B*(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]) + (b*c - a*d)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]) + 2*d*(-(b*c) + a
*d)*(a + b*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]) - 2*d^2*(a + b*x)^2*Log[a + b*x]*(A + B*Log[(e*(c + d*x
)^2)/(a + b*x)^2]) + 2*d^2*(a + b*x)^2*Log[c + d*x]*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]) - 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*c - a*d)^2)/(b*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. \(638\) vs. \(2(297)=594\).
time = 0.56, size = 639, normalized size = 2.14 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1010 vs. \(2 (301) = 602\).
time = 0.37, size = 1010, normalized size = 3.38 \begin {gather*} -{\left ({\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} \log \left (\frac {d^{2} x^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d x e}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + \frac {b^{2} c^{2} - 8 \, a b c d + 7 \, a^{2} d^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (d x + c\right )^{2} - 6 \, {\left (b^{2} c d - a b d^{2}\right )} x - 6 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 6 \, a b d^{2} x + 3 \, a^{2} d^{2} - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a^{2} b^{3} c^{2} g^{3} - 2 \, a^{3} b^{2} c d g^{3} + a^{4} b d^{2} g^{3} + {\left (b^{5} c^{2} g^{3} - 2 \, a b^{4} c d g^{3} + a^{2} b^{3} d^{2} g^{3}\right )} x^{2} + 2 \, {\left (a b^{4} c^{2} g^{3} - 2 \, a^{2} b^{3} c d g^{3} + a^{3} b^{2} d^{2} g^{3}\right )} x}\right )} B^{2} - A B {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} + \frac {\log \left (\frac {d^{2} x^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d x e}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac {B^{2} \log \left (\frac {d^{2} x^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d x e}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )^{2}}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} - \frac {A^{2}}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.34, size = 409, normalized size = 1.37 \begin {gather*} -\frac {{\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b^{2} c^{2} - 2 \, {\left (A^{2} - 4 \, A B + 8 \, B^{2}\right )} a b c d + {\left (A^{2} - 6 \, A B + 14 \, B^{2}\right )} a^{2} d^{2} - {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} a b d^{2} x - B^{2} b^{2} c^{2} + 2 \, B^{2} a b c d\right )} \log \left (\frac {{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )^{2} + 4 \, {\left ({\left (A B - 3 \, B^{2}\right )} b^{2} c d - {\left (A B - 3 \, B^{2}\right )} a b d^{2}\right )} x - 2 \, {\left ({\left (A B - 3 \, B^{2}\right )} b^{2} d^{2} x^{2} - {\left (A B - B^{2}\right )} b^{2} c^{2} + 2 \, {\left (A B - 2 \, B^{2}\right )} a b c d - 2 \, {\left (B^{2} b^{2} c d - {\left (A B - 2 \, B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{2 \, {\left ({\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} g^{3} x + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} g^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((A^2 - 2*A*B + 2*B^2)*b^2*c^2 - 2*(A^2 - 4*A*B + 8*B^2)*a*b*c*d + (A^2 - 6*A*B + 14*B^2)*a^2*d^2 - (B^2*
b^2*d^2*x^2 + 2*B^2*a*b*d^2*x - B^2*b^2*c^2 + 2*B^2*a*b*c*d)*log((d^2*x^2 + 2*c*d*x + c^2)*e/(b^2*x^2 + 2*a*b*
x + a^2))^2 + 4*((A*B - 3*B^2)*b^2*c*d - (A*B - 3*B^2)*a*b*d^2)*x - 2*((A*B - 3*B^2)*b^2*d^2*x^2 - (A*B - B^2)
*b^2*c^2 + 2*(A*B - 2*B^2)*a*b*c*d - 2*(B^2*b^2*c*d - (A*B - 2*B^2)*a*b*d^2)*x)*log((d^2*x^2 + 2*c*d*x + c^2)*
e/(b^2*x^2 + 2*a*b*x + a^2)))/((b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*g^3*x^2 + 2*(a*b^4*c^2 - 2*a^2*b^3*c*d +
a^3*b^2*d^2)*g^3*x + (a^2*b^3*c^2 - 2*a^3*b^2*c*d + a^4*b*d^2)*g^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 877 vs. \(2 (279) = 558\).
time = 5.22, size = 877, normalized size = 2.93 \begin {gather*} \frac {2 B d^{2} \left (A - 3 B\right ) \log {\left (x + \frac {2 A B a d^{3} + 2 A B b c d^{2} - 6 B^{2} a d^{3} - 6 B^{2} b c d^{2} - \frac {2 B a^{3} d^{5} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {6 B a^{2} b c d^{4} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {6 B a b^{2} c^{2} d^{3} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {2 B b^{3} c^{3} d^{2} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}}}{4 A B b d^{3} - 12 B^{2} b d^{3}} \right )}}{b g^{3} \left (a d - b c\right )^{2}} - \frac {2 B d^{2} \left (A - 3 B\right ) \log {\left (x + \frac {2 A B a d^{3} + 2 A B b c d^{2} - 6 B^{2} a d^{3} - 6 B^{2} b c d^{2} + \frac {2 B a^{3} d^{5} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {6 B a^{2} b c d^{4} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {6 B a b^{2} c^{2} d^{3} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {2 B b^{3} c^{3} d^{2} \left (A - 3 B\right )}{\left (a d - b c\right )^{2}}}{4 A B b d^{3} - 12 B^{2} b d^{3}} \right )}}{b g^{3} \left (a d - b c\right )^{2}} + \frac {\left (2 B^{2} a c d + 2 B^{2} a d^{2} x - B^{2} b c^{2} + B^{2} b d^{2} x^{2}\right ) \log {\left (\frac {e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )}^{2}}{2 a^{4} d^{2} g^{3} - 4 a^{3} b c d g^{3} + 4 a^{3} b d^{2} g^{3} x + 2 a^{2} b^{2} c^{2} g^{3} - 8 a^{2} b^{2} c d g^{3} x + 2 a^{2} b^{2} d^{2} g^{3} x^{2} + 4 a b^{3} c^{2} g^{3} x - 4 a b^{3} c d g^{3} x^{2} + 2 b^{4} c^{2} g^{3} x^{2}} + \frac {\left (- A B a d + A B b c + 3 B^{2} a d - B^{2} b c + 2 B^{2} b d x\right ) \log {\left (\frac {e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )}}{a^{3} b d g^{3} - a^{2} b^{2} c g^{3} + 2 a^{2} b^{2} d g^{3} x - 2 a b^{3} c g^{3} x + a b^{3} d g^{3} x^{2} - b^{4} c g^{3} x^{2}} + \frac {- A^{2} a d + A^{2} b c + 6 A B a d - 2 A B b c - 14 B^{2} a d + 2 B^{2} b c + x \left (4 A B b d - 12 B^{2} b d\right )}{2 a^{3} b d g^{3} - 2 a^{2} b^{2} c g^{3} + x^{2} \cdot \left (2 a b^{3} d g^{3} - 2 b^{4} c g^{3}\right ) + x \left (4 a^{2} b^{2} d g^{3} - 4 a b^{3} c g^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*B*d**2*(A - 3*B)*log(x + (2*A*B*a*d**3 + 2*A*B*b*c*d**2 - 6*B**2*a*d**3 - 6*B**2*b*c*d**2 - 2*B*a**3*d**5*(A
- 3*B)/(a*d - b*c)**2 + 6*B*a**2*b*c*d**4*(A - 3*B)/(a*d - b*c)**2 - 6*B*a*b**2*c**2*d**3*(A - 3*B)/(a*d - b*
c)**2 + 2*B*b**3*c**3*d**2*(A - 3*B)/(a*d - b*c)**2)/(4*A*B*b*d**3 - 12*B**2*b*d**3))/(b*g**3*(a*d - b*c)**2)
- 2*B*d**2*(A - 3*B)*log(x + (2*A*B*a*d**3 + 2*A*B*b*c*d**2 - 6*B**2*a*d**3 - 6*B**2*b*c*d**2 + 2*B*a**3*d**5*
(A - 3*B)/(a*d - b*c)**2 - 6*B*a**2*b*c*d**4*(A - 3*B)/(a*d - b*c)**2 + 6*B*a*b**2*c**2*d**3*(A - 3*B)/(a*d -
b*c)**2 - 2*B*b**3*c**3*d**2*(A - 3*B)/(a*d - b*c)**2)/(4*A*B*b*d**3 - 12*B**2*b*d**3))/(b*g**3*(a*d - b*c)**2
) + (2*B**2*a*c*d + 2*B**2*a*d**2*x - B**2*b*c**2 + B**2*b*d**2*x**2)*log(e*(c + d*x)**2/(a + b*x)**2)**2/(2*a
**4*d**2*g**3 - 4*a**3*b*c*d*g**3 + 4*a**3*b*d**2*g**3*x + 2*a**2*b**2*c**2*g**3 - 8*a**2*b**2*c*d*g**3*x + 2*
a**2*b**2*d**2*g**3*x**2 + 4*a*b**3*c**2*g**3*x - 4*a*b**3*c*d*g**3*x**2 + 2*b**4*c**2*g**3*x**2) + (-A*B*a*d
+ A*B*b*c + 3*B**2*a*d - B**2*b*c + 2*B**2*b*d*x)*log(e*(c + d*x)**2/(a + b*x)**2)/(a**3*b*d*g**3 - a**2*b**2*
c*g**3 + 2*a**2*b**2*d*g**3*x - 2*a*b**3*c*g**3*x + a*b**3*d*g**3*x**2 - b**4*c*g**3*x**2) + (-A**2*a*d + A**2
*b*c + 6*A*B*a*d - 2*A*B*b*c - 14*B**2*a*d + 2*B**2*b*c + x*(4*A*B*b*d - 12*B**2*b*d))/(2*a**3*b*d*g**3 - 2*a*
*2*b**2*c*g**3 + x**2*(2*a*b**3*d*g**3 - 2*b**4*c*g**3) + x*(4*a**2*b**2*d*g**3 - 4*a*b**3*c*g**3))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 6.71, size = 504, normalized size = 1.69 \begin {gather*} \frac {\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\,\left (\frac {2\,B^2\,x\,\left (a\,d-b\,c\right )}{b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {A\,B}{b^2\,d\,g^3}+\frac {B^2\,d^2\,\left (\frac {2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2}{b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{b\,d^2}\right )}{b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )}{\frac {b\,x^2}{d}+\frac {a^2}{b\,d}+\frac {2\,a\,x}{d}}-{\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )}^2\,\left (\frac {B^2}{2\,b^2\,g^3\,\left (2\,a\,x+b\,x^2+\frac {a^2}{b}\right )}-\frac {B^2\,d^2}{2\,b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {\frac {A^2\,a\,d-A^2\,b\,c+14\,B^2\,a\,d-2\,B^2\,b\,c-6\,A\,B\,a\,d+2\,A\,B\,b\,c}{2\,\left (a\,d-b\,c\right )}+\frac {2\,x\,\left (3\,B^2\,b\,d-A\,B\,b\,d\right )}{a\,d-b\,c}}{a^2\,b\,g^3+2\,a\,b^2\,g^3\,x+b^3\,g^3\,x^2}-\frac {B\,d^2\,\mathrm {atan}\left (\frac {B\,d^2\,\left (2\,b\,d\,x-\frac {b^3\,c^2\,g^3-a^2\,b\,d^2\,g^3}{b\,g^3\,\left (a\,d-b\,c\right )}\right )\,\left (A-3\,B\right )\,2{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (6\,B^2\,d^2-2\,A\,B\,d^2\right )}\right )\,\left (A-3\,B\right )\,4{}\mathrm {i}}{b\,g^3\,{\left (a\,d-b\,c\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((e*(c + d*x)^2)/(a + b*x)^2)*((2*B^2*x*(a*d - b*c))/(b*g^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (A*B)/(b^2*
d*g^3) + (B^2*d^2*((2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)/(b*d^3) + (a*(a*d - b*c))/(b*d^2)))/(b*g^3*(a^2*d^2 + b^2
*c^2 - 2*a*b*c*d))))/((b*x^2)/d + a^2/(b*d) + (2*a*x)/d) - log((e*(c + d*x)^2)/(a + b*x)^2)^2*(B^2/(2*b^2*g^3*
(2*a*x + b*x^2 + a^2/b)) - (B^2*d^2)/(2*b*g^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) - ((A^2*a*d - A^2*b*c + 14*B^2
*a*d - 2*B^2*b*c - 6*A*B*a*d + 2*A*B*b*c)/(2*(a*d - b*c)) + (2*x*(3*B^2*b*d - A*B*b*d))/(a*d - b*c))/(a^2*b*g^
3 + b^3*g^3*x^2 + 2*a*b^2*g^3*x) - (B*d^2*atan((B*d^2*(2*b*d*x - (b^3*c^2*g^3 - a^2*b*d^2*g^3)/(b*g^3*(a*d - b
*c)))*(A - 3*B)*2i)/((a*d - b*c)*(6*B^2*d^2 - 2*A*B*d^2)))*(A - 3*B)*4i)/(b*g^3*(a*d - b*c)^2)

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