3.161 \(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^2}{(a+b x)^3} \, dx\)

Optimal. Leaf size=274 \[ -\frac {b B n (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{2 (a+b x)^2 (b c-a d)^2}+\frac {2 B d n (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(a+b x) (b c-a d)^2}-\frac {b (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 (a+b x)^2 (b c-a d)^2}+\frac {d (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(a+b x) (b c-a d)^2}-\frac {b B^2 n^2 (c+d x)^2}{4 (a+b x)^2 (b c-a d)^2}+\frac {2 B^2 d n^2 (c+d x)}{(a+b x) (b c-a d)^2} \]

[Out]

2*B^2*d*n^2*(d*x+c)/(-a*d+b*c)^2/(b*x+a)-1/4*b*B^2*n^2*(d*x+c)^2/(-a*d+b*c)^2/(b*x+a)^2+2*B*d*n*(d*x+c)*(A+B*l
n(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^2/(b*x+a)-1/2*b*B*n*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b
*c)^2/(b*x+a)^2+d*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(-a*d+b*c)^2/(b*x+a)-1/2*b*(d*x+c)^2*(A+B*ln(e*(
b*x+a)^n/((d*x+c)^n)))^2/(-a*d+b*c)^2/(b*x+a)^2

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Rubi [A]  time = 0.42, antiderivative size = 411, normalized size of antiderivative = 1.50, number of steps used = 12, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {6742, 2492, 44, 2491, 2490, 32, 2509, 37} \[ -\frac {A^2}{2 b (a+b x)^2}+\frac {A B d^2 n \log (a+b x)}{b (b c-a d)^2}-\frac {A B d^2 n \log (c+d x)}{b (b c-a d)^2}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^2}+\frac {A B d n}{b (a+b x) (b c-a d)}-\frac {A B n}{2 b (a+b x)^2}-\frac {b B^2 (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (a+b x)^2 (b c-a d)^2}+\frac {B^2 d (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (b c-a d)^2}-\frac {b B^2 n (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (a+b x)^2 (b c-a d)^2}+\frac {2 B^2 d n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (b c-a d)^2}-\frac {b B^2 n^2 (c+d x)^2}{4 (a+b x)^2 (b c-a d)^2}+\frac {2 B^2 d n^2}{b (a+b x) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2490

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)/((g_.) + (h_.)*(x_))^
2, x_Symbol] :> Simp[((a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/((b*g - a*h)*(g + h*x)), x] - Dist[(p*
r*s*(b*c - a*d))/(b*g - a*h), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/((c + d*x)*(g + h*x)), x], x] /
; FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[b*g - a*h, 0] &&
IGtQ[s, 0]

Rule 2491

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_)/((g_.) + (h_.)*(x_))^3
, x_Symbol] :> Dist[d/(d*g - c*h), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(g + h*x)^2, x], x] - Dist[h/(d*
g - c*h), Int[((c + d*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(g + h*x)^3, x], x] /; FreeQ[{a, b, c, d, e,
f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && EqQ[b*g - a*h, 0] && NeQ[d*g - c*h, 0] && IG
tQ[s, 0]

Rule 2492

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*((g_.) + (h_.)*(x_))^
(m_.), x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(h*(m + 1)), x] - Dist[(p*
r*s*(b*c - a*d))/(h*(m + 1)), Int[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*
(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0]
&& IGtQ[s, 0] && NeQ[m, -1]

Rule 2509

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*((a_.) + (b_.)*(x_))^
(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1)*Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^s)/((m + 1)*(b*c - a*d)), x] - Dist[(p*r*s*(b*c - a*d))/((m + 1)*(b*c - a*d)), Int[(a + b*x)^m
*(c + d*x)^n*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r, s
}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && EqQ[m + n + 2, 0] && NeQ[m, -1] && IGtQ[s, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(a+b x)^3} \, dx &=\int \left (\frac {A^2}{(a+b x)^3}+\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3}+\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3}\right ) \, dx\\ &=-\frac {A^2}{2 b (a+b x)^2}+(2 A B) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx+B^2 \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx\\ &=-\frac {A^2}{2 b (a+b x)^2}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^2}+\frac {\left (b B^2\right ) \int \frac {(c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{b c-a d}-\frac {\left (B^2 d\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{b c-a d}+\frac {(A B (b c-a d) n) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b}\\ &=-\frac {A^2}{2 b (a+b x)^2}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^2}+\frac {B^2 d (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac {b B^2 (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}+\frac {\left (b B^2 n\right ) \int \frac {(c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{b c-a d}-\frac {\left (2 B^2 d n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{b c-a d}+\frac {(A B (b c-a d) n) \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}\\ &=-\frac {A^2}{2 b (a+b x)^2}-\frac {A B n}{2 b (a+b x)^2}+\frac {A B d n}{b (b c-a d) (a+b x)}+\frac {A B d^2 n \log (a+b x)}{b (b c-a d)^2}-\frac {A B d^2 n \log (c+d x)}{b (b c-a d)^2}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^2}+\frac {2 B^2 d n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac {b B^2 n (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}+\frac {B^2 d (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac {b B^2 (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}+\frac {\left (b B^2 n^2\right ) \int \frac {c+d x}{(a+b x)^3} \, dx}{2 (b c-a d)}-\frac {\left (2 B^2 d n^2\right ) \int \frac {1}{(a+b x)^2} \, dx}{b c-a d}\\ &=-\frac {A^2}{2 b (a+b x)^2}-\frac {A B n}{2 b (a+b x)^2}+\frac {A B d n}{b (b c-a d) (a+b x)}+\frac {2 B^2 d n^2}{b (b c-a d) (a+b x)}-\frac {b B^2 n^2 (c+d x)^2}{4 (b c-a d)^2 (a+b x)^2}+\frac {A B d^2 n \log (a+b x)}{b (b c-a d)^2}-\frac {A B d^2 n \log (c+d x)}{b (b c-a d)^2}-\frac {A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^2}+\frac {2 B^2 d n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac {b B^2 n (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}+\frac {B^2 d (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)}-\frac {b B^2 (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^2 (a+b x)^2}\\ \end {align*}

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Mathematica [A]  time = 0.52, size = 332, normalized size = 1.21 \[ -\frac {(b c-a d) \left (2 A^2 (b c-a d)+2 B (2 A (b c-a d)+B n (-3 a d+b c-2 b d x)) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 A B n (-3 a d+b c-2 b d x)+2 B^2 (b c-a d) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+B^2 n^2 (-7 a d+b c-6 b d x)\right )+2 B d^2 n (a+b x)^2 \log (c+d x) \left (2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 A+3 B n\right )-2 B d^2 n (a+b x)^2 \log (a+b x) \left (2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 A+2 B n \log (c+d x)+3 B n\right )+2 B^2 d^2 n^2 (a+b x)^2 \log ^2(c+d x)+2 B^2 d^2 n^2 (a+b x)^2 \log ^2(a+b x)}{4 b (a+b x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(2*B^2*d^2*n^2*(a + b*x)^2*Log[a + b*x]^2 + 2*B^2*d^2*n^2*(a + b*x)^2*Log[c + d*x]^2 + 2*B*d^2*n*(a + b*x
)^2*Log[c + d*x]*(2*A + 3*B*n + 2*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) - 2*B*d^2*n*(a + b*x)^2*Log[a + b*x]*(2*
A + 3*B*n + 2*B*n*Log[c + d*x] + 2*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + (b*c - a*d)*(2*A^2*(b*c - a*d) + B^2*
n^2*(b*c - 7*a*d - 6*b*d*x) + 2*A*B*n*(b*c - 3*a*d - 2*b*d*x) + 2*B*(2*A*(b*c - a*d) + B*n*(b*c - 3*a*d - 2*b*
d*x))*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 2*B^2*(b*c - a*d)*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2))/(b*(b*c - a*d)
^2*(a + b*x)^2)

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fricas [B]  time = 0.77, size = 919, normalized size = 3.35 \[ -\frac {2 \, A^{2} b^{2} c^{2} - 4 \, A^{2} a b c d + 2 \, A^{2} a^{2} d^{2} + {\left (B^{2} b^{2} c^{2} - 8 \, B^{2} a b c d + 7 \, B^{2} a^{2} d^{2}\right )} n^{2} - 2 \, {\left (B^{2} b^{2} d^{2} n^{2} x^{2} + 2 \, B^{2} a b d^{2} n^{2} x - {\left (B^{2} b^{2} c^{2} - 2 \, B^{2} a b c d\right )} n^{2}\right )} \log \left (b x + a\right )^{2} - 2 \, {\left (B^{2} b^{2} d^{2} n^{2} x^{2} + 2 \, B^{2} a b d^{2} n^{2} x - {\left (B^{2} b^{2} c^{2} - 2 \, B^{2} a b c d\right )} n^{2}\right )} \log \left (d x + c\right )^{2} + 2 \, {\left (B^{2} b^{2} c^{2} - 2 \, B^{2} a b c d + B^{2} a^{2} d^{2}\right )} \log \relax (e)^{2} + 2 \, {\left (A B b^{2} c^{2} - 4 \, A B a b c d + 3 \, A B a^{2} d^{2}\right )} n - 2 \, {\left (3 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n^{2} + 2 \, {\left (A B b^{2} c d - A B a b d^{2}\right )} n\right )} x + 2 \, {\left ({\left (B^{2} b^{2} c^{2} - 4 \, B^{2} a b c d\right )} n^{2} - {\left (3 \, B^{2} b^{2} d^{2} n^{2} + 2 \, A B b^{2} d^{2} n\right )} x^{2} + 2 \, {\left (A B b^{2} c^{2} - 2 \, A B a b c d\right )} n - 2 \, {\left (2 \, A B a b d^{2} n + {\left (B^{2} b^{2} c d + 2 \, B^{2} a b d^{2}\right )} n^{2}\right )} x - 2 \, {\left (B^{2} b^{2} d^{2} n x^{2} + 2 \, B^{2} a b d^{2} n x - {\left (B^{2} b^{2} c^{2} - 2 \, B^{2} a b c d\right )} n\right )} \log \relax (e)\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (B^{2} b^{2} c^{2} - 4 \, B^{2} a b c d\right )} n^{2} - {\left (3 \, B^{2} b^{2} d^{2} n^{2} + 2 \, A B b^{2} d^{2} n\right )} x^{2} + 2 \, {\left (A B b^{2} c^{2} - 2 \, A B a b c d\right )} n - 2 \, {\left (2 \, A B a b d^{2} n + {\left (B^{2} b^{2} c d + 2 \, B^{2} a b d^{2}\right )} n^{2}\right )} x - 2 \, {\left (B^{2} b^{2} d^{2} n^{2} x^{2} + 2 \, B^{2} a b d^{2} n^{2} x - {\left (B^{2} b^{2} c^{2} - 2 \, B^{2} a b c d\right )} n^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left (B^{2} b^{2} d^{2} n x^{2} + 2 \, B^{2} a b d^{2} n x - {\left (B^{2} b^{2} c^{2} - 2 \, B^{2} a b c d\right )} n\right )} \log \relax (e)\right )} \log \left (d x + c\right ) + 2 \, {\left (2 \, A B b^{2} c^{2} - 4 \, A B a b c d + 2 \, A B a^{2} d^{2} - 2 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n x + {\left (B^{2} b^{2} c^{2} - 4 \, B^{2} a b c d + 3 \, B^{2} a^{2} d^{2}\right )} n\right )} \log \relax (e)}{4 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} + {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{2} + 2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}{{\left (b x + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 3.36, size = 17300, normalized size = 63.14 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [B]  time = 1.85, size = 899, normalized size = 3.28 \[ \frac {1}{4} \, B^{2} {\left (\frac {2 \, {\left (\frac {2 \, d^{2} e n \log \left (b x + a\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} - \frac {2 \, d^{2} e n \log \left (d x + c\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} + \frac {2 \, b d e n x - b c e n + 3 \, a d e n}{a^{2} b^{2} c - a^{3} b d + {\left (b^{4} c - a b^{3} d\right )} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} x}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{e} - \frac {b^{2} c^{2} e^{2} n^{2} - 8 \, a b c d e^{2} n^{2} + 7 \, a^{2} d^{2} e^{2} n^{2} + 2 \, {\left (b^{2} d^{2} e^{2} n^{2} x^{2} + 2 \, a b d^{2} e^{2} n^{2} x + a^{2} d^{2} e^{2} n^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} e^{2} n^{2} x^{2} + 2 \, a b d^{2} e^{2} n^{2} x + a^{2} d^{2} e^{2} n^{2}\right )} \log \left (d x + c\right )^{2} - 6 \, {\left (b^{2} c d e^{2} n^{2} - a b d^{2} e^{2} n^{2}\right )} x - 6 \, {\left (b^{2} d^{2} e^{2} n^{2} x^{2} + 2 \, a b d^{2} e^{2} n^{2} x + a^{2} d^{2} e^{2} n^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, b^{2} d^{2} e^{2} n^{2} x^{2} + 6 \, a b d^{2} e^{2} n^{2} x + 3 \, a^{2} d^{2} e^{2} n^{2} - 2 \, {\left (b^{2} d^{2} e^{2} n^{2} x^{2} + 2 \, a b d^{2} e^{2} n^{2} x + a^{2} d^{2} e^{2} n^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{{\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} + {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{2} + 2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x\right )} e^{2}}\right )} - \frac {B^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2}}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} + \frac {{\left (\frac {2 \, d^{2} e n \log \left (b x + a\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} - \frac {2 \, d^{2} e n \log \left (d x + c\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} + \frac {2 \, b d e n x - b c e n + 3 \, a d e n}{a^{2} b^{2} c - a^{3} b d + {\left (b^{4} c - a b^{3} d\right )} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} x}\right )} A B}{2 \, e} - \frac {A B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b} - \frac {A^{2}}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.32, size = 444, normalized size = 1.62 \[ -{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^2\,\left (\frac {B^2}{2\,b\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}-\frac {B^2\,d^2}{2\,b\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {\frac {2\,A^2\,a\,d-2\,A^2\,b\,c+7\,B^2\,a\,d\,n^2-B^2\,b\,c\,n^2+6\,A\,B\,a\,d\,n-2\,A\,B\,b\,c\,n}{2\,\left (a\,d-b\,c\right )}+\frac {d\,x\,\left (3\,b\,B^2\,n^2+2\,A\,b\,B\,n\right )}{a\,d-b\,c}}{2\,a^2\,b+4\,a\,b^2\,x+2\,b^3\,x^2}-\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\,\left (\frac {A\,B}{a^2\,b+2\,a\,b^2\,x+b^3\,x^2}+\frac {B^2\,d^2\,\left (\frac {b\,n\,\left (a\,d-b\,c\right )\,\left (2\,a\,d-b\,c\right )}{2\,d^2}+\frac {b^2\,n\,x\,\left (a\,d-b\,c\right )}{d}+\frac {a\,b\,n\,\left (a\,d-b\,c\right )}{2\,d}\right )}{b\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (a^2\,b+2\,a\,b^2\,x+b^3\,x^2\right )}\right )-\frac {B\,d^2\,n\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x-\frac {2\,b^3\,c^2-2\,a^2\,b\,d^2}{2\,b\,\left (a\,d-b\,c\right )}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (2\,A+3\,B\,n\right )\,1{}\mathrm {i}}{b\,{\left (a\,d-b\,c\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- log((e*(a + b*x)^n)/(c + d*x)^n)^2*(B^2/(2*b*(a^2 + b^2*x^2 + 2*a*b*x)) - (B^2*d^2)/(2*b*(a^2*d^2 + b^2*c^2
- 2*a*b*c*d))) - ((2*A^2*a*d - 2*A^2*b*c + 7*B^2*a*d*n^2 - B^2*b*c*n^2 + 6*A*B*a*d*n - 2*A*B*b*c*n)/(2*(a*d -
b*c)) + (d*x*(3*B^2*b*n^2 + 2*A*B*b*n))/(a*d - b*c))/(2*a^2*b + 2*b^3*x^2 + 4*a*b^2*x) - log((e*(a + b*x)^n)/(
c + d*x)^n)*((A*B)/(a^2*b + b^3*x^2 + 2*a*b^2*x) + (B^2*d^2*((b*n*(a*d - b*c)*(2*a*d - b*c))/(2*d^2) + (b^2*n*
x*(a*d - b*c))/d + (a*b*n*(a*d - b*c))/(2*d)))/(b*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(a^2*b + b^3*x^2 + 2*a*b^2*x
))) - (B*d^2*n*atan(((2*b*d*x - (2*b^3*c^2 - 2*a^2*b*d^2)/(2*b*(a*d - b*c)))*1i)/(a*d - b*c))*(2*A + 3*B*n)*1i
)/(b*(a*d - b*c)^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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