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

Optimal. Leaf size=137 \[ \frac{2 B^2 n^2 (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b d}+\frac{2 B n (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b d}+\frac{(a+b x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{b} \]

[Out]

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

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Rubi [A]  time = 0.320797, antiderivative size = 195, normalized size of antiderivative = 1.42, number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {6742, 2486, 31, 2488, 2411, 2343, 2333, 2315} \[ \frac{2 B^2 n^2 (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b d}+\frac{2 A B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{2 A B n (b c-a d) \log (c+d x)}{b d}+\frac{B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac{2 B^2 n (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+A^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6742

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

Rule 2486

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2488

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

Rule 2411

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

Rule 2343

Int[((a_.) + Log[(c_.)*(x_)^(n_)]*(b_.))/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Dist[1/n, Subst[Int[(a
+ b*Log[c*x])/(x*(d + e*x^(r/n))), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[r/n]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)/(x_))^(q_.)*(x_)^(m_.), x_Symbol] :> Int[(e + d*
x)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[m, q] && IntegerQ[q]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.147173, size = 217, normalized size = 1.58 \[ \frac{B^2 n (b c-a d) \left (2 n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )-\log \left (\frac{b c-a d}{b c+b d x}\right ) \left (-2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 n \log \left (\frac{d (a+b x)}{a d-b c}\right )+n \log \left (\frac{b c-a d}{b c+b d x}\right )\right )\right )+2 A B d (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )-2 A B n (b c-a d) \log (c+d x)+B^2 d (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+A^2 b d x}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 1.311, size = 4749, normalized size = 34.7 \begin{align*} \text{output too large to display} \end{align*}

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,x)

[Out]

A^2*x-2/d*n*B^2*ln((b*x+a)^n)*c*ln(d*x+c)-I*x*ln((b*x+a)^n)*B^2*Pi*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-1/4*B^2*Pi^
2*x*csgn(I*(b*x+a)^n/((d*x+c)^n))^6+ln(e)^2*x*B^2+2/b*A*B*ln(b*x+a)*a*n-1/4*B^2*Pi^2*x*csgn(I/((d*x+c)^n))^2*c
sgn(I*(b*x+a)^n/((d*x+c)^n))^4+1/2*B^2*Pi^2*x*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^5+1/2*B^2*Pi^2
*x*csgn(I*(b*x+a)^n/((d*x+c)^n))^4*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-1/2*B^2*Pi^2*x*csgn(I*(b*x+a)^n/((d*x+c)^
n))^3*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-1/4*B^2*Pi^2*x*csgn(I*(b*x+a)^n/((d*x+c)^n))^2*csgn(I*e/((d*x+c)^n)*(b
*x+a)^n)^4+B^2*x*ln((d*x+c)^n)^2-I*x*ln((b*x+a)^n)*B^2*Pi*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-B^2*Pi^2*x*csgn(I*
(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^4-1/2*B^2*Pi^2*x*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)
^n/((d*x+c)^n))^3*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/2*B^2*Pi^2*x*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)
^n))^2*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-1/2*B^2*Pi^2*x*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^3*cs
gn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+B^2*x*ln((b*x+a)^n)^2-n^2*B^2*c/d*ln(d*x+c)^2-1/2*B^2*Pi^2*x*csgn(I*e)*csgn(I*
(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/2*B^2*Pi^2*x*csgn(I*e)*csgn(I/(
(d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^3*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)+I*B^2*c*n/d*ln(d*x+c)*Pi*csgn(I*(b
*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))-I*A*B*Pi*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*cs
gn(I*e/((d*x+c)^n)*(b*x+a)^n)*x-I*A*B*Pi*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))*x
-I/b*B^2*ln(b*x+a)*Pi*a*n*csgn(I*(b*x+a)^n/((d*x+c)^n))^3+2*ln(e)*x*B*A+I/b*B^2*ln(b*x+a)*Pi*a*n*csgn(I/((d*x+
c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-I*B^2*c*n/d*ln(d*x+c)*Pi*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-I*
B^2*c*n/d*ln(d*x+c)*Pi*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-2*B*c*n/d*ln(d*x+c)*A+2/b*B^2*a*n^2*l
n(b*x+a)*ln((-a*d+b*c+d*(b*x+a))/(-a*d+b*c))-I*B^2*ln(e)*Pi*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/(
(d*x+c)^n)*(b*x+a)^n)*x-I*B^2*ln(e)*Pi*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))*x+2
*B*x*ln((b*x+a)^n)*A+B^2*a/b*ln((b*x+a)^n)^2+2*x*ln((b*x+a)^n)*B^2*ln(e)+(-2*B^2*x*ln((b*x+a)^n)-B*(-I*B*Pi*b*
d*x*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)+I*B*Pi*b*d*x*csgn(I*e)*csgn(I*e/((
d*x+c)^n)*(b*x+a)^n)^2-I*B*Pi*b*d*x*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))+I*B*Pi
*b*d*x*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+I*B*Pi*b*d*x*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d
*x+c)^n))^2-I*B*Pi*b*d*x*csgn(I*(b*x+a)^n/((d*x+c)^n))^3+I*B*Pi*b*d*x*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/(
(d*x+c)^n)*(b*x+a)^n)^2-I*B*Pi*b*d*x*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3+2*B*ln(e)*b*d*x+2*B*a*d*n*ln(b*x+a)-2*B
*ln(d*x+c)*b*c*n+2*A*b*d*x)/b/d)*ln((d*x+c)^n)-1/4*B^2*Pi^2*x*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^6-2*B^2/b*a*n*ln
((b*x+a)^n)+1/2*B^2*Pi^2*x*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^5-1/4*B^2*Pi^2*x*csgn(I*(b*x+a)^n)^2*csgn
(I*(b*x+a)^n/((d*x+c)^n))^4+1/2*B^2*Pi^2*x*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^5-I*A*B*Pi*csgn(I*e
/((d*x+c)^n)*(b*x+a)^n)^3*x+I*B^2*ln(e)*Pi*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2*x+I*B^2*ln(e)*Pi*csgn(I
*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2*x+I*B^2*ln(e)*Pi*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))
^2*x-2*n^2*B^2*c/d+I*B^2*ln(e)*Pi*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2*x+2/b*B^2*ln
(b*x+a)*ln(e)*a*n-2*B^2*c*n/d*ln(d*x+c)*ln(e)+2*n^2*B^2*c/d*dilog((b*(d*x+c)+a*d-b*c)/(a*d-b*c))+2*n^2*B^2/b*a
*ln(b*(d*x+c)+a*d-b*c)-I*x*ln((b*x+a)^n)*B^2*Pi*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c
)^n))+I/b*B^2*ln(b*x+a)*Pi*a*n*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+I/b*B^2*ln(b*x+a)*Pi*a*n*csgn(I*(b*
x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+I/b*B^2*ln(b*x+a)*Pi*a*n*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+
c)^n)*(b*x+a)^n)^2-I*B^2*c*n/d*ln(d*x+c)*Pi*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-I*B^2*c*n/d*ln
(d*x+c)*Pi*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+2/b*B^2*a*n^2*dilog((-a*d+b*c+d*(b*
x+a))/(-a*d+b*c))+2*B^2*a*n^2/b+1/2*B^2*Pi^2*x*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))^3*csgn(I*e/((d*x+c)^n)*
(b*x+a)^n)^2+1/2*B^2*Pi^2*x*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-B^2*Pi
^2*x*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^4-1/4*B^2*Pi^2*x*csgn(I*(b*x+a)^n
)^2*csgn(I/((d*x+c)^n))^2*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/2*B^2*Pi^2*x*csgn(I*(b*x+a)^n)^2*csgn(I/((d*x+c)^n
))*csgn(I*(b*x+a)^n/((d*x+c)^n))^3+1/2*B^2*Pi^2*x*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))^2*csgn(I*(b*x+a)^n/((d
*x+c)^n))^3-I*B^2*ln(e)*Pi*csgn(I*(b*x+a)^n/((d*x+c)^n))^3*x-I*B^2*ln(e)*Pi*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3*
x-I*A*B*Pi*csgn(I*(b*x+a)^n/((d*x+c)^n))^3*x+I*x*ln((b*x+a)^n)*B^2*Pi*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x
+c)^n))^2+I*x*ln((b*x+a)^n)*B^2*Pi*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/2*B^2*Pi^2*x*csgn(I*(
b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^5-1/4*B^2*Pi^2*x*csgn(I*e)^2*csgn(I*e/((d*x+c)^n)*(b*x+a
)^n)^4+I*x*ln((b*x+a)^n)*B^2*Pi*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+I*x*ln((b*x+a)
^n)*B^2*Pi*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/2*B^2*Pi^2*x*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d
*x+c)^n))^2*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-1/4*B^2*Pi^2*x*csgn(I*e)^2*csgn(I*(b*x+a)^n/((d*x+c)^n))^2*csgn(
I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/2*B^2*Pi^2*x*csgn(I*e)^2*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*
x+a)^n)^3-1/2*B^2*Pi^2*x*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))^4*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)-I/b*B^2*ln(
b*x+a)*Pi*a*n*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)-I/b*B^2*ln(b*x+a)*Pi*a*n
*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))+I*B^2*c*n/d*ln(d*x+c)*Pi*csgn(I*e)*csgn(I
*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)-1/2*B^2*Pi^2*x*csgn(I*e)*csgn(I/((d*x+c)^n))*csgn(I*(b
*x+a)^n/((d*x+c)^n))^2*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/2*B^2*Pi^2*x*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*
csgn(I*(b*x+a)^n/((d*x+c)^n))^2*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-1/2*B^2*Pi^2*x*csgn(I*(b*x+a)^n)*csgn(I/((d*
x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3+2*n^2*B^2*c/d*ln(d*x+c)*ln((b*(d*x+c)
+a*d-b*c)/(a*d-b*c))+I*B^2*c*n/d*ln(d*x+c)*Pi*csgn(I*(b*x+a)^n/((d*x+c)^n))^3+I*B^2*c*n/d*ln(d*x+c)*Pi*csgn(I*
e/((d*x+c)^n)*(b*x+a)^n)^3-1/2*B^2*Pi^2*x*csgn(I*e)*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d
*x+c)^n))^2*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)+1/2*B^2*Pi^2*x*csgn(I*e)*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csg
n(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-I*x*ln((b*x+a)^n)*B^2*Pi*csgn(I*e)*csgn(I*(b*x+a)
^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)+I*A*B*Pi*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b
*x+a)^n)^2*x+I*A*B*Pi*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2*x+I*A*B*Pi*csgn(I/((d*x+c)^n))*csgn(I*
(b*x+a)^n/((d*x+c)^n))^2*x-I/b*B^2*ln(b*x+a)*Pi*a*n*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3+I*A*B*Pi*csgn(I*e)*csgn(
I*e/((d*x+c)^n)*(b*x+a)^n)^2*x+1/2*B^2*Pi^2*x*csgn(I*e)*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^3*csgn
(I*e/((d*x+c)^n)*(b*x+a)^n)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 2 \, A B x \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A^{2} x + B^{2}{\left (\frac{2 \, b c n^{2} \log \left (b x + a\right ) \log \left (d x + c\right ) - b c n^{2} \log \left (d x + c\right )^{2} + b d x \log \left ({\left (b x + a\right )}^{n}\right )^{2} + b d x \log \left ({\left (d x + c\right )}^{n}\right )^{2} + 2 \,{\left (a d n \log \left (b x + a\right ) - b c n \log \left (d x + c\right ) + b d x \log \left (e\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \,{\left (a d n \log \left (b x + a\right ) - b c n \log \left (d x + c\right ) + b d x \log \left ({\left (b x + a\right )}^{n}\right ) + b d x \log \left (e\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{b d} - \int -\frac{b^{2} d x^{2} \log \left (e\right )^{2} + a b c \log \left (e\right )^{2} -{\left ({\left (2 \, n \log \left (e\right ) - \log \left (e\right )^{2}\right )} b^{2} c -{\left (2 \, n \log \left (e\right ) + \log \left (e\right )^{2}\right )} a b d\right )} x - 2 \,{\left (b^{2} c n^{2} x + 2 \, a b c n^{2} - a^{2} d n^{2}\right )} \log \left (b x + a\right )}{b^{2} d x^{2} + a b c +{\left (b^{2} c + a b d\right )} x}\,{d x}\right )} + \frac{2 \,{\left (\frac{a e n \log \left (b x + a\right )}{b} - \frac{c e n \log \left (d x + c\right )}{d}\right )} A B}{e} \end{align*}

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,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (B^{2} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 2 \, A B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A^{2}, x\right ) \end{align*}

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,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}\,{d x} \end{align*}

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,x, algorithm="giac")

[Out]

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