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

Optimal. Leaf size=263 \[ \frac{2 B^2 n^2 (b c-a d)^3 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{3 b d^3}+\frac{B n (b c-a d)^3 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 A+3 B n\right )}{3 b d^3}+\frac{B n (a+b x) (b c-a d)^2 \left (2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 A+B n\right )}{3 b d^2}-\frac{B n (a+b x)^2 (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{3 b d}+\frac{(a+b x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{3 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.625638, antiderivative size = 427, normalized size of antiderivative = 1.62, number of steps used = 18, number of rules used = 11, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6742, 2492, 43, 2514, 2486, 31, 2488, 2411, 2343, 2333, 2315} \[ \frac{2 B^2 n^2 (b c-a d)^3 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{3 b d^3}+\frac{A^2 (a+b x)^3}{3 b}+\frac{2 A B n x (b c-a d)^2}{3 d^2}-\frac{2 A B n (b c-a d)^3 \log (c+d x)}{3 b d^3}+\frac{2 A B (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b}-\frac{A B n (a+b x)^2 (b c-a d)}{3 b d}+\frac{2 B^2 n (b c-a d)^3 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b d^3}+\frac{2 B^2 n (a+b x) (b c-a d)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b d^2}+\frac{B^2 n^2 x (b c-a d)^2}{3 d^2}-\frac{B^2 n^2 (b c-a d)^3 \log (c+d x)}{b d^3}+\frac{B^2 (a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b}-\frac{B^2 n (a+b x)^2 (b c-a d) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6742

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

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2514

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(RFx_), x_Symbol] :>
With[{u = ExpandIntegrand[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a,
b, c, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x] && IGtQ[s, 0]

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

Mathematica [B]  time = 1.02683, size = 1149, normalized size = 4.37 \[ \frac{A^2 d^3 x^3 b^3-A B c d^2 n x^2 b^3-B^2 c^3 n^2 \log ^2(c+d x) b^3+B^2 d^3 x^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) b^3+B^2 c^2 d n^2 x b^3+2 A B c^2 d n x b^3-3 B^2 c^3 n^2 \log (c+d x) b^3-2 A B c^3 n \log (c+d x) b^3+2 A B d^3 x^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^3-B^2 c d^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^3+2 B^2 c^2 d n x \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^3-2 B^2 c^3 n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^3-2 a B^2 c^2 d n^2 b^2+3 a A^2 d^3 x^2 b^2+a A B d^3 n x^2 b^2+3 a B^2 c^2 d n^2 \log ^2(c+d x) b^2+3 a B^2 d^3 x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) b^2-2 a B^2 c d^2 n^2 x b^2-6 a A B c d^2 n x b^2+7 a B^2 c^2 d n^2 \log (c+d x) b^2+6 a A B c^2 d n \log (c+d x) b^2+6 a A B d^3 x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^2+a B^2 d^3 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^2-6 a B^2 c d^2 n x \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^2+6 a B^2 c^2 d n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^2+6 a^2 B^2 c d^2 n^2 b-3 a^2 B^2 c d^2 n^2 \log ^2(c+d x) b+3 a^2 B^2 d^3 x \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) b+3 a^2 A^2 d^3 x b+a^2 B^2 d^3 n^2 x b+4 a^2 A B d^3 n x b-4 a^2 B^2 c d^2 n^2 \log (c+d x) b-6 a^2 A B c d^2 n \log (c+d x) b+6 a^2 A B d^3 x \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b+4 a^2 B^2 d^3 n x \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b-6 a^2 B^2 c d^2 n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b-6 a^3 B^2 d^3 n^2-a^3 B^2 d^3 n^2 \log ^2(a+b x)-6 a^3 A B d^3 n-6 a^3 B^2 d^3 n^2 \log (c+d x)-6 a^3 B^2 d^3 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )+B n \log (a+b x) \left (-2 B n \log \left (\frac{b (c+d x)}{b c-a d}\right ) (b c-a d)^3+2 b B c \left (b^2 c^2-3 a b d c+3 a^2 d^2\right ) n \log (c+d x)+a d \left (2 b^2 B n c^2-5 a b B d n c+a^2 d^2 (2 A+9 B n)+2 a^2 B d^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )-2 B^2 (b c-a d)^3 n^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )}{3 b d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*a^3*A*B*d^3*n - 2*a*b^2*B^2*c^2*d*n^2 + 6*a^2*b*B^2*c*d^2*n^2 - 6*a^3*B^2*d^3*n^2 + 3*a^2*A^2*b*d^3*x + 2*
A*b^3*B*c^2*d*n*x - 6*a*A*b^2*B*c*d^2*n*x + 4*a^2*A*b*B*d^3*n*x + b^3*B^2*c^2*d*n^2*x - 2*a*b^2*B^2*c*d^2*n^2*
x + a^2*b*B^2*d^3*n^2*x + 3*a*A^2*b^2*d^3*x^2 - A*b^3*B*c*d^2*n*x^2 + a*A*b^2*B*d^3*n*x^2 + A^2*b^3*d^3*x^3 -
a^3*B^2*d^3*n^2*Log[a + b*x]^2 - 2*A*b^3*B*c^3*n*Log[c + d*x] + 6*a*A*b^2*B*c^2*d*n*Log[c + d*x] - 6*a^2*A*b*B
*c*d^2*n*Log[c + d*x] - 3*b^3*B^2*c^3*n^2*Log[c + d*x] + 7*a*b^2*B^2*c^2*d*n^2*Log[c + d*x] - 4*a^2*b*B^2*c*d^
2*n^2*Log[c + d*x] - 6*a^3*B^2*d^3*n^2*Log[c + d*x] - b^3*B^2*c^3*n^2*Log[c + d*x]^2 + 3*a*b^2*B^2*c^2*d*n^2*L
og[c + d*x]^2 - 3*a^2*b*B^2*c*d^2*n^2*Log[c + d*x]^2 - 6*a^3*B^2*d^3*n*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6*a^
2*A*b*B*d^3*x*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 2*b^3*B^2*c^2*d*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 6*a*b^
2*B^2*c*d^2*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 4*a^2*b*B^2*d^3*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6*a*
A*b^2*B*d^3*x^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] - b^3*B^2*c*d^2*n*x^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + a*b^
2*B^2*d^3*n*x^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 2*A*b^3*B*d^3*x^3*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 2*b^3*
B^2*c^3*n*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6*a*b^2*B^2*c^2*d*n*Log[c + d*x]*Log[(e*(a + b*x)^n)
/(c + d*x)^n] - 6*a^2*b*B^2*c*d^2*n*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 3*a^2*b*B^2*d^3*x*Log[(e*(
a + b*x)^n)/(c + d*x)^n]^2 + 3*a*b^2*B^2*d^3*x^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + b^3*B^2*d^3*x^3*Log[(e*(
a + b*x)^n)/(c + d*x)^n]^2 + B*n*Log[a + b*x]*(2*b*B*c*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*n*Log[c + d*x] - 2*B*
(b*c - a*d)^3*n*Log[(b*(c + d*x))/(b*c - a*d)] + a*d*(2*b^2*B*c^2*n - 5*a*b*B*c*d*n + a^2*d^2*(2*A + 9*B*n) +
2*a^2*B*d^2*Log[(e*(a + b*x)^n)/(c + d*x)^n])) - 2*B^2*(b*c - a*d)^3*n^2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*
d)])/(3*b*d^3)

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Maple [C]  time = 2.135, size = 19969, normalized size = 75.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [B]  time = 3.73536, size = 1733, normalized size = 6.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(A^2*b^2*x^2 + 2*A^2*a*b*x + A^2*a^2 + (B^2*b^2*x^2 + 2*B^2*a*b*x + B^2*a^2)*log((b*x + a)^n*e/(d*x +
c)^n)^2 + 2*(A*B*b^2*x^2 + 2*A*B*a*b*x + A*B*a^2)*log((b*x + a)^n*e/(d*x + c)^n), 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((b*x+a)**2*(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 x + a\right )}^{2}{\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((b*x+a)^2*(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^2,x, algorithm="giac")

[Out]

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