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

Optimal. Leaf size=322 \[ -\frac{B^2 n^2 (b c-a d)^4 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{2 b d^4}-\frac{B n (b c-a d)^4 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 A+11 B n\right )}{12 b d^4}-\frac{B n (a+b x) (b c-a d)^3 \left (6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 A+5 B n\right )}{12 b d^3}+\frac{B n (a+b x)^2 (b c-a d)^2 \left (3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+3 A+B n\right )}{12 b d^2}-\frac{B n (a+b x)^3 (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{6 b d}+\frac{(a+b x)^4 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{4 b} \]

[Out]

-(B*(b*c - a*d)*n*(a + b*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(6*b*d) + ((a + b*x)^4*(A + B*Log[(e*(
a + b*x)^n)/(c + d*x)^n])^2)/(4*b) + (B*(b*c - a*d)^2*n*(a + b*x)^2*(3*A + B*n + 3*B*Log[(e*(a + b*x)^n)/(c +
d*x)^n]))/(12*b*d^2) - (B*(b*c - a*d)^3*n*(a + b*x)*(6*A + 5*B*n + 6*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(12*
b*d^3) - (B*(b*c - a*d)^4*n*Log[(b*c - a*d)/(b*(c + d*x))]*(6*A + 11*B*n + 6*B*Log[(e*(a + b*x)^n)/(c + d*x)^n
]))/(12*b*d^4) - (B^2*(b*c - a*d)^4*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(2*b*d^4)

________________________________________________________________________________________

Rubi [A]  time = 0.771862, antiderivative size = 542, normalized size of antiderivative = 1.68, number of steps used = 21, 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} \[ -\frac{B^2 n^2 (b c-a d)^4 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{2 b d^4}+\frac{A^2 (a+b x)^4}{4 b}-\frac{A B n x (b c-a d)^3}{2 d^3}+\frac{A B n (a+b x)^2 (b c-a d)^2}{4 b d^2}+\frac{A B n (b c-a d)^4 \log (c+d x)}{2 b d^4}+\frac{A B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{A B n (a+b x)^3 (b c-a d)}{6 b d}-\frac{B^2 n (b c-a d)^4 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d^4}-\frac{B^2 n (a+b x) (b c-a d)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d^3}+\frac{B^2 n (a+b x)^2 (b c-a d)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}-\frac{5 B^2 n^2 x (b c-a d)^3}{12 d^3}+\frac{B^2 n^2 (a+b x)^2 (b c-a d)^2}{12 b d^2}+\frac{11 B^2 n^2 (b c-a d)^4 \log (c+d x)}{12 b d^4}+\frac{B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{B^2 n (a+b x)^3 (b c-a d) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A*B*(b*c - a*d)^3*n*x)/(2*d^3) - (5*B^2*(b*c - a*d)^3*n^2*x)/(12*d^3) + (A*B*(b*c - a*d)^2*n*(a + b*x)^2)/(4
*b*d^2) + (B^2*(b*c - a*d)^2*n^2*(a + b*x)^2)/(12*b*d^2) - (A*B*(b*c - a*d)*n*(a + b*x)^3)/(6*b*d) + (A^2*(a +
b*x)^4)/(4*b) + (A*B*(b*c - a*d)^4*n*Log[c + d*x])/(2*b*d^4) + (11*B^2*(b*c - a*d)^4*n^2*Log[c + d*x])/(12*b*
d^4) - (B^2*(b*c - a*d)^3*n*(a + b*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(2*b*d^3) + (B^2*(b*c - a*d)^2*n*(a +
b*x)^2*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(4*b*d^2) - (B^2*(b*c - a*d)*n*(a + b*x)^3*Log[(e*(a + b*x)^n)/(c + d
*x)^n])/(6*b*d) + (A*B*(a + b*x)^4*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(2*b) - (B^2*(b*c - a*d)^4*n*Log[(b*c - a
*d)/(b*(c + d*x))]*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(2*b*d^4) + (B^2*(a + b*x)^4*Log[(e*(a + b*x)^n)/(c + d*x
)^n]^2)/(4*b) - (B^2*(b*c - a*d)^4*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(2*b*d^4)

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)^3 \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)^3+2 A B (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+B^2 (a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=\frac{A^2 (a+b x)^4}{4 b}+(2 A B) \int (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx+B^2 \int (a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=\frac{A^2 (a+b x)^4}{4 b}+\frac{A B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}+\frac{B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{(A B (b c-a d) n) \int \frac{(a+b x)^3}{c+d x} \, dx}{2 b}-\frac{\left (B^2 (b c-a d) n\right ) \int \frac{(a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{2 b}\\ &=\frac{A^2 (a+b x)^4}{4 b}+\frac{A B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}+\frac{B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{(A B (b c-a d) n) \int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{2 b}-\frac{\left (B^2 (b c-a d) n\right ) \int \left (\frac{b (b c-a d)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3}-\frac{b (b c-a d) (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^2}+\frac{b (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d}+\frac{(-b c+a d)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^3 (c+d x)}\right ) \, dx}{2 b}\\ &=-\frac{A B (b c-a d)^3 n x}{2 d^3}+\frac{A B (b c-a d)^2 n (a+b x)^2}{4 b d^2}-\frac{A B (b c-a d) n (a+b x)^3}{6 b d}+\frac{A^2 (a+b x)^4}{4 b}+\frac{A B (b c-a d)^4 n \log (c+d x)}{2 b d^4}+\frac{A B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}+\frac{B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{\left (B^2 (b c-a d) n\right ) \int (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{2 d}+\frac{\left (B^2 (b c-a d)^2 n\right ) \int (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{2 d^2}-\frac{\left (B^2 (b c-a d)^3 n\right ) \int \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{2 d^3}+\frac{\left (B^2 (b c-a d)^4 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{2 b d^3}\\ &=-\frac{A B (b c-a d)^3 n x}{2 d^3}+\frac{A B (b c-a d)^2 n (a+b x)^2}{4 b d^2}-\frac{A B (b c-a d) n (a+b x)^3}{6 b d}+\frac{A^2 (a+b x)^4}{4 b}+\frac{A B (b c-a d)^4 n \log (c+d x)}{2 b d^4}-\frac{B^2 (b c-a d)^3 n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d^3}+\frac{B^2 (b c-a d)^2 n (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}-\frac{B^2 (b c-a d) n (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b d}+\frac{A B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{B^2 (b c-a d)^4 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 )}{2 b d^4}+\frac{B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}+\frac{\left (B^2 (b c-a d)^2 n^2\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{6 b d}-\frac{\left (B^2 (b c-a d)^3 n^2\right ) \int \frac{a+b x}{c+d x} \, dx}{4 b d^2}+\frac{\left (B^2 (b c-a d)^4 n^2\right ) \int \frac{1}{c+d x} \, dx}{2 b d^3}+\frac{\left (B^2 (b c-a d)^5 n^2\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{2 b d^4}\\ &=-\frac{A B (b c-a d)^3 n x}{2 d^3}+\frac{A B (b c-a d)^2 n (a+b x)^2}{4 b d^2}-\frac{A B (b c-a d) n (a+b x)^3}{6 b d}+\frac{A^2 (a+b x)^4}{4 b}+\frac{A B (b c-a d)^4 n \log (c+d x)}{2 b d^4}+\frac{B^2 (b c-a d)^4 n^2 \log (c+d x)}{2 b d^4}-\frac{B^2 (b c-a d)^3 n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d^3}+\frac{B^2 (b c-a d)^2 n (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}-\frac{B^2 (b c-a d) n (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b d}+\frac{A B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{B^2 (b c-a d)^4 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 )}{2 b d^4}+\frac{B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}+\frac{\left (B^2 (b c-a d)^2 n^2\right ) \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}{6 b d}-\frac{\left (B^2 (b c-a d)^3 n^2\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{4 b d^2}+\frac{\left (B^2 (b c-a d)^5 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 )}{2 b d^5}\\ &=-\frac{A B (b c-a d)^3 n x}{2 d^3}-\frac{5 B^2 (b c-a d)^3 n^2 x}{12 d^3}+\frac{A B (b c-a d)^2 n (a+b x)^2}{4 b d^2}+\frac{B^2 (b c-a d)^2 n^2 (a+b x)^2}{12 b d^2}-\frac{A B (b c-a d) n (a+b x)^3}{6 b d}+\frac{A^2 (a+b x)^4}{4 b}+\frac{A B (b c-a d)^4 n \log (c+d x)}{2 b d^4}+\frac{11 B^2 (b c-a d)^4 n^2 \log (c+d x)}{12 b d^4}-\frac{B^2 (b c-a d)^3 n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d^3}+\frac{B^2 (b c-a d)^2 n (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}-\frac{B^2 (b c-a d) n (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b d}+\frac{A B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{B^2 (b c-a d)^4 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 )}{2 b d^4}+\frac{B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{\left (B^2 (b c-a d)^5 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 )}{2 b d^5}\\ &=-\frac{A B (b c-a d)^3 n x}{2 d^3}-\frac{5 B^2 (b c-a d)^3 n^2 x}{12 d^3}+\frac{A B (b c-a d)^2 n (a+b x)^2}{4 b d^2}+\frac{B^2 (b c-a d)^2 n^2 (a+b x)^2}{12 b d^2}-\frac{A B (b c-a d) n (a+b x)^3}{6 b d}+\frac{A^2 (a+b x)^4}{4 b}+\frac{A B (b c-a d)^4 n \log (c+d x)}{2 b d^4}+\frac{11 B^2 (b c-a d)^4 n^2 \log (c+d x)}{12 b d^4}-\frac{B^2 (b c-a d)^3 n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d^3}+\frac{B^2 (b c-a d)^2 n (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}-\frac{B^2 (b c-a d) n (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b d}+\frac{A B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{B^2 (b c-a d)^4 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 )}{2 b d^4}+\frac{B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{\left (B^2 (b c-a d)^5 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 )}{2 b d^5}\\ &=-\frac{A B (b c-a d)^3 n x}{2 d^3}-\frac{5 B^2 (b c-a d)^3 n^2 x}{12 d^3}+\frac{A B (b c-a d)^2 n (a+b x)^2}{4 b d^2}+\frac{B^2 (b c-a d)^2 n^2 (a+b x)^2}{12 b d^2}-\frac{A B (b c-a d) n (a+b x)^3}{6 b d}+\frac{A^2 (a+b x)^4}{4 b}+\frac{A B (b c-a d)^4 n \log (c+d x)}{2 b d^4}+\frac{11 B^2 (b c-a d)^4 n^2 \log (c+d x)}{12 b d^4}-\frac{B^2 (b c-a d)^3 n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b d^3}+\frac{B^2 (b c-a d)^2 n (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b d^2}-\frac{B^2 (b c-a d) n (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{6 b d}+\frac{A B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{B^2 (b c-a d)^4 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 )}{2 b d^4}+\frac{B^2 (a+b x)^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{B^2 (b c-a d)^4 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{2 b d^4}\\ \end{align*}

Mathematica [B]  time = 1.551, size = 1709, normalized size = 5.31 \[ \frac{3 A^2 d^4 x^4 b^4-2 A B c d^3 n x^3 b^4+B^2 c^2 d^2 n^2 x^2 b^4+3 A B c^2 d^2 n x^2 b^4+3 B^2 c^4 n^2 \log ^2(c+d x) b^4+3 B^2 d^4 x^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) b^4-5 B^2 c^3 d n^2 x b^4-6 A B c^3 d n x b^4+11 B^2 c^4 n^2 \log (c+d x) b^4+6 A B c^4 n \log (c+d x) b^4+6 A B d^4 x^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^4-2 B^2 c d^3 n x^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^4+3 B^2 c^2 d^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^4-6 B^2 c^3 d n x \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^4+6 B^2 c^4 n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^4+12 a A^2 d^4 x^3 b^3+2 a A B d^4 n x^3 b^3+6 a B^2 c^3 d n^2 b^3-2 a B^2 c d^3 n^2 x^2 b^3-12 a A B c d^3 n x^2 b^3-12 a B^2 c^3 d n^2 \log ^2(c+d x) b^3+12 a B^2 d^4 x^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) b^3+17 a B^2 c^2 d^2 n^2 x b^3+24 a A B c^2 d^2 n x b^3-38 a B^2 c^3 d n^2 \log (c+d x) b^3-24 a A B c^3 d n \log (c+d x) b^3+24 a A B d^4 x^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^3+2 a B^2 d^4 n x^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^3-12 a B^2 c d^3 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^3+24 a B^2 c^2 d^2 n x \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^3-24 a B^2 c^3 d n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^3-24 a^2 B^2 c^2 d^2 n^2 b^2+18 a^2 A^2 d^4 x^2 b^2+a^2 B^2 d^4 n^2 x^2 b^2+9 a^2 A B d^4 n x^2 b^2+18 a^2 B^2 c^2 d^2 n^2 \log ^2(c+d x) b^2+18 a^2 B^2 d^4 x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) b^2-19 a^2 B^2 c d^3 n^2 x b^2-36 a^2 A B c d^3 n x b^2+45 a^2 B^2 c^2 d^2 n^2 \log (c+d x) b^2+36 a^2 A B c^2 d^2 n \log (c+d x) b^2+36 a^2 A B d^4 x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^2+9 a^2 B^2 d^4 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^2-36 a^2 B^2 c d^3 n x \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^2+36 a^2 B^2 c^2 d^2 n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^2+36 a^3 B^2 c d^3 n^2 b-12 a^3 B^2 c d^3 n^2 \log ^2(c+d x) b+12 a^3 B^2 d^4 x \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) b+12 a^3 A^2 d^4 x b+7 a^3 B^2 d^4 n^2 x b+18 a^3 A B d^4 n x b-18 a^3 B^2 c d^3 n^2 \log (c+d x) b-24 a^3 A B c d^3 n \log (c+d x) b+24 a^3 A B d^4 x \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b+18 a^3 B^2 d^4 n x \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b-24 a^3 B^2 c d^3 n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b-24 a^4 B^2 d^4 n^2-3 a^4 B^2 d^4 n^2 \log ^2(a+b x)-24 a^4 A B d^4 n-24 a^4 B^2 d^4 n^2 \log (c+d x)-24 a^4 B^2 d^4 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )+B n \log (a+b x) \left (6 B n \log \left (\frac{b (c+d x)}{b c-a d}\right ) (b c-a d)^4-6 b B c \left (b^3 c^3-4 a b^2 d c^2+6 a^2 b d^2 c-4 a^3 d^3\right ) n \log (c+d x)+a d \left (-6 b^3 B n c^3+21 a b^2 B d n c^2-26 a^2 b B d^2 n c+a^3 d^3 (6 A+35 B n)+6 a^3 B d^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )+6 B^2 (b c-a d)^4 n^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )}{12 b d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-24*a^4*A*B*d^4*n + 6*a*b^3*B^2*c^3*d*n^2 - 24*a^2*b^2*B^2*c^2*d^2*n^2 + 36*a^3*b*B^2*c*d^3*n^2 - 24*a^4*B^2*
d^4*n^2 + 12*a^3*A^2*b*d^4*x - 6*A*b^4*B*c^3*d*n*x + 24*a*A*b^3*B*c^2*d^2*n*x - 36*a^2*A*b^2*B*c*d^3*n*x + 18*
a^3*A*b*B*d^4*n*x - 5*b^4*B^2*c^3*d*n^2*x + 17*a*b^3*B^2*c^2*d^2*n^2*x - 19*a^2*b^2*B^2*c*d^3*n^2*x + 7*a^3*b*
B^2*d^4*n^2*x + 18*a^2*A^2*b^2*d^4*x^2 + 3*A*b^4*B*c^2*d^2*n*x^2 - 12*a*A*b^3*B*c*d^3*n*x^2 + 9*a^2*A*b^2*B*d^
4*n*x^2 + b^4*B^2*c^2*d^2*n^2*x^2 - 2*a*b^3*B^2*c*d^3*n^2*x^2 + a^2*b^2*B^2*d^4*n^2*x^2 + 12*a*A^2*b^3*d^4*x^3
- 2*A*b^4*B*c*d^3*n*x^3 + 2*a*A*b^3*B*d^4*n*x^3 + 3*A^2*b^4*d^4*x^4 - 3*a^4*B^2*d^4*n^2*Log[a + b*x]^2 + 6*A*
b^4*B*c^4*n*Log[c + d*x] - 24*a*A*b^3*B*c^3*d*n*Log[c + d*x] + 36*a^2*A*b^2*B*c^2*d^2*n*Log[c + d*x] - 24*a^3*
A*b*B*c*d^3*n*Log[c + d*x] + 11*b^4*B^2*c^4*n^2*Log[c + d*x] - 38*a*b^3*B^2*c^3*d*n^2*Log[c + d*x] + 45*a^2*b^
2*B^2*c^2*d^2*n^2*Log[c + d*x] - 18*a^3*b*B^2*c*d^3*n^2*Log[c + d*x] - 24*a^4*B^2*d^4*n^2*Log[c + d*x] + 3*b^4
*B^2*c^4*n^2*Log[c + d*x]^2 - 12*a*b^3*B^2*c^3*d*n^2*Log[c + d*x]^2 + 18*a^2*b^2*B^2*c^2*d^2*n^2*Log[c + d*x]^
2 - 12*a^3*b*B^2*c*d^3*n^2*Log[c + d*x]^2 - 24*a^4*B^2*d^4*n*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 24*a^3*A*b*B*d
^4*x*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 6*b^4*B^2*c^3*d*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 24*a*b^3*B^2*c^
2*d^2*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 36*a^2*b^2*B^2*c*d^3*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 18*a^
3*b*B^2*d^4*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 36*a^2*A*b^2*B*d^4*x^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 3
*b^4*B^2*c^2*d^2*n*x^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 12*a*b^3*B^2*c*d^3*n*x^2*Log[(e*(a + b*x)^n)/(c + d*
x)^n] + 9*a^2*b^2*B^2*d^4*n*x^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 24*a*A*b^3*B*d^4*x^3*Log[(e*(a + b*x)^n)/(c
+ d*x)^n] - 2*b^4*B^2*c*d^3*n*x^3*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 2*a*b^3*B^2*d^4*n*x^3*Log[(e*(a + b*x)^n
)/(c + d*x)^n] + 6*A*b^4*B*d^4*x^4*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 6*b^4*B^2*c^4*n*Log[c + d*x]*Log[(e*(a +
b*x)^n)/(c + d*x)^n] - 24*a*b^3*B^2*c^3*d*n*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 36*a^2*b^2*B^2*c^
2*d^2*n*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 24*a^3*b*B^2*c*d^3*n*Log[c + d*x]*Log[(e*(a + b*x)^n)/
(c + d*x)^n] + 12*a^3*b*B^2*d^4*x*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 18*a^2*b^2*B^2*d^4*x^2*Log[(e*(a + b*x)
^n)/(c + d*x)^n]^2 + 12*a*b^3*B^2*d^4*x^3*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 3*b^4*B^2*d^4*x^4*Log[(e*(a + b
*x)^n)/(c + d*x)^n]^2 + B*n*Log[a + b*x]*(-6*b*B*c*(b^3*c^3 - 4*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 4*a^3*d^3)*n*Log
[c + d*x] + 6*B*(b*c - a*d)^4*n*Log[(b*(c + d*x))/(b*c - a*d)] + a*d*(-6*b^3*B*c^3*n + 21*a*b^2*B*c^2*d*n - 26
*a^2*b*B*c*d^2*n + a^3*d^3*(6*A + 35*B*n) + 6*a^3*B*d^3*Log[(e*(a + b*x)^n)/(c + d*x)^n])) + 6*B^2*(b*c - a*d)
^4*n^2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])/(12*b*d^4)

________________________________________________________________________________________

Maple [C]  time = 2.524, size = 26948, normalized size = 83.7 \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)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2,x)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [B]  time = 3.86044, size = 2526, normalized size = 7.84 \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)^3*(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^2,x, algorithm="maxima")

[Out]

1/2*A*B*b^3*x^4*log((b*x + a)^n*e/(d*x + c)^n) + 1/4*A^2*b^3*x^4 + 2*A*B*a*b^2*x^3*log((b*x + a)^n*e/(d*x + c)
^n) + A^2*a*b^2*x^3 + 3*A*B*a^2*b*x^2*log((b*x + a)^n*e/(d*x + c)^n) + 3/2*A^2*a^2*b*x^2 + 2*A*B*a^3*x*log((b*
x + a)^n*e/(d*x + c)^n) + A^2*a^3*x + 2*(a*e*n*log(b*x + a)/b - c*e*n*log(d*x + c)/d)*A*B*a^3/e - 3*(a^2*e*n*l
og(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^2*b/e + (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*a*b^2/e - 1/12*(6*a^4*e*n*log(b*x + a)/b^4 - 6*c^4*e*n*log(d*x + c)/d^4 + (2*(b^3*c*d^2*e*n - a*b^2*
d^3*e*n)*x^3 - 3*(b^3*c^2*d*e*n - a^2*b*d^3*e*n)*x^2 + 6*(b^3*c^3*e*n - a^3*d^3*e*n)*x)/(b^3*d^3))*A*B*b^3/e +
1/12*((11*n^2 + 6*n*log(e))*b^3*c^4 - 2*(19*n^2 + 12*n*log(e))*a*b^2*c^3*d + 9*(5*n^2 + 4*n*log(e))*a^2*b*c^2
*d^2 - 6*(3*n^2 + 4*n*log(e))*a^3*c*d^3)*B^2*log(d*x + c)/d^4 + 1/2*(b^4*c^4*n^2 - 4*a*b^3*c^3*d*n^2 + 6*a^2*b
^2*c^2*d^2*n^2 - 4*a^3*b*c*d^3*n^2 + a^4*d^4*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^4) + 1/12*(3*B^2*b^4*d^4*x^4*log(e)^2 - 3*B^2*a^4*d^4*n^2*log(b*x + a)^2 -
2*(b^4*c*d^3*n*log(e) - (n*log(e) + 6*log(e)^2)*a*b^3*d^4)*B^2*x^3 + ((n^2 + 3*n*log(e))*b^4*c^2*d^2 - 2*(n^2
+ 6*n*log(e))*a*b^3*c*d^3 + (n^2 + 9*n*log(e) + 18*log(e)^2)*a^2*b^2*d^4)*B^2*x^2 - 6*(b^4*c^4*n^2 - 4*a*b^3*c
^3*d*n^2 + 6*a^2*b^2*c^2*d^2*n^2 - 4*a^3*b*c*d^3*n^2)*B^2*log(b*x + a)*log(d*x + c) + 3*(b^4*c^4*n^2 - 4*a*b^3
*c^3*d*n^2 + 6*a^2*b^2*c^2*d^2*n^2 - 4*a^3*b*c*d^3*n^2)*B^2*log(d*x + c)^2 - ((5*n^2 + 6*n*log(e))*b^4*c^3*d -
(17*n^2 + 24*n*log(e))*a*b^3*c^2*d^2 + (19*n^2 + 36*n*log(e))*a^2*b^2*c*d^3 - (7*n^2 + 18*n*log(e) + 12*log(e
)^2)*a^3*b*d^4)*B^2*x - (6*a*b^3*c^3*d*n^2 - 21*a^2*b^2*c^2*d^2*n^2 + 26*a^3*b*c*d^3*n^2 - (11*n^2 + 6*n*log(e
))*a^4*d^4)*B^2*log(b*x + a) + 3*(B^2*b^4*d^4*x^4 + 4*B^2*a*b^3*d^4*x^3 + 6*B^2*a^2*b^2*d^4*x^2 + 4*B^2*a^3*b*
d^4*x)*log((b*x + a)^n)^2 + 3*(B^2*b^4*d^4*x^4 + 4*B^2*a*b^3*d^4*x^3 + 6*B^2*a^2*b^2*d^4*x^2 + 4*B^2*a^3*b*d^4
*x)*log((d*x + c)^n)^2 + (6*B^2*b^4*d^4*x^4*log(e) + 6*B^2*a^4*d^4*n*log(b*x + a) + 2*(a*b^3*d^4*(n + 12*log(e
)) - b^4*c*d^3*n)*B^2*x^3 + 3*(3*a^2*b^2*d^4*(n + 4*log(e)) + b^4*c^2*d^2*n - 4*a*b^3*c*d^3*n)*B^2*x^2 + 6*(a^
3*b*d^4*(3*n + 4*log(e)) - b^4*c^3*d*n + 4*a*b^3*c^2*d^2*n - 6*a^2*b^2*c*d^3*n)*B^2*x + 6*(b^4*c^4*n - 4*a*b^3
*c^3*d*n + 6*a^2*b^2*c^2*d^2*n - 4*a^3*b*c*d^3*n)*B^2*log(d*x + c))*log((b*x + a)^n) - (6*B^2*b^4*d^4*x^4*log(
e) + 6*B^2*a^4*d^4*n*log(b*x + a) + 2*(a*b^3*d^4*(n + 12*log(e)) - b^4*c*d^3*n)*B^2*x^3 + 3*(3*a^2*b^2*d^4*(n
+ 4*log(e)) + b^4*c^2*d^2*n - 4*a*b^3*c*d^3*n)*B^2*x^2 + 6*(a^3*b*d^4*(3*n + 4*log(e)) - b^4*c^3*d*n + 4*a*b^3
*c^2*d^2*n - 6*a^2*b^2*c*d^3*n)*B^2*x + 6*(b^4*c^4*n - 4*a*b^3*c^3*d*n + 6*a^2*b^2*c^2*d^2*n - 4*a^3*b*c*d^3*n
)*B^2*log(d*x + c) + 6*(B^2*b^4*d^4*x^4 + 4*B^2*a*b^3*d^4*x^3 + 6*B^2*a^2*b^2*d^4*x^2 + 4*B^2*a^3*b*d^4*x)*log
((b*x + a)^n))*log((d*x + c)^n))/(b*d^4)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} b^{3} x^{3} + 3 \, A^{2} a b^{2} x^{2} + 3 \, A^{2} a^{2} b x + A^{2} a^{3} +{\left (B^{2} b^{3} x^{3} + 3 \, B^{2} a b^{2} x^{2} + 3 \, B^{2} a^{2} b x + B^{2} a^{3}\right )} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 2 \,{\left (A B b^{3} x^{3} + 3 \, A B a b^{2} x^{2} + 3 \, A B a^{2} b x + A B a^{3}\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)^3*(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^2,x, algorithm="fricas")

[Out]

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

[Out]

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