3.2.57 \(\int (a+b x)^2 (A+B \log (e (a+b x)^n (c+d x)^{-n}))^2 \, dx\) [157]

Optimal. Leaf size=263 \[ -\frac {B (b c-a d) n (a+b x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 b d}+\frac {(a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{3 b}+\frac {B (b c-a d)^2 n (a+b x) \left (2 A+B n+2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 b d^2}+\frac {B (b c-a d)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 A+3 B n+2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 b d^3}+\frac {2 B^2 (b c-a d)^3 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{3 b d^3} \]

[Out]

-1/3*B*(-a*d+b*c)*n*(b*x+a)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b/d+1/3*(b*x+a)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)
^n)))^2/b+1/3*B*(-a*d+b*c)^2*n*(b*x+a)*(2*A+B*n+2*B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b/d^2+1/3*B*(-a*d+b*c)^3*n*ln
((-a*d+b*c)/b/(d*x+c))*(2*A+3*B*n+2*B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b/d^3+2/3*B^2*(-a*d+b*c)^3*n^2*polylog(2,d*
(b*x+a)/b/(d*x+c))/b/d^3

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Rubi [A]
time = 0.23, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2573, 2549, 2381, 2384, 2354, 2438} \begin {gather*} \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} \end {gather*}

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]

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

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2381

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

Rule 2384

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(f*x
)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])/(e*(q + 1))), x] - Dist[f/(e*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x)^(
q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]

Rule 2438

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

Rule 2549

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.), x_Symbol] :> Dist[(b*c - a*d)^(m + 1)*(g/b)^m, Subst[Int[x^m*((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] && NeQ[b*c - a*d, 0] && IntegersQ[m,
p] && EqQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])

Rule 2573

Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^
n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; FreeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !I
ntegerQ[n]

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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1149\) vs. \(2(263)=526\).
time = 0.48, size = 1149, normalized size = 4.37 \begin {gather*} \frac {-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 ^2(a+b x)-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 ^2(c+d x)+3 a b^2 B^2 c^2 d n^2 \log ^2(c+d x)-3 a^2 b B^2 c d^2 n^2 \log ^2(c+d x)-6 a^3 B^2 d^3 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 a^2 A b B d^3 x \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 b^3 B^2 c^2 d n x \log \left (e (a+b x)^n (c+d x)^{-n}\right )-6 a b^2 B^2 c d^2 n x \log \left (e (a+b x)^n (c+d x)^{-n}\right )+4 a^2 b B^2 d^3 n x \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 a A b^2 B d^3 x^2 \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 )+a b^2 B^2 d^3 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 A b^3 B d^3 x^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )-2 b^3 B^2 c^3 n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 a b^2 B^2 c^2 d n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )-6 a^2 b B^2 c d^2 n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+3 a^2 b B^2 d^3 x \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+3 a b^2 B^2 d^3 x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+b^3 B^2 d^3 x^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+B n \log (a+b x) \left (2 b B c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) n \log (c+d x)-2 B (b c-a d)^3 n \log \left (\frac {b (c+d x)}{b c-a d}\right )+a d \left (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 \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )-2 B^2 (b c-a d)^3 n^2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )}{3 b d^3} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.09, size = 19970, normalized size = 75.93

method	result	size
risch	\(\text {Expression too large to display}\)	\(19970\)

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,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1244 vs. \(2 (256) = 512\).
time = 0.88, size = 1244, normalized size = 4.73 \begin {gather*} \frac {2}{3} \, A B b^{2} x^{3} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + \frac {1}{3} \, A^{2} b^{2} x^{3} + 2 \, A B a b x^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A^{2} a b x^{2} + 2 \, {\left (\frac {a n e \log \left (b x + a\right )}{b} - \frac {c n e \log \left (d x + c\right )}{d}\right )} A B a^{2} e^{\left (-1\right )} - 2 \, {\left (\frac {a^{2} n e \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} n e \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c n - a d n\right )} x e}{b d}\right )} A B a b e^{\left (-1\right )} + \frac {1}{3} \, {\left (\frac {2 \, a^{3} n e \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} n e \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d n - a b d^{2} n\right )} x^{2} e - 2 \, {\left (b^{2} c^{2} n - a^{2} d^{2} n\right )} x e}{b^{2} d^{2}}\right )} A B b^{2} e^{\left (-1\right )} + 2 \, A B a^{2} x \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A^{2} a^{2} x - \frac {{\left ({\left (3 \, n^{2} + 2 \, n\right )} b^{2} c^{3} - {\left (7 \, n^{2} + 6 \, n\right )} a b c^{2} d + 2 \, {\left (2 \, n^{2} + 3 \, n\right )} a^{2} c d^{2}\right )} B^{2} \log \left (d x + c\right )}{3 \, d^{3}} - \frac {2 \, {\left (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}\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B^{2}}{3 \, b d^{3}} - \frac {B^{2} a^{3} d^{3} n^{2} \log \left (b x + a\right )^{2} - B^{2} b^{3} d^{3} x^{3} - {\left (a b^{2} d^{3} {\left (n + 3\right )} - b^{3} c d^{2} n\right )} B^{2} x^{2} - 2 \, {\left (b^{3} c^{3} n^{2} - 3 \, a b^{2} c^{2} d n^{2} + 3 \, a^{2} b c d^{2} n^{2}\right )} B^{2} \log \left (b x + a\right ) \log \left (d x + c\right ) + {\left (b^{3} c^{3} n^{2} - 3 \, a b^{2} c^{2} d n^{2} + 3 \, a^{2} b c d^{2} n^{2}\right )} B^{2} \log \left (d x + c\right )^{2} - {\left ({\left (n^{2} + 2 \, n\right )} b^{3} c^{2} d - 2 \, {\left (n^{2} + 3 \, n\right )} a b^{2} c d^{2} + {\left (n^{2} + 4 \, n + 3\right )} a^{2} b d^{3}\right )} B^{2} x - {\left (2 \, a b^{2} c^{2} d n^{2} - 5 \, a^{2} b c d^{2} n^{2} + {\left (3 \, n^{2} + 2 \, n\right )} a^{3} d^{3}\right )} B^{2} \log \left (b x + a\right ) - {\left (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\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} - {\left (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\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2} - {\left (2 \, B^{2} b^{3} d^{3} x^{3} + 2 \, B^{2} a^{3} d^{3} n \log \left (b x + a\right ) + {\left (a b^{2} d^{3} {\left (n + 6\right )} - b^{3} c d^{2} n\right )} B^{2} x^{2} + 2 \, {\left (a^{2} b d^{3} {\left (2 \, n + 3\right )} + b^{3} c^{2} d n - 3 \, a b^{2} c d^{2} n\right )} B^{2} x - 2 \, {\left (b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n\right )} B^{2} \log \left (d x + c\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) + {\left (2 \, B^{2} b^{3} d^{3} x^{3} + 2 \, B^{2} a^{3} d^{3} n \log \left (b x + a\right ) + {\left (a b^{2} d^{3} {\left (n + 6\right )} - b^{3} c d^{2} n\right )} B^{2} x^{2} + 2 \, {\left (a^{2} b d^{3} {\left (2 \, n + 3\right )} + b^{3} c^{2} d n - 3 \, a b^{2} c d^{2} n\right )} B^{2} x - 2 \, {\left (b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n\right )} B^{2} \log \left (d x + c\right ) + 2 \, {\left (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\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{3 \, b d^{3}} \end {gather*}

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*n*e*log(b*x + a)/b - c*n*e*log(d*x + c)/d)*A*B*a^2*e^(-1) - 2*(a^2*n*e*log(b*x + a)/b^2
- c^2*n*e*log(d*x + c)/d^2 + (b*c*n - a*d*n)*x*e/(b*d))*A*B*a*b*e^(-1) + 1/3*(2*a^3*n*e*log(b*x + a)/b^3 - 2*
c^3*n*e*log(d*x + c)/d^3 - ((b^2*c*d*n - a*b*d^2*n)*x^2*e - 2*(b^2*c^2*n - a^2*d^2*n)*x*e)/(b^2*d^2))*A*B*b^2*
e^(-1) + 2*A*B*a^2*x*log((b*x + a)^n*e/(d*x + c)^n) + A^2*a^2*x - 1/3*((3*n^2 + 2*n)*b^2*c^3 - (7*n^2 + 6*n)*a
*b*c^2*d + 2*(2*n^2 + 3*n)*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*a^3*d^3*n^2*log(b*x + a)^2 - B^2*b^3*d^3*x^3 - (a*b^2*d^3*(n + 3) - b^3*c*d^2*n)*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)*b^3*c^2*d - 2*(n^2 + 3*n)*a*b^2*c*d^2 + (
n^2 + 4*n + 3)*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)*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 + 2*B^2*a^3*d^3*n*log(b*x + a) +
(a*b^2*d^3*(n + 6) - b^3*c*d^2*n)*B^2*x^2 + 2*(a^2*b*d^3*(2*n + 3) + 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 + 2*B
^2*a^3*d^3*n*log(b*x + a) + (a*b^2*d^3*(n + 6) - b^3*c*d^2*n)*B^2*x^2 + 2*(a^2*b*d^3*(2*n + 3) + 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 = 0.00, 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((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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

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]

Exception raised: HeuristicGCDFailed >> no luck

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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((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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^2\,{\left (a+b\,x\right )}^2 \,d x \end {gather*}

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)^2,x)

[Out]

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

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