3.2.56 \(\int (a+b x)^3 (A+B \log (e (a+b x)^n (c+d x)^{-n}))^2 \, dx\) [156]

Optimal. Leaf size=322 \[ -\frac {B (b c-a d) n (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{6 b d}+\frac {(a+b x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{4 b}+\frac {B (b c-a d)^2 n (a+b x)^2 \left (3 A+B n+3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{12 b d^2}-\frac {B (b c-a d)^3 n (a+b x) \left (6 A+5 B n+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{12 b d^3}-\frac {B (b c-a d)^4 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 A+11 B n+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{12 b d^4}-\frac {B^2 (b c-a d)^4 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{2 b d^4} \]

[Out]

-1/6*B*(-a*d+b*c)*n*(b*x+a)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b/d+1/4*(b*x+a)^4*(A+B*ln(e*(b*x+a)^n/((d*x+c)
^n)))^2/b+1/12*B*(-a*d+b*c)^2*n*(b*x+a)^2*(3*A+B*n+3*B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b/d^2-1/12*B*(-a*d+b*c)^3*
n*(b*x+a)*(6*A+5*B*n+6*B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b/d^3-1/12*B*(-a*d+b*c)^4*n*ln((-a*d+b*c)/b/(d*x+c))*(6*
A+11*B*n+6*B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b/d^4-1/2*B^2*(-a*d+b*c)^4*n^2*polylog(2,d*(b*x+a)/b/(d*x+c))/b/d^4

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Rubi [A]
time = 0.29, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {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} \end {gather*}

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]

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

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)^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 ) \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 )}{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 ) \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 )}{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 ) \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 )}{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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1709\) vs. \(2(322)=644\).
time = 0.70, size = 1709, normalized size = 5.31 \begin {gather*} \frac {-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 ^2(a+b x)+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 ^2(c+d x)-12 a b^3 B^2 c^3 d n^2 \log ^2(c+d x)+18 a^2 b^2 B^2 c^2 d^2 n^2 \log ^2(c+d x)-12 a^3 b B^2 c d^3 n^2 \log ^2(c+d x)-24 a^4 B^2 d^4 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )+24 a^3 A b B d^4 x \log \left (e (a+b x)^n (c+d x)^{-n}\right )-6 b^4 B^2 c^3 d n x \log \left (e (a+b x)^n (c+d x)^{-n}\right )+24 a b^3 B^2 c^2 d^2 n x \log \left (e (a+b x)^n (c+d x)^{-n}\right )-36 a^2 b^2 B^2 c d^3 n x \log \left (e (a+b x)^n (c+d x)^{-n}\right )+18 a^3 b B^2 d^4 n x \log \left (e (a+b x)^n (c+d x)^{-n}\right )+36 a^2 A b^2 B d^4 x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+3 b^4 B^2 c^2 d^2 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )-12 a b^3 B^2 c d^3 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+9 a^2 b^2 B^2 d^4 n x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+24 a A b^3 B d^4 x^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )-2 b^4 B^2 c d^3 n x^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 a b^3 B^2 d^4 n x^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 A b^4 B d^4 x^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 b^4 B^2 c^4 n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )-24 a b^3 B^2 c^3 d n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+36 a^2 b^2 B^2 c^2 d^2 n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )-24 a^3 b B^2 c d^3 n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+12 a^3 b B^2 d^4 x \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+18 a^2 b^2 B^2 d^4 x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+12 a b^3 B^2 d^4 x^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+3 b^4 B^2 d^4 x^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+B n \log (a+b x) \left (-6 b B c \left (b^3 c^3-4 a b^2 c^2 d+6 a^2 b c d^2-4 a^3 d^3\right ) n \log (c+d x)+6 B (b c-a d)^4 n \log \left (\frac {b (c+d x)}{b c-a d}\right )+a d \left (-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 \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )+6 B^2 (b c-a d)^4 n^2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )}{12 b d^4} \end {gather*}

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)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.60, size = 26938, normalized size = 83.66

method	result	size
risch	\(\text {Expression too large to display}\)	\(26938\)

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,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. 1809 vs. \(2 (314) = 628\).
time = 0.82, size = 1809, normalized size = 5.62 \begin {gather*} \text {Too large to display} \end {gather*}

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*n*e*log(b*x +
a)/b - c*n*e*log(d*x + c)/d)*A*B*a^3*e^(-1) - 3*(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^2*b*e^(-1) + (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*a*b^2*e^(-1) - 1/12*(6*a^4*n*e*log(b*x + a)
/b^4 - 6*c^4*n*e*log(d*x + c)/d^4 + (2*(b^3*c*d^2*n - a*b^2*d^3*n)*x^3*e - 3*(b^3*c^2*d*n - a^2*b*d^3*n)*x^2*e
+ 6*(b^3*c^3*n - a^3*d^3*n)*x*e)/(b^3*d^3))*A*B*b^3*e^(-1) + 2*A*B*a^3*x*log((b*x + a)^n*e/(d*x + c)^n) + A^2
*a^3*x + 1/12*((11*n^2 + 6*n)*b^3*c^4 - 2*(19*n^2 + 12*n)*a*b^2*c^3*d + 9*(5*n^2 + 4*n)*a^2*b*c^2*d^2 - 6*(3*n
^2 + 4*n)*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 - 3*B^2*a^4*d^4*n^2*log(b*x + a)^2 + 2*(a*b^3*d^4*(n + 6) - b^4*c*
d^3*n)*B^2*x^3 + ((n^2 + 3*n)*b^4*c^2*d^2 - 2*(n^2 + 6*n)*a*b^3*c*d^3 + (n^2 + 9*n + 18)*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)*b^4*c^3*d - (17*n^2 + 24*n)*a*b^3*c^2*d^2 + (19*n^2 + 36*n)*a^2*b^2*c*d^3 - (7*n^2 + 18*n + 12)*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)*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 + 6*B^2*a^4*d^4*n*log(b*x + a) + 2*(a*b^3*d^4*(n + 12) - b^4*c*d^3*n)*B^2*x^3 + 3*
(3*a^2*b^2*d^4*(n + 4) + 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) - 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 + 6*B^2*a^4*d^4*n*log(b*x + a) + 2*(a*b^3*d^4*(n
+ 12) - b^4*c*d^3*n)*B^2*x^3 + 3*(3*a^2*b^2*d^4*(n + 4) + 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) - 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 = 0.00, 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)^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(-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)**3*(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)^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)

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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 )}^3 \,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)^3,x)

[Out]

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

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