3.4.11 \(\int (A+B \log (e (a+b x)^n (c+d x)^{-n}))^3 \, dx\) [311]

Optimal. Leaf size=203 \[ \frac {3 B (b c-a d) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{b d}+\frac {(a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{b}+\frac {6 B^2 (b c-a d) n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b d}-\frac {6 B^3 (b c-a d) n^3 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b d} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2536, 2573, 2551, 2354, 2421, 6724} \begin {gather*} \frac {6 B^2 n^2 (b c-a d) \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b d}-\frac {6 B^3 n^3 (b c-a d) \text {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{b d}+\frac {3 B n (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{b d}+\frac {(a+b x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2421

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

Rule 2536

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

Rule 2551

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/d)^m, Subst[Int[(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[d*f - c*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]

Rule 6724

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

Rubi steps

\begin {align*} \int \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx &=\int \left (A^3+3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=A^3 x+\left (3 A^2 B\right ) \int \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx+\left (3 A B^2\right ) \int \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx+B^3 \int \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=A^3 x+\frac {3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac {\left (3 A^2 B (b c-a d) n\right ) \int \frac {1}{c+d x} \, dx}{b}-\frac {\left (6 A B^2 (b c-a d) n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{b}-\frac {\left (3 B^3 (b c-a d) n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{b}\\ &=A^3 x-\frac {3 A^2 B (b c-a d) n \log (c+d x)}{b d}+\frac {3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {6 A B^2 (b c-a d) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac {3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {3 B^3 (b c-a d) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac {B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac {\left (6 A B^2 (b c-a d)^2 n^2\right ) \int \frac {\log \left (-\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{b d}-\frac {\left (6 B^3 (b c-a d)^2 n^2\right ) \int \frac {\log \left (-\frac {-b c+a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{b d}\\ &=A^3 x-\frac {3 A^2 B (b c-a d) n \log (c+d x)}{b d}+\frac {3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {6 A B^2 (b c-a d) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac {3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {3 B^3 (b c-a d) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac {B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {6 B^3 (b c-a d) n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b d}-\frac {\left (6 A B^2 (b c-a d)^2 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 )}{b d^2}-\frac {\left (6 B^3 (b c-a d)^2 n^3\right ) \int \frac {\text {Li}_2\left (1+\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{b d}\\ &=A^3 x-\frac {3 A^2 B (b c-a d) n \log (c+d x)}{b d}+\frac {3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {6 A B^2 (b c-a d) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac {3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {3 B^3 (b c-a d) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac {B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {6 B^3 (b c-a d) n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b d}-\frac {6 B^3 (b c-a d) n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b d}+\frac {\left (6 A B^2 (b c-a d)^2 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 )}{b d^2}\\ &=A^3 x-\frac {3 A^2 B (b c-a d) n \log (c+d x)}{b d}+\frac {3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {6 A B^2 (b c-a d) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac {3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {3 B^3 (b c-a d) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac {B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {6 B^3 (b c-a d) n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b d}-\frac {6 B^3 (b c-a d) n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b d}+\frac {\left (6 A B^2 (b c-a d)^2 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 )}{b d^2}\\ &=A^3 x-\frac {3 A^2 B (b c-a d) n \log (c+d x)}{b d}+\frac {3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {6 A B^2 (b c-a d) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac {3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {3 B^3 (b c-a d) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac {B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {6 A B^2 (b c-a d) n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b d}+\frac {6 B^3 (b c-a d) n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b d}-\frac {6 B^3 (b c-a d) n^3 \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b d}\\ \end {align*}

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Mathematica [A]
time = 0.33, size = 372, normalized size = 1.83 \begin {gather*} A^3 x+\frac {B \left (-3 A^2 (b c-a d) n \log (c+d x)+3 A^2 d (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+3 A B d (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+B^2 d (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )+3 A B (b c-a d) n \left (-\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 n \log \left (\frac {d (a+b x)}{-b c+a d}\right )-2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+n \log \left (\frac {b c-a d}{b c+b d x}\right )\right )+2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )+3 B^2 (b c-a d) n \left (\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )-2 n^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )\right )}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3,x, algorithm="maxima")

[Out]

3*(a*n*e*log(b*x + a)/b - c*n*e*log(d*x + c)/d)*A^2*B*e^(-1) + 3*A^2*B*x*log((b*x + a)^n*e/(d*x + c)^n) + A^3*
x - (B^3*b*d*x*log((d*x + c)^n)^3 - 3*(B^3*a*d*n*log(b*x + a) - B^3*b*c*n*log(d*x + c) + B^3*b*d*x*log((b*x +
a)^n) + (A*B^2*b*d + B^3*b*d)*x)*log((d*x + c)^n)^2)/(b*d) - integrate(-(3*A*B^2*b*c + B^3*b*c + (B^3*b*d*x +
B^3*b*c)*log((b*x + a)^n)^3 + 3*(A*B^2*b*c + B^3*b*c + (A*B^2*b*d + B^3*b*d)*x)*log((b*x + a)^n)^2 + (3*A*B^2*
b*d + B^3*b*d)*x + 3*(2*A*B^2*b*c + B^3*b*c + (2*A*B^2*b*d + B^3*b*d)*x)*log((b*x + a)^n) - 3*(2*B^3*a*d*n^2*l
og(b*x + a) - 2*B^3*b*c*n^2*log(d*x + c) + 2*A*B^2*b*c + B^3*b*c + (B^3*b*d*x + B^3*b*c)*log((b*x + a)^n)^2 +
(B^3*b*d*(2*n + 1) + 2*A*B^2*b*d*(n + 1))*x + 2*(A*B^2*b*c + B^3*b*c + (B^3*b*d*(n + 1) + A*B^2*b*d)*x)*log((b
*x + a)^n))*log((d*x + c)^n))/(b*d*x + b*c), x)

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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((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3,x, algorithm="fricas")

[Out]

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

[Out]

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

[Out]

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

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