3.4.43 \(\int \frac {\log ^3(c (a+b x)^n)}{d x+e x^2} \, dx\) [343]

Optimal. Leaf size=238 \[ \frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (1+\frac {b x}{a}\right )}{d}+\frac {6 n^2 \log \left (c (a+b x)^n\right ) \text {Li}_3\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {6 n^2 \log \left (c (a+b x)^n\right ) \text {Li}_3\left (1+\frac {b x}{a}\right )}{d}-\frac {6 n^3 \text {Li}_4\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {6 n^3 \text {Li}_4\left (1+\frac {b x}{a}\right )}{d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.25, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1607, 2463, 2443, 2481, 2421, 2430, 6724} \begin {gather*} \frac {6 n^2 \log \left (c (a+b x)^n\right ) \text {PolyLog}\left (3,-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {6 n^2 \text {PolyLog}\left (3,\frac {b x}{a}+1\right ) \log \left (c (a+b x)^n\right )}{d}+\frac {3 n \text {PolyLog}\left (2,\frac {b x}{a}+1\right ) \log ^2\left (c (a+b x)^n\right )}{d}-\frac {6 n^3 \text {PolyLog}\left (4,-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {6 n^3 \text {PolyLog}\left (4,\frac {b x}{a}+1\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}+\frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[-((b*x)/a)]*Log[c*(a + b*x)^n]^3)/d - (Log[c*(a + b*x)^n]^3*Log[(b*(d + e*x))/(b*d - a*e)])/d - (3*n*Log[
c*(a + b*x)^n]^2*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/d + (3*n*Log[c*(a + b*x)^n]^2*PolyLog[2, 1 + (b*x)/
a])/d + (6*n^2*Log[c*(a + b*x)^n]*PolyLog[3, -((e*(a + b*x))/(b*d - a*e))])/d - (6*n^2*Log[c*(a + b*x)^n]*Poly
Log[3, 1 + (b*x)/a])/d - (6*n^3*PolyLog[4, -((e*(a + b*x))/(b*d - a*e))])/d + (6*n^3*PolyLog[4, 1 + (b*x)/a])/
d

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q), x] - Dist[b*n*(p/q), Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(
p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 2443

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

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

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 \frac {\log ^3\left (c (a+b x)^n\right )}{d x+e x^2} \, dx &=\int \frac {\log ^3\left (c (a+b x)^n\right )}{x (d+e x)} \, dx\\ &=\int \left (\frac {\log ^3\left (c (a+b x)^n\right )}{d x}-\frac {e \log ^3\left (c (a+b x)^n\right )}{d (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\log ^3\left (c (a+b x)^n\right )}{x} \, dx}{d}-\frac {e \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x} \, dx}{d}\\ &=\frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {(3 b n) \int \frac {\log \left (-\frac {b x}{a}\right ) \log ^2\left (c (a+b x)^n\right )}{a+b x} \, dx}{d}+\frac {(3 b n) \int \frac {\log ^2\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{a+b x} \, dx}{d}\\ &=\frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {(3 n) \text {Subst}\left (\int \frac {\log ^2\left (c x^n\right ) \log \left (-\frac {b \left (-\frac {a}{b}+\frac {x}{b}\right )}{a}\right )}{x} \, dx,x,a+b x\right )}{d}+\frac {(3 n) \text {Subst}\left (\int \frac {\log ^2\left (c x^n\right ) \log \left (\frac {b \left (\frac {b d-a e}{b}+\frac {e x}{b}\right )}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{d}\\ &=\frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (1+\frac {b x}{a}\right )}{d}-\frac {\left (6 n^2\right ) \text {Subst}\left (\int \frac {\log \left (c x^n\right ) \text {Li}_2\left (\frac {x}{a}\right )}{x} \, dx,x,a+b x\right )}{d}+\frac {\left (6 n^2\right ) \text {Subst}\left (\int \frac {\log \left (c x^n\right ) \text {Li}_2\left (-\frac {e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{d}\\ &=\frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (1+\frac {b x}{a}\right )}{d}+\frac {6 n^2 \log \left (c (a+b x)^n\right ) \text {Li}_3\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {6 n^2 \log \left (c (a+b x)^n\right ) \text {Li}_3\left (1+\frac {b x}{a}\right )}{d}+\frac {\left (6 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {x}{a}\right )}{x} \, dx,x,a+b x\right )}{d}-\frac {\left (6 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{d}\\ &=\frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (1+\frac {b x}{a}\right )}{d}+\frac {6 n^2 \log \left (c (a+b x)^n\right ) \text {Li}_3\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {6 n^2 \log \left (c (a+b x)^n\right ) \text {Li}_3\left (1+\frac {b x}{a}\right )}{d}-\frac {6 n^3 \text {Li}_4\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {6 n^3 \text {Li}_4\left (1+\frac {b x}{a}\right )}{d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(494\) vs. \(2(238)=476\).
time = 0.14, size = 494, normalized size = 2.08 \begin {gather*} \frac {-\log (x) \left (n \log (a+b x)-\log \left (c (a+b x)^n\right )\right )^3+\left (n \log (a+b x)-\log \left (c (a+b x)^n\right )\right )^3 \log (d+e x)+3 n \left (-n \log (a+b x)+\log \left (c (a+b x)^n\right )\right )^2 \left (\log (x) \left (\log (a+b x)-\log \left (1+\frac {b x}{a}\right )\right )-\log (a+b x) \log \left (\frac {b (d+e x)}{b d-a e}\right )-\text {Li}_2\left (-\frac {b x}{a}\right )-\text {Li}_2\left (\frac {e (a+b x)}{-b d+a e}\right )\right )-3 n^2 \left (n \log (a+b x)-\log \left (c (a+b x)^n\right )\right ) \left (\log \left (-\frac {b x}{a}\right ) \log ^2(a+b x)-\log ^2(a+b x) \log \left (\frac {b (d+e x)}{b d-a e}\right )-2 \log (a+b x) \text {Li}_2\left (\frac {e (a+b x)}{-b d+a e}\right )+2 \log (a+b x) \text {Li}_2\left (1+\frac {b x}{a}\right )+2 \text {Li}_3\left (\frac {e (a+b x)}{-b d+a e}\right )-2 \text {Li}_3\left (1+\frac {b x}{a}\right )\right )+n^3 \left (\log \left (-\frac {b x}{a}\right ) \log ^3(a+b x)-\log ^3(a+b x) \log \left (\frac {b (d+e x)}{b d-a e}\right )-3 \log ^2(a+b x) \text {Li}_2\left (\frac {e (a+b x)}{-b d+a e}\right )+3 \log ^2(a+b x) \text {Li}_2\left (1+\frac {b x}{a}\right )+6 \log (a+b x) \text {Li}_3\left (\frac {e (a+b x)}{-b d+a e}\right )-6 \log (a+b x) \text {Li}_3\left (1+\frac {b x}{a}\right )-6 \text {Li}_4\left (\frac {e (a+b x)}{-b d+a e}\right )+6 \text {Li}_4\left (1+\frac {b x}{a}\right )\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Log[x]*(n*Log[a + b*x] - Log[c*(a + b*x)^n])^3) + (n*Log[a + b*x] - Log[c*(a + b*x)^n])^3*Log[d + e*x] + 3*
n*(-(n*Log[a + b*x]) + Log[c*(a + b*x)^n])^2*(Log[x]*(Log[a + b*x] - Log[1 + (b*x)/a]) - Log[a + b*x]*Log[(b*(
d + e*x))/(b*d - a*e)] - PolyLog[2, -((b*x)/a)] - PolyLog[2, (e*(a + b*x))/(-(b*d) + a*e)]) - 3*n^2*(n*Log[a +
b*x] - Log[c*(a + b*x)^n])*(Log[-((b*x)/a)]*Log[a + b*x]^2 - Log[a + b*x]^2*Log[(b*(d + e*x))/(b*d - a*e)] -
2*Log[a + b*x]*PolyLog[2, (e*(a + b*x))/(-(b*d) + a*e)] + 2*Log[a + b*x]*PolyLog[2, 1 + (b*x)/a] + 2*PolyLog[3
, (e*(a + b*x))/(-(b*d) + a*e)] - 2*PolyLog[3, 1 + (b*x)/a]) + n^3*(Log[-((b*x)/a)]*Log[a + b*x]^3 - Log[a + b
*x]^3*Log[(b*(d + e*x))/(b*d - a*e)] - 3*Log[a + b*x]^2*PolyLog[2, (e*(a + b*x))/(-(b*d) + a*e)] + 3*Log[a + b
*x]^2*PolyLog[2, 1 + (b*x)/a] + 6*Log[a + b*x]*PolyLog[3, (e*(a + b*x))/(-(b*d) + a*e)] - 6*Log[a + b*x]*PolyL
og[3, 1 + (b*x)/a] - 6*PolyLog[4, (e*(a + b*x))/(-(b*d) + a*e)] + 6*PolyLog[4, 1 + (b*x)/a]))/d

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.00, size = 12205, normalized size = 51.28

method	result	size
risch	\(\text {Expression too large to display}\)	\(12205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x+a)^n)^3/(e*x^2+d*x),x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

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(log(c*(b*x+a)^n)^3/(e*x^2+d*x),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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(log(c*(b*x+a)^n)^3/(e*x^2+d*x),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c \left (a + b x\right )^{n} \right )}^{3}}{x \left (d + e x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(b*x+a)**n)**3/(e*x**2+d*x),x)

[Out]

Integral(log(c*(a + b*x)**n)**3/(x*(d + e*x)), x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^3}{e\,x^2+d\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________