3.123 \(\int x^3 \log (d+e (f^{c (a+b x)})^n) \, dx\)

Optimal. Leaf size=193 \[ \frac{3 x^2 \text{PolyLog}\left (3,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{6 x \text{PolyLog}\left (4,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac{6 \text{PolyLog}\left (5,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^4 c^4 n^4 \log ^4(f)}-\frac{x^3 \text{PolyLog}\left (2,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{1}{4} x^4 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac{1}{4} x^4 \log \left (\frac{e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]

[Out]

(x^4*Log[d + e*(f^(c*(a + b*x)))^n])/4 - (x^4*Log[1 + (e*(f^(c*(a + b*x)))^n)/d])/4 - (x^3*PolyLog[2, -((e*(f^
(c*(a + b*x)))^n)/d)])/(b*c*n*Log[f]) + (3*x^2*PolyLog[3, -((e*(f^(c*(a + b*x)))^n)/d)])/(b^2*c^2*n^2*Log[f]^2
) - (6*x*PolyLog[4, -((e*(f^(c*(a + b*x)))^n)/d)])/(b^3*c^3*n^3*Log[f]^3) + (6*PolyLog[5, -((e*(f^(c*(a + b*x)
))^n)/d)])/(b^4*c^4*n^4*Log[f]^4)

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Rubi [A]  time = 0.129465, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2532, 2531, 6609, 2282, 6589} \[ \frac{3 x^2 \text{PolyLog}\left (3,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{6 x \text{PolyLog}\left (4,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac{6 \text{PolyLog}\left (5,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^4 c^4 n^4 \log ^4(f)}-\frac{x^3 \text{PolyLog}\left (2,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{1}{4} x^4 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac{1}{4} x^4 \log \left (\frac{e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*Log[d + e*(f^(c*(a + b*x)))^n])/4 - (x^4*Log[1 + (e*(f^(c*(a + b*x)))^n)/d])/4 - (x^3*PolyLog[2, -((e*(f^
(c*(a + b*x)))^n)/d)])/(b*c*n*Log[f]) + (3*x^2*PolyLog[3, -((e*(f^(c*(a + b*x)))^n)/d)])/(b^2*c^2*n^2*Log[f]^2
) - (6*x*PolyLog[4, -((e*(f^(c*(a + b*x)))^n)/d)])/(b^3*c^3*n^3*Log[f]^3) + (6*PolyLog[5, -((e*(f^(c*(a + b*x)
))^n)/d)])/(b^4*c^4*n^4*Log[f]^4)

Rule 2532

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

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6609

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

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

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 x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right ) \, dx &=\frac{1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac{1}{4} x^4 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )+\int x^3 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx\\ &=\frac{1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac{1}{4} x^4 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac{x^3 \text{Li}_2\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{3 \int x^2 \text{Li}_2\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx}{b c n \log (f)}\\ &=\frac{1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac{1}{4} x^4 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac{x^3 \text{Li}_2\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{3 x^2 \text{Li}_3\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{6 \int x \text{Li}_3\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx}{b^2 c^2 n^2 \log ^2(f)}\\ &=\frac{1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac{1}{4} x^4 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac{x^3 \text{Li}_2\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{3 x^2 \text{Li}_3\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{6 x \text{Li}_4\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac{6 \int \text{Li}_4\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx}{b^3 c^3 n^3 \log ^3(f)}\\ &=\frac{1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac{1}{4} x^4 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac{x^3 \text{Li}_2\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{3 x^2 \text{Li}_3\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{6 x \text{Li}_4\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac{6 \operatorname{Subst}\left (\int \frac{\text{Li}_4\left (-\frac{e x^n}{d}\right )}{x} \, dx,x,f^{c (a+b x)}\right )}{b^4 c^4 n^3 \log ^4(f)}\\ &=\frac{1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac{1}{4} x^4 \log \left (1+\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac{x^3 \text{Li}_2\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{3 x^2 \text{Li}_3\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{6 x \text{Li}_4\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac{6 \text{Li}_5\left (-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^4 c^4 n^4 \log ^4(f)}\\ \end{align*}

Mathematica [A]  time = 0.0087122, size = 193, normalized size = 1. \[ \frac{3 x^2 \text{PolyLog}\left (3,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{6 x \text{PolyLog}\left (4,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac{6 \text{PolyLog}\left (5,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^4 c^4 n^4 \log ^4(f)}-\frac{x^3 \text{PolyLog}\left (2,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac{1}{4} x^4 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac{1}{4} x^4 \log \left (\frac{e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*Log[d + e*(f^(c*(a + b*x)))^n])/4 - (x^4*Log[1 + (e*(f^(c*(a + b*x)))^n)/d])/4 - (x^3*PolyLog[2, -((e*(f^
(c*(a + b*x)))^n)/d)])/(b*c*n*Log[f]) + (3*x^2*PolyLog[3, -((e*(f^(c*(a + b*x)))^n)/d)])/(b^2*c^2*n^2*Log[f]^2
) - (6*x*PolyLog[4, -((e*(f^(c*(a + b*x)))^n)/d)])/(b^3*c^3*n^3*Log[f]^3) + (6*PolyLog[5, -((e*(f^(c*(a + b*x)
))^n)/d)])/(b^4*c^4*n^4*Log[f]^4)

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Maple [B]  time = 0.101, size = 1364, normalized size = 7.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*ln(d+e*(f^(c*(b*x+a)))^n)-3/2/c^2/b^2/ln(f)^2*ln(1+e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a)))
))/d)*x^2*ln(f^(c*(b*x+a)))^2+2/c^3/b^3/ln(f)^3*ln(1+e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a)))))/d)*
x*ln(f^(c*(b*x+a)))^3-1/c/b/ln(f)*ln((d+e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a))))))/d)*x^3*ln(f^(c*
(b*x+a)))+3/c^2/b^2/ln(f)^2*ln((d+e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a))))))/d)*x^2*ln(f^(c*(b*x+a
)))^2-3/c^3/b^3/ln(f)^3*ln((d+e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a))))))/d)*x*ln(f^(c*(b*x+a)))^3-
3/c^2/b^2/ln(f)^2/n*polylog(2,-e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a)))))/d)*ln(f^(c*(b*x+a)))*x^2+
3/c^3/b^3/ln(f)^3/n*polylog(2,-e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a)))))/d)*ln(f^(c*(b*x+a)))^2*x+
3/c^2/b^2/ln(f)^2/n*dilog((d+e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a))))))/d)*ln(f^(c*(b*x+a)))*x^2-3
/c^3/b^3/ln(f)^3/n*dilog((d+e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a))))))/d)*ln(f^(c*(b*x+a)))^2*x+1/
c/b/ln(f)*ln(d+e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a))))))*ln(f^(c*(b*x+a)))*x^3-3/2/c^2/b^2/ln(f)^
2*ln(d+e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a))))))*ln(f^(c*(b*x+a)))^2*x^2+1/c^3/b^3/ln(f)^3*ln(d+e
*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a))))))*ln(f^(c*(b*x+a)))^3*x+3/c^2/b^2/ln(f)^2/n^2*polylog(3,-e
*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a)))))/d)*x^2-6/c^3/b^3/ln(f)^3/n^3*polylog(4,-e*f^(b*c*n*x)*exp
(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a)))))/d)*x-1/c^4/b^4/ln(f)^4/n*polylog(2,-e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(
f^(c*(b*x+a)))))/d)*ln(f^(c*(b*x+a)))^3-1/c/b/ln(f)/n*dilog((d+e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+
a))))))/d)*x^3+1/c^4/b^4/ln(f)^4/n*dilog((d+e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a))))))/d)*ln(f^(c*
(b*x+a)))^3+1/c^4/b^4/ln(f)^4*ln((d+e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a))))))/d)*ln(f^(c*(b*x+a))
)^4-1/4*ln(d+e*f^(b*c*n*x)*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a))))))*x^4-1/4/c^4/b^4/ln(f)^4*ln(d+e*f^(b*c*n*x)
*exp(-n*(ln(f)*b*c*x-ln(f^(c*(b*x+a))))))*ln(f^(c*(b*x+a)))^4-3/4/c^4/b^4/ln(f)^4*ln(1+e*f^(b*c*n*x)*exp(-n*(l
n(f)*b*c*x-ln(f^(c*(b*x+a)))))/d)*ln(f^(c*(b*x+a)))^4+6/c^4/b^4/ln(f)^4/n^4*polylog(5,-e*f^(b*c*n*x)*exp(-n*(l
n(f)*b*c*x-ln(f^(c*(b*x+a)))))/d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{20} \, b c n x^{5} \log \left (f\right ) + b c d n \int \frac{x^{4}}{4 \,{\left (e{\left (f^{b c x}\right )}^{n}{\left (f^{a c}\right )}^{n} + d\right )}}\,{d x} \log \left (f\right ) + \frac{1}{4} \, x^{4} \log \left (e{\left (f^{b c x}\right )}^{n}{\left (f^{a c}\right )}^{n} + d\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*b*c*n*x^5*log(f) + b*c*d*n*integrate(1/4*x^4/(e*(f^(b*c*x))^n*(f^(a*c))^n + d), x)*log(f) + 1/4*x^4*log(
e*(f^(b*c*x))^n*(f^(a*c))^n + d)

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Fricas [C]  time = 2.21145, size = 564, normalized size = 2.92 \begin{align*} -\frac{4 \, b^{3} c^{3} n^{3} x^{3}{\rm Li}_2\left (-\frac{e f^{b c n x + a c n} + d}{d} + 1\right ) \log \left (f\right )^{3} - 12 \, b^{2} c^{2} n^{2} x^{2} \log \left (f\right )^{2}{\rm polylog}\left (3, -\frac{e f^{b c n x + a c n}}{d}\right ) -{\left (b^{4} c^{4} n^{4} x^{4} - a^{4} c^{4} n^{4}\right )} \log \left (e f^{b c n x + a c n} + d\right ) \log \left (f\right )^{4} +{\left (b^{4} c^{4} n^{4} x^{4} - a^{4} c^{4} n^{4}\right )} \log \left (f\right )^{4} \log \left (\frac{e f^{b c n x + a c n} + d}{d}\right ) + 24 \, b c n x \log \left (f\right ){\rm polylog}\left (4, -\frac{e f^{b c n x + a c n}}{d}\right ) - 24 \,{\rm polylog}\left (5, -\frac{e f^{b c n x + a c n}}{d}\right )}{4 \, b^{4} c^{4} n^{4} \log \left (f\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*b^3*c^3*n^3*x^3*dilog(-(e*f^(b*c*n*x + a*c*n) + d)/d + 1)*log(f)^3 - 12*b^2*c^2*n^2*x^2*log(f)^2*polyl
og(3, -e*f^(b*c*n*x + a*c*n)/d) - (b^4*c^4*n^4*x^4 - a^4*c^4*n^4)*log(e*f^(b*c*n*x + a*c*n) + d)*log(f)^4 + (b
^4*c^4*n^4*x^4 - a^4*c^4*n^4)*log(f)^4*log((e*f^(b*c*n*x + a*c*n) + d)/d) + 24*b*c*n*x*log(f)*polylog(4, -e*f^
(b*c*n*x + a*c*n)/d) - 24*polylog(5, -e*f^(b*c*n*x + a*c*n)/d))/(b^4*c^4*n^4*log(f)^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{b c e n e^{a c n \log{\left (f \right )}} \log{\left (f \right )} \int \frac{x^{4} e^{b c n x \log{\left (f \right )}}}{d + e e^{a c n \log{\left (f \right )}} e^{b c n x \log{\left (f \right )}}}\, dx}{4} + \frac{x^{4} \log{\left (d + e \left (f^{c \left (a + b x\right )}\right )^{n} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*c*e*n*exp(a*c*n*log(f))*log(f)*Integral(x**4*exp(b*c*n*x*log(f))/(d + e*exp(a*c*n*log(f))*exp(b*c*n*x*log(f
))), x)/4 + x**4*log(d + e*(f**(c*(a + b*x)))**n)/4

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \log \left (e{\left (f^{{\left (b x + a\right )} c}\right )}^{n} + d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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