3.2.23 \(\int x^3 \log (d+e (f^{c (a+b x)})^n) \, dx\) [123]

Optimal. Leaf size=193 \[ \frac {1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{4} x^4 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^3 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {3 x^2 \text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {6 x \text {Li}_4\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac {6 \text {Li}_5\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^4 c^4 n^4 \log ^4(f)} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2612, 2611, 6744, 2320, 6724} \begin {gather*} \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 {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 {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 {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 ) \end {gather*}

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 2320

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 2611

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 2612

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/d)*(F^(c*(a +
b*x)))^n], x] - Simp[(f + g*x)^(m + 1)*(Log[1 + (e/d)*(F^(c*(a + b*x)))^n]/(g*(m + 1))), x]) /; FreeQ[{F, a,
b, c, d, e, f, g, n}, x] && GtQ[m, 0] && NeQ[d, 1]

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]

Rule 6744

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]

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 \text {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*}

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Mathematica [A]
time = 0.01, size = 193, normalized size = 1.00 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(1275\) vs. \(2(189)=378\).
time = 0.04, size = 1276, normalized size = 6.61

method	result	size
risch	\(\text {Expression too large to display}\)	\(1276\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 215, normalized size = 1.11 \begin {gather*} \frac {1}{4} \, x^{4} \log \left (f^{{\left (b x + a\right )} c n} e + d\right ) - \frac {b^{4} c^{4} n^{4} x^{4} \log \left (f\right )^{4} \log \left (\frac {f^{a c n} e^{\left (b c n x \log \left (f\right ) + 1\right )}}{d} + 1\right ) + 4 \, b^{3} c^{3} n^{3} x^{3} {\rm Li}_2\left (-\frac {f^{a c n} e^{\left (b c n x \log \left (f\right ) + 1\right )}}{d}\right ) \log \left (f\right )^{3} - 12 \, b^{2} c^{2} n^{2} x^{2} \log \left (f\right )^{2} {\rm Li}_{3}(-\frac {f^{a c n} e^{\left (b c n x \log \left (f\right ) + 1\right )}}{d}) + 24 \, b c n x \log \left (f\right ) {\rm Li}_{4}(-\frac {f^{a c n} e^{\left (b c n x \log \left (f\right ) + 1\right )}}{d}) - 24 \, {\rm Li}_{5}(-\frac {f^{a c n} e^{\left (b c n x \log \left (f\right ) + 1\right )}}{d})}{4 \, b^{4} c^{4} n^{4} \log \left (f\right )^{4}} \end {gather*}

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]

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

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Fricas [A]
time = 0.38, size = 251, normalized size = 1.30 \begin {gather*} -\frac {4 \, b^{3} c^{3} n^{3} x^{3} {\rm Li}_2\left (-\frac {f^{b c n x + a c n} e + 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 {f^{b c n x + a c n} e}{d}\right ) - {\left (b^{4} c^{4} n^{4} x^{4} - a^{4} c^{4} n^{4}\right )} \log \left (f^{b c n x + a c n} e + 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 {f^{b c n x + a c n} e + d}{d}\right ) + 24 \, b c n x \log \left (f\right ) {\rm polylog}\left (4, -\frac {f^{b c n x + a c n} e}{d}\right ) - 24 \, {\rm polylog}\left (5, -\frac {f^{b c n x + a c n} e}{d}\right )}{4 \, b^{4} c^{4} n^{4} \log \left (f\right )^{4}} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 {gather*}

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 = 0.00, 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(x^3*log(d+e*(f^(c*(b*x+a)))^n),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,\ln \left (d+e\,{\left (f^{c\,\left (a+b\,x\right )}\right )}^n\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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