3.1.1 \(\int \frac {x^3}{a+b e^{c+d x}} \, dx\) [1]

Optimal. Leaf size=110 \[ \frac {x^4}{4 a}-\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a d}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a d^2}+\frac {6 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a}\right )}{a d^3}-\frac {6 \text {Li}_4\left (-\frac {b e^{c+d x}}{a}\right )}{a d^4} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2215, 2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {6 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a}\right )}{a d^4}+\frac {6 x \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{a d^3}-\frac {3 x^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{a d^2}-\frac {x^3 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{a d}+\frac {x^4}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*E^(c + d*x)),x]

[Out]

x^4/(4*a) - (x^3*Log[1 + (b*E^(c + d*x))/a])/(a*d) - (3*x^2*PolyLog[2, -((b*E^(c + d*x))/a)])/(a*d^2) + (6*x*P
olyLog[3, -((b*E^(c + d*x))/a)])/(a*d^3) - (6*PolyLog[4, -((b*E^(c + d*x))/a)])/(a*d^4)

Rule 2215

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

Rule 2221

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

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

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Mathematica [A]
time = 0.04, size = 112, normalized size = 1.02 \begin {gather*} -\frac {x^3 \log \left (1+\frac {a e^{-c-d x}}{b}\right )}{a d}+\frac {3 x^2 \text {Li}_2\left (-\frac {a e^{-c-d x}}{b}\right )}{a d^2}+\frac {6 x \text {Li}_3\left (-\frac {a e^{-c-d x}}{b}\right )}{a d^3}+\frac {6 \text {Li}_4\left (-\frac {a e^{-c-d x}}{b}\right )}{a d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*E^(c + d*x)),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(294\) vs. \(2(104)=208\).
time = 0.04, size = 295, normalized size = 2.68

method	result	size
risch	\(\frac {x^{4}}{4 a}+\frac {c^{3} x}{d^{3} a}+\frac {3 c^{4}}{4 d^{4} a}-\frac {x^{3} \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a d}-\frac {\ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right ) c^{3}}{d^{4} a}-\frac {3 x^{2} \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a \,d^{2}}+\frac {6 x \polylog \left (3, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a \,d^{3}}-\frac {6 \polylog \left (4, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a \,d^{4}}-\frac {c^{3} \ln \left ({\mathrm e}^{d x +c}\right )}{d^{4} a}+\frac {c^{3} \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{d^{4} a}\)	\(191\)
derivativedivides	\(\frac {\frac {\left (d x +c \right )^{4}}{4 a}-\frac {\left (d x +c \right )^{3} \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}-\frac {3 \left (d x +c \right )^{2} \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}+\frac {6 \left (d x +c \right ) \polylog \left (3, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}-\frac {6 \polylog \left (4, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}-\frac {c^{3} \ln \left ({\mathrm e}^{d x +c}\right )}{a}+\frac {c^{3} \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a}-\frac {c \left (d x +c \right )^{3}}{a}+\frac {3 c \left (d x +c \right )^{2} \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}+\frac {6 c \left (d x +c \right ) \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}-\frac {6 c \polylog \left (3, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}+\frac {3 c^{2} \left (d x +c \right )^{2}}{2 a}-\frac {3 c^{2} \left (d x +c \right ) \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}-\frac {3 c^{2} \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}}{d^{4}}\)	\(295\)
default	\(\frac {\frac {\left (d x +c \right )^{4}}{4 a}-\frac {\left (d x +c \right )^{3} \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}-\frac {3 \left (d x +c \right )^{2} \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}+\frac {6 \left (d x +c \right ) \polylog \left (3, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}-\frac {6 \polylog \left (4, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}-\frac {c^{3} \ln \left ({\mathrm e}^{d x +c}\right )}{a}+\frac {c^{3} \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a}-\frac {c \left (d x +c \right )^{3}}{a}+\frac {3 c \left (d x +c \right )^{2} \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}+\frac {6 c \left (d x +c \right ) \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}-\frac {6 c \polylog \left (3, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}+\frac {3 c^{2} \left (d x +c \right )^{2}}{2 a}-\frac {3 c^{2} \left (d x +c \right ) \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}-\frac {3 c^{2} \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a}}{d^{4}}\)	\(295\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+b*exp(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.32, size = 94, normalized size = 0.85 \begin {gather*} \frac {x^{4}}{4 \, a} - \frac {d^{3} x^{3} \log \left (\frac {b e^{\left (d x + c\right )}}{a} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )}}{a}\right ) - 6 \, d x {\rm Li}_{3}(-\frac {b e^{\left (d x + c\right )}}{a}) + 6 \, {\rm Li}_{4}(-\frac {b e^{\left (d x + c\right )}}{a})}{a d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*exp(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 120, normalized size = 1.09 \begin {gather*} \frac {d^{4} x^{4} - 12 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )} + a}{a} + 1\right ) + 4 \, c^{3} \log \left (b e^{\left (d x + c\right )} + a\right ) + 24 \, d x {\rm polylog}\left (3, -\frac {b e^{\left (d x + c\right )}}{a}\right ) - 4 \, {\left (d^{3} x^{3} + c^{3}\right )} \log \left (\frac {b e^{\left (d x + c\right )} + a}{a}\right ) - 24 \, {\rm polylog}\left (4, -\frac {b e^{\left (d x + c\right )}}{a}\right )}{4 \, a d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*exp(d*x+c)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a+b*exp(d*x+c)),x)

[Out]

Integral(x**3/(a + b*exp(c)*exp(d*x)), x)

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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/(a+b*exp(d*x+c)),x, algorithm="giac")

[Out]

integrate(x^3/(b*e^(d*x + c) + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3}{a+b\,{\mathrm {e}}^{c+d\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*exp(c + d*x)),x)

[Out]

int(x^3/(a + b*exp(c + d*x)), x)

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