3.1 \(\int \frac{x^3}{a+b e^{c+d x}} \, dx\)

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

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

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Rubi [A]  time = 0.307828, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{6 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a}\right )}{a d^4}+\frac{6 x \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a}\right )}{a d^3}-\frac{3 x^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a d^2}-\frac{x^3 \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a d}+\frac{x^4}{4 a} \]

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

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Rubi in Sympy [A]  time = 21.5877, size = 92, normalized size = 0.84 \[ - \frac{x^{3} \log{\left (\frac{a e^{- c - d x}}{b} + 1 \right )}}{a d} + \frac{3 x^{2} \operatorname{Li}_{2}\left (- \frac{a e^{- c - d x}}{b}\right )}{a d^{2}} + \frac{6 x \operatorname{Li}_{3}\left (- \frac{a e^{- c - d x}}{b}\right )}{a d^{3}} + \frac{6 \operatorname{Li}_{4}\left (- \frac{a e^{- c - d x}}{b}\right )}{a d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(a+b*exp(d*x+c)),x)

[Out]

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

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Mathematica [A]  time = 0.0195039, size = 110, normalized size = 1. \[ -\frac{6 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a}\right )}{a d^4}+\frac{6 x \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a}\right )}{a d^3}-\frac{3 x^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a d^2}-\frac{x^3 \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a d}+\frac{x^4}{4 a} \]

Antiderivative was successfully verified.

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

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Maple [A]  time = 0.02, size = 191, normalized size = 1.7 \[{\frac{{x}^{4}}{4\,a}}+{\frac{x{c}^{3}}{{d}^{3}a}}+{\frac{3\,{c}^{4}}{4\,{d}^{4}a}}-{\frac{{x}^{3}}{ad}\ln \left ( 1+{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }-{\frac{{c}^{3}}{{d}^{4}a}\ln \left ( 1+{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }-3\,{\frac{{x}^{2}}{a{d}^{2}}{\it polylog} \left ( 2,-{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }+6\,{\frac{x}{{d}^{3}a}{\it polylog} \left ( 3,-{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }-6\,{\frac{1}{{d}^{4}a}{\it polylog} \left ( 4,-{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }-{\frac{{c}^{3}\ln \left ({{\rm e}^{dx+c}} \right ) }{{d}^{4}a}}+{\frac{{c}^{3}\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{{d}^{4}a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b*exp(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 0.799311, size = 127, normalized size = 1.15 \[ \frac{x^{4}}{4 \, a} - \frac{d^{3} x^{3} \log \left (\frac{b e^{\left (d x + c\right )}}{a} + 1\right ) + 3 \, d^{2} x^{2}{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )}}{a}\right ) - 6 \, d x{\rm Li}_{3}(-\frac{b e^{\left (d x + c\right )}}{a}) + 6 \,{\rm Li}_{4}(-\frac{b e^{\left (d x + c\right )}}{a})}{a d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*e^(d*x + c) + a),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 = 0.249467, size = 162, normalized size = 1.47 \[ \frac{d^{4} x^{4} - 12 \, d^{2} x^{2}{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )} + a}{a} + 1\right ) + 4 \, c^{3} \log \left (b e^{\left (d x + c\right )} + a\right ) + 24 \, d x{\rm Li}_{3}(-\frac{b e^{\left (d x + c\right )}}{a}) - 4 \,{\left (d^{3} x^{3} + c^{3}\right )} \log \left (\frac{b e^{\left (d x + c\right )} + a}{a}\right ) - 24 \,{\rm Li}_{4}(-\frac{b e^{\left (d x + c\right )}}{a})}{4 \, a d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*e^(d*x + c) + a),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., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{a + b e^{c} e^{d x}}\, dx \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{b e^{\left (d x + c\right )} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*e^(d*x + c) + a),x, algorithm="giac")

[Out]

integrate(x^3/(b*e^(d*x + c) + a), x)