3.9 \(\int \frac{x^3}{\left (a+b e^{c+d x}\right )^2} \, dx\)

Optimal. Leaf size=217 \[ -\frac{6 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a}\right )}{a^2 d^4}-\frac{6 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a}\right )}{a^2 d^4}+\frac{6 x \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a^2 d^3}+\frac{6 x \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a}\right )}{a^2 d^3}-\frac{3 x^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a^2 d^2}+\frac{3 x^2 \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a^2 d^2}-\frac{x^3 \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a^2 d}-\frac{x^3}{a^2 d}+\frac{x^4}{4 a^2}+\frac{x^3}{a d \left (a+b e^{c+d x}\right )} \]

[Out]

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

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Rubi [A]  time = 0.789913, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ -\frac{6 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a}\right )}{a^2 d^4}-\frac{6 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a}\right )}{a^2 d^4}+\frac{6 x \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a^2 d^3}+\frac{6 x \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a}\right )}{a^2 d^3}-\frac{3 x^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a^2 d^2}+\frac{3 x^2 \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a^2 d^2}-\frac{x^3 \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a^2 d}-\frac{x^3}{a^2 d}+\frac{x^4}{4 a^2}+\frac{x^3}{a d \left (a+b e^{c+d x}\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*E^(c + d*x))^2,x]

[Out]

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

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: PolificationFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(a+b*exp(d*x+c))**2,x)

[Out]

Exception raised: PolificationFailed

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Mathematica [A]  time = 0.277544, size = 158, normalized size = 0.73 \[ \frac{\frac{24 (d x-1) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a}\right )}{d^4}-\frac{24 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a}\right )}{d^4}-\frac{12 x (d x-2) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{d^3}+\frac{12 x^2 \log \left (\frac{b e^{c+d x}}{a}+1\right )}{d^2}+\frac{4 a x^3}{a d+b d e^{c+d x}}-\frac{4 x^3 \log \left (\frac{b e^{c+d x}}{a}+1\right )}{d}-\frac{4 x^3}{d}+x^4}{4 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b*E^(c + d*x))^2,x]

[Out]

((-4*x^3)/d + (4*a*x^3)/(a*d + b*d*E^(c + d*x)) + x^4 + (12*x^2*Log[1 + (b*E^(c
+ d*x))/a])/d^2 - (4*x^3*Log[1 + (b*E^(c + d*x))/a])/d - (12*x*(-2 + d*x)*PolyLo
g[2, -((b*E^(c + d*x))/a)])/d^3 + (24*(-1 + d*x)*PolyLog[3, -((b*E^(c + d*x))/a)
])/d^4 - (24*PolyLog[4, -((b*E^(c + d*x))/a)])/d^4)/(4*a^2)

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Maple [A]  time = 0.072, size = 382, normalized size = 1.8 \[{\frac{{x}^{3}}{da \left ( a+b{{\rm e}^{dx+c}} \right ) }}+{\frac{3\,{c}^{4}}{4\,{a}^{2}{d}^{4}}}-6\,{\frac{1}{{a}^{2}{d}^{4}}{\it polylog} \left ( 4,-{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }-6\,{\frac{1}{{a}^{2}{d}^{4}}{\it polylog} \left ( 3,-{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }+2\,{\frac{{c}^{3}}{{a}^{2}{d}^{4}}}-3\,{\frac{{c}^{2}\ln \left ({{\rm e}^{dx+c}} \right ) }{{a}^{2}{d}^{4}}}+3\,{\frac{{c}^{2}\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{{a}^{2}{d}^{4}}}-{\frac{{c}^{3}\ln \left ({{\rm e}^{dx+c}} \right ) }{{a}^{2}{d}^{4}}}+{\frac{{c}^{3}\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{{a}^{2}{d}^{4}}}+{\frac{{x}^{4}}{4\,{a}^{2}}}-{\frac{{x}^{3}}{{a}^{2}d}}-{\frac{{c}^{3}}{{a}^{2}{d}^{4}}\ln \left ( 1+{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }-3\,{\frac{{c}^{2}}{{a}^{2}{d}^{4}}\ln \left ( 1+{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }-3\,{\frac{{x}^{2}}{{a}^{2}{d}^{2}}{\it polylog} \left ( 2,-{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }-{\frac{{x}^{3}}{{a}^{2}d}\ln \left ( 1+{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }+3\,{\frac{{x}^{2}}{{a}^{2}{d}^{2}}\ln \left ( 1+{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }+6\,{\frac{x}{{a}^{2}{d}^{3}}{\it polylog} \left ( 2,-{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }+6\,{\frac{x}{{a}^{2}{d}^{3}}{\it polylog} \left ( 3,-{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }+3\,{\frac{{c}^{2}x}{{a}^{2}{d}^{3}}}+{\frac{{c}^{3}x}{{a}^{2}{d}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b*exp(d*x+c))^2,x)

[Out]

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

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Maxima [A]  time = 0.797398, size = 263, normalized size = 1.21 \[ \frac{x^{3}}{a b d e^{\left (d x + c\right )} + a^{2} d} + \frac{d^{4} x^{4} - 4 \, d^{3} x^{3}}{4 \, a^{2} d^{4}} - \frac{d^{3} x^{3} \log \left (\frac{b e^{\left (d x + c\right )}}{a} + 1\right ) + 3 \, d^{2} x^{2}{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )}}{a}\right ) - 6 \, d x{\rm Li}_{3}(-\frac{b e^{\left (d x + c\right )}}{a}) + 6 \,{\rm Li}_{4}(-\frac{b e^{\left (d x + c\right )}}{a})}{a^{2} d^{4}} + \frac{3 \,{\left (d^{2} x^{2} \log \left (\frac{b e^{\left (d x + c\right )}}{a} + 1\right ) + 2 \, d x{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )}}{a}\right ) - 2 \,{\rm Li}_{3}(-\frac{b e^{\left (d x + c\right )}}{a})\right )}}{a^{2} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*e^(d*x + c) + a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.260912, size = 446, normalized size = 2.06 \[ \frac{a d^{4} x^{4} - a c^{4} - 4 \, a c^{3} - 12 \,{\left (a d^{2} x^{2} - 2 \, a d x +{\left (b d^{2} x^{2} - 2 \, b d x\right )} e^{\left (d x + c\right )}\right )}{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )} + a}{a} + 1\right ) +{\left (b d^{4} x^{4} - 4 \, b d^{3} x^{3} - b c^{4} - 4 \, b c^{3}\right )} e^{\left (d x + c\right )} + 4 \,{\left (a c^{3} + 3 \, a c^{2} +{\left (b c^{3} + 3 \, b c^{2}\right )} e^{\left (d x + c\right )}\right )} \log \left (b e^{\left (d x + c\right )} + a\right ) - 4 \,{\left (a d^{3} x^{3} - 3 \, a d^{2} x^{2} + a c^{3} + 3 \, a c^{2} +{\left (b d^{3} x^{3} - 3 \, b d^{2} x^{2} + b c^{3} + 3 \, b c^{2}\right )} e^{\left (d x + c\right )}\right )} \log \left (\frac{b e^{\left (d x + c\right )} + a}{a}\right ) - 24 \,{\left (b e^{\left (d x + c\right )} + a\right )}{\rm Li}_{4}(-\frac{b e^{\left (d x + c\right )}}{a}) + 24 \,{\left (a d x +{\left (b d x - b\right )} e^{\left (d x + c\right )} - a\right )}{\rm Li}_{3}(-\frac{b e^{\left (d x + c\right )}}{a})}{4 \,{\left (a^{2} b d^{4} e^{\left (d x + c\right )} + a^{3} d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*e^(d*x + c) + a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{3}}{a^{2} d + a b d e^{c + d x}} + \frac{\int \left (- \frac{3 x^{2}}{a + b e^{c} e^{d x}}\right )\, dx + \int \frac{d x^{3}}{a + b e^{c} e^{d x}}\, dx}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(a+b*exp(d*x+c))**2,x)

[Out]

x**3/(a**2*d + a*b*d*exp(c + d*x)) + (Integral(-3*x**2/(a + b*exp(c)*exp(d*x)),
x) + Integral(d*x**3/(a + b*exp(c)*exp(d*x)), x))/(a*d)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (b e^{\left (d x + c\right )} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*e^(d*x + c) + a)^2,x, algorithm="giac")

[Out]

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