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

Optimal. Leaf size=159 \[ -\frac{\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a^3 d^2}+\frac{3 \log \left (a+b e^{c+d x}\right )}{2 a^3 d^2}-\frac{x \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a^3 d}-\frac{3 x}{2 a^3 d}+\frac{x^2}{2 a^3}-\frac{1}{2 a^2 d^2 \left (a+b e^{c+d x}\right )}+\frac{x}{a^2 d \left (a+b e^{c+d x}\right )}+\frac{x}{2 a d \left (a+b e^{c+d x}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.534349, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.733 \[ -\frac{\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a^3 d^2}+\frac{3 \log \left (a+b e^{c+d x}\right )}{2 a^3 d^2}-\frac{x \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a^3 d}-\frac{3 x}{2 a^3 d}+\frac{x^2}{2 a^3}-\frac{1}{2 a^2 d^2 \left (a+b e^{c+d x}\right )}+\frac{x}{a^2 d \left (a+b e^{c+d x}\right )}+\frac{x}{2 a d \left (a+b e^{c+d x}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*E^(c + d*x))^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x e^{- c - d x} e^{c + d x}}{2 a d \left (a + b e^{c + d x}\right )^{2}} + \frac{x e^{- c - d x} e^{c + d x}}{a^{2} d \left (a + b e^{c + d x}\right )} - \frac{1}{2 a^{2} d^{2} \left (a + b e^{c + d x}\right )} + \frac{\int x\, dx}{a^{3}} - \frac{x e^{- c - d x} e^{c + d x} \log{\left (a + b e^{c + d x} \right )}}{a^{3} d} + \frac{x e^{- c - d x} e^{c + d x} \log{\left (e^{c + d x} \right )}}{a^{3} d} - \frac{x \log{\left (1 + \frac{b e^{c + d x}}{a} \right )}}{a^{3} d} + \frac{x \log{\left (a + b e^{c + d x} \right )}}{a^{3} d} - \frac{x \log{\left (e^{c + d x} \right )}}{a^{3} d} + \frac{3 \log{\left (a + b e^{c + d x} \right )}}{2 a^{3} d^{2}} - \frac{3 \log{\left (e^{c + d x} \right )}}{2 a^{3} d^{2}} - \frac{\operatorname{Li}_{2}\left (- \frac{b e^{c + d x}}{a}\right )}{a^{3} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*exp(-c - d*x)*exp(c + d*x)/(2*a*d*(a + b*exp(c + d*x))**2) + x*exp(-c - d*x)*e
xp(c + d*x)/(a**2*d*(a + b*exp(c + d*x))) - 1/(2*a**2*d**2*(a + b*exp(c + d*x)))
+ Integral(x, x)/a**3 - x*exp(-c - d*x)*exp(c + d*x)*log(a + b*exp(c + d*x))/(a
**3*d) + x*exp(-c - d*x)*exp(c + d*x)*log(exp(c + d*x))/(a**3*d) - x*log(1 + b*e
xp(c + d*x)/a)/(a**3*d) + x*log(a + b*exp(c + d*x))/(a**3*d) - x*log(exp(c + d*x
))/(a**3*d) + 3*log(a + b*exp(c + d*x))/(2*a**3*d**2) - 3*log(exp(c + d*x))/(2*a
**3*d**2) - polylog(2, -b*exp(c + d*x)/a)/(a**3*d**2)

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Mathematica [A]  time = 0.182841, size = 120, normalized size = 0.75 \[ \frac{\frac{d x \left (d x-2 \log \left (\frac{b e^{c+d x}}{a}+1\right )\right )-2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a}+\frac{a d x}{\left (a+b e^{c+d x}\right )^2}+\frac{2 d x-1}{a+b e^{c+d x}}+\frac{3 \log \left (\frac{b e^{c+d x}}{a}+1\right )-3 d x}{a}}{2 a^2 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.036, size = 393, normalized size = 2.5 \[ -{\frac{1}{2\,{a}^{2}{d}^{2} \left ( a+b{{\rm e}^{dx+c}} \right ) }}+{\frac{3\,\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{2\,{a}^{3}{d}^{2}}}-{\frac{{b}^{2} \left ({{\rm e}^{dx+c}} \right ) ^{2}x}{2\,d{a}^{3} \left ( a+b{{\rm e}^{dx+c}} \right ) ^{2}}}-{\frac{{b}^{2} \left ({{\rm e}^{dx+c}} \right ) ^{2}c}{2\,{a}^{3}{d}^{2} \left ( a+b{{\rm e}^{dx+c}} \right ) ^{2}}}-{\frac{b{{\rm e}^{dx+c}}x}{d{a}^{2} \left ( a+b{{\rm e}^{dx+c}} \right ) ^{2}}}-{\frac{b{{\rm e}^{dx+c}}c}{{a}^{2}{d}^{2} \left ( a+b{{\rm e}^{dx+c}} \right ) ^{2}}}+{\frac{{x}^{2}}{2\,{a}^{3}}}+{\frac{xc}{d{a}^{3}}}+{\frac{{c}^{2}}{2\,{a}^{3}{d}^{2}}}-{\frac{b{{\rm e}^{dx+c}}x}{d{a}^{3} \left ( a+b{{\rm e}^{dx+c}} \right ) }}-{\frac{b{{\rm e}^{dx+c}}c}{{a}^{3}{d}^{2} \left ( a+b{{\rm e}^{dx+c}} \right ) }}-{\frac{1}{{a}^{3}{d}^{2}}{\it dilog} \left ({\frac{a+b{{\rm e}^{dx+c}}}{a}} \right ) }-{\frac{x}{d{a}^{3}}\ln \left ({\frac{a+b{{\rm e}^{dx+c}}}{a}} \right ) }-{\frac{c}{{a}^{3}{d}^{2}}\ln \left ({\frac{a+b{{\rm e}^{dx+c}}}{a}} \right ) }-{\frac{c\ln \left ({{\rm e}^{dx+c}} \right ) }{{a}^{3}{d}^{2}}}+{\frac{c\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{{a}^{3}{d}^{2}}}-{\frac{c}{{a}^{2}{d}^{2} \left ( a+b{{\rm e}^{dx+c}} \right ) }}-{\frac{c}{2\,{d}^{2}a \left ( a+b{{\rm e}^{dx+c}} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*exp(d*x+c))^3,x)

[Out]

-1/2/a^2/d^2/(a+b*exp(d*x+c))+3/2*ln(a+b*exp(d*x+c))/a^3/d^2-1/2/d/a^3*b^2*exp(d
*x+c)^2/(a+b*exp(d*x+c))^2*x-1/2/d^2/a^3*b^2*exp(d*x+c)^2/(a+b*exp(d*x+c))^2*c-1
/d/a^2*b*exp(d*x+c)/(a+b*exp(d*x+c))^2*x-1/d^2/a^2*b*exp(d*x+c)/(a+b*exp(d*x+c))
^2*c+1/2*x^2/a^3+1/d/a^3*x*c+1/2/d^2/a^3*c^2-1/d/a^3*b*exp(d*x+c)/(a+b*exp(d*x+c
))*x-1/d^2/a^3*b*exp(d*x+c)/(a+b*exp(d*x+c))*c-1/d^2/a^3*dilog((a+b*exp(d*x+c))/
a)-1/d/a^3*ln((a+b*exp(d*x+c))/a)*x-1/d^2/a^3*ln((a+b*exp(d*x+c))/a)*c-1/d^2*c/a
^3*ln(exp(d*x+c))+1/d^2*c/a^3*ln(a+b*exp(d*x+c))-1/d^2*c/a^2/(a+b*exp(d*x+c))-1/
2/d^2*c/a/(a+b*exp(d*x+c))^2

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Maxima [A]  time = 0.923559, size = 201, normalized size = 1.26 \[ \frac{3 \, a d x +{\left (2 \, b d x e^{c} - b e^{c}\right )} e^{\left (d x\right )} - a}{2 \,{\left (a^{2} b^{2} d^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} b d^{2} e^{\left (d x + c\right )} + a^{4} d^{2}\right )}} + \frac{x^{2}}{2 \, a^{3}} - \frac{3 \, x}{2 \, a^{3} d} - \frac{d x \log \left (\frac{b e^{\left (d x + c\right )}}{a} + 1\right ) +{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )}}{a}\right )}{a^{3} d^{2}} + \frac{3 \, \log \left (b e^{\left (d x + c\right )} + a\right )}{2 \, a^{3} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.255066, size = 456, normalized size = 2.87 \[ \frac{a^{2} d^{2} x^{2} - a^{2} c^{2} - 3 \, a^{2} c - a^{2} - 2 \,{\left (b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b e^{\left (d x + c\right )} + a^{2}\right )}{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )} + a}{a} + 1\right ) +{\left (b^{2} d^{2} x^{2} - b^{2} c^{2} - 3 \, b^{2} d x - 3 \, b^{2} c\right )} e^{\left (2 \, d x + 2 \, c\right )} +{\left (2 \, a b d^{2} x^{2} - 2 \, a b c^{2} - 4 \, a b d x - 6 \, a b c - a b\right )} e^{\left (d x + c\right )} +{\left (2 \, a^{2} c + 3 \, a^{2} +{\left (2 \, b^{2} c + 3 \, b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \,{\left (2 \, a b c + 3 \, a b\right )} e^{\left (d x + c\right )}\right )} \log \left (b e^{\left (d x + c\right )} + a\right ) - 2 \,{\left (a^{2} d x + a^{2} c +{\left (b^{2} d x + b^{2} c\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \,{\left (a b d x + a b c\right )} e^{\left (d x + c\right )}\right )} \log \left (\frac{b e^{\left (d x + c\right )} + a}{a}\right )}{2 \,{\left (a^{3} b^{2} d^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{4} b d^{2} e^{\left (d x + c\right )} + a^{5} d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(a^2*d^2*x^2 - a^2*c^2 - 3*a^2*c - a^2 - 2*(b^2*e^(2*d*x + 2*c) + 2*a*b*e^(d
*x + c) + a^2)*dilog(-(b*e^(d*x + c) + a)/a + 1) + (b^2*d^2*x^2 - b^2*c^2 - 3*b^
2*d*x - 3*b^2*c)*e^(2*d*x + 2*c) + (2*a*b*d^2*x^2 - 2*a*b*c^2 - 4*a*b*d*x - 6*a*
b*c - a*b)*e^(d*x + c) + (2*a^2*c + 3*a^2 + (2*b^2*c + 3*b^2)*e^(2*d*x + 2*c) +
2*(2*a*b*c + 3*a*b)*e^(d*x + c))*log(b*e^(d*x + c) + a) - 2*(a^2*d*x + a^2*c + (
b^2*d*x + b^2*c)*e^(2*d*x + 2*c) + 2*(a*b*d*x + a*b*c)*e^(d*x + c))*log((b*e^(d*
x + c) + a)/a))/(a^3*b^2*d^2*e^(2*d*x + 2*c) + 2*a^4*b*d^2*e^(d*x + c) + a^5*d^2
)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a d x - a + \left (2 b d x - b\right ) e^{c + d x}}{2 a^{4} d^{2} + 4 a^{3} b d^{2} e^{c + d x} + 2 a^{2} b^{2} d^{2} e^{2 c + 2 d x}} + \frac{\int \frac{2 d x}{a + b e^{c} e^{d x}}\, dx + \int \left (- \frac{3}{a + b e^{c} e^{d x}}\right )\, dx}{2 a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(3*a*d*x - a + (2*b*d*x - b)*exp(c + d*x))/(2*a**4*d**2 + 4*a**3*b*d**2*exp(c +
d*x) + 2*a**2*b**2*d**2*exp(2*c + 2*d*x)) + (Integral(2*d*x/(a + b*exp(c)*exp(d*
x)), x) + Integral(-3/(a + b*exp(c)*exp(d*x)), x))/(2*a**2*d)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*e^(d*x + c) + a)^3, x)