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

Optimal. Leaf size=243 \[ \frac{3 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a^3 d^3}+\frac{2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a}\right )}{a^3 d^3}-\frac{2 x \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a^3 d^2}-\frac{\log \left (a+b e^{c+d x}\right )}{a^3 d^3}+\frac{3 x \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a^3 d^2}-\frac{x^2 \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a^3 d}+\frac{x}{a^3 d^2}-\frac{3 x^2}{2 a^3 d}+\frac{x^3}{3 a^3}-\frac{x}{a^2 d^2 \left (a+b e^{c+d x}\right )}+\frac{x^2}{a^2 d \left (a+b e^{c+d x}\right )}+\frac{x^2}{2 a d \left (a+b e^{c+d x}\right )^2} \]

[Out]

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

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Rubi [A]  time = 1.19326, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 12, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.706 \[ \frac{3 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a^3 d^3}+\frac{2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a}\right )}{a^3 d^3}-\frac{2 x \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a^3 d^2}-\frac{\log \left (a+b e^{c+d x}\right )}{a^3 d^3}+\frac{3 x \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a^3 d^2}-\frac{x^2 \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a^3 d}+\frac{x}{a^3 d^2}-\frac{3 x^2}{2 a^3 d}+\frac{x^3}{3 a^3}-\frac{x}{a^2 d^2 \left (a+b e^{c+d x}\right )}+\frac{x^2}{a^2 d \left (a+b e^{c+d x}\right )}+\frac{x^2}{2 a d \left (a+b e^{c+d x}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.229566, size = 203, normalized size = 0.84 \[ \frac{-\frac{6 (2 d x-3) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{d^3}+\frac{12 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a}\right )}{d^3}+\frac{3 a^2 x^2}{d \left (a+b e^{c+d x}\right )^2}-\frac{6 \log \left (\frac{b e^{c+d x}}{a}+1\right )}{d^3}-\frac{6 a x}{d^2 \left (a+b e^{c+d x}\right )}+\frac{18 x \log \left (\frac{b e^{c+d x}}{a}+1\right )}{d^2}+\frac{6 a x^2}{a d+b d e^{c+d x}}-\frac{6 x^2 \log \left (\frac{b e^{c+d x}}{a}+1\right )}{d}+\frac{6 x}{d^2}-\frac{9 x^2}{d}+2 x^3}{6 a^3} \]

Antiderivative was successfully verified.

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

[Out]

((6*x)/d^2 - (6*a*x)/(d^2*(a + b*E^(c + d*x))) - (9*x^2)/d + (3*a^2*x^2)/(d*(a +
b*E^(c + d*x))^2) + (6*a*x^2)/(a*d + b*d*E^(c + d*x)) + 2*x^3 - (6*Log[1 + (b*E
^(c + d*x))/a])/d^3 + (18*x*Log[1 + (b*E^(c + d*x))/a])/d^2 - (6*x^2*Log[1 + (b*
E^(c + d*x))/a])/d - (6*(-3 + 2*d*x)*PolyLog[2, -((b*E^(c + d*x))/a)])/d^3 + (12
*PolyLog[3, -((b*E^(c + d*x))/a)])/d^3)/(6*a^3)

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Maple [A]  time = 0.078, size = 385, normalized size = 1.6 \[{\frac{ \left ( 2\,xbd{{\rm e}^{dx+c}}+3\,xda-2\,b{{\rm e}^{dx+c}}-2\,a \right ) x}{2\,{d}^{2}{a}^{2} \left ( a+b{{\rm e}^{dx+c}} \right ) ^{2}}}+{\frac{\ln \left ({{\rm e}^{dx+c}} \right ) }{{a}^{3}{d}^{3}}}-{\frac{\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{{a}^{3}{d}^{3}}}+{\frac{{c}^{2}\ln \left ({{\rm e}^{dx+c}} \right ) }{{a}^{3}{d}^{3}}}-{\frac{{c}^{2}\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{{a}^{3}{d}^{3}}}+{\frac{{x}^{3}}{3\,{a}^{3}}}-{\frac{{c}^{2}x}{{a}^{3}{d}^{2}}}-{\frac{2\,{c}^{3}}{3\,{a}^{3}{d}^{3}}}-{\frac{{x}^{2}}{{a}^{3}d}\ln \left ( 1+{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }+{\frac{{c}^{2}}{{a}^{3}{d}^{3}}\ln \left ( 1+{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }-2\,{\frac{x}{{a}^{3}{d}^{2}}{\it polylog} \left ( 2,-{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }+2\,{\frac{1}{{a}^{3}{d}^{3}}{\it polylog} \left ( 3,-{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }+3\,{\frac{c\ln \left ({{\rm e}^{dx+c}} \right ) }{{a}^{3}{d}^{3}}}-3\,{\frac{c\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{{a}^{3}{d}^{3}}}-{\frac{3\,{x}^{2}}{2\,{a}^{3}d}}-3\,{\frac{xc}{{a}^{3}{d}^{2}}}-{\frac{3\,{c}^{2}}{2\,{a}^{3}{d}^{3}}}+3\,{\frac{x}{{a}^{3}{d}^{2}}\ln \left ( 1+{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }+3\,{\frac{c}{{a}^{3}{d}^{3}}\ln \left ( 1+{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }+3\,{\frac{1}{{a}^{3}{d}^{3}}{\it polylog} \left ( 2,-{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b*exp(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 0.981267, size = 316, normalized size = 1.3 \[ \frac{3 \, a d x^{2} - 2 \, a x + 2 \,{\left (b d x^{2} e^{c} - b x e^{c}\right )} e^{\left (d x\right )}}{2 \,{\left (a^{2} b^{2} d^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} b d^{2} e^{\left (d x + c\right )} + a^{4} d^{2}\right )}} + \frac{x}{a^{3} d^{2}} + \frac{2 \, d^{3} x^{3} - 9 \, d^{2} x^{2}}{6 \, a^{3} d^{3}} - \frac{d^{2} x^{2} \log \left (\frac{b e^{\left (d x + c\right )}}{a} + 1\right ) + 2 \, d x{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )}}{a}\right ) - 2 \,{\rm Li}_{3}(-\frac{b e^{\left (d x + c\right )}}{a})}{a^{3} d^{3}} + \frac{3 \,{\left (d x \log \left (\frac{b e^{\left (d x + c\right )}}{a} + 1\right ) +{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )}}{a}\right )\right )}}{a^{3} d^{3}} - \frac{\log \left (b e^{\left (d x + c\right )} + a\right )}{a^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(3*a*d*x^2 - 2*a*x + 2*(b*d*x^2*e^c - b*x*e^c)*e^(d*x))/(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) + x/(a^3*d^2) + 1/6*(2*d^3*x^3 - 9
*d^2*x^2)/(a^3*d^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^3*d^3) + 3*(d*x*log(b*e^(d*x + c)/a
+ 1) + dilog(-b*e^(d*x + c)/a))/(a^3*d^3) - log(b*e^(d*x + c) + a)/(a^3*d^3)

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Fricas [A]  time = 0.270182, size = 703, normalized size = 2.89 \[ \frac{2 \, a^{2} d^{3} x^{3} + 2 \, a^{2} c^{3} + 9 \, a^{2} c^{2} + 6 \, a^{2} c - 6 \,{\left (2 \, a^{2} d x - 3 \, a^{2} +{\left (2 \, b^{2} d x - 3 \, b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \,{\left (2 \, a b d x - 3 \, a b\right )} e^{\left (d x + c\right )}\right )}{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )} + a}{a} + 1\right ) +{\left (2 \, b^{2} d^{3} x^{3} - 9 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{3} + 9 \, b^{2} c^{2} + 6 \, b^{2} d x + 6 \, b^{2} c\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \,{\left (2 \, a b d^{3} x^{3} - 6 \, a b d^{2} x^{2} + 2 \, a b c^{3} + 9 \, a b c^{2} + 3 \, a b d x + 6 \, a b c\right )} e^{\left (d x + c\right )} - 6 \,{\left (a^{2} c^{2} + 3 \, a^{2} c + a^{2} +{\left (b^{2} c^{2} + 3 \, b^{2} c + b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \,{\left (a b c^{2} + 3 \, a b c + a b\right )} e^{\left (d x + c\right )}\right )} \log \left (b e^{\left (d x + c\right )} + a\right ) - 6 \,{\left (a^{2} d^{2} x^{2} - a^{2} c^{2} - 3 \, a^{2} d x - 3 \, a^{2} c +{\left (b^{2} d^{2} x^{2} - b^{2} c^{2} - 3 \, b^{2} d x - 3 \, b^{2} c\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \,{\left (a b d^{2} x^{2} - a b c^{2} - 3 \, a b d x - 3 \, a b c\right )} e^{\left (d x + c\right )}\right )} \log \left (\frac{b e^{\left (d x + c\right )} + a}{a}\right ) + 12 \,{\left (b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b e^{\left (d x + c\right )} + a^{2}\right )}{\rm Li}_{3}(-\frac{b e^{\left (d x + c\right )}}{a})}{6 \,{\left (a^{3} b^{2} d^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{4} b d^{3} e^{\left (d x + c\right )} + a^{5} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*a^2*d^3*x^3 + 2*a^2*c^3 + 9*a^2*c^2 + 6*a^2*c - 6*(2*a^2*d*x - 3*a^2 + (2
*b^2*d*x - 3*b^2)*e^(2*d*x + 2*c) + 2*(2*a*b*d*x - 3*a*b)*e^(d*x + c))*dilog(-(b
*e^(d*x + c) + a)/a + 1) + (2*b^2*d^3*x^3 - 9*b^2*d^2*x^2 + 2*b^2*c^3 + 9*b^2*c^
2 + 6*b^2*d*x + 6*b^2*c)*e^(2*d*x + 2*c) + 2*(2*a*b*d^3*x^3 - 6*a*b*d^2*x^2 + 2*
a*b*c^3 + 9*a*b*c^2 + 3*a*b*d*x + 6*a*b*c)*e^(d*x + c) - 6*(a^2*c^2 + 3*a^2*c +
a^2 + (b^2*c^2 + 3*b^2*c + b^2)*e^(2*d*x + 2*c) + 2*(a*b*c^2 + 3*a*b*c + a*b)*e^
(d*x + c))*log(b*e^(d*x + c) + a) - 6*(a^2*d^2*x^2 - a^2*c^2 - 3*a^2*d*x - 3*a^2
*c + (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 - a*b*c^2 - 3*a*b*d*x - 3*a*b*c)*e^(d*x + c))*log((b*e^(d*x + c) + a)/a) + 1
2*(b^2*e^(2*d*x + 2*c) + 2*a*b*e^(d*x + c) + a^2)*polylog(3, -b*e^(d*x + c)/a))/
(a^3*b^2*d^3*e^(2*d*x + 2*c) + 2*a^4*b*d^3*e^(d*x + c) + a^5*d^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a d x^{2} - 2 a x + \left (2 b d x^{2} - 2 b x\right ) e^{c + d x}}{2 a^{4} d^{2} + 4 a^{3} b d^{2} e^{c + d x} + 2 a^{2} b^{2} d^{2} e^{2 c + 2 d x}} + \frac{\int \left (- \frac{3 d x}{a + b e^{c} e^{d x}}\right )\, dx + \int \frac{d^{2} x^{2}}{a + b e^{c} e^{d x}}\, dx + \int \frac{1}{a + b e^{c} e^{d x}}\, dx}{a^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(3*a*d*x**2 - 2*a*x + (2*b*d*x**2 - 2*b*x)*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(-3*d*x/(a +
b*exp(c)*exp(d*x)), x) + Integral(d**2*x**2/(a + b*exp(c)*exp(d*x)), x) + Integr
al(1/(a + b*exp(c)*exp(d*x)), x))/(a**2*d**2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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