3.539 \(\int \frac{x^2}{a+b e^{-x}+c e^x} \, dx\)

Optimal. Leaf size=244 \[ \frac{2 x \text{PolyLog}\left (2,-\frac{2 c e^x}{a-\sqrt{a^2-4 b c}}\right )}{\sqrt{a^2-4 b c}}-\frac{2 x \text{PolyLog}\left (2,-\frac{2 c e^x}{\sqrt{a^2-4 b c}+a}\right )}{\sqrt{a^2-4 b c}}-\frac{2 \text{PolyLog}\left (3,-\frac{2 c e^x}{a-\sqrt{a^2-4 b c}}\right )}{\sqrt{a^2-4 b c}}+\frac{2 \text{PolyLog}\left (3,-\frac{2 c e^x}{\sqrt{a^2-4 b c}+a}\right )}{\sqrt{a^2-4 b c}}+\frac{x^2 \log \left (\frac{2 c e^x}{a-\sqrt{a^2-4 b c}}+1\right )}{\sqrt{a^2-4 b c}}-\frac{x^2 \log \left (\frac{2 c e^x}{\sqrt{a^2-4 b c}+a}+1\right )}{\sqrt{a^2-4 b c}} \]

[Out]

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

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Rubi [A]  time = 0.797101, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{2 x \text{PolyLog}\left (2,-\frac{2 c e^x}{a-\sqrt{a^2-4 b c}}\right )}{\sqrt{a^2-4 b c}}-\frac{2 x \text{PolyLog}\left (2,-\frac{2 c e^x}{\sqrt{a^2-4 b c}+a}\right )}{\sqrt{a^2-4 b c}}-\frac{2 \text{PolyLog}\left (3,-\frac{2 c e^x}{a-\sqrt{a^2-4 b c}}\right )}{\sqrt{a^2-4 b c}}+\frac{2 \text{PolyLog}\left (3,-\frac{2 c e^x}{\sqrt{a^2-4 b c}+a}\right )}{\sqrt{a^2-4 b c}}+\frac{x^2 \log \left (\frac{2 c e^x}{a-\sqrt{a^2-4 b c}}+1\right )}{\sqrt{a^2-4 b c}}-\frac{x^2 \log \left (\frac{2 c e^x}{\sqrt{a^2-4 b c}+a}+1\right )}{\sqrt{a^2-4 b c}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b/E^x + c*E^x),x]

[Out]

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

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Rubi in Sympy [A]  time = 82.7921, size = 230, normalized size = 0.94 \[ \frac{x^{2} \log{\left (\frac{2 c e^{x}}{a - \sqrt{a^{2} - 4 b c}} + 1 \right )}}{\sqrt{a^{2} - 4 b c}} - \frac{x^{2} \log{\left (\frac{2 c e^{x}}{a + \sqrt{a^{2} - 4 b c}} + 1 \right )}}{\sqrt{a^{2} - 4 b c}} + \frac{2 x \operatorname{Li}_{2}\left (- \frac{2 c e^{x}}{a - \sqrt{a^{2} - 4 b c}}\right )}{\sqrt{a^{2} - 4 b c}} - \frac{2 x \operatorname{Li}_{2}\left (- \frac{2 c e^{x}}{a + \sqrt{a^{2} - 4 b c}}\right )}{\sqrt{a^{2} - 4 b c}} - \frac{2 \operatorname{Li}_{3}\left (- \frac{2 c e^{x}}{a - \sqrt{a^{2} - 4 b c}}\right )}{\sqrt{a^{2} - 4 b c}} + \frac{2 \operatorname{Li}_{3}\left (- \frac{2 c e^{x}}{a + \sqrt{a^{2} - 4 b c}}\right )}{\sqrt{a^{2} - 4 b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0755944, size = 185, normalized size = 0.76 \[ \frac{2 x \text{PolyLog}\left (2,\frac{2 c e^x}{\sqrt{a^2-4 b c}-a}\right )-2 x \text{PolyLog}\left (2,-\frac{2 c e^x}{\sqrt{a^2-4 b c}+a}\right )-2 \text{PolyLog}\left (3,\frac{2 c e^x}{\sqrt{a^2-4 b c}-a}\right )+2 \text{PolyLog}\left (3,-\frac{2 c e^x}{\sqrt{a^2-4 b c}+a}\right )+x^2 \log \left (\frac{2 c e^x}{a-\sqrt{a^2-4 b c}}+1\right )-x^2 \log \left (\frac{2 c e^x}{\sqrt{a^2-4 b c}+a}+1\right )}{\sqrt{a^2-4 b c}} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*Log[1 + (2*c*E^x)/(a - Sqrt[a^2 - 4*b*c])] - x^2*Log[1 + (2*c*E^x)/(a + Sqr
t[a^2 - 4*b*c])] + 2*x*PolyLog[2, (2*c*E^x)/(-a + Sqrt[a^2 - 4*b*c])] - 2*x*Poly
Log[2, (-2*c*E^x)/(a + Sqrt[a^2 - 4*b*c])] - 2*PolyLog[3, (2*c*E^x)/(-a + Sqrt[a
^2 - 4*b*c])] + 2*PolyLog[3, (-2*c*E^x)/(a + Sqrt[a^2 - 4*b*c])])/Sqrt[a^2 - 4*b
*c]

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \[ \int{{x}^{2} \left ( a+{\frac{b}{{{\rm e}^{x}}}}+c{{\rm e}^{x}} \right ) ^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b/exp(x)+c*exp(x)),x)

[Out]

int(x^2/(a+b/exp(x)+c*exp(x)),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266704, size = 500, normalized size = 2.05 \[ -\frac{b x^{2} \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} \log \left (\frac{2 \, c e^{x} + b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a}{b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a}\right ) - b x^{2} \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} \log \left (-\frac{2 \, c e^{x} - b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a}{b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} - a}\right ) + 2 \, b x \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}}{\rm Li}_2\left (-\frac{2 \, c e^{x} + b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a}{b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a} + 1\right ) - 2 \, b x \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}}{\rm Li}_2\left (\frac{2 \, c e^{x} - b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a}{b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} - a} + 1\right ) - 2 \, b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}}{\rm Li}_{3}(-\frac{2 \, c e^{x}}{b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} + a}) + 2 \, b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}}{\rm Li}_{3}(\frac{2 \, c e^{x}}{b \sqrt{\frac{a^{2} - 4 \, b c}{b^{2}}} - a})}{a^{2} - 4 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} e^{x}}{a e^{x} + b + c e^{2 x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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