3.45 \(\int \frac{f^x x^2}{a+b f^{2 x}} \, dx\)

Optimal. Leaf size=184 \[ \frac{i \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}-\frac{i \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}-\frac{i x \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^2(f)}+\frac{i x \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^2(f)}+\frac{x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \]

[Out]

(x^2*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Log[f]) - (I*x*PolyLog[2, (
(-I)*Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Log[f]^2) + (I*x*PolyLog[2, (I*Sqrt
[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Log[f]^2) + (I*PolyLog[3, ((-I)*Sqrt[b]*f^x)
/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Log[f]^3) - (I*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/
(Sqrt[a]*Sqrt[b]*Log[f]^3)

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Rubi [A]  time = 0.307652, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ \frac{i \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}-\frac{i \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}-\frac{i x \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^2(f)}+\frac{i x \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^2(f)}+\frac{x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[(f^x*x^2)/(a + b*f^(2*x)),x]

[Out]

(x^2*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Log[f]) - (I*x*PolyLog[2, (
(-I)*Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Log[f]^2) + (I*x*PolyLog[2, (I*Sqrt
[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Log[f]^2) + (I*PolyLog[3, ((-I)*Sqrt[b]*f^x)
/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Log[f]^3) - (I*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/
(Sqrt[a]*Sqrt[b]*Log[f]^3)

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Rubi in Sympy [A]  time = 87.8661, size = 173, normalized size = 0.94 \[ \frac{x^{2} \operatorname{atan}{\left (\frac{\sqrt{b} f^{x}}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{b} \log{\left (f \right )}} - \frac{i x \operatorname{Li}_{2}\left (- \frac{i \sqrt{b} f^{x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{2}} + \frac{i x \operatorname{Li}_{2}\left (\frac{i \sqrt{b} f^{x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{2}} + \frac{i \operatorname{Li}_{3}\left (- \frac{i \sqrt{b} f^{x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{3}} - \frac{i \operatorname{Li}_{3}\left (\frac{i \sqrt{b} f^{x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*atan(sqrt(b)*f**x/sqrt(a))/(sqrt(a)*sqrt(b)*log(f)) - I*x*polylog(2, -I*sqr
t(b)*f**x/sqrt(a))/(sqrt(a)*sqrt(b)*log(f)**2) + I*x*polylog(2, I*sqrt(b)*f**x/s
qrt(a))/(sqrt(a)*sqrt(b)*log(f)**2) + I*polylog(3, -I*sqrt(b)*f**x/sqrt(a))/(sqr
t(a)*sqrt(b)*log(f)**3) - I*polylog(3, I*sqrt(b)*f**x/sqrt(a))/(sqrt(a)*sqrt(b)*
log(f)**3)

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Mathematica [A]  time = 0.0446261, size = 168, normalized size = 0.91 \[ \frac{i \left (2 \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )-2 \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )-2 x \log (f) \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )+2 x \log (f) \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )+x^2 \log ^2(f) \log \left (1-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )-x^2 \log ^2(f) \log \left (1+\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )\right )}{2 \sqrt{a} \sqrt{b} \log ^3(f)} \]

Antiderivative was successfully verified.

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

[Out]

((I/2)*(x^2*Log[f]^2*Log[1 - (I*Sqrt[b]*f^x)/Sqrt[a]] - x^2*Log[f]^2*Log[1 + (I*
Sqrt[b]*f^x)/Sqrt[a]] - 2*x*Log[f]*PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]] + 2*x*
Log[f]*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]] + 2*PolyLog[3, ((-I)*Sqrt[b]*f^x)/Sqr
t[a]] - 2*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]]))/(Sqrt[a]*Sqrt[b]*Log[f]^3)

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int{\frac{{f}^{x}{x}^{2}}{a+b{f}^{2\,x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^x*x^2/(a+b*f^(2*x)),x)

[Out]

int(f^x*x^2/(a+b*f^(2*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(f^x*x^2/(b*f^(2*x) + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274217, size = 338, normalized size = 1.84 \[ \frac{x^{2} \sqrt{-\frac{b}{a}} \log \left (f\right )^{2} \log \left (\frac{b f^{x} + a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}}\right ) - x^{2} \sqrt{-\frac{b}{a}} \log \left (f\right )^{2} \log \left (-\frac{b f^{x} - a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}}\right ) + 2 \, x \sqrt{-\frac{b}{a}}{\rm Li}_2\left (-\frac{b f^{x} + a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}} + 1\right ) \log \left (f\right ) - 2 \, x \sqrt{-\frac{b}{a}}{\rm Li}_2\left (\frac{b f^{x} - a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}} + 1\right ) \log \left (f\right ) + 2 \, \sqrt{-\frac{b}{a}}{\rm Li}_{3}(\frac{b f^{x}}{a \sqrt{-\frac{b}{a}}}) - 2 \, \sqrt{-\frac{b}{a}}{\rm Li}_{3}(-\frac{b f^{x}}{a \sqrt{-\frac{b}{a}}})}{2 \, b \log \left (f\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(x^2*sqrt(-b/a)*log(f)^2*log((b*f^x + a*sqrt(-b/a))/(a*sqrt(-b/a))) - x^2*sq
rt(-b/a)*log(f)^2*log(-(b*f^x - a*sqrt(-b/a))/(a*sqrt(-b/a))) + 2*x*sqrt(-b/a)*d
ilog(-(b*f^x + a*sqrt(-b/a))/(a*sqrt(-b/a)) + 1)*log(f) - 2*x*sqrt(-b/a)*dilog((
b*f^x - a*sqrt(-b/a))/(a*sqrt(-b/a)) + 1)*log(f) + 2*sqrt(-b/a)*polylog(3, b*f^x
/(a*sqrt(-b/a))) - 2*sqrt(-b/a)*polylog(3, -b*f^x/(a*sqrt(-b/a))))/(b*log(f)^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(f**x*x**2/(a + b*f**(2*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{x} x^{2}}{b f^{2 \, x} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^x*x^2/(b*f^(2*x) + a), x)