3.50 \(\int \frac{f^x x^3}{\left (a+b f^{2 x}\right )^2} \, dx\)

Optimal. Leaf size=501 \[ -\frac{3 i x^2 \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{3 i x^2 \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}-\frac{3 i \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}+\frac{3 i \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}-\frac{3 i \text{PolyLog}\left (4,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}+\frac{3 i \text{PolyLog}\left (4,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}+\frac{3 i x \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i x \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}+\frac{3 i x \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i x \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}-\frac{3 x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{x^3 f^x}{2 a \log (f) \left (a+b f^{2 x}\right )} \]

[Out]

(-3*x^2*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]^2) + (f^x*x^3)/
(2*a*(a + b*f^(2*x))*Log[f]) + (x^3*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sq
rt[b]*Log[f]) + (((3*I)/2)*x*PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sq
rt[b]*Log[f]^3) - (((3*I)/4)*x^2*PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2
)*Sqrt[b]*Log[f]^2) - (((3*I)/2)*x*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)
*Sqrt[b]*Log[f]^3) + (((3*I)/4)*x^2*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2
)*Sqrt[b]*Log[f]^2) - (((3*I)/2)*PolyLog[3, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2
)*Sqrt[b]*Log[f]^4) + (((3*I)/2)*x*PolyLog[3, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3
/2)*Sqrt[b]*Log[f]^3) + (((3*I)/2)*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)
*Sqrt[b]*Log[f]^4) - (((3*I)/2)*x*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*
Sqrt[b]*Log[f]^3) - (((3*I)/2)*PolyLog[4, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*
Sqrt[b]*Log[f]^4) + (((3*I)/2)*PolyLog[4, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sqr
t[b]*Log[f]^4)

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Rubi [A]  time = 0.909373, antiderivative size = 501, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611 \[ -\frac{3 i x^2 \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{3 i x^2 \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b} \log ^2(f)}-\frac{3 i \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}+\frac{3 i \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}-\frac{3 i \text{PolyLog}\left (4,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}+\frac{3 i \text{PolyLog}\left (4,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^4(f)}+\frac{3 i x \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i x \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}+\frac{3 i x \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}-\frac{3 i x \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^3(f)}+\frac{x^3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log (f)}-\frac{3 x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} \log ^2(f)}+\frac{x^3 f^x}{2 a \log (f) \left (a+b f^{2 x}\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-3*x^2*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]^2) + (f^x*x^3)/
(2*a*(a + b*f^(2*x))*Log[f]) + (x^3*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sq
rt[b]*Log[f]) + (((3*I)/2)*x*PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sq
rt[b]*Log[f]^3) - (((3*I)/4)*x^2*PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2
)*Sqrt[b]*Log[f]^2) - (((3*I)/2)*x*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)
*Sqrt[b]*Log[f]^3) + (((3*I)/4)*x^2*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2
)*Sqrt[b]*Log[f]^2) - (((3*I)/2)*PolyLog[3, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2
)*Sqrt[b]*Log[f]^4) + (((3*I)/2)*x*PolyLog[3, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3
/2)*Sqrt[b]*Log[f]^3) + (((3*I)/2)*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)
*Sqrt[b]*Log[f]^4) - (((3*I)/2)*x*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*
Sqrt[b]*Log[f]^3) - (((3*I)/2)*PolyLog[4, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*
Sqrt[b]*Log[f]^4) + (((3*I)/2)*PolyLog[4, (I*Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sqr
t[b]*Log[f]^4)

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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(f**x*x**3/(a+b*f**(2*x))**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.419888, size = 434, normalized size = 0.87 \[ \frac{-\frac{6 i \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{6 i \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{6 i \text{PolyLog}\left (4,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{6 i \text{PolyLog}\left (4,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{3 i x \log (f) (x \log (f)-2) \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{3 i x \log (f) (x \log (f)-2) \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{6 i x \log (f) \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{6 i x \log (f) \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{i x^3 \log ^3(f) \log \left (1-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{i x^3 \log ^3(f) \log \left (1+\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{3 i x^2 \log ^2(f) \log \left (1-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{3 i x^2 \log ^2(f) \log \left (1+\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{2 \sqrt{a} x^3 f^x \log ^3(f)}{a+b f^{2 x}}}{4 a^{3/2} \log ^4(f)} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a]*f^x*x^3*Log[f]^3)/(a + b*f^(2*x)) - ((3*I)*x^2*Log[f]^2*Log[1 - (I*S
qrt[b]*f^x)/Sqrt[a]])/Sqrt[b] + (I*x^3*Log[f]^3*Log[1 - (I*Sqrt[b]*f^x)/Sqrt[a]]
)/Sqrt[b] + ((3*I)*x^2*Log[f]^2*Log[1 + (I*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] - (I*x
^3*Log[f]^3*Log[1 + (I*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] - ((3*I)*x*Log[f]*(-2 + x*
Log[f])*PolyLog[2, ((-I)*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] + ((3*I)*x*Log[f]*(-2 +
x*Log[f])*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] - ((6*I)*PolyLog[3, ((-I)
*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] + ((6*I)*x*Log[f]*PolyLog[3, ((-I)*Sqrt[b]*f^x)/
Sqrt[a]])/Sqrt[b] + ((6*I)*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] - ((6*I)
*x*Log[f]*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] - ((6*I)*PolyLog[4, ((-I)
*Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b] + ((6*I)*PolyLog[4, (I*Sqrt[b]*f^x)/Sqrt[a]])/Sq
rt[b])/(4*a^(3/2)*Log[f]^4)

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Maple [F]  time = 0.105, size = 0, normalized size = 0. \[ \int{\frac{{f}^{x}{x}^{3}}{ \left ( a+b{f}^{2\,x} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^x*x^3/(a+b*f^(2*x))^2,x)

[Out]

int(f^x*x^3/(a+b*f^(2*x))^2,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^3/(b*f^(2*x) + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.260864, size = 852, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(2*b*f^x*x^3*log(f)^3 + 3*((b*x^2*log(f)^2 - 2*b*x*log(f))*f^(2*x)*sqrt(-b/a
) + (a*x^2*log(f)^2 - 2*a*x*log(f))*sqrt(-b/a))*dilog(-(b*f^x + a*sqrt(-b/a))/(a
*sqrt(-b/a)) + 1) - 3*((b*x^2*log(f)^2 - 2*b*x*log(f))*f^(2*x)*sqrt(-b/a) + (a*x
^2*log(f)^2 - 2*a*x*log(f))*sqrt(-b/a))*dilog((b*f^x - a*sqrt(-b/a))/(a*sqrt(-b/
a)) + 1) + ((b*x^3*log(f)^3 - 3*b*x^2*log(f)^2)*f^(2*x)*sqrt(-b/a) + (a*x^3*log(
f)^3 - 3*a*x^2*log(f)^2)*sqrt(-b/a))*log((b*f^x + a*sqrt(-b/a))/(a*sqrt(-b/a)))
- ((b*x^3*log(f)^3 - 3*b*x^2*log(f)^2)*f^(2*x)*sqrt(-b/a) + (a*x^3*log(f)^3 - 3*
a*x^2*log(f)^2)*sqrt(-b/a))*log(-(b*f^x - a*sqrt(-b/a))/(a*sqrt(-b/a))) - 6*(b*f
^(2*x)*sqrt(-b/a) + a*sqrt(-b/a))*polylog(4, b*f^x/(a*sqrt(-b/a))) + 6*(b*f^(2*x
)*sqrt(-b/a) + a*sqrt(-b/a))*polylog(4, -b*f^x/(a*sqrt(-b/a))) + 6*((b*x*log(f)
- b)*f^(2*x)*sqrt(-b/a) + (a*x*log(f) - a)*sqrt(-b/a))*polylog(3, b*f^x/(a*sqrt(
-b/a))) - 6*((b*x*log(f) - b)*f^(2*x)*sqrt(-b/a) + (a*x*log(f) - a)*sqrt(-b/a))*
polylog(3, -b*f^x/(a*sqrt(-b/a))))/(a*b^2*f^(2*x)*log(f)^4 + a^2*b*log(f)^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**x*x**3/(a+b*f**(2*x))**2,x)

[Out]

f**x*x**3/(2*a**2*log(f) + 2*a*b*f**(2*x)*log(f)) + (Integral(-3*f**x*x**2/(a +
b*f**(2*x)), x) + Integral(f**x*x**3*log(f)/(a + b*f**(2*x)), x))/(2*a*log(f))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^x*x^3/(b*f^(2*x) + a)^2, x)