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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]