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

Optimal. Leaf size=84 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}+\frac{3 f^x}{8 a^2 \log (f) \left (a+b f^{2 x}\right )}+\frac{f^x}{4 a \log (f) \left (a+b f^{2 x}\right )^2} \]

[Out]

f^x/(4*a*(a + b*f^(2*x))^2*Log[f]) + (3*f^x)/(8*a^2*(a + b*f^(2*x))*Log[f]) + (3
*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*Log[f])

_______________________________________________________________________________________

Rubi [A]  time = 0.0984949, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}+\frac{3 f^x}{8 a^2 \log (f) \left (a+b f^{2 x}\right )}+\frac{f^x}{4 a \log (f) \left (a+b f^{2 x}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[f^x/(a + b*f^(2*x))^3,x]

[Out]

f^x/(4*a*(a + b*f^(2*x))^2*Log[f]) + (3*f^x)/(8*a^2*(a + b*f^(2*x))*Log[f]) + (3
*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*Log[f])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.3973, size = 73, normalized size = 0.87 \[ \frac{f^{x}}{4 a \left (a + b f^{2 x}\right )^{2} \log{\left (f \right )}} + \frac{3 f^{x}}{8 a^{2} \left (a + b f^{2 x}\right ) \log{\left (f \right )}} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt{b} f^{x}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} \sqrt{b} \log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f**x/(4*a*(a + b*f**(2*x))**2*log(f)) + 3*f**x/(8*a**2*(a + b*f**(2*x))*log(f))
+ 3*atan(sqrt(b)*f**x/sqrt(a))/(8*a**(5/2)*sqrt(b)*log(f))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0851824, size = 68, normalized size = 0.81 \[ \frac{\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{\sqrt{a} f^x \left (5 a+3 b f^{2 x}\right )}{\left (a+b f^{2 x}\right )^2}}{8 a^{5/2} \log (f)} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a]*f^x*(5*a + 3*b*f^(2*x)))/(a + b*f^(2*x))^2 + (3*ArcTan[(Sqrt[b]*f^x)/S
qrt[a]])/Sqrt[b])/(8*a^(5/2)*Log[f])

_______________________________________________________________________________________

Maple [A]  time = 0.061, size = 94, normalized size = 1.1 \[{\frac{ \left ( 3\,b \left ({f}^{x} \right ) ^{2}+5\,a \right ){f}^{x}}{8\,\ln \left ( f \right ){a}^{2} \left ( a+b \left ({f}^{x} \right ) ^{2} \right ) ^{2}}}-{\frac{3}{16\,\ln \left ( f \right ){a}^{2}}\ln \left ({f}^{x}-{a{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{3}{16\,\ln \left ( f \right ){a}^{2}}\ln \left ({f}^{x}+{a{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^x/(a+b*f^(2*x))^3,x)

[Out]

1/8*f^x*(3*b*(f^x)^2+5*a)/ln(f)/a^2/(a+b*(f^x)^2)^2-3/16/(-a*b)^(1/2)/a^2/ln(f)*
ln(f^x-1/(-a*b)^(1/2)*a)+3/16/(-a*b)^(1/2)/a^2/ln(f)*ln(f^x+1/(-a*b)^(1/2)*a)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.271917, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, \sqrt{-a b} b f^{3 \, x} + 10 \, \sqrt{-a b} a f^{x} + 3 \,{\left (b^{2} f^{4 \, x} + 2 \, a b f^{2 \, x} + a^{2}\right )} \log \left (\frac{2 \, a b f^{x} + \sqrt{-a b} b f^{2 \, x} - \sqrt{-a b} a}{b f^{2 \, x} + a}\right )}{16 \,{\left (\sqrt{-a b} a^{2} b^{2} f^{4 \, x} \log \left (f\right ) + 2 \, \sqrt{-a b} a^{3} b f^{2 \, x} \log \left (f\right ) + \sqrt{-a b} a^{4} \log \left (f\right )\right )}}, \frac{3 \, \sqrt{a b} b f^{3 \, x} + 5 \, \sqrt{a b} a f^{x} - 3 \,{\left (b^{2} f^{4 \, x} + 2 \, a b f^{2 \, x} + a^{2}\right )} \arctan \left (\frac{a}{\sqrt{a b} f^{x}}\right )}{8 \,{\left (\sqrt{a b} a^{2} b^{2} f^{4 \, x} \log \left (f\right ) + 2 \, \sqrt{a b} a^{3} b f^{2 \, x} \log \left (f\right ) + \sqrt{a b} a^{4} \log \left (f\right )\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(6*sqrt(-a*b)*b*f^(3*x) + 10*sqrt(-a*b)*a*f^x + 3*(b^2*f^(4*x) + 2*a*b*f^(
2*x) + a^2)*log((2*a*b*f^x + sqrt(-a*b)*b*f^(2*x) - sqrt(-a*b)*a)/(b*f^(2*x) + a
)))/(sqrt(-a*b)*a^2*b^2*f^(4*x)*log(f) + 2*sqrt(-a*b)*a^3*b*f^(2*x)*log(f) + sqr
t(-a*b)*a^4*log(f)), 1/8*(3*sqrt(a*b)*b*f^(3*x) + 5*sqrt(a*b)*a*f^x - 3*(b^2*f^(
4*x) + 2*a*b*f^(2*x) + a^2)*arctan(a/(sqrt(a*b)*f^x)))/(sqrt(a*b)*a^2*b^2*f^(4*x
)*log(f) + 2*sqrt(a*b)*a^3*b*f^(2*x)*log(f) + sqrt(a*b)*a^4*log(f))]

_______________________________________________________________________________________

Sympy [A]  time = 0.689219, size = 85, normalized size = 1.01 \[ \frac{5 a f^{x} + 3 b f^{3 x}}{8 a^{4} \log{\left (f \right )} + 16 a^{3} b f^{2 x} \log{\left (f \right )} + 8 a^{2} b^{2} f^{4 x} \log{\left (f \right )}} + \frac{\operatorname{RootSum}{\left (256 z^{2} a^{5} b + 9, \left ( i \mapsto i \log{\left (\frac{16 i a^{3}}{3} + f^{x} \right )} \right )\right )}}{\log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(5*a*f**x + 3*b*f**(3*x))/(8*a**4*log(f) + 16*a**3*b*f**(2*x)*log(f) + 8*a**2*b*
*2*f**(4*x)*log(f)) + RootSum(256*_z**2*a**5*b + 9, Lambda(_i, _i*log(16*_i*a**3
/3 + f**x)))/log(f)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.234863, size = 82, normalized size = 0.98 \[ \frac{3 \, \arctan \left (\frac{b f^{x}}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2}{\rm ln}\left (f\right )} + \frac{3 \, b f^{3 \, x} + 5 \, a f^{x}}{8 \,{\left (b f^{2 \, x} + a\right )}^{2} a^{2}{\rm ln}\left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*arctan(b*f^x/sqrt(a*b))/(sqrt(a*b)*a^2*ln(f)) + 1/8*(3*b*f^(3*x) + 5*a*f^x)/
((b*f^(2*x) + a)^2*a^2*ln(f))