Optimal. Leaf size=27 \[ -\frac{1}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]
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Rubi [A] time = 0.0578369, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{1}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]
Antiderivative was successfully verified.
[In] Int[F^(c + d*x)/(a + b*F^(c + d*x))^3,x]
[Out]
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Rubi in Sympy [A] time = 8.52864, size = 22, normalized size = 0.81 \[ - \frac{1}{2 b d \left (F^{c + d x} b + a\right )^{2} \log{\left (F \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(d*x+c)/(a+b*F**(d*x+c))**3,x)
[Out]
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Mathematica [A] time = 0.0171485, size = 27, normalized size = 1. \[ -\frac{1}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[F^(c + d*x)/(a + b*F^(c + d*x))^3,x]
[Out]
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Maple [A] time = 0.003, size = 26, normalized size = 1. \[ -{\frac{1}{2\,bd \left ( a+b{F}^{dx+c} \right ) ^{2}\ln \left ( F \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(d*x+c)/(a+b*F^(d*x+c))^3,x)
[Out]
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Maxima [A] time = 0.771134, size = 34, normalized size = 1.26 \[ -\frac{1}{2 \,{\left (F^{d x + c} b + a\right )}^{2} b d \log \left (F\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^(d*x + c)/(F^(d*x + c)*b + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.255177, size = 62, normalized size = 2.3 \[ -\frac{1}{2 \,{\left (2 \, F^{d x + c} a b^{2} d \log \left (F\right ) + F^{2 \, d x + 2 \, c} b^{3} d \log \left (F\right ) + a^{2} b d \log \left (F\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^(d*x + c)/(F^(d*x + c)*b + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.299489, size = 53, normalized size = 1.96 \[ - \frac{1}{4 F^{c + d x} a b^{2} d \log{\left (F \right )} + 2 F^{2 c + 2 d x} b^{3} d \log{\left (F \right )} + 2 a^{2} b d \log{\left (F \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F**(d*x+c)/(a+b*F**(d*x+c))**3,x)
[Out]
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GIAC/XCAS [A] time = 0.239818, size = 34, normalized size = 1.26 \[ -\frac{1}{2 \,{\left (F^{d x + c} b + a\right )}^{2} b d{\rm ln}\left (F\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^(d*x + c)/(F^(d*x + c)*b + a)^3,x, algorithm="giac")
[Out]