3.85 \(\int \frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^2} \, dx\)

Optimal. Leaf size=25 \[ -\frac{1}{b d \log (F) \left (a+b F^{c+d x}\right )} \]

[Out]

-(1/(b*d*(a + b*F^(c + d*x))*Log[F]))

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Rubi [A]  time = 0.0571669, antiderivative size = 25, 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 \[ -\frac{1}{b d \log (F) \left (a+b F^{c+d x}\right )} \]

Antiderivative was successfully verified.

[In]  Int[F^(c + d*x)/(a + b*F^(c + d*x))^2,x]

[Out]

-(1/(b*d*(a + b*F^(c + d*x))*Log[F]))

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Rubi in Sympy [A]  time = 7.62534, size = 19, normalized size = 0.76 \[ - \frac{1}{b d \left (F^{c + d x} b + a\right ) \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(d*x+c)/(a+b*F**(d*x+c))**2,x)

[Out]

-1/(b*d*(F**(c + d*x)*b + a)*log(F))

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Mathematica [A]  time = 0.0199145, size = 25, normalized size = 1. \[ -\frac{1}{b d \log (F) \left (a+b F^{c+d x}\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(c + d*x)/(a + b*F^(c + d*x))^2,x]

[Out]

-(1/(b*d*(a + b*F^(c + d*x))*Log[F]))

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Maple [A]  time = 0.003, size = 26, normalized size = 1. \[ -{\frac{1}{bd \left ( a+b{F}^{dx+c} \right ) \ln \left ( F \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(d*x+c)/(a+b*F^(d*x+c))^2,x)

[Out]

-1/b/d/(a+b*F^(d*x+c))/ln(F)

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Maxima [A]  time = 0.776674, size = 34, normalized size = 1.36 \[ -\frac{1}{{\left (F^{d x + c} b + a\right )} b d \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(d*x + c)/(F^(d*x + c)*b + a)^2,x, algorithm="maxima")

[Out]

-1/((F^(d*x + c)*b + a)*b*d*log(F))

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Fricas [A]  time = 0.256049, size = 34, normalized size = 1.36 \[ -\frac{1}{F^{d x + c} b^{2} d \log \left (F\right ) + a b d \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(F^(d*x + c)*b^2*d*log(F) + a*b*d*log(F))

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Sympy [A]  time = 0.267698, size = 26, normalized size = 1.04 \[ - \frac{1}{F^{c + d x} b^{2} d \log{\left (F \right )} + a b d \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(d*x+c)/(a+b*F**(d*x+c))**2,x)

[Out]

-1/(F**(c + d*x)*b**2*d*log(F) + a*b*d*log(F))

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GIAC/XCAS [A]  time = 0.227116, size = 34, normalized size = 1.36 \[ -\frac{1}{{\left (F^{d x + c} b + a\right )} b d{\rm ln}\left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/((F^(d*x + c)*b + a)*b*d*ln(F))