3.9 \(\int \frac{F^{d x}}{a+b F^{c+d x}} \, dx\)

Optimal. Leaf size=28 \[ \frac{F^{-c} \log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]

[Out]

Log[a + b*F^(c + d*x)]/(b*d*F^c*Log[F])

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Rubi [A]  time = 0.112887, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{F^{-c} \log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*F^(c + d*x)]/(b*d*F^c*Log[F])

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Rubi in Sympy [A]  time = 13.347, size = 20, normalized size = 0.71 \[ \frac{F^{- c} \log{\left (F^{c + d x} b + a \right )}}{b d \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00792758, size = 28, normalized size = 1. \[ \frac{F^{-c} \log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*F^(c + d*x)]/(b*d*F^c*Log[F])

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Maple [A]  time = 0.016, size = 33, normalized size = 1.2 \[{\frac{\ln \left ( a+b{{\rm e}^{c\ln \left ( F \right ) }}{{\rm e}^{d\ln \left ( F \right ) x}} \right ) }{{F}^{c}b\ln \left ( F \right ) d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(d*x)/(a+b*F^(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 0.965649, size = 38, normalized size = 1.36 \[ \frac{\log \left (F^{d x + c} b + a\right )}{F^{c} b d \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.282102, size = 38, normalized size = 1.36 \[ \frac{\log \left (F^{d x + c} b + a\right )}{F^{c} b d \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.33169, size = 24, normalized size = 0.86 \[ \frac{e^{- c \log{\left (F \right )}} \log{\left (F^{c + d x} + \frac{a}{b} \right )}}{b d \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.227087, size = 41, normalized size = 1.46 \[ \frac{{\rm ln}\left ({\left | F^{d x} F^{c} b + a \right |}\right )}{F^{c} b d{\rm ln}\left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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