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

Optimal. Leaf size=23 \[ \frac{\log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]

[Out]

Log[a + b*F^(c + d*x)]/(b*d*Log[F])

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Rubi [A]  time = 0.0576683, antiderivative size = 23, 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 \[ \frac{\log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*F^(c + d*x)]/(b*d*Log[F])

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Rubi in Sympy [A]  time = 13.4553, size = 17, normalized size = 0.74 \[ \frac{\log{\left (F^{c + d x} b + a \right )}}{b d \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(F**(c + d*x)*b + a)/(b*d*log(F))

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Mathematica [A]  time = 0.00519428, size = 23, normalized size = 1. \[ \frac{\log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*F^(c + d*x)]/(b*d*Log[F])

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Maple [A]  time = 0.003, size = 24, normalized size = 1. \[{\frac{\ln \left ( a+b{F}^{dx+c} \right ) }{bd\ln \left ( F \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.789124, size = 31, normalized size = 1.35 \[ \frac{\log \left (F^{d x + c} b + a\right )}{b d \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(F^(d*x + c)*b + a)/(b*d*log(F))

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Fricas [A]  time = 0.250574, size = 31, normalized size = 1.35 \[ \frac{\log \left (F^{d x + c} b + a\right )}{b d \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(F^(d*x + c)*b + a)/(b*d*log(F))

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Sympy [A]  time = 0.285326, size = 17, normalized size = 0.74 \[ \frac{\log{\left (F^{c + d x} + \frac{a}{b} \right )}}{b d \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(F**(c + d*x) + a/b)/(b*d*log(F))

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GIAC/XCAS [A]  time = 0.251633, size = 32, normalized size = 1.39 \[ \frac{{\rm ln}\left ({\left | F^{d x + c} b + a \right |}\right )}{b d{\rm ln}\left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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