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

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

[Out]

-(F^(a + b/(c + d*x))/(b*d*Log[F]))

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Rubi [A]  time = 0.0659469, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{F^{a+\frac{b}{c+d x}}}{b d \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

-(F^(a + b/(c + d*x))/(b*d*Log[F]))

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Rubi in Sympy [A]  time = 5.35887, size = 17, normalized size = 0.68 \[ - \frac{F^{a + \frac{b}{c + d x}}}{b d \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-F**(a + b/(c + d*x))/(b*d*log(F))

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Mathematica [A]  time = 0.0106558, size = 25, normalized size = 1. \[ -\frac{F^{a+\frac{b}{c+d x}}}{b d \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

-(F^(a + b/(c + d*x))/(b*d*Log[F]))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.775547, size = 34, normalized size = 1.36 \[ -\frac{F^{a + \frac{b}{d x + c}}}{b d \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-F^(a + b/(d*x + c))/(b*d*log(F))

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Fricas [A]  time = 0.260342, size = 42, normalized size = 1.68 \[ -\frac{F^{\frac{a d x + a c + b}{d x + c}}}{b d \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.712625, size = 34, normalized size = 1.36 \[ \begin{cases} - \frac{F^{a + \frac{b}{c + d x}}}{b d \log{\left (F \right )}} & \text{for}\: b d \log{\left (F \right )} \neq 0 \\- \frac{1}{c d + d^{2} x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-F**(a + b/(c + d*x))/(b*d*log(F)), Ne(b*d*log(F), 0)), (-1/(c*d + d*
*2*x), True))

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GIAC/XCAS [A]  time = 0.254895, size = 34, normalized size = 1.36 \[ -\frac{F^{a + \frac{b}{d x + c}}}{b d{\rm ln}\left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-F^(a + b/(d*x + c))/(b*d*ln(F))