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

Optimal. Leaf size=69 \[ -\frac{\log \left (a+b F^{c+d x}\right )}{a b d^2 \log ^2(F)}-\frac{x}{b d \log (F) \left (a+b F^{c+d x}\right )}+\frac{x}{a b d \log (F)} \]

[Out]

x/(a*b*d*Log[F]) - x/(b*d*(a + b*F^(c + d*x))*Log[F]) - Log[a + b*F^(c + d*x)]/(
a*b*d^2*Log[F]^2)

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Rubi [A]  time = 0.127374, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\log \left (a+b F^{c+d x}\right )}{a b d^2 \log ^2(F)}-\frac{x}{b d \log (F) \left (a+b F^{c+d x}\right )}+\frac{x}{a b d \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

x/(a*b*d*Log[F]) - x/(b*d*(a + b*F^(c + d*x))*Log[F]) - Log[a + b*F^(c + d*x)]/(
a*b*d^2*Log[F]^2)

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Rubi in Sympy [A]  time = 22.4891, size = 92, normalized size = 1.33 \[ \frac{F^{- c - d x} F^{c + d x} \log{\left (F^{c + d x} \right )}}{a b d^{2} \log{\left (F \right )}^{2}} - \frac{F^{- c - d x} F^{c + d x} \log{\left (F^{c + d x} b + a \right )}}{a b d^{2} \log{\left (F \right )}^{2}} - \frac{x}{b d \left (F^{c + d x} b + a\right ) \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0787814, size = 54, normalized size = 0.78 \[ \frac{\frac{d x \log (F) F^{c+d x}}{a+b F^{c+d x}}-\frac{\log \left (a+b F^{c+d x}\right )}{b}}{a d^2 \log ^2(F)} \]

Antiderivative was successfully verified.

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

[Out]

((d*F^(c + d*x)*x*Log[F])/(a + b*F^(c + d*x)) - Log[a + b*F^(c + d*x)]/b)/(a*d^2
*Log[F]^2)

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Maple [A]  time = 0.019, size = 67, normalized size = 1. \[{\frac{x{{\rm e}^{ \left ( dx+c \right ) \ln \left ( F \right ) }}}{\ln \left ( F \right ) ad \left ( a+b{{\rm e}^{ \left ( dx+c \right ) \ln \left ( F \right ) }} \right ) }}-{\frac{\ln \left ( a+b{{\rm e}^{ \left ( dx+c \right ) \ln \left ( F \right ) }} \right ) }{ \left ( \ln \left ( F \right ) \right ) ^{2}ab{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.851064, size = 97, normalized size = 1.41 \[ \frac{F^{d x} F^{c} x}{F^{d x} F^{c} a b d \log \left (F\right ) + a^{2} d \log \left (F\right )} - \frac{\log \left (\frac{F^{d x} F^{c} b + a}{F^{c} b}\right )}{a b d^{2} \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.246253, size = 100, normalized size = 1.45 \[ \frac{F^{d x + c} b d x \log \left (F\right ) -{\left (F^{d x + c} b + a\right )} \log \left (F^{d x + c} b + a\right )}{F^{d x + c} a b^{2} d^{2} \log \left (F\right )^{2} + a^{2} b d^{2} \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.402991, size = 58, normalized size = 0.84 \[ - \frac{x}{F^{c + d x} b^{2} d \log{\left (F \right )} + a b d \log{\left (F \right )}} + \frac{x}{a b d \log{\left (F \right )}} - \frac{\log{\left (F^{c + d x} + \frac{a}{b} \right )}}{a b d^{2} \log{\left (F \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{d x + c} x}{{\left (F^{d x + c} b + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(F^(d*x + c)*x/(F^(d*x + c)*b + a)^2, x)