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

Optimal. Leaf size=54 \[ \frac{\text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}+\frac{x \log \left (\frac{b F^{c+d x}}{a}+1\right )}{b d \log (F)} \]

[Out]

(x*Log[1 + (b*F^(c + d*x))/a])/(b*d*Log[F]) + PolyLog[2, -((b*F^(c + d*x))/a)]/(
b*d^2*Log[F]^2)

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Rubi [A]  time = 0.105195, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}+\frac{x \log \left (\frac{b F^{c+d x}}{a}+1\right )}{b d \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

(x*Log[1 + (b*F^(c + d*x))/a])/(b*d*Log[F]) + PolyLog[2, -((b*F^(c + d*x))/a)]/(
b*d^2*Log[F]^2)

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Rubi in Sympy [A]  time = 18.6001, size = 44, normalized size = 0.81 \[ \frac{x \log{\left (\frac{F^{c + d x} b}{a} + 1 \right )}}{b d \log{\left (F \right )}} + \frac{\operatorname{Li}_{2}\left (- \frac{F^{c + d x} b}{a}\right )}{b d^{2} \log{\left (F \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0228365, size = 47, normalized size = 0.87 \[ \frac{\text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )+d x \log (F) \log \left (\frac{b F^{c+d x}}{a}+1\right )}{b d^2 \log ^2(F)} \]

Antiderivative was successfully verified.

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

[Out]

(d*x*Log[F]*Log[1 + (b*F^(c + d*x))/a] + PolyLog[2, -((b*F^(c + d*x))/a)])/(b*d^
2*Log[F]^2)

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Maple [B]  time = 0.023, size = 148, normalized size = 2.7 \[ -{\frac{cx}{bd}}-{\frac{{c}^{2}}{2\,b{d}^{2}}}+{\frac{x}{bd\ln \left ( F \right ) }\ln \left ( 1+{\frac{b{F}^{dx+c}}{a}} \right ) }+{\frac{c}{{d}^{2}\ln \left ( F \right ) b}\ln \left ( 1+{\frac{b{F}^{dx+c}}{a}} \right ) }+{\frac{1}{b{d}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}{\it polylog} \left ( 2,-{\frac{b{F}^{dx+c}}{a}} \right ) }+{\frac{c\ln \left ({F}^{dx+c} \right ) }{{d}^{2}\ln \left ( F \right ) b}}-{\frac{c\ln \left ( a+b{F}^{dx+c} \right ) }{{d}^{2}\ln \left ( F \right ) b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.797333, size = 107, normalized size = 1.98 \[ \frac{x^{2}}{2 \, b} - \frac{\log \left (F^{d x}\right )^{2}}{2 \, b d^{2} \log \left (F\right )^{2}} + \frac{\log \left (\frac{F^{d x} F^{c} b}{a} + 1\right ) \log \left (F^{d x}\right ) +{\rm Li}_2\left (-\frac{F^{d x} F^{c} b}{a}\right )}{b d^{2} \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.27164, size = 101, normalized size = 1.87 \[ -\frac{c \log \left (F^{d x + c} b + a\right ) \log \left (F\right ) -{\left (d x + c\right )} \log \left (F\right ) \log \left (\frac{F^{d x + c} b + a}{a}\right ) -{\rm Li}_2\left (-\frac{F^{d x + c} b + a}{a} + 1\right )}{b d^{2} \log \left (F\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{c + d x} x}{F^{c} F^{d x} b + a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(F**(c + d*x)*x/(F**c*F**(d*x)*b + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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