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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]