Optimal. Leaf size=115 \[ \frac{6 \text{PolyLog}\left (4,-\frac{b F^{c+d x}}{a}\right )}{b d^4 \log ^4(F)}-\frac{6 x \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )}{b d^3 \log ^3(F)}+\frac{3 x^2 \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}+\frac{x^3 \log \left (\frac{b F^{c+d x}}{a}+1\right )}{b d \log (F)} \]
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Rubi [A] time = 0.216895, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{6 \text{PolyLog}\left (4,-\frac{b F^{c+d x}}{a}\right )}{b d^4 \log ^4(F)}-\frac{6 x \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )}{b d^3 \log ^3(F)}+\frac{3 x^2 \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}+\frac{x^3 \log \left (\frac{b F^{c+d x}}{a}+1\right )}{b d \log (F)} \]
Antiderivative was successfully verified.
[In] Int[(F^(c + d*x)*x^3)/(a + b*F^(c + d*x)),x]
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Rubi in Sympy [A] time = 32.533, size = 104, normalized size = 0.9 \[ \frac{x^{3} \log{\left (\frac{F^{c + d x} b}{a} + 1 \right )}}{b d \log{\left (F \right )}} + \frac{3 x^{2} \operatorname{Li}_{2}\left (- \frac{F^{c + d x} b}{a}\right )}{b d^{2} \log{\left (F \right )}^{2}} - \frac{6 x \operatorname{Li}_{3}\left (- \frac{F^{c + d x} b}{a}\right )}{b d^{3} \log{\left (F \right )}^{3}} + \frac{6 \operatorname{Li}_{4}\left (- \frac{F^{c + d x} b}{a}\right )}{b d^{4} \log{\left (F \right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(d*x+c)*x**3/(a+b*F**(d*x+c)),x)
[Out]
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Mathematica [A] time = 0.0513483, size = 103, normalized size = 0.9 \[ \frac{3 d^2 x^2 \log ^2(F) \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )+6 \text{PolyLog}\left (4,-\frac{b F^{c+d x}}{a}\right )-6 d x \log (F) \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )+d^3 x^3 \log ^3(F) \log \left (\frac{b F^{c+d x}}{a}+1\right )}{b d^4 \log ^4(F)} \]
Antiderivative was successfully verified.
[In] Integrate[(F^(c + d*x)*x^3)/(a + b*F^(c + d*x)),x]
[Out]
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Maple [A] time = 0.04, size = 217, normalized size = 1.9 \[ -{\frac{{c}^{3}x}{b{d}^{3}}}-{\frac{3\,{c}^{4}}{4\,b{d}^{4}}}+{\frac{{x}^{3}}{bd\ln \left ( F \right ) }\ln \left ( 1+{\frac{b{F}^{dx+c}}{a}} \right ) }+{\frac{{c}^{3}}{{d}^{4}\ln \left ( F \right ) b}\ln \left ( 1+{\frac{b{F}^{dx+c}}{a}} \right ) }+3\,{\frac{{x}^{2}}{b{d}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}{\it polylog} \left ( 2,-{\frac{b{F}^{dx+c}}{a}} \right ) }-6\,{\frac{x}{b{d}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}}{\it polylog} \left ( 3,-{\frac{b{F}^{dx+c}}{a}} \right ) }+6\,{\frac{1}{b{d}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}}{\it polylog} \left ( 4,-{\frac{b{F}^{dx+c}}{a}} \right ) }+{\frac{{c}^{3}\ln \left ({F}^{dx+c} \right ) }{{d}^{4}\ln \left ( F \right ) b}}-{\frac{{c}^{3}\ln \left ( a+b{F}^{dx+c} \right ) }{{d}^{4}\ln \left ( F \right ) b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(d*x+c)*x^3/(a+b*F^(d*x+c)),x)
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Maxima [A] time = 0.815698, size = 180, normalized size = 1.57 \[ \frac{x^{4}}{4 \, b} - \frac{\log \left (F^{d x}\right )^{4}}{4 \, b d^{4} \log \left (F\right )^{4}} + \frac{\log \left (\frac{F^{d x} F^{c} b}{a} + 1\right ) \log \left (F^{d x}\right )^{3} + 3 \,{\rm Li}_2\left (-\frac{F^{d x} F^{c} b}{a}\right ) \log \left (F^{d x}\right )^{2} - 6 \, \log \left (F^{d x}\right ){\rm Li}_{3}(-\frac{F^{d x} F^{c} b}{a}) + 6 \,{\rm Li}_{4}(-\frac{F^{d x} F^{c} b}{a})}{b d^{4} \log \left (F\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^(d*x + c)*x^3/(F^(d*x + c)*b + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.300729, size = 181, normalized size = 1.57 \[ \frac{3 \, d^{2} x^{2}{\rm Li}_2\left (-\frac{F^{d x + c} b + a}{a} + 1\right ) \log \left (F\right )^{2} - c^{3} \log \left (F^{d x + c} b + a\right ) \log \left (F\right )^{3} +{\left (d^{3} x^{3} + c^{3}\right )} \log \left (F\right )^{3} \log \left (\frac{F^{d x + c} b + a}{a}\right ) - 6 \, d x \log \left (F\right ){\rm Li}_{3}(-\frac{F^{d x + c} b}{a}) + 6 \,{\rm Li}_{4}(-\frac{F^{d x + c} b}{a})}{b d^{4} \log \left (F\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^(d*x + c)*x^3/(F^(d*x + c)*b + a),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{c + d x} x^{3}}{F^{c} F^{d x} b + a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F**(d*x+c)*x**3/(a+b*F**(d*x+c)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{d x + c} x^{3}}{F^{d x + c} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^(d*x + c)*x^3/(F^(d*x + c)*b + a),x, algorithm="giac")
[Out]