3.76 \(\int \frac{F^{c+d x} x^3}{a+b F^{c+d x}} \, dx\)

Optimal. Leaf size=115 \[ \frac{6 \text{PolyLog}\left (4,-\frac{b F^{c+d x}}{a}\right )}{b d^4 \log ^4(F)}-\frac{6 x \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )}{b d^3 \log ^3(F)}+\frac{3 x^2 \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}+\frac{x^3 \log \left (\frac{b F^{c+d x}}{a}+1\right )}{b d \log (F)} \]

[Out]

(x^3*Log[1 + (b*F^(c + d*x))/a])/(b*d*Log[F]) + (3*x^2*PolyLog[2, -((b*F^(c + d*
x))/a)])/(b*d^2*Log[F]^2) - (6*x*PolyLog[3, -((b*F^(c + d*x))/a)])/(b*d^3*Log[F]
^3) + (6*PolyLog[4, -((b*F^(c + d*x))/a)])/(b*d^4*Log[F]^4)

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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]

[Out]

(x^3*Log[1 + (b*F^(c + d*x))/a])/(b*d*Log[F]) + (3*x^2*PolyLog[2, -((b*F^(c + d*
x))/a)])/(b*d^2*Log[F]^2) - (6*x*PolyLog[3, -((b*F^(c + d*x))/a)])/(b*d^3*Log[F]
^3) + (6*PolyLog[4, -((b*F^(c + d*x))/a)])/(b*d^4*Log[F]^4)

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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]

x**3*log(F**(c + d*x)*b/a + 1)/(b*d*log(F)) + 3*x**2*polylog(2, -F**(c + d*x)*b/
a)/(b*d**2*log(F)**2) - 6*x*polylog(3, -F**(c + d*x)*b/a)/(b*d**3*log(F)**3) + 6
*polylog(4, -F**(c + d*x)*b/a)/(b*d**4*log(F)**4)

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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]

(d^3*x^3*Log[F]^3*Log[1 + (b*F^(c + d*x))/a] + 3*d^2*x^2*Log[F]^2*PolyLog[2, -((
b*F^(c + d*x))/a)] - 6*d*x*Log[F]*PolyLog[3, -((b*F^(c + d*x))/a)] + 6*PolyLog[4
, -((b*F^(c + d*x))/a)])/(b*d^4*Log[F]^4)

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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)

[Out]

-1/d^3/b*c^3*x-3/4/d^4/b*c^4+x^3*ln(1+b*F^(d*x+c)/a)/b/d/ln(F)+1/d^4/ln(F)/b*ln(
1+b*F^(d*x+c)/a)*c^3+3*x^2*polylog(2,-b*F^(d*x+c)/a)/b/d^2/ln(F)^2-6*x*polylog(3
,-b*F^(d*x+c)/a)/b/d^3/ln(F)^3+6*polylog(4,-b*F^(d*x+c)/a)/b/d^4/ln(F)^4+1/d^4/l
n(F)/b*c^3*ln(F^(d*x+c))-1/d^4/ln(F)/b*c^3*ln(a+b*F^(d*x+c))

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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]

1/4*x^4/b - 1/4*log(F^(d*x))^4/(b*d^4*log(F)^4) + (log(F^(d*x)*F^c*b/a + 1)*log(
F^(d*x))^3 + 3*dilog(-F^(d*x)*F^c*b/a)*log(F^(d*x))^2 - 6*log(F^(d*x))*polylog(3
, -F^(d*x)*F^c*b/a) + 6*polylog(4, -F^(d*x)*F^c*b/a))/(b*d^4*log(F)^4)

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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")

[Out]

(3*d^2*x^2*dilog(-(F^(d*x + c)*b + a)/a + 1)*log(F)^2 - c^3*log(F^(d*x + c)*b +
a)*log(F)^3 + (d^3*x^3 + c^3)*log(F)^3*log((F^(d*x + c)*b + a)/a) - 6*d*x*log(F)
*polylog(3, -F^(d*x + c)*b/a) + 6*polylog(4, -F^(d*x + c)*b/a))/(b*d^4*log(F)^4)

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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]

Integral(F**(c + d*x)*x**3/(F**c*F**(d*x)*b + a), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{d x + c} x^{3}}{F^{d x + c} b + a}\,{d x} \]

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]

integrate(F^(d*x + c)*x^3/(F^(d*x + c)*b + a), x)