3.82 \(\int \frac{F^{c+d x} x^3}{\left (a+b F^{c+d x}\right )^2} \, dx\)

Optimal. Leaf size=140 \[ \frac{6 \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )}{a b d^4 \log ^4(F)}-\frac{6 x \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a b d^3 \log ^3(F)}-\frac{3 x^2 \log \left (\frac{b F^{c+d x}}{a}+1\right )}{a b d^2 \log ^2(F)}-\frac{x^3}{b d \log (F) \left (a+b F^{c+d x}\right )}+\frac{x^3}{a b d \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.396219, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{6 \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )}{a b d^4 \log ^4(F)}-\frac{6 x \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a b d^3 \log ^3(F)}-\frac{3 x^2 \log \left (\frac{b F^{c+d x}}{a}+1\right )}{a b d^2 \log ^2(F)}-\frac{x^3}{b d \log (F) \left (a+b F^{c+d x}\right )}+\frac{x^3}{a b d \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[(F^(c + d*x)*x^3)/(a + b*F^(c + d*x))^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 39.0274, size = 110, normalized size = 0.79 \[ - \frac{x^{3}}{b d \left (F^{c + d x} b + a\right ) \log{\left (F \right )}} - \frac{3 x^{2} \log{\left (\frac{F^{- c - d x} a}{b} + 1 \right )}}{a b d^{2} \log{\left (F \right )}^{2}} + \frac{6 x \operatorname{Li}_{2}\left (- \frac{F^{- c - d x} a}{b}\right )}{a b d^{3} \log{\left (F \right )}^{3}} + \frac{6 \operatorname{Li}_{3}\left (- \frac{F^{- c - d x} a}{b}\right )}{a b d^{4} \log{\left (F \right )}^{4}} \]

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))**2,x)

[Out]

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

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Mathematica [A]  time = 0.168359, size = 137, normalized size = 0.98 \[ \frac{3 \left (\frac{2 \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )}{a d^3 \log ^3(F)}-\frac{2 x \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a d^2 \log ^2(F)}-\frac{x^2 \log \left (\frac{b F^{c+d x}}{a}+1\right )}{a d \log (F)}+\frac{x^3}{3 a}\right )}{b d \log (F)}-\frac{x^3}{b d \log (F) \left (a+b F^{c+d x}\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(F^(c + d*x)*x^3)/(a + b*F^(c + d*x))^2,x]

[Out]

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

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Maple [A]  time = 0.032, size = 267, normalized size = 1.9 \[ -{\frac{{x}^{3}}{bd \left ( a+b{F}^{dx+c} \right ) \ln \left ( F \right ) }}+{\frac{{x}^{3}}{\ln \left ( F \right ) abd}}-3\,{\frac{{c}^{2}x}{\ln \left ( F \right ){d}^{3}ba}}-2\,{\frac{{c}^{3}}{\ln \left ( F \right ){d}^{4}ba}}-3\,{\frac{{x}^{2}}{ab{d}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}\ln \left ( 1+{\frac{b{F}^{dx+c}}{a}} \right ) }+3\,{\frac{{c}^{2}}{ \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{4}ba}\ln \left ( 1+{\frac{b{F}^{dx+c}}{a}} \right ) }-6\,{\frac{x}{ab{d}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}}{\it polylog} \left ( 2,-{\frac{b{F}^{dx+c}}{a}} \right ) }+6\,{\frac{1}{ab{d}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}}{\it polylog} \left ( 3,-{\frac{b{F}^{dx+c}}{a}} \right ) }+3\,{\frac{{c}^{2}\ln \left ({F}^{dx+c} \right ) }{ \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{4}ba}}-3\,{\frac{{c}^{2}\ln \left ( a+b{F}^{dx+c} \right ) }{ \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{4}ba}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(d*x+c)*x^3/(a+b*F^(d*x+c))^2,x)

[Out]

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

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Maxima [A]  time = 0.811206, size = 181, normalized size = 1.29 \[ -\frac{x^{3}}{F^{d x} F^{c} b^{2} d \log \left (F\right ) + a b d \log \left (F\right )} + \frac{\log \left (F^{d x}\right )^{3}}{a b d^{4} \log \left (F\right )^{4}} - \frac{3 \,{\left (\log \left (\frac{F^{d x} F^{c} b}{a} + 1\right ) \log \left (F^{d x}\right )^{2} + 2 \,{\rm Li}_2\left (-\frac{F^{d x} F^{c} b}{a}\right ) \log \left (F^{d x}\right ) - 2 \,{\rm Li}_{3}(-\frac{F^{d x} F^{c} b}{a})\right )}}{a b d^{4} \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(d*x + c)*x^3/(F^(d*x + c)*b + a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.240121, size = 332, normalized size = 2.37 \[ \frac{a c^{3} \log \left (F\right )^{3} +{\left (b d^{3} x^{3} + b c^{3}\right )} F^{d x + c} \log \left (F\right )^{3} - 6 \,{\left (F^{d x + c} b d x \log \left (F\right ) + a d x \log \left (F\right )\right )}{\rm Li}_2\left (-\frac{F^{d x + c} b + a}{a} + 1\right ) - 3 \,{\left (F^{d x + c} b c^{2} \log \left (F\right )^{2} + a c^{2} \log \left (F\right )^{2}\right )} \log \left (F^{d x + c} b + a\right ) - 3 \,{\left ({\left (b d^{2} x^{2} - b c^{2}\right )} F^{d x + c} \log \left (F\right )^{2} +{\left (a d^{2} x^{2} - a c^{2}\right )} \log \left (F\right )^{2}\right )} \log \left (\frac{F^{d x + c} b + a}{a}\right ) + 6 \,{\left (F^{d x + c} b + a\right )}{\rm Li}_{3}(-\frac{F^{d x + c} b}{a})}{F^{d x + c} a b^{2} d^{4} \log \left (F\right )^{4} + a^{2} b d^{4} \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(d*x + c)*x^3/(F^(d*x + c)*b + a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{x^{3}}{F^{c + d x} b^{2} d \log{\left (F \right )} + a b d \log{\left (F \right )}} + \frac{3 \int \frac{x^{2}}{a + b e^{c \log{\left (F \right )}} e^{d x \log{\left (F \right )}}}\, dx}{b d \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(d*x+c)*x**3/(a+b*F**(d*x+c))**2,x)

[Out]

-x**3/(F**(c + d*x)*b**2*d*log(F) + a*b*d*log(F)) + 3*Integral(x**2/(a + b*exp(c
*log(F))*exp(d*x*log(F))), x)/(b*d*log(F))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{d x + c} x^{3}}{{\left (F^{d x + c} b + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(d*x + c)*x^3/(F^(d*x + c)*b + a)^2,x, algorithm="giac")

[Out]

integrate(F^(d*x + c)*x^3/(F^(d*x + c)*b + a)^2, x)