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

Optimal. Leaf size=107 \[ -\frac{2 \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a b d^3 \log ^3(F)}-\frac{2 x \log \left (\frac{b F^{c+d x}}{a}+1\right )}{a b d^2 \log ^2(F)}-\frac{x^2}{b d \log (F) \left (a+b F^{c+d x}\right )}+\frac{x^2}{a b d \log (F)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.286074, antiderivative size = 107, 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 \[ -\frac{2 \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a b d^3 \log ^3(F)}-\frac{2 x \log \left (\frac{b F^{c+d x}}{a}+1\right )}{a b d^2 \log ^2(F)}-\frac{x^2}{b d \log (F) \left (a+b F^{c+d x}\right )}+\frac{x^2}{a b d \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 29.0544, size = 78, normalized size = 0.73 \[ - \frac{x^{2}}{b d \left (F^{c + d x} b + a\right ) \log{\left (F \right )}} - \frac{2 x \log{\left (\frac{F^{- c - d x} a}{b} + 1 \right )}}{a b d^{2} \log{\left (F \right )}^{2}} + \frac{2 \operatorname{Li}_{2}\left (- \frac{F^{- c - d x} a}{b}\right )}{a b d^{3} \log{\left (F \right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.114668, size = 103, normalized size = 0.96 \[ \frac{d x \log (F) \left (b d x \log (F) F^{c+d x}-2 \left (a+b F^{c+d x}\right ) \log \left (\frac{b F^{c+d x}}{a}+1\right )\right )-2 \left (a+b F^{c+d x}\right ) \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a b d^3 \log ^3(F) \left (a+b F^{c+d x}\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.028, size = 225, normalized size = 2.1 \[ -{\frac{{x}^{2}}{bd \left ( a+b{F}^{dx+c} \right ) \ln \left ( F \right ) }}+{\frac{{x}^{2}}{\ln \left ( F \right ) abd}}+2\,{\frac{cx}{\ln \left ( F \right ){d}^{2}ba}}+{\frac{{c}^{2}}{\ln \left ( F \right ){d}^{3}ba}}-2\,{\frac{x}{ab{d}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}\ln \left ( 1+{\frac{b{F}^{dx+c}}{a}} \right ) }-2\,{\frac{c}{ \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{3}ba}\ln \left ( 1+{\frac{b{F}^{dx+c}}{a}} \right ) }-2\,{\frac{1}{ab{d}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}}{\it polylog} \left ( 2,-{\frac{b{F}^{dx+c}}{a}} \right ) }-2\,{\frac{c\ln \left ({F}^{dx+c} \right ) }{ \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{3}ba}}+2\,{\frac{c\ln \left ( a+b{F}^{dx+c} \right ) }{ \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{3}ba}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.802561, size = 143, normalized size = 1.34 \[ -\frac{x^{2}}{F^{d x} F^{c} b^{2} d \log \left (F\right ) + a b d \log \left (F\right )} + \frac{\log \left (F^{d x}\right )^{2}}{a b d^{3} \log \left (F\right )^{3}} - \frac{2 \,{\left (\log \left (\frac{F^{d x} F^{c} b}{a} + 1\right ) \log \left (F^{d x}\right ) +{\rm Li}_2\left (-\frac{F^{d x} F^{c} b}{a}\right )\right )}}{a b d^{3} \log \left (F\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.254642, size = 251, normalized size = 2.35 \[ -\frac{a c^{2} \log \left (F\right )^{2} -{\left (b d^{2} x^{2} - b c^{2}\right )} F^{d x + c} \log \left (F\right )^{2} + 2 \,{\left (F^{d x + c} b + a\right )}{\rm Li}_2\left (-\frac{F^{d x + c} b + a}{a} + 1\right ) - 2 \,{\left (F^{d x + c} b c \log \left (F\right ) + a c \log \left (F\right )\right )} \log \left (F^{d x + c} b + a\right ) + 2 \,{\left ({\left (b d x + b c\right )} F^{d x + c} \log \left (F\right ) +{\left (a d x + a c\right )} \log \left (F\right )\right )} \log \left (\frac{F^{d x + c} b + a}{a}\right )}{F^{d x + c} a b^{2} d^{3} \log \left (F\right )^{3} + a^{2} b d^{3} \log \left (F\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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