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

Optimal. Leaf size=182 \[ -\frac{\text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}+\frac{\log \left (a+b F^{c+d x}\right )}{a^2 b d^3 \log ^3(F)}-\frac{x \log \left (\frac{b F^{c+d x}}{a}+1\right )}{a^2 b d^2 \log ^2(F)}-\frac{x}{a^2 b d^2 \log ^2(F)}+\frac{x^2}{2 a^2 b d \log (F)}+\frac{x}{a b d^2 \log ^2(F) \left (a+b F^{c+d x}\right )}-\frac{x^2}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.485859, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{\text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}+\frac{\log \left (a+b F^{c+d x}\right )}{a^2 b d^3 \log ^3(F)}-\frac{x \log \left (\frac{b F^{c+d x}}{a}+1\right )}{a^2 b d^2 \log ^2(F)}-\frac{x}{a^2 b d^2 \log ^2(F)}+\frac{x^2}{2 a^2 b d \log (F)}+\frac{x}{a b d^2 \log ^2(F) \left (a+b F^{c+d x}\right )}-\frac{x^2}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{F^{- c - d x} F^{c + d x} x}{a b d^{2} \left (F^{c + d x} b + a\right ) \log{\left (F \right )}^{2}} + \frac{F^{- c - d x} F^{c + d x} x \log{\left (F^{c + d x} \right )}}{a^{2} b d^{2} \log{\left (F \right )}^{2}} - \frac{F^{- c - d x} F^{c + d x} x \log{\left (F^{c + d x} b + a \right )}}{a^{2} b d^{2} \log{\left (F \right )}^{2}} - \frac{x^{2}}{2 b d \left (F^{c + d x} b + a\right )^{2} \log{\left (F \right )}} + \frac{\int x\, dx}{a^{2} b d \log{\left (F \right )}} - \frac{x \log{\left (F^{c + d x} \right )}}{a^{2} b d^{2} \log{\left (F \right )}^{2}} + \frac{x \log{\left (F^{c + d x} b + a \right )}}{a^{2} b d^{2} \log{\left (F \right )}^{2}} - \frac{x \log{\left (\frac{F^{c + d x} b}{a} + 1 \right )}}{a^{2} b d^{2} \log{\left (F \right )}^{2}} - \frac{\log{\left (F^{c + d x} \right )}}{a^{2} b d^{3} \log{\left (F \right )}^{3}} + \frac{\log{\left (F^{c + d x} b + a \right )}}{a^{2} b d^{3} \log{\left (F \right )}^{3}} - \frac{\operatorname{Li}_{2}\left (- \frac{F^{c + d x} b}{a}\right )}{a^{2} b d^{3} \log{\left (F \right )}^{3}} \]

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))**3,x)

[Out]

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

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Mathematica [A]  time = 0.215485, size = 177, normalized size = 0.97 \[ \frac{-2 \left (a+b F^{c+d x}\right )^2 \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )+b d^2 x^2 \log ^2(F) F^{c+d x} \left (2 a+b F^{c+d x}\right )+2 \left (a+b F^{c+d x}\right )^2 \log \left (\frac{b F^{c+d x}}{a}+1\right )-2 d x \log (F) \left (a+b F^{c+d x}\right ) \left (\left (a+b F^{c+d x}\right ) \log \left (\frac{b F^{c+d x}}{a}+1\right )+b F^{c+d x}\right )}{2 a^2 b d^3 \log ^3(F) \left (a+b F^{c+d x}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.04, size = 295, normalized size = 1.6 \[ -{\frac{ \left ( \ln \left ( F \right ) adx-2\,b{F}^{dx+c}-2\,a \right ) x}{2\, \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{2}ab \left ( a+b{F}^{dx+c} \right ) ^{2}}}+{\frac{{x}^{2}}{2\,{a}^{2}bd\ln \left ( F \right ) }}+{\frac{cx}{{a}^{2}b{d}^{2}\ln \left ( F \right ) }}+{\frac{{c}^{2}}{2\,{a}^{2}b{d}^{3}\ln \left ( F \right ) }}-{\frac{x}{{a}^{2}b{d}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}\ln \left ( 1+{\frac{b{F}^{dx+c}}{a}} \right ) }-{\frac{c}{{a}^{2}b{d}^{3} \left ( \ln \left ( F \right ) \right ) ^{2}}\ln \left ( 1+{\frac{b{F}^{dx+c}}{a}} \right ) }-{\frac{1}{{a}^{2}b{d}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}}{\it polylog} \left ( 2,-{\frac{b{F}^{dx+c}}{a}} \right ) }-{\frac{\ln \left ({F}^{dx+c} \right ) }{{a}^{2}b{d}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}}}+{\frac{\ln \left ( a+b{F}^{dx+c} \right ) }{{a}^{2}b{d}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}}}-{\frac{c\ln \left ({F}^{dx+c} \right ) }{{a}^{2}b{d}^{3} \left ( \ln \left ( F \right ) \right ) ^{2}}}+{\frac{c\ln \left ( a+b{F}^{dx+c} \right ) }{{a}^{2}b{d}^{3} \left ( \ln \left ( F \right ) \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.799338, size = 289, normalized size = 1.59 \[ -\frac{a d x^{2} \log \left (F\right ) - 2 \, F^{d x} F^{c} b x - 2 \, a x}{2 \,{\left (2 \, F^{d x} F^{c} a^{2} b^{2} d^{2} \log \left (F\right )^{2} + F^{2 \, d x} F^{2 \, c} a b^{3} d^{2} \log \left (F\right )^{2} + a^{3} b d^{2} \log \left (F\right )^{2}\right )}} + \frac{\log \left (F^{d x}\right )^{2}}{2 \, a^{2} b d^{3} \log \left (F\right )^{3}} - \frac{\log \left (\frac{F^{d x} F^{c} b}{a} + 1\right ) \log \left (F^{d x}\right ) +{\rm Li}_2\left (-\frac{F^{d x} F^{c} b}{a}\right )}{a^{2} b d^{3} \log \left (F\right )^{3}} + \frac{\log \left (F^{d x} F^{c} b + a\right )}{a^{2} b d^{3} \log \left (F\right )^{3}} - \frac{\log \left (F^{d x}\right )}{a^{2} b d^{3} \log \left (F\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.266591, size = 512, normalized size = 2.81 \[ -\frac{a^{2} c^{2} \log \left (F\right )^{2} + 2 \, a^{2} c \log \left (F\right ) -{\left ({\left (b^{2} d^{2} x^{2} - b^{2} c^{2}\right )} \log \left (F\right )^{2} - 2 \,{\left (b^{2} d x + b^{2} c\right )} \log \left (F\right )\right )} F^{2 \, d x + 2 \, c} - 2 \,{\left ({\left (a b d^{2} x^{2} - a b c^{2}\right )} \log \left (F\right )^{2} -{\left (a b d x + 2 \, a b c\right )} \log \left (F\right )\right )} F^{d x + c} + 2 \,{\left (2 \, F^{d x + c} a b + F^{2 \, d x + 2 \, c} b^{2} + a^{2}\right )}{\rm Li}_2\left (-\frac{F^{d x + c} b + a}{a} + 1\right ) - 2 \,{\left (a^{2} c \log \left (F\right ) +{\left (b^{2} c \log \left (F\right ) + b^{2}\right )} F^{2 \, d x + 2 \, c} + 2 \,{\left (a b c \log \left (F\right ) + a b\right )} F^{d x + c} + a^{2}\right )} \log \left (F^{d x + c} b + a\right ) + 2 \,{\left ({\left (b^{2} d x + b^{2} c\right )} F^{2 \, d x + 2 \, c} \log \left (F\right ) + 2 \,{\left (a b d x + a b c\right )} F^{d x + c} \log \left (F\right ) +{\left (a^{2} d x + a^{2} c\right )} \log \left (F\right )\right )} \log \left (\frac{F^{d x + c} b + a}{a}\right )}{2 \,{\left (2 \, F^{d x + c} a^{3} b^{2} d^{3} \log \left (F\right )^{3} + F^{2 \, d x + 2 \, c} a^{2} b^{3} d^{3} \log \left (F\right )^{3} + a^{4} b d^{3} \log \left (F\right )^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 F^{c + d x} b x - a d x^{2} \log{\left (F \right )} + 2 a x}{4 F^{c + d x} a^{2} b^{2} d^{2} \log{\left (F \right )}^{2} + 2 F^{2 c + 2 d x} a b^{3} d^{2} \log{\left (F \right )}^{2} + 2 a^{3} b d^{2} \log{\left (F \right )}^{2}} + \frac{\int \frac{d x \log{\left (F \right )}}{a + b e^{c \log{\left (F \right )}} e^{d x \log{\left (F \right )}}}\, dx + \int \left (- \frac{1}{a + b e^{c \log{\left (F \right )}} e^{d x \log{\left (F \right )}}}\right )\, dx}{a b d^{2} \log{\left (F \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*F**(c + d*x)*b*x - a*d*x**2*log(F) + 2*a*x)/(4*F**(c + d*x)*a**2*b**2*d**2*lo
g(F)**2 + 2*F**(2*c + 2*d*x)*a*b**3*d**2*log(F)**2 + 2*a**3*b*d**2*log(F)**2) +
(Integral(d*x*log(F)/(a + b*exp(c*log(F))*exp(d*x*log(F))), x) + Integral(-1/(a
+ b*exp(c*log(F))*exp(d*x*log(F))), x))/(a*b*d**2*log(F)**2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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