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

Optimal. Leaf size=261 \[ \frac{3 \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}+\frac{3 \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}-\frac{3 x \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}+\frac{3 x \log \left (\frac{b F^{c+d x}}{a}+1\right )}{a^2 b d^3 \log ^3(F)}-\frac{3 x^2 \log \left (\frac{b F^{c+d x}}{a}+1\right )}{2 a^2 b d^2 \log ^2(F)}-\frac{3 x^2}{2 a^2 b d^2 \log ^2(F)}+\frac{x^3}{2 a^2 b d \log (F)}+\frac{3 x^2}{2 a b d^2 \log ^2(F) \left (a+b F^{c+d x}\right )}-\frac{x^3}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.804055, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{3 \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}+\frac{3 \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}-\frac{3 x \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}+\frac{3 x \log \left (\frac{b F^{c+d x}}{a}+1\right )}{a^2 b d^3 \log ^3(F)}-\frac{3 x^2 \log \left (\frac{b F^{c+d x}}{a}+1\right )}{2 a^2 b d^2 \log ^2(F)}-\frac{3 x^2}{2 a^2 b d^2 \log ^2(F)}+\frac{x^3}{2 a^2 b d \log (F)}+\frac{3 x^2}{2 a b d^2 \log ^2(F) \left (a+b F^{c+d x}\right )}-\frac{x^3}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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))**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.457701, size = 220, normalized size = 0.84 \[ \frac{6 \left (a+b F^{c+d x}\right )^2 \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )-6 (d x \log (F)-1) \left (a+b F^{c+d x}\right )^2 \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )+d x \log (F) \left (b d^2 x^2 \log ^2(F) F^{c+d x} \left (2 a+b F^{c+d x}\right )+6 \left (a+b F^{c+d x}\right )^2 \log \left (\frac{b F^{c+d x}}{a}+1\right )-3 d x \log (F) \left (a+b F^{c+d x}\right ) \left (\left (a+b F^{c+d x}\right ) \log \left (\frac{b F^{c+d x}}{a}+1\right )+b F^{c+d x}\right )\right )}{2 a^2 b d^4 \log ^4(F) \left (a+b F^{c+d x}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.053, size = 488, normalized size = 1.9 \[ -{\frac{ \left ( \ln \left ( F \right ) adx-3\,b{F}^{dx+c}-3\,a \right ){x}^{2}}{2\, \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{2}ab \left ( a+b{F}^{dx+c} \right ) ^{2}}}+{\frac{{x}^{3}}{2\,{a}^{2}bd\ln \left ( F \right ) }}-{\frac{3\,{c}^{2}x}{2\,{a}^{2}b{d}^{3}\ln \left ( F \right ) }}-{\frac{{c}^{3}}{{a}^{2}b{d}^{4}\ln \left ( F \right ) }}-{\frac{3\,{x}^{2}}{2\,{a}^{2}b{d}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}\ln \left ( 1+{\frac{b{F}^{dx+c}}{a}} \right ) }+{\frac{3\,{c}^{2}}{2\,{a}^{2}b{d}^{4} \left ( \ln \left ( F \right ) \right ) ^{2}}\ln \left ( 1+{\frac{b{F}^{dx+c}}{a}} \right ) }-3\,{\frac{x}{{a}^{2}b{d}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}}{\it polylog} \left ( 2,-{\frac{b{F}^{dx+c}}{a}} \right ) }+3\,{\frac{1}{{a}^{2}b{d}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}}{\it polylog} \left ( 3,-{\frac{b{F}^{dx+c}}{a}} \right ) }+{\frac{3\,{c}^{2}\ln \left ({F}^{dx+c} \right ) }{2\,{a}^{2}b{d}^{4} \left ( \ln \left ( F \right ) \right ) ^{2}}}-{\frac{3\,{c}^{2}\ln \left ( a+b{F}^{dx+c} \right ) }{2\,{a}^{2}b{d}^{4} \left ( \ln \left ( F \right ) \right ) ^{2}}}-{\frac{3\,{x}^{2}}{2\,{a}^{2}b{d}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}}-3\,{\frac{cx}{{a}^{2}b{d}^{3} \left ( \ln \left ( F \right ) \right ) ^{2}}}-{\frac{3\,{c}^{2}}{2\,{a}^{2}b{d}^{4} \left ( \ln \left ( F \right ) \right ) ^{2}}}+3\,{\frac{x}{{a}^{2}b{d}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}}\ln \left ( 1+{\frac{b{F}^{dx+c}}{a}} \right ) }+3\,{\frac{c}{{a}^{2}b{d}^{4} \left ( \ln \left ( F \right ) \right ) ^{3}}\ln \left ( 1+{\frac{b{F}^{dx+c}}{a}} \right ) }+3\,{\frac{1}{{a}^{2}b{d}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}}{\it polylog} \left ( 2,-{\frac{b{F}^{dx+c}}{a}} \right ) }+3\,{\frac{c\ln \left ({F}^{dx+c} \right ) }{{a}^{2}b{d}^{4} \left ( \ln \left ( F \right ) \right ) ^{3}}}-3\,{\frac{c\ln \left ( a+b{F}^{dx+c} \right ) }{{a}^{2}b{d}^{4} \left ( \ln \left ( F \right ) \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.812794, size = 355, normalized size = 1.36 \[ -\frac{a d x^{3} \log \left (F\right ) - 3 \, F^{d x} F^{c} b x^{2} - 3 \, a x^{2}}{2 \,{\left (2 \, F^{d x} F^{c} a^{2} b^{2} d^{2} \log \left (F\right )^{2} + F^{2 \, d x} F^{2 \, c} a b^{3} d^{2} \log \left (F\right )^{2} + a^{3} b d^{2} \log \left (F\right )^{2}\right )}} - \frac{3 \,{\left (\log \left (\frac{F^{d x} F^{c} b}{a} + 1\right ) \log \left (F^{d x}\right )^{2} + 2 \,{\rm Li}_2\left (-\frac{F^{d x} F^{c} b}{a}\right ) \log \left (F^{d x}\right ) - 2 \,{\rm Li}_{3}(-\frac{F^{d x} F^{c} b}{a})\right )}}{2 \, a^{2} b d^{4} \log \left (F\right )^{4}} + \frac{\log \left (F^{d x}\right )^{3} - 3 \, \log \left (F^{d x}\right )^{2}}{2 \, a^{2} b d^{4} \log \left (F\right )^{4}} + \frac{3 \,{\left (\log \left (\frac{F^{d x} F^{c} b}{a} + 1\right ) \log \left (F^{d x}\right ) +{\rm Li}_2\left (-\frac{F^{d x} F^{c} b}{a}\right )\right )}}{a^{2} b d^{4} \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.27339, size = 779, normalized size = 2.98 \[ \frac{a^{2} c^{3} \log \left (F\right )^{3} + 3 \, a^{2} c^{2} \log \left (F\right )^{2} +{\left ({\left (b^{2} d^{3} x^{3} + b^{2} c^{3}\right )} \log \left (F\right )^{3} - 3 \,{\left (b^{2} d^{2} x^{2} - b^{2} c^{2}\right )} \log \left (F\right )^{2}\right )} F^{2 \, d x + 2 \, c} +{\left (2 \,{\left (a b d^{3} x^{3} + a b c^{3}\right )} \log \left (F\right )^{3} - 3 \,{\left (a b d^{2} x^{2} - 2 \, a b c^{2}\right )} \log \left (F\right )^{2}\right )} F^{d x + c} - 6 \,{\left (a^{2} d x \log \left (F\right ) +{\left (b^{2} d x \log \left (F\right ) - b^{2}\right )} F^{2 \, d x + 2 \, c} + 2 \,{\left (a b d x \log \left (F\right ) - a b\right )} F^{d x + c} - a^{2}\right )}{\rm Li}_2\left (-\frac{F^{d x + c} b + a}{a} + 1\right ) - 3 \,{\left (a^{2} c^{2} \log \left (F\right )^{2} + 2 \, a^{2} c \log \left (F\right ) +{\left (b^{2} c^{2} \log \left (F\right )^{2} + 2 \, b^{2} c \log \left (F\right )\right )} F^{2 \, d x + 2 \, c} + 2 \,{\left (a b c^{2} \log \left (F\right )^{2} + 2 \, a b c \log \left (F\right )\right )} F^{d x + c}\right )} \log \left (F^{d x + c} b + a\right ) - 3 \,{\left ({\left (a^{2} d^{2} x^{2} - a^{2} c^{2}\right )} \log \left (F\right )^{2} +{\left ({\left (b^{2} d^{2} x^{2} - b^{2} c^{2}\right )} \log \left (F\right )^{2} - 2 \,{\left (b^{2} d x + b^{2} c\right )} \log \left (F\right )\right )} F^{2 \, d x + 2 \, c} + 2 \,{\left ({\left (a b d^{2} x^{2} - a b c^{2}\right )} \log \left (F\right )^{2} - 2 \,{\left (a b d x + a b c\right )} \log \left (F\right )\right )} F^{d x + c} - 2 \,{\left (a^{2} d x + a^{2} c\right )} \log \left (F\right )\right )} \log \left (\frac{F^{d x + c} b + a}{a}\right ) + 6 \,{\left (2 \, F^{d x + c} a b + F^{2 \, d x + 2 \, c} b^{2} + a^{2}\right )}{\rm Li}_{3}(-\frac{F^{d x + c} b}{a})}{2 \,{\left (2 \, F^{d x + c} a^{3} b^{2} d^{4} \log \left (F\right )^{4} + F^{2 \, d x + 2 \, c} a^{2} b^{3} d^{4} \log \left (F\right )^{4} + a^{4} b d^{4} \log \left (F\right )^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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