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

Optimal. Leaf size=65 \[ -\frac{\text{Int}\left (\frac{1}{x^2 \left (a+b F^{c+d x}\right )^2},x\right )}{2 b d \log (F)}-\frac{1}{2 b d x \log (F) \left (a+b F^{c+d x}\right )^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.182751, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x},x\right ) \]

Verification is Not applicable to the result.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.832427, size = 0, normalized size = 0. \[ \int \frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.091, size = 0, normalized size = 0. \[ \int{\frac{{F}^{dx+c}}{ \left ( a+b{F}^{dx+c} \right ) ^{3}x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{F^{d x + c}}{3 \, F^{d x + c} a^{2} b x + 3 \, F^{2 \, d x + 2 \, c} a b^{2} x + F^{3 \, d x + 3 \, c} b^{3} x + a^{3} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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