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

Optimal. Leaf size=62 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^3 \left (a+b F^{c+d x}\right )^2},x\right )}{b d \log (F)}-\frac{1}{2 b d x^2 \log (F) \left (a+b F^{c+d x}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.113154, 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., Rules used = {} \[ \int \frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx &=-\frac{1}{2 b d \left (a+b F^{c+d x}\right )^2 x^2 \log (F)}-\frac{\int \frac{1}{\left (a+b F^{c+d x}\right )^2 x^3} \, dx}{b d \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.72888, size = 0, normalized size = 0. \[ \int \frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.185, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{dx+c}}{ \left ( a+b{F}^{dx+c} \right ) ^{3}{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{d x + c}}{{\left (F^{d x + c} b + a\right )}^{3} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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