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

Optimal. Leaf size=60 \[ -\frac{2 \text{Unintegrable}\left (\frac{1}{x^3 \left (a+b F^{c+d x}\right )},x\right )}{b d \log (F)}-\frac{1}{b d x^2 \log (F) \left (a+b F^{c+d x}\right )} \]

[Out]

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

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Rubi [A]  time = 0.117562, 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 = {} \[ \int \frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Mathematica [A]  time = 0.153874, size = 0, normalized size = 0. \[ \int \frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^2 x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{dx+c}}{ \left ( a+b{F}^{dx+c} \right ) ^{2}{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))^2/x^2,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{F^{d x} F^{c} b^{2} d x^{2} \log \left (F\right ) + a b d x^{2} \log \left (F\right )} - 2 \, \int \frac{1}{F^{d x} F^{c} b^{2} d x^{3} \log \left (F\right ) + a b d x^{3} \log \left (F\right )}\,{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))^2/x^2,x, algorithm="maxima")

[Out]

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

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

[Out]

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

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

[Out]

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

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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 )}^{2} 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))^2/x^2,x, algorithm="giac")

[Out]

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