3.764 \(\int \frac{\text{F0}(x)}{(x+\text{F0}(x))^2} \, dx\)

Optimal. Leaf size=21 \[ \text{CannotIntegrate}\left (\frac{1}{\text{F0}(x)+x},x\right )-\text{CannotIntegrate}\left (\frac{x}{(\text{F0}(x)+x)^2},x\right ) \]

[Out]

-CannotIntegrate[x/(x + F0[x])^2, x] + CannotIntegrate[(x + F0[x])^(-1), x]

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Rubi [A]  time = 0.0561132, 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 = {} \[ \int \frac{\text{F0}(x)}{(x+\text{F0}(x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[F0[x]/(x + F0[x])^2,x]

[Out]

-Defer[Int][x/(x + F0[x])^2, x] + Defer[Int][(x + F0[x])^(-1), x]

Rubi steps

\begin{align*} \int \frac{\text{F0}(x)}{(x+\text{F0}(x))^2} \, dx &=\int \left (-\frac{x}{(x+\text{F0}(x))^2}+\frac{1}{x+\text{F0}(x)}\right ) \, dx\\ &=-\int \frac{x}{(x+\text{F0}(x))^2} \, dx+\int \frac{1}{x+\text{F0}(x)} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0169089, size = 0, normalized size = 0. \[ \int \frac{\text{F0}(x)}{(x+\text{F0}(x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[F0[x]/(x + F0[x])^2,x]

[Out]

Integrate[F0[x]/(x + F0[x])^2, x]

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Maple [A]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it F0} \left ( x \right ) }{ \left ( x+{\it F0} \left ( x \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F0(x)/(x+F0(x))^2,x)

[Out]

int(F0(x)/(x+F0(x))^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F_{0}\left (x\right )}{{\left (x + F_{0}\left (x\right )\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F0(x)/(x+F0(x))^2,x, algorithm="maxima")

[Out]

integrate(F0(x)/(x + F0(x))^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F0(x)/(x+F0(x))^2,x, algorithm="fricas")

[Out]

integral(F0(x)/(x^2 + 2*x*F0(x) + F0(x)^2), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{F_{0}}{\left (x \right )}}{\left (x + \operatorname{F_{0}}{\left (x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F0(x)/(x+F0(x))**2,x)

[Out]

Integral(F0(x)/(x + F0(x))**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F0(x)/(x+F0(x))^2,x, algorithm="giac")

[Out]

integrate(F0(x)/(x + F0(x))^2, x)