3.197 \(\int \frac{x^5}{(b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{2 c \left (b+c x^2\right )} \]

[Out]

-1/(2*c*(b + c*x^2))

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Rubi [A]  time = 0.0093297, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1584, 261} \[ -\frac{1}{2 c \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(b*x^2 + c*x^4)^2,x]

[Out]

-1/(2*c*(b + c*x^2))

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{x^5}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{x}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac{1}{2 c \left (b+c x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.0020596, size = 16, normalized size = 1. \[ -\frac{1}{2 c \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(b*x^2 + c*x^4)^2,x]

[Out]

-1/(2*c*(b + c*x^2))

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Maple [A]  time = 0.042, size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,c \left ( c{x}^{2}+b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(c*x^4+b*x^2)^2,x)

[Out]

-1/2/c/(c*x^2+b)

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Maxima [A]  time = 0.970984, size = 20, normalized size = 1.25 \begin{align*} -\frac{1}{2 \,{\left (c^{2} x^{2} + b c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^4+b*x^2)^2,x, algorithm="maxima")

[Out]

-1/2/(c^2*x^2 + b*c)

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Fricas [A]  time = 1.40502, size = 30, normalized size = 1.88 \begin{align*} -\frac{1}{2 \,{\left (c^{2} x^{2} + b c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^4+b*x^2)^2,x, algorithm="fricas")

[Out]

-1/2/(c^2*x^2 + b*c)

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Sympy [A]  time = 0.337265, size = 15, normalized size = 0.94 \begin{align*} - \frac{1}{2 b c + 2 c^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(c*x**4+b*x**2)**2,x)

[Out]

-1/(2*b*c + 2*c**2*x**2)

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Giac [A]  time = 1.26203, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{2 \,{\left (c x^{2} + b\right )} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(c*x^4+b*x^2)^2,x, algorithm="giac")

[Out]

-1/2/((c*x^2 + b)*c)