3.662 \(\int \frac{x^3}{(a+c x^4)^2} \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{4 c \left (a+c x^4\right )} \]

[Out]

-1/(4*c*(a + c*x^4))

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Rubi [A]  time = 0.0034657, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ -\frac{1}{4 c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + c*x^4)^2,x]

[Out]

-1/(4*c*(a + c*x^4))

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^3}{\left (a+c x^4\right )^2} \, dx &=-\frac{1}{4 c \left (a+c x^4\right )}\\ \end{align*}

Mathematica [A]  time = 0.0034458, size = 16, normalized size = 1. \[ -\frac{1}{4 c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + c*x^4)^2,x]

[Out]

-1/(4*c*(a + c*x^4))

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Maple [A]  time = 0.001, size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,c \left ( c{x}^{4}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(c*x^4+a)^2,x)

[Out]

-1/4/c/(c*x^4+a)

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Maxima [A]  time = 1.04328, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{4 \,{\left (c x^{4} + a\right )} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4+a)^2,x, algorithm="maxima")

[Out]

-1/4/((c*x^4 + a)*c)

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Fricas [A]  time = 1.67147, size = 30, normalized size = 1.88 \begin{align*} -\frac{1}{4 \,{\left (c^{2} x^{4} + a c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4+a)^2,x, algorithm="fricas")

[Out]

-1/4/(c^2*x^4 + a*c)

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Sympy [A]  time = 0.522334, size = 15, normalized size = 0.94 \begin{align*} - \frac{1}{4 a c + 4 c^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(c*x**4+a)**2,x)

[Out]

-1/(4*a*c + 4*c**2*x**4)

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Giac [A]  time = 1.11633, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{4 \,{\left (c x^{4} + a\right )} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4+a)^2,x, algorithm="giac")

[Out]

-1/4/((c*x^4 + a)*c)