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

Optimal. Leaf size=16 \[ -\frac{1}{8 c \left (a+c x^4\right )^2} \]

[Out]

-1/(8*c*(a + c*x^4)^2)

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Rubi [A]  time = 0.0032871, 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} \[ -\frac{1}{8 c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + c*x^4)^3,x]

[Out]

-1/(8*c*(a + c*x^4)^2)

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 )^3} \, dx &=-\frac{1}{8 c \left (a+c x^4\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0037017, size = 16, normalized size = 1. \[ -\frac{1}{8 c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + c*x^4)^3,x]

[Out]

-1/(8*c*(a + c*x^4)^2)

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Maple [A]  time = 0.001, size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{8\,c \left ( c{x}^{4}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/c/(c*x^4+a)^2

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Maxima [A]  time = 0.955873, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{8 \,{\left (c x^{4} + a\right )}^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/((c*x^4 + a)^2*c)

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Fricas [A]  time = 1.59647, size = 51, normalized size = 3.19 \begin{align*} -\frac{1}{8 \,{\left (c^{3} x^{8} + 2 \, a c^{2} x^{4} + a^{2} c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/(c^3*x^8 + 2*a*c^2*x^4 + a^2*c)

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Sympy [A]  time = 1.36612, size = 27, normalized size = 1.69 \begin{align*} - \frac{1}{8 a^{2} c + 16 a c^{2} x^{4} + 8 c^{3} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(8*a**2*c + 16*a*c**2*x**4 + 8*c**3*x**8)

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Giac [A]  time = 1.13548, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{8 \,{\left (c x^{4} + a\right )}^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/((c*x^4 + a)^2*c)