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

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 = 0.00971756, 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 \[ -\frac{1}{4 c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + c*x^4)^2,x]

[Out]

-1/(4*c*(a + c*x^4))

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Rubi in Sympy [A]  time = 2.13711, size = 12, normalized size = 0.75 \[ - \frac{1}{4 c \left (a + c x^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(c*x**4+a)**2,x)

[Out]

-1/(4*c*(a + c*x**4))

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Mathematica [A]  time = 0.00764503, 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 \[ -{\frac{1}{4\,c \left ( c{x}^{4}+a \right ) }} \]

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 = 1.4462, size = 19, normalized size = 1.19 \[ -\frac{1}{4 \,{\left (c x^{4} + a\right )} c} \]

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 = 0.220257, size = 20, normalized size = 1.25 \[ -\frac{1}{4 \,{\left (c^{2} x^{4} + a c\right )}} \]

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 = 1.61802, size = 15, normalized size = 0.94 \[ - \frac{1}{4 a c + 4 c^{2} x^{4}} \]

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/XCAS [A]  time = 0.217045, size = 19, normalized size = 1.19 \[ -\frac{1}{4 \,{\left (c x^{4} + a\right )} c} \]

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)