3.672 \(\int \frac{x^7}{\left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=19 \[ \frac{x^8}{8 a \left (a+c x^4\right )^2} \]

[Out]

x^8/(8*a*(a + c*x^4)^2)

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Rubi [A]  time = 0.0164148, antiderivative size = 19, 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 \[ \frac{x^8}{8 a \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + c*x^4)^3,x]

[Out]

x^8/(8*a*(a + c*x^4)^2)

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Rubi in Sympy [A]  time = 2.84427, size = 14, normalized size = 0.74 \[ \frac{x^{8}}{8 a \left (a + c x^{4}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(c*x**4+a)**3,x)

[Out]

x**8/(8*a*(a + c*x**4)**2)

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Mathematica [A]  time = 0.0164769, size = 24, normalized size = 1.26 \[ -\frac{a+2 c x^4}{8 c^2 \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + c*x^4)^3,x]

[Out]

-(a + 2*c*x^4)/(8*c^2*(a + c*x^4)^2)

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Maple [A]  time = 0.013, size = 31, normalized size = 1.6 \[ -{\frac{1}{ \left ( 4\,c{x}^{4}+4\,a \right ){c}^{2}}}+{\frac{a}{8\,{c}^{2} \left ( c{x}^{4}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(c*x^4+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.43852, size = 49, normalized size = 2.58 \[ -\frac{2 \, c x^{4} + a}{8 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(c*x^4 + a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.217046, size = 49, normalized size = 2.58 \[ -\frac{2 \, c x^{4} + a}{8 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(c*x^4 + a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.81274, size = 36, normalized size = 1.89 \[ - \frac{a + 2 c x^{4}}{8 a^{2} c^{2} + 16 a c^{3} x^{4} + 8 c^{4} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(c*x**4+a)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.220026, size = 30, normalized size = 1.58 \[ -\frac{2 \, c x^{4} + a}{8 \,{\left (c x^{4} + a\right )}^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(c*x^4 + a)^3,x, algorithm="giac")

[Out]

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