3.342 \(\int \frac{x^5}{\left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=19 \[ \frac{x^6}{6 a \left (a+b x^3\right )^2} \]

[Out]

x^6/(6*a*(a + b*x^3)^2)

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Rubi [A]  time = 0.0165156, 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 \[ \frac{x^6}{6 a \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^3)^3,x]

[Out]

x^6/(6*a*(a + b*x^3)^2)

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Rubi in Sympy [A]  time = 2.95561, size = 14, normalized size = 0.74 \[ \frac{x^{6}}{6 a \left (a + b x^{3}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**3+a)**3,x)

[Out]

x**6/(6*a*(a + b*x**3)**2)

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Mathematica [A]  time = 0.0130143, size = 24, normalized size = 1.26 \[ -\frac{a+2 b x^3}{6 b^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x^3)^3,x]

[Out]

-(a + 2*b*x^3)/(6*b^2*(a + b*x^3)^2)

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Maple [A]  time = 0.006, size = 31, normalized size = 1.6 \[{\frac{a}{6\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{1}{ \left ( 3\,b{x}^{3}+3\,a \right ){b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^3+a)^3,x)

[Out]

1/6*a/b^2/(b*x^3+a)^2-1/3/(b*x^3+a)/b^2

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Maxima [A]  time = 1.43355, size = 49, normalized size = 2.58 \[ -\frac{2 \, b x^{3} + a}{6 \,{\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

-1/6*(2*b*x^3 + a)/(b^4*x^6 + 2*a*b^3*x^3 + a^2*b^2)

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Fricas [A]  time = 0.245814, size = 49, normalized size = 2.58 \[ -\frac{2 \, b x^{3} + a}{6 \,{\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^3 + a)^3,x, algorithm="fricas")

[Out]

-1/6*(2*b*x^3 + a)/(b^4*x^6 + 2*a*b^3*x^3 + a^2*b^2)

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Sympy [A]  time = 2.25337, size = 36, normalized size = 1.89 \[ - \frac{a + 2 b x^{3}}{6 a^{2} b^{2} + 12 a b^{3} x^{3} + 6 b^{4} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**3+a)**3,x)

[Out]

-(a + 2*b*x**3)/(6*a**2*b**2 + 12*a*b**3*x**3 + 6*b**4*x**6)

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GIAC/XCAS [A]  time = 0.219196, size = 30, normalized size = 1.58 \[ -\frac{2 \, b x^{3} + a}{6 \,{\left (b x^{3} + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^3 + a)^3,x, algorithm="giac")

[Out]

-1/6*(2*b*x^3 + a)/((b*x^3 + a)^2*b^2)