3.100 \(\int \frac{x^2}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=38 \[ -\frac{1}{6 b \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

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

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Rubi [A]  time = 0.0716736, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{1}{6 b \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.28141, size = 36, normalized size = 0.95 \[ - \frac{2 a + 2 b x^{3}}{12 b \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0196767, size = 27, normalized size = 0.71 \[ -\frac{a+b x^3}{6 b \left (\left (a+b x^3\right )^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 24, normalized size = 0.6 \[ -{\frac{b{x}^{3}+a}{6\,b} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.806821, size = 24, normalized size = 0.63 \[ -\frac{1}{6 \,{\left (x^{3} + \frac{a}{b}\right )}^{2}{\left (b^{2}\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2),x, algorithm="maxima")

[Out]

-1/6/((x^3 + a/b)^2*(b^2)^(3/2))

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Fricas [A]  time = 0.261232, size = 35, normalized size = 0.92 \[ -\frac{1}{6 \,{\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2),x, algorithm="fricas")

[Out]

-1/6/(b^3*x^6 + 2*a*b^2*x^3 + a^2*b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)

[Out]

Integral(x**2/((a + b*x**3)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.740985, size = 4, normalized size = 0.11 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x