3.111 \(\int \frac{x^2}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx\)

Optimal. Leaf size=38 \[ -\frac{1}{12 b \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \]

[Out]

-1/(12*b*(a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2))

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Rubi [A]  time = 0.0716311, 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 \[ -\frac{1}{12 b \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

-1/(12*b*(a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2))

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Rubi in Sympy [A]  time = 9.26603, size = 36, normalized size = 0.95 \[ - \frac{2 a + 2 b x^{3}}{24 b \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}} \]

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)**(5/2),x)

[Out]

-(2*a + 2*b*x**3)/(24*b*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2))

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Mathematica [A]  time = 0.0215019, size = 27, normalized size = 0.71 \[ -\frac{a+b x^3}{12 b \left (\left (a+b x^3\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

-(a + b*x^3)/(12*b*((a + b*x^3)^2)^(5/2))

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Maple [A]  time = 0.011, size = 24, normalized size = 0.6 \[ -{\frac{b{x}^{3}+a}{12\,b} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

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)^(5/2),x)

[Out]

-1/12*(b*x^3+a)/b/((b*x^3+a)^2)^(5/2)

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Maxima [A]  time = 0.791476, size = 24, normalized size = 0.63 \[ -\frac{1}{12 \,{\left (x^{3} + \frac{a}{b}\right )}^{4}{\left (b^{2}\right )}^{\frac{5}{2}}} \]

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)^(5/2),x, algorithm="maxima")

[Out]

-1/12/((x^3 + a/b)^4*(b^2)^(5/2))

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Fricas [A]  time = 0.264969, size = 65, normalized size = 1.71 \[ -\frac{1}{12 \,{\left (b^{5} x^{12} + 4 \, a b^{4} x^{9} + 6 \, a^{2} b^{3} x^{6} + 4 \, a^{3} b^{2} x^{3} + a^{4} b\right )}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

-1/12/(b^5*x^12 + 4*a*b^4*x^9 + 6*a^2*b^3*x^6 + 4*a^3*b^2*x^3 + a^4*b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

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)**(5/2),x)

[Out]

Integral(x**2/((a + b*x**3)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.708335, 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)^(5/2),x, algorithm="giac")

[Out]

sage0*x