3.567 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^9} \, dx\)

Optimal. Leaf size=41 \[ -\frac{\left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 a x^8} \]

[Out]

-((a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*a*x^8)

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Rubi [A]  time = 0.105306, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{\left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^9,x]

[Out]

-((a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*a*x^8)

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Rubi in Sympy [A]  time = 8.19731, size = 39, normalized size = 0.95 \[ - \frac{\left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{16 a x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/2)/x**9,x)

[Out]

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

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Mathematica [A]  time = 0.0268799, size = 59, normalized size = 1.44 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (a^3+4 a^2 b x^2+6 a b^2 x^4+4 b^3 x^6\right )}{8 x^8 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^9,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(a^3 + 4*a^2*b*x^2 + 6*a*b^2*x^4 + 4*b^3*x^6))/(8*x^8*(a +
b*x^2))

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Maple [A]  time = 0.01, size = 56, normalized size = 1.4 \[ -{\frac{4\,{b}^{3}{x}^{6}+6\,a{x}^{4}{b}^{2}+4\,{a}^{2}b{x}^{2}+{a}^{3}}{8\,{x}^{8} \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^(3/2)/x^9,x)

[Out]

-1/8*(4*b^3*x^6+6*a*b^2*x^4+4*a^2*b*x^2+a^3)*((b*x^2+a)^2)^(3/2)/x^8/(b*x^2+a)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.26178, size = 47, normalized size = 1.15 \[ -\frac{4 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} + a^{3}}{8 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/x^9,x, algorithm="fricas")

[Out]

-1/8*(4*b^3*x^6 + 6*a*b^2*x^4 + 4*a^2*b*x^2 + a^3)/x^8

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}{x^{9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/2)/x**9,x)

[Out]

Integral(((a + b*x**2)**2)**(3/2)/x**9, x)

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GIAC/XCAS [A]  time = 0.269549, size = 92, normalized size = 2.24 \[ -\frac{4 \, b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 6 \, a b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 4 \, a^{2} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + a^{3}{\rm sign}\left (b x^{2} + a\right )}{8 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/x^9,x, algorithm="giac")

[Out]

-1/8*(4*b^3*x^6*sign(b*x^2 + a) + 6*a*b^2*x^4*sign(b*x^2 + a) + 4*a^2*b*x^2*sign
(b*x^2 + a) + a^3*sign(b*x^2 + a))/x^8