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

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

[Out]

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

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Rubi [A]  time = 0.131287, antiderivative size = 69, normalized size of antiderivative = 1.68, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{a}{4 b^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + a/(4*b^2*(a + b*x^2)*Sqrt[a^2 + 2*a
*b*x^2 + b^2*x^4])

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Rubi in Sympy [A]  time = 8.38562, size = 37, normalized size = 0.9 \[ \frac{x^{4} \left (2 a + 2 b x^{2}\right )}{8 a \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0227758, size = 39, normalized size = 0.95 \[ \frac{-a-2 b x^2}{4 b^2 \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 32, normalized size = 0.8 \[ -{\frac{ \left ( b{x}^{2}+a \right ) \left ( 2\,b{x}^{2}+a \right ) }{4\,{b}^{2}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.705099, size = 65, normalized size = 1.59 \[ -\frac{1}{2 \, \sqrt{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}} b^{2}} + \frac{a}{4 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.257924, size = 49, normalized size = 1.2 \[ -\frac{2 \, b x^{2} + a}{4 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.659879, size = 4, normalized size = 0.1 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x