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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]