Optimal. Leaf size=36 \[ \frac{\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{8 b} \]
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Rubi [A] time = 0.0648148, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{8 b} \]
Antiderivative was successfully verified.
[In] Int[x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 8.17256, size = 34, normalized size = 0.94 \[ \frac{\left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{16 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0222523, size = 27, normalized size = 0.75 \[ \frac{\left (a+b x^2\right ) \left (\left (a+b x^2\right )^2\right )^{3/2}}{8 b} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.007, size = 57, normalized size = 1.6 \[{\frac{{x}^{2} \left ({b}^{3}{x}^{6}+4\,a{b}^{2}{x}^{4}+6\,{a}^{2}b{x}^{2}+4\,{a}^{3} \right ) }{8\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)
[Out]
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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,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.265308, size = 47, normalized size = 1.31 \[ \frac{1}{8} \, b^{3} x^{8} + \frac{1}{2} \, a b^{2} x^{6} + \frac{3}{4} \, a^{2} b x^{4} + \frac{1}{2} \, a^{3} x^{2} \]
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,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.269433, size = 59, normalized size = 1.64 \[ \frac{1}{8} \,{\left (b^{3} x^{8} + 4 \, a b^{2} x^{6} + 6 \, a^{2} b x^{4} + 4 \, a^{3} x^{2}\right )}{\rm sign}\left (b x^{2} + a\right ) \]
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,x, algorithm="giac")
[Out]