Optimal. Leaf size=36 \[ \frac{\left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 b} \]
[Out]
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Rubi [A] time = 0.0656612, 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 \[ \frac{\left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 b} \]
Antiderivative was successfully verified.
[In] Int[x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{\left (a + b x^{2}\right )^{2}} \int ^{a + b x^{2}} x\, dx}{2 b \left (a + b x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*((b*x**2+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.012167, size = 38, normalized size = 1.06 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (2 a x^2+b x^4\right )}{4 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]
[Out]
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Maple [A] time = 0.062, size = 35, normalized size = 1. \[{\frac{{x}^{2} \left ( b{x}^{2}+2\,a \right ) }{4\,b{x}^{2}+4\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*((b*x^2+a)^2)^(1/2),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(sqrt((b*x^2 + a)^2)*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.258766, size = 18, normalized size = 0.5 \[ \frac{1}{4} \, b x^{4} + \frac{1}{2} \, a x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^2 + a)^2)*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.167063, size = 12, normalized size = 0.33 \[ \frac{a x^{2}}{2} + \frac{b x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*((b*x**2+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.269295, size = 30, normalized size = 0.83 \[ \frac{1}{4} \,{\left (b x^{4} + 2 \, a x^{2}\right )}{\rm sign}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^2 + a)^2)*x,x, algorithm="giac")
[Out]