Optimal. Leaf size=74 \[ \frac{a x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{b x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]
[Out]
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Rubi [A] time = 0.0344427, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{a x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac{b x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (a + b x^{3}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((b*x**3+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0152645, size = 36, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (4 a x+b x^4\right )}{4 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]
[Out]
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Maple [A] time = 0.003, size = 33, normalized size = 0.5 \[{\frac{x \left ( b{x}^{3}+4\,a \right ) }{4\,b{x}^{3}+4\,a}\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((b*x^3+a)^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.815274, size = 14, normalized size = 0.19 \[ \frac{1}{4} \, b x^{4} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^3 + a)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.264772, size = 14, normalized size = 0.19 \[ \frac{1}{4} \, b x^{4} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^3 + a)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.195398, size = 8, normalized size = 0.11 \[ a x + \frac{b x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x**3+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.2681, size = 27, normalized size = 0.36 \[ \frac{1}{4} \,{\left (b x^{4} + 4 \, a x\right )}{\rm sign}\left (b x^{3} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^3 + a)^2),x, algorithm="giac")
[Out]