Optimal. Leaf size=79 \[ \frac{b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac{a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]
[Out]
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Rubi [A] time = 0.0537178, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{b x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac{a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Int[x*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. \[ \int x \sqrt{\left (a + b x^{3}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*((b*x**3+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0133414, size = 39, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (5 a x^2+2 b x^5\right )}{10 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]
[Out]
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Maple [A] time = 0.004, size = 36, normalized size = 0.5 \[{\frac{{x}^{2} \left ( 2\,b{x}^{3}+5\,a \right ) }{10\,b{x}^{3}+10\,a}\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*((b*x^3+a)^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.816568, size = 18, normalized size = 0.23 \[ \frac{1}{5} \, b x^{5} + \frac{1}{2} \, a x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^3 + a)^2)*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.26646, size = 18, normalized size = 0.23 \[ \frac{1}{5} \, b x^{5} + \frac{1}{2} \, a x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^3 + a)^2)*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.207347, size = 12, normalized size = 0.15 \[ \frac{a x^{2}}{2} + \frac{b x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*((b*x**3+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.277304, size = 39, normalized size = 0.49 \[ \frac{1}{5} \, b x^{5}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{2} \, a x^{2}{\rm sign}\left (b x^{3} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^3 + a)^2)*x,x, algorithm="giac")
[Out]