Optimal. Leaf size=61 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2}-\frac{a (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b^2} \]
[Out]
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Rubi [A] time = 0.0561733, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2}-\frac{a (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b^2} \]
Antiderivative was successfully verified.
[In] Int[x*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 8.19547, size = 60, normalized size = 0.98 \[ - \frac{a \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 b^{2}} + \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0136479, size = 33, normalized size = 0.54 \[ \frac{x^2 \sqrt{(a+b x)^2} (3 a+2 b x)}{6 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[x*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.004, size = 30, normalized size = 0.5 \[{\frac{{x}^{2} \left ( 2\,bx+3\,a \right ) }{6\,bx+6\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*((b*x+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 + a)^2)*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222649, size = 18, normalized size = 0.3 \[ \frac{1}{3} \, b x^{3} + \frac{1}{2} \, a x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.140569, size = 12, normalized size = 0.2 \[ \frac{a x^{2}}{2} + \frac{b x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.208473, size = 53, normalized size = 0.87 \[ \frac{1}{3} \, b x^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a x^{2}{\rm sign}\left (b x + a\right ) - \frac{a^{3}{\rm sign}\left (b x + a\right )}{6 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*x,x, algorithm="giac")
[Out]