Optimal. Leaf size=71 \[ \frac{b x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{a x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]
[Out]
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Rubi [A] time = 0.0764171, antiderivative size = 71, 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 \[ \frac{b x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{a x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 9.13858, size = 58, normalized size = 0.82 \[ \frac{a x^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{12 \left (a + b x\right )} + \frac{x^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0145099, size = 33, normalized size = 0.46 \[ \frac{x^3 \sqrt{(a+b x)^2} (4 a+3 b x)}{12 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.003, size = 30, normalized size = 0.4 \[{\frac{{x}^{3} \left ( 3\,bx+4\,a \right ) }{12\,bx+12\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*((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^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22712, size = 18, normalized size = 0.25 \[ \frac{1}{4} \, b x^{4} + \frac{1}{3} \, a x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.140304, size = 12, normalized size = 0.17 \[ \frac{a x^{3}}{3} + \frac{b x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.209526, size = 53, normalized size = 0.75 \[ \frac{1}{4} \, b x^{4}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, a x^{3}{\rm sign}\left (b x + a\right ) + \frac{a^{4}{\rm sign}\left (b x + a\right )}{12 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*x^2,x, algorithm="giac")
[Out]