3.3 \(\int x^2 \sqrt{b x^2} \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{4} x^3 \sqrt{b x^2} \]

[Out]

(x^3*Sqrt[b*x^2])/4

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Rubi [A]  time = 0.00663837, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{4} x^3 \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[b*x^2],x]

[Out]

(x^3*Sqrt[b*x^2])/4

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Rubi in Sympy [A]  time = 1.93493, size = 12, normalized size = 0.75 \[ \frac{x^{3} \sqrt{b x^{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x**2)**(1/2),x)

[Out]

x**3*sqrt(b*x**2)/4

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Mathematica [A]  time = 0.00193494, size = 16, normalized size = 1. \[ \frac{1}{4} x^3 \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Sqrt[b*x^2],x]

[Out]

(x^3*Sqrt[b*x^2])/4

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Maple [A]  time = 0.004, size = 13, normalized size = 0.8 \[{\frac{{x}^{3}}{4}\sqrt{b{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^2)^(1/2),x)

[Out]

1/4*x^3*(b*x^2)^(1/2)

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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)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233431, size = 16, normalized size = 1. \[ \frac{1}{4} \, \sqrt{b x^{2}} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2)*x^2,x, algorithm="fricas")

[Out]

1/4*sqrt(b*x^2)*x^3

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Sympy [A]  time = 0.670121, size = 15, normalized size = 0.94 \[ \frac{\sqrt{b} x^{3} \sqrt{x^{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x**2)**(1/2),x)

[Out]

sqrt(b)*x**3*sqrt(x**2)/4

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GIAC/XCAS [A]  time = 0.212732, size = 14, normalized size = 0.88 \[ \frac{1}{4} \, \sqrt{b} x^{4}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2)*x^2,x, algorithm="giac")

[Out]

1/4*sqrt(b)*x^4*sign(x)