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

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

[Out]

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

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Rubi [A]  time = 0.00729945, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{3} x^2 \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[b*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 2.11791, size = 10, normalized size = 0.62 \[ \frac{\left (b x^{2}\right )^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00201301, size = 16, normalized size = 1. \[ \frac{1}{3} x^2 \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[b*x^2],x]

[Out]

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

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \[{\frac{{x}^{2}}{3}\sqrt{b{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.44502, size = 16, normalized size = 1. \[ \frac{\left (b x^{2}\right )^{\frac{3}{2}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.224265, size = 16, normalized size = 1. \[ \frac{1}{3} \, \sqrt{b x^{2}} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.507729, size = 15, normalized size = 0.94 \[ \frac{\sqrt{b} x^{2} \sqrt{x^{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218895, size = 14, normalized size = 0.88 \[ \frac{1}{3} \, \sqrt{b} x^{3}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(b)*x^3*sign(x)