3.2 \(\int x^3 \sqrt{b x^2} \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{5} x^4 \sqrt{b x^2} \]

[Out]

(x^4*Sqrt[b*x^2])/5

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Rubi [A]  time = 0.00647422, 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 \[ \frac{1}{5} x^4 \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3*Sqrt[b*x^2],x]

[Out]

(x^4*Sqrt[b*x^2])/5

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Rubi in Sympy [A]  time = 2.56971, size = 12, normalized size = 0.75 \[ \frac{\left (b x^{2}\right )^{\frac{5}{2}}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2)**(1/2),x)

[Out]

(b*x**2)**(5/2)/(5*b**2)

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Mathematica [A]  time = 0.00312335, size = 16, normalized size = 1. \[ \frac{1}{5} x^4 \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*Sqrt[b*x^2],x]

[Out]

(x^4*Sqrt[b*x^2])/5

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Maple [A]  time = 0.004, size = 13, normalized size = 0.8 \[{\frac{{x}^{4}}{5}\sqrt{b{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2)^(1/2),x)

[Out]

1/5*x^4*(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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229734, size = 16, normalized size = 1. \[ \frac{1}{5} \, \sqrt{b x^{2}} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2)*x^3,x, algorithm="fricas")

[Out]

1/5*sqrt(b*x^2)*x^4

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Sympy [A]  time = 0.917808, size = 15, normalized size = 0.94 \[ \frac{\sqrt{b} x^{4} \sqrt{x^{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**2)**(1/2),x)

[Out]

sqrt(b)*x**4*sqrt(x**2)/5

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GIAC/XCAS [A]  time = 0.217069, size = 14, normalized size = 0.88 \[ \frac{1}{5} \, \sqrt{b} x^{5}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2)*x^3,x, algorithm="giac")

[Out]

1/5*sqrt(b)*x^5*sign(x)