3.16 \(\int \frac{\left (b x^2\right )^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=15 \[ \frac{1}{2} b x \sqrt{b x^2} \]

[Out]

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

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Rubi [A]  time = 0.00760952, antiderivative size = 15, 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 \[ \frac{1}{2} b x \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2)^(3/2)/x^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b \sqrt{b x^{2}} \int x\, dx}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*sqrt(b*x**2)*Integral(x, x)/x

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Mathematica [A]  time = 0.00164343, size = 15, normalized size = 1. \[ \frac{1}{2} b x \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^2)^(3/2)/x^2,x]

[Out]

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

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \[{\frac{1}{2\,x} \left ( b{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2)^(3/2)/x^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.244749, size = 15, normalized size = 1. \[ \frac{1}{2} \, \sqrt{b x^{2}} b x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.17319, size = 14, normalized size = 0.93 \[ \frac{b^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.212813, size = 14, normalized size = 0.93 \[ \frac{1}{2} \, b^{\frac{3}{2}} x^{2}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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