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

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

[Out]

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

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Rubi [A]  time = 0.00733273, 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 \[ \frac{x^2}{b \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(b*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.58571, size = 10, normalized size = 0.62 \[ \frac{\sqrt{b x^{2}}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00212405, size = 13, normalized size = 0.81 \[ \frac{x^4}{\left (b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(b*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.002, size = 12, normalized size = 0.8 \[{{x}^{4} \left ( b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.44046, size = 19, normalized size = 1.19 \[ \frac{x^{2}}{\sqrt{b x^{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.232458, size = 15, normalized size = 0.94 \[ \frac{\sqrt{b x^{2}}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.9739, size = 14, normalized size = 0.88 \[ \frac{x^{4}}{b^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213096, size = 15, normalized size = 0.94 \[ \frac{\sqrt{b x^{2}}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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