3.48 \(\int \frac{x^4}{\left (b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=19 \[ \frac{x^3}{2 b \sqrt{b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.00866098, antiderivative size = 19, 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 \[ \frac{x^3}{2 b \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(b*x^2)^(3/2),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**2)**(3/2),x)

[Out]

sqrt(b*x**2)*Integral(x, x)/(b**2*x)

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Mathematica [A]  time = 0.00224436, size = 16, normalized size = 0.84 \[ \frac{x^5}{2 \left (b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(b*x^2)^(3/2),x]

[Out]

x^5/(2*(b*x^2)^(3/2))

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Maple [A]  time = 0.001, size = 13, normalized size = 0.7 \[{\frac{{x}^{5}}{2} \left ( b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2)^(3/2),x)

[Out]

1/2*x^5/(b*x^2)^(3/2)

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Maxima [A]  time = 1.42246, size = 20, normalized size = 1.05 \[ \frac{x^{3}}{2 \, \sqrt{b x^{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.260822, size = 18, normalized size = 0.95 \[ \frac{\sqrt{b x^{2}} x}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.34851, size = 15, normalized size = 0.79 \[ \frac{x^{5}}{2 b^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**2)**(3/2),x)

[Out]

x**5/(2*b**(3/2)*(x**2)**(3/2))

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GIAC/XCAS [A]  time = 0.218145, size = 18, normalized size = 0.95 \[ \frac{\sqrt{b x^{2}} x}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2)^(3/2),x, algorithm="giac")

[Out]

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