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

Optimal. Leaf size=17 \[ \frac{1}{6} b x^5 \sqrt{b x^2} \]

[Out]

(b*x^5*Sqrt[b*x^2])/6

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Rubi [A]  time = 0.00803669, antiderivative size = 17, 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 \[ \frac{1}{6} b x^5 \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(b*x^2)^(3/2),x]

[Out]

(b*x^5*Sqrt[b*x^2])/6

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Rubi in Sympy [A]  time = 2.28528, size = 14, normalized size = 0.82 \[ \frac{b x^{5} \sqrt{b x^{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x**5*sqrt(b*x**2)/6

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Mathematica [A]  time = 0.00311503, size = 16, normalized size = 0.94 \[ \frac{1}{6} x^3 \left (b x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(b*x^2)^(3/2),x]

[Out]

(x^3*(b*x^2)^(3/2))/6

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \[{\frac{{x}^{3}}{6} \left ( b{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^2)^(3/2),x)

[Out]

1/6*x^3*(b*x^2)^(3/2)

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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((b*x^2)^(3/2)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28918, size = 18, normalized size = 1.06 \[ \frac{1}{6} \, \sqrt{b x^{2}} b x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(b*x^2)*b*x^5

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Sympy [A]  time = 1.96922, size = 15, normalized size = 0.88 \[ \frac{b^{\frac{3}{2}} x^{3} \left (x^{2}\right )^{\frac{3}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**(3/2)*x**3*(x**2)**(3/2)/6

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GIAC/XCAS [A]  time = 0.214332, size = 14, normalized size = 0.82 \[ \frac{1}{6} \, b^{\frac{3}{2}} x^{6}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b^(3/2)*x^6*sign(x)