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

Optimal. Leaf size=17 \[ \frac{1}{5} b x^4 \sqrt{b x^2} \]

[Out]

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

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Rubi [A]  time = 0.00780087, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{5} b x^4 \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]  Int[x*(b*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.14279, size = 10, normalized size = 0.59 \[ \frac{\left (b x^{2}\right )^{\frac{5}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00272786, size = 16, normalized size = 0.94 \[ \frac{1}{5} x^2 \left (b x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(b*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \[{\frac{{x}^{2}}{5} \left ( b{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.45459, size = 16, normalized size = 0.94 \[ \frac{\left (b x^{2}\right )^{\frac{5}{2}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.265858, size = 18, normalized size = 1.06 \[ \frac{1}{5} \, \sqrt{b x^{2}} b x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.56005, size = 15, normalized size = 0.88 \[ \frac{b^{\frac{3}{2}} x^{2} \left (x^{2}\right )^{\frac{3}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x**2)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.212536, size = 14, normalized size = 0.82 \[ \frac{1}{5} \, b^{\frac{3}{2}} x^{5}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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