3.2270 \(\int \frac{x^2}{\left (a+b x^{3/2}\right )^{2/3}} \, dx\)

Optimal. Leaf size=40 \[ \frac{\left (a+b x^{3/2}\right )^{4/3}}{2 b^2}-\frac{2 a \sqrt [3]{a+b x^{3/2}}}{b^2} \]

[Out]

(-2*a*(a + b*x^(3/2))^(1/3))/b^2 + (a + b*x^(3/2))^(4/3)/(2*b^2)

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Rubi [A]  time = 0.0624712, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\left (a+b x^{3/2}\right )^{4/3}}{2 b^2}-\frac{2 a \sqrt [3]{a+b x^{3/2}}}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x^(3/2))^(2/3),x]

[Out]

(-2*a*(a + b*x^(3/2))^(1/3))/b^2 + (a + b*x^(3/2))^(4/3)/(2*b^2)

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Rubi in Sympy [A]  time = 7.14871, size = 34, normalized size = 0.85 \[ - \frac{2 a \sqrt [3]{a + b x^{\frac{3}{2}}}}{b^{2}} + \frac{\left (a + b x^{\frac{3}{2}}\right )^{\frac{4}{3}}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*(a + b*x**(3/2))**(1/3)/b**2 + (a + b*x**(3/2))**(4/3)/(2*b**2)

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Mathematica [A]  time = 0.0222897, size = 31, normalized size = 0.78 \[ \frac{\left (b x^{3/2}-3 a\right ) \sqrt [3]{a+b x^{3/2}}}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b*x^(3/2))^(2/3),x]

[Out]

((-3*a + b*x^(3/2))*(a + b*x^(3/2))^(1/3))/(2*b^2)

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Maple [A]  time = 0.004, size = 30, normalized size = 0.8 \[ 2\,{\frac{1/4\, \left ( a+b{x}^{3/2} \right ) ^{4/3}-a\sqrt [3]{a+b{x}^{3/2}}}{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b*x^(3/2))^(2/3),x)

[Out]

2/b^2*(1/4*(a+b*x^(3/2))^(4/3)-a*(a+b*x^(3/2))^(1/3))

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Maxima [A]  time = 1.44101, size = 41, normalized size = 1.02 \[ \frac{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{4}{3}}}{2 \, b^{2}} - \frac{2 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}} a}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^(3/2) + a)^(4/3)/b^2 - 2*(b*x^(3/2) + a)^(1/3)*a/b^2

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Fricas [A]  time = 0.43604, size = 31, normalized size = 0.78 \[ \frac{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}}{\left (b x^{\frac{3}{2}} - 3 \, a\right )}}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^(3/2) + a)^(1/3)*(b*x^(3/2) - 3*a)/b^2

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Sympy [A]  time = 5.14262, size = 49, normalized size = 1.22 \[ \begin{cases} - \frac{3 a \sqrt [3]{a + b x^{\frac{3}{2}}}}{2 b^{2}} + \frac{x^{\frac{3}{2}} \sqrt [3]{a + b x^{\frac{3}{2}}}}{2 b} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{\frac{2}{3}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-3*a*(a + b*x**(3/2))**(1/3)/(2*b**2) + x**(3/2)*(a + b*x**(3/2))**(1
/3)/(2*b), Ne(b, 0)), (x**3/(3*a**(2/3)), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(b*x^(3/2) + a)^(2/3), x)