3.2937 \(\int x^2 \sqrt{a+b \left (c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=36 \[ \frac{2 x^3 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b \left (c x^2\right )^{3/2}} \]

[Out]

(2*x^3*(a + b*(c*x^2)^(3/2))^(3/2))/(9*b*(c*x^2)^(3/2))

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Rubi [A]  time = 0.0481334, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{2 x^3 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^3*(a + b*(c*x^2)^(3/2))^(3/2))/(9*b*(c*x^2)^(3/2))

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Rubi in Sympy [A]  time = 5.13997, size = 31, normalized size = 0.86 \[ \frac{2 x^{3} \left (a + b \left (c x^{2}\right )^{\frac{3}{2}}\right )^{\frac{3}{2}}}{9 b \left (c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(a+b*(c*x**2)**(3/2))**(1/2),x)

[Out]

2*x**3*(a + b*(c*x**2)**(3/2))**(3/2)/(9*b*(c*x**2)**(3/2))

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Mathematica [A]  time = 0.0365523, size = 37, normalized size = 1.03 \[ \frac{2 x \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(a + b*(c*x^2)^(3/2))^(3/2))/(9*b*c*Sqrt[c*x^2])

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Maple [A]  time = 0.009, size = 29, normalized size = 0.8 \[{\frac{2\,{x}^{3}}{9\,b} \left ( a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}} \right ) ^{{\frac{3}{2}}} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a+b*(c*x^2)^(3/2))^(1/2),x)

[Out]

2/9*x^3*(a+b*(c*x^2)^(3/2))^(3/2)/b/(c*x^2)^(3/2)

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Maxima [A]  time = 1.3573, size = 81, normalized size = 2.25 \[ \frac{2 \,{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{3}{2}}{\left (c - \sqrt{c}\right )}}{9 \, b{\left (c + 1\right )} c^{\frac{3}{2}}} + \frac{{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{3}{2}}}{3 \,{\left (c^{2} + c\right )} b \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(b*c^(3/2)*x^3 + a)^(3/2)*(c - sqrt(c))/(b*(c + 1)*c^(3/2)) + 1/3*(b*c^(3/2)
*x^3 + a)^(3/2)/((c^2 + c)*b*sqrt(c))

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Fricas [A]  time = 0.212305, size = 62, normalized size = 1.72 \[ \frac{2 \,{\left (b c^{2} x^{4} + \sqrt{c x^{2}} a\right )} \sqrt{\sqrt{c x^{2}} b c x^{2} + a}}{9 \, b c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(b*c^2*x^4 + sqrt(c*x^2)*a)*sqrt(sqrt(c*x^2)*b*c*x^2 + a)/(b*c^2*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{2} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(a+b*(c*x**2)**(3/2))**(1/2),x)

[Out]

Integral(x**2*sqrt(a + b*(c*x**2)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.218206, size = 27, normalized size = 0.75 \[ \frac{2 \,{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{3}{2}}}{9 \, b c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(b*c^(3/2)*x^3 + a)^(3/2)/(b*c^(3/2))