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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]