Optimal. Leaf size=46 \[ \frac{2 \sqrt{a+b x^3} (A b-a B)}{3 b^2}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^2} \]
[Out]
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Rubi [A] time = 0.132885, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2 \sqrt{a+b x^3} (A b-a B)}{3 b^2}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^2} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(A + B*x^3))/Sqrt[a + b*x^3],x]
[Out]
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Rubi in Sympy [A] time = 12.175, size = 41, normalized size = 0.89 \[ \frac{2 B \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 b^{2}} + \frac{2 \sqrt{a + b x^{3}} \left (A b - B a\right )}{3 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(B*x**3+A)/(b*x**3+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0299018, size = 33, normalized size = 0.72 \[ \frac{2 \sqrt{a+b x^3} \left (-2 a B+3 A b+b B x^3\right )}{9 b^2} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(A + B*x^3))/Sqrt[a + b*x^3],x]
[Out]
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Maple [A] time = 0.009, size = 30, normalized size = 0.7 \[{\frac{2\,bB{x}^{3}+6\,Ab-4\,Ba}{9\,{b}^{2}}\sqrt{b{x}^{3}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(B*x^3+A)/(b*x^3+a)^(1/2),x)
[Out]
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Maxima [A] time = 1.37796, size = 65, normalized size = 1.41 \[ \frac{2}{9} \, B{\left (\frac{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{b^{2}} - \frac{3 \, \sqrt{b x^{3} + a} a}{b^{2}}\right )} + \frac{2 \, \sqrt{b x^{3} + a} A}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^2/sqrt(b*x^3 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.248998, size = 39, normalized size = 0.85 \[ \frac{2 \,{\left (B b x^{3} - 2 \, B a + 3 \, A b\right )} \sqrt{b x^{3} + a}}{9 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^2/sqrt(b*x^3 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.64358, size = 75, normalized size = 1.63 \[ \begin{cases} \frac{2 A \sqrt{a + b x^{3}}}{3 b} - \frac{4 B a \sqrt{a + b x^{3}}}{9 b^{2}} + \frac{2 B x^{3} \sqrt{a + b x^{3}}}{9 b} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{3}}{3} + \frac{B x^{6}}{6}}{\sqrt{a}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(B*x**3+A)/(b*x**3+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.211573, size = 58, normalized size = 1.26 \[ \frac{2 \,{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} B - 3 \, \sqrt{b x^{3} + a} B a + 3 \, \sqrt{b x^{3} + a} A b\right )}}{9 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^2/sqrt(b*x^3 + a),x, algorithm="giac")
[Out]