3.215 \(\int \frac{x^2 \left (A+B x^3\right )}{\sqrt{a+b x^3}} \, dx\)

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]

(2*(A*b - a*B)*Sqrt[a + b*x^3])/(3*b^2) + (2*B*(a + b*x^3)^(3/2))/(9*b^2)

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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]

(2*(A*b - a*B)*Sqrt[a + b*x^3])/(3*b^2) + (2*B*(a + b*x^3)^(3/2))/(9*b^2)

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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]

2*B*(a + b*x**3)**(3/2)/(9*b**2) + 2*sqrt(a + b*x**3)*(A*b - B*a)/(3*b**2)

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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]

(2*Sqrt[a + b*x^3]*(3*A*b - 2*a*B + b*B*x^3))/(9*b^2)

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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]

2/9*(b*x^3+a)^(1/2)*(B*b*x^3+3*A*b-2*B*a)/b^2

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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]

2/9*B*((b*x^3 + a)^(3/2)/b^2 - 3*sqrt(b*x^3 + a)*a/b^2) + 2/3*sqrt(b*x^3 + a)*A/
b

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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]

2/9*(B*b*x^3 - 2*B*a + 3*A*b)*sqrt(b*x^3 + a)/b^2

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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]

Piecewise((2*A*sqrt(a + b*x**3)/(3*b) - 4*B*a*sqrt(a + b*x**3)/(9*b**2) + 2*B*x*
*3*sqrt(a + b*x**3)/(9*b), Ne(b, 0)), ((A*x**3/3 + B*x**6/6)/sqrt(a), True))

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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]

2/9*((b*x^3 + a)^(3/2)*B - 3*sqrt(b*x^3 + a)*B*a + 3*sqrt(b*x^3 + a)*A*b)/b^2