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

Optimal. Leaf size=46 \[ \frac{2 B \sqrt{a+b x^3}}{3 b^2}-\frac{2 (A b-a B)}{3 b^2 \sqrt{a+b x^3}} \]

[Out]

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

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Rubi [A]  time = 0.131313, 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 \[ \frac{2 B \sqrt{a+b x^3}}{3 b^2}-\frac{2 (A b-a B)}{3 b^2 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.0311, size = 42, normalized size = 0.91 \[ \frac{2 B \sqrt{a + b x^{3}}}{3 b^{2}} - \frac{2 \left (A b - B a\right )}{3 b^{2} \sqrt{a + b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0303689, size = 33, normalized size = 0.72 \[ \frac{2 \left (2 a B-A b+b B x^3\right )}{3 b^2 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 30, normalized size = 0.7 \[ -{\frac{-2\,bB{x}^{3}+2\,Ab-4\,Ba}{3\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{3}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.36576, size = 63, normalized size = 1.37 \[ \frac{2}{3} \, B{\left (\frac{\sqrt{b x^{3} + a}}{b^{2}} + \frac{a}{\sqrt{b x^{3} + a} b^{2}}\right )} - \frac{2 \, A}{3 \, \sqrt{b x^{3} + a} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.249605, size = 39, normalized size = 0.85 \[ \frac{2 \,{\left (B b x^{3} + 2 \, B a - A b\right )}}{3 \, \sqrt{b x^{3} + a} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.07461, size = 75, normalized size = 1.63 \[ \begin{cases} - \frac{2 A}{3 b \sqrt{a + b x^{3}}} + \frac{4 B a}{3 b^{2} \sqrt{a + b x^{3}}} + \frac{2 B x^{3}}{3 b \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{3}}{3} + \frac{B x^{6}}{6}}{a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213419, size = 47, normalized size = 1.02 \[ \frac{2 \,{\left (\sqrt{b x^{3} + a} B + \frac{B a - A b}{\sqrt{b x^{3} + a}}\right )}}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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