3.246 \(\int \frac{x^2 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx\)

Optimal. Leaf size=46 \[ -\frac{2 (A b-a B)}{9 b^2 \left (a+b x^3\right )^{3/2}}-\frac{2 B}{3 b^2 \sqrt{a+b x^3}} \]

[Out]

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

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Rubi [A]  time = 0.134879, 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 \[ -\frac{2 (A b-a B)}{9 b^2 \left (a+b x^3\right )^{3/2}}-\frac{2 B}{3 b^2 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(A + B*x^3))/(a + b*x^3)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.2656, size = 44, normalized size = 0.96 \[ - \frac{2 B}{3 b^{2} \sqrt{a + b x^{3}}} - \frac{2 \left (A b - B a\right )}{9 b^{2} \left (a + b x^{3}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(B*x**3+A)/(b*x**3+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0336187, size = 33, normalized size = 0.72 \[ -\frac{2 \left (2 a B+A b+3 b B x^3\right )}{9 b^2 \left (a+b x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(A + B*x^3))/(a + b*x^3)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.009, size = 30, normalized size = 0.7 \[ -{\frac{6\,bB{x}^{3}+2\,Ab+4\,Ba}{9\,{b}^{2}} \left ( b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(B*x^3+A)/(b*x^3+a)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.39064, size = 66, normalized size = 1.43 \[ -\frac{2}{9} \, B{\left (\frac{3}{\sqrt{b x^{3} + a} b^{2}} - \frac{a}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} b^{2}}\right )} - \frac{2 \, A}{9 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^2/(b*x^3 + a)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.324719, size = 55, normalized size = 1.2 \[ -\frac{2 \,{\left (3 \, B b x^{3} + 2 \, B a + A b\right )}}{9 \,{\left (b^{3} x^{3} + a b^{2}\right )} \sqrt{b x^{3} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^2/(b*x^3 + a)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.86405, size = 144, normalized size = 3.13 \[ \begin{cases} - \frac{2 A b}{9 a b^{2} \sqrt{a + b x^{3}} + 9 b^{3} x^{3} \sqrt{a + b x^{3}}} - \frac{4 B a}{9 a b^{2} \sqrt{a + b x^{3}} + 9 b^{3} x^{3} \sqrt{a + b x^{3}}} - \frac{6 B b x^{3}}{9 a b^{2} \sqrt{a + b x^{3}} + 9 b^{3} x^{3} \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{3}}{3} + \frac{B x^{6}}{6}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(B*x**3+A)/(b*x**3+a)**(5/2),x)

[Out]

Piecewise((-2*A*b/(9*a*b**2*sqrt(a + b*x**3) + 9*b**3*x**3*sqrt(a + b*x**3)) - 4
*B*a/(9*a*b**2*sqrt(a + b*x**3) + 9*b**3*x**3*sqrt(a + b*x**3)) - 6*B*b*x**3/(9*
a*b**2*sqrt(a + b*x**3) + 9*b**3*x**3*sqrt(a + b*x**3)), Ne(b, 0)), ((A*x**3/3 +
B*x**6/6)/a**(5/2), True))

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GIAC/XCAS [A]  time = 0.215101, size = 43, normalized size = 0.93 \[ -\frac{2 \,{\left (3 \,{\left (b x^{3} + a\right )} B - B a + A b\right )}}{9 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^2/(b*x^3 + a)^(5/2),x, algorithm="giac")

[Out]

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