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

Optimal. Leaf size=32 \[ -\frac{\left (A+B x^3\right )^2}{6 \left (a+b x^3\right )^2 (A b-a B)} \]

[Out]

-(A + B*x^3)^2/(6*(A*b - a*B)*(a + b*x^3)^2)

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Rubi [A]  time = 0.08007, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{\left (A+B x^3\right )^2}{6 \left (a+b x^3\right )^2 (A b-a B)} \]

Antiderivative was successfully verified.

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

[Out]

-(A + B*x^3)^2/(6*(A*b - a*B)*(a + b*x^3)^2)

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Rubi in Sympy [A]  time = 8.11514, size = 26, normalized size = 0.81 \[ - \frac{\left (A + B x^{3}\right )^{2}}{6 \left (a + b x^{3}\right )^{2} \left (A b - B a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(A + B*x**3)**2/(6*(a + b*x**3)**2*(A*b - B*a))

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Mathematica [A]  time = 0.0263372, size = 30, normalized size = 0.94 \[ -\frac{B \left (a+2 b x^3\right )+A b}{6 b^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(A*b + B*(a + 2*b*x^3))/(6*b^2*(a + b*x^3)^2)

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Maple [A]  time = 0.007, size = 39, normalized size = 1.2 \[ -{\frac{Ab-Ba}{6\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{B}{ \left ( 3\,b{x}^{3}+3\,a \right ){b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(A*b-B*a)/b^2/(b*x^3+a)^2-1/3*B/(b*x^3+a)/b^2

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Maxima [A]  time = 1.36835, size = 57, normalized size = 1.78 \[ -\frac{2 \, B b x^{3} + B a + A b}{6 \,{\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(2*B*b*x^3 + B*a + A*b)/(b^4*x^6 + 2*a*b^3*x^3 + a^2*b^2)

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Fricas [A]  time = 0.216746, size = 57, normalized size = 1.78 \[ -\frac{2 \, B b x^{3} + B a + A b}{6 \,{\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(2*B*b*x^3 + B*a + A*b)/(b^4*x^6 + 2*a*b^3*x^3 + a^2*b^2)

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Sympy [A]  time = 4.34779, size = 42, normalized size = 1.31 \[ - \frac{A b + B a + 2 B b x^{3}}{6 a^{2} b^{2} + 12 a b^{3} x^{3} + 6 b^{4} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(A*b + B*a + 2*B*b*x**3)/(6*a**2*b**2 + 12*a*b**3*x**3 + 6*b**4*x**6)

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GIAC/XCAS [A]  time = 0.219386, size = 38, normalized size = 1.19 \[ -\frac{2 \, B b x^{3} + B a + A b}{6 \,{\left (b x^{3} + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(2*B*b*x^3 + B*a + A*b)/((b*x^3 + a)^2*b^2)