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

Optimal. Leaf size=32 \[ -\frac{\left (A+B x^2\right )^2}{4 \left (a+b x^2\right )^2 (A b-a B)} \]

[Out]

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

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Rubi [A]  time = 0.0621845, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{\left (A+B x^2\right )^2}{4 \left (a+b x^2\right )^2 (A b-a B)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.50241, size = 26, normalized size = 0.81 \[ - \frac{\left (A + B x^{2}\right )^{2}}{4 \left (a + b x^{2}\right )^{2} \left (A b - B a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.02174, size = 30, normalized size = 0.94 \[ -\frac{B \left (a+2 b x^2\right )+A b}{4 b^2 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 39, normalized size = 1.2 \[ -{\frac{Ab-Ba}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{B}{ \left ( 2\,b{x}^{2}+2\,a \right ){b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x^2+A)/(b*x^2+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.34172, size = 57, normalized size = 1.78 \[ -\frac{2 \, B b x^{2} + B a + A b}{4 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.219274, size = 57, normalized size = 1.78 \[ -\frac{2 \, B b x^{2} + B a + A b}{4 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.70544, size = 42, normalized size = 1.31 \[ - \frac{A b + B a + 2 B b x^{2}}{4 a^{2} b^{2} + 8 a b^{3} x^{2} + 4 b^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(A*b + B*a + 2*B*b*x**2)/(4*a**2*b**2 + 8*a*b**3*x**2 + 4*b**4*x**4)

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GIAC/XCAS [A]  time = 0.222838, size = 38, normalized size = 1.19 \[ -\frac{2 \, B b x^{2} + B a + A b}{4 \,{\left (b x^{2} + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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