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

Optimal. Leaf size=71 \[ -\frac{(3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{x (A b-a B)}{2 a^2 \left (a+b x^2\right )}-\frac{A}{a^2 x} \]

[Out]

-(A/(a^2*x)) - ((A*b - a*B)*x)/(2*a^2*(a + b*x^2)) - ((3*A*b - a*B)*ArcTan[(Sqrt
[b]*x)/Sqrt[a]])/(2*a^(5/2)*Sqrt[b])

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Rubi [A]  time = 0.148648, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{(3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{x (A b-a B)}{2 a^2 \left (a+b x^2\right )}-\frac{A}{a^2 x} \]

Antiderivative was successfully verified.

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

[Out]

-(A/(a^2*x)) - ((A*b - a*B)*x)/(2*a^2*(a + b*x^2)) - ((3*A*b - a*B)*ArcTan[(Sqrt
[b]*x)/Sqrt[a]])/(2*a^(5/2)*Sqrt[b])

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Rubi in Sympy [A]  time = 19.3615, size = 61, normalized size = 0.86 \[ - \frac{A}{a^{2} x} - \frac{x \left (A b - B a\right )}{2 a^{2} \left (a + b x^{2}\right )} - \frac{\left (3 A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(a**2*x) - x*(A*b - B*a)/(2*a**2*(a + b*x**2)) - (3*A*b - B*a)*atan(sqrt(b)*x
/sqrt(a))/(2*a**(5/2)*sqrt(b))

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Mathematica [A]  time = 0.0556774, size = 70, normalized size = 0.99 \[ \frac{(a B-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}+\frac{x (a B-A b)}{2 a^2 \left (a+b x^2\right )}-\frac{A}{a^2 x} \]

Antiderivative was successfully verified.

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

[Out]

-(A/(a^2*x)) + ((-(A*b) + a*B)*x)/(2*a^2*(a + b*x^2)) + ((-3*A*b + a*B)*ArcTan[(
Sqrt[b]*x)/Sqrt[a]])/(2*a^(5/2)*Sqrt[b])

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Maple [A]  time = 0.014, size = 85, normalized size = 1.2 \[ -{\frac{A}{{a}^{2}x}}-{\frac{Axb}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{xB}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{3\,Ab}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243319, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{3} +{\left (B a^{2} - 3 \, A a b\right )} x\right )} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left ({\left (B a - 3 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt{-a b}}{4 \,{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{-a b}}, \frac{{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{3} +{\left (B a^{2} - 3 \, A a b\right )} x\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left ({\left (B a - 3 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt{a b}}{2 \,{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(((B*a*b - 3*A*b^2)*x^3 + (B*a^2 - 3*A*a*b)*x)*log(-(2*a*b*x - (b*x^2 - a)
*sqrt(-a*b))/(b*x^2 + a)) - 2*((B*a - 3*A*b)*x^2 - 2*A*a)*sqrt(-a*b))/((a^2*b*x^
3 + a^3*x)*sqrt(-a*b)), 1/2*(((B*a*b - 3*A*b^2)*x^3 + (B*a^2 - 3*A*a*b)*x)*arcta
n(sqrt(a*b)*x/a) + ((B*a - 3*A*b)*x^2 - 2*A*a)*sqrt(a*b))/((a^2*b*x^3 + a^3*x)*s
qrt(a*b))]

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Sympy [A]  time = 2.50535, size = 114, normalized size = 1.61 \[ - \frac{\sqrt{- \frac{1}{a^{5} b}} \left (- 3 A b + B a\right ) \log{\left (- a^{3} \sqrt{- \frac{1}{a^{5} b}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{5} b}} \left (- 3 A b + B a\right ) \log{\left (a^{3} \sqrt{- \frac{1}{a^{5} b}} + x \right )}}{4} + \frac{- 2 A a + x^{2} \left (- 3 A b + B a\right )}{2 a^{3} x + 2 a^{2} b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(a**5*b))*(-3*A*b + B*a)*log(-a**3*sqrt(-1/(a**5*b)) + x)/4 + sqrt(-1/(
a**5*b))*(-3*A*b + B*a)*log(a**3*sqrt(-1/(a**5*b)) + x)/4 + (-2*A*a + x**2*(-3*A
*b + B*a))/(2*a**3*x + 2*a**2*b*x**3)

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GIAC/XCAS [A]  time = 0.222502, size = 84, normalized size = 1.18 \[ \frac{{\left (B a - 3 \, A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2}} + \frac{B a x^{2} - 3 \, A b x^{2} - 2 \, A a}{2 \,{\left (b x^{3} + a x\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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