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

Optimal. Leaf size=63 \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{3/2}}+\frac{x (A b-a B)}{2 a b \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.061669, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{3/2}}+\frac{x (A b-a B)}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.44616, size = 51, normalized size = 0.81 \[ \frac{x \left (A b - B a\right )}{2 a b \left (a + b x^{2}\right )} + \frac{\left (A b + B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0812478, size = 63, normalized size = 1. \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{3/2}}-\frac{x (a B-A b)}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 68, normalized size = 1.1 \[{\frac{ \left ( Ab-Ba \right ) x}{2\,ab \left ( b{x}^{2}+a \right ) }}+{\frac{A}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/(b*x^2+a)^2,x)

[Out]

1/2*(A*b-B*a)*x/a/b/(b*x^2+a)+1/2/a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*A+1/2/b/
(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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23813, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \,{\left (B a - A b\right )} \sqrt{-a b} x -{\left (B a^{2} + A a b +{\left (B a b + A b^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right )}{4 \,{\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt{-a b}}, -\frac{{\left (B a - A b\right )} \sqrt{a b} x -{\left (B a^{2} + A a b +{\left (B a b + A b^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{2 \,{\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(2*(B*a - A*b)*sqrt(-a*b)*x - (B*a^2 + A*a*b + (B*a*b + A*b^2)*x^2)*log((2
*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)))/((a*b^2*x^2 + a^2*b)*sqrt(-a*b)),
-1/2*((B*a - A*b)*sqrt(a*b)*x - (B*a^2 + A*a*b + (B*a*b + A*b^2)*x^2)*arctan(sq
rt(a*b)*x/a))/((a*b^2*x^2 + a^2*b)*sqrt(a*b))]

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Sympy [A]  time = 2.11941, size = 112, normalized size = 1.78 \[ - \frac{x \left (- A b + B a\right )}{2 a^{2} b + 2 a b^{2} x^{2}} - \frac{\sqrt{- \frac{1}{a^{3} b^{3}}} \left (A b + B a\right ) \log{\left (- a^{2} b \sqrt{- \frac{1}{a^{3} b^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} b^{3}}} \left (A b + B a\right ) \log{\left (a^{2} b \sqrt{- \frac{1}{a^{3} b^{3}}} + x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.225755, size = 77, normalized size = 1.22 \[ \frac{{\left (B a + A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a b} - \frac{B a x - A b x}{2 \,{\left (b x^{2} + a\right )} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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