3.171 \(\int \frac{a+\frac{b}{x^2}}{c+\frac{d}{x^2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{c^{3/2} \sqrt{d}}+\frac{a x}{c} \]

[Out]

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

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Rubi [A]  time = 0.0718176, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{c^{3/2} \sqrt{d}}+\frac{a x}{c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)/(c + d/x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.336, size = 34, normalized size = 0.87 \[ \frac{a x}{c} - \frac{\left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{c^{\frac{3}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)/(c+d/x**2),x)

[Out]

a*x/c - (a*d - b*c)*atan(sqrt(c)*x/sqrt(d))/(c**(3/2)*sqrt(d))

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Mathematica [A]  time = 0.0442703, size = 40, normalized size = 1.03 \[ \frac{a x}{c}-\frac{(a d-b c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{c^{3/2} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)/(c + d/x^2),x]

[Out]

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

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Maple [A]  time = 0.007, size = 45, normalized size = 1.2 \[{\frac{ax}{c}}-{\frac{ad}{c}\arctan \left ({cx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{b\arctan \left ({cx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)/(c+d/x^2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234813, size = 1, normalized size = 0.03 \[ \left [\frac{2 \, \sqrt{-c d} a x -{\left (b c - a d\right )} \log \left (-\frac{2 \, c d x -{\left (c x^{2} - d\right )} \sqrt{-c d}}{c x^{2} + d}\right )}{2 \, \sqrt{-c d} c}, \frac{\sqrt{c d} a x +{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{c d} x}{d}\right )}{\sqrt{c d} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/(c + d/x^2),x, algorithm="fricas")

[Out]

[1/2*(2*sqrt(-c*d)*a*x - (b*c - a*d)*log(-(2*c*d*x - (c*x^2 - d)*sqrt(-c*d))/(c*
x^2 + d)))/(sqrt(-c*d)*c), (sqrt(c*d)*a*x + (b*c - a*d)*arctan(sqrt(c*d)*x/d))/(
sqrt(c*d)*c)]

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Sympy [A]  time = 1.66471, size = 82, normalized size = 2.1 \[ \frac{a x}{c} + \frac{\sqrt{- \frac{1}{c^{3} d}} \left (a d - b c\right ) \log{\left (- c d \sqrt{- \frac{1}{c^{3} d}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{c^{3} d}} \left (a d - b c\right ) \log{\left (c d \sqrt{- \frac{1}{c^{3} d}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)/(c+d/x**2),x)

[Out]

a*x/c + sqrt(-1/(c**3*d))*(a*d - b*c)*log(-c*d*sqrt(-1/(c**3*d)) + x)/2 - sqrt(-
1/(c**3*d))*(a*d - b*c)*log(c*d*sqrt(-1/(c**3*d)) + x)/2

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GIAC/XCAS [A]  time = 0.213993, size = 45, normalized size = 1.15 \[ \frac{a x}{c} + \frac{{\left (b c - a d\right )} \arctan \left (\frac{c x}{\sqrt{c d}}\right )}{\sqrt{c d} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/(c + d/x^2),x, algorithm="giac")

[Out]

a*x/c + (b*c - a*d)*arctan(c*x/sqrt(c*d))/(sqrt(c*d)*c)