3.52 \(\int \frac{A+B x^2}{b x^2-c x^4} \, dx\)

Optimal. Leaf size=41 \[ \frac{(A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}-\frac{A}{b x} \]

[Out]

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

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Rubi [A]  time = 0.0735945, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{(A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}-\frac{A}{b x} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(b*x^2 - c*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.6369, size = 34, normalized size = 0.83 \[ - \frac{A}{b x} + \frac{\left (A c + B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{b^{\frac{3}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/(-c*x**4+b*x**2),x)

[Out]

-A/(b*x) + (A*c + B*b)*atanh(sqrt(c)*x/sqrt(b))/(b**(3/2)*sqrt(c))

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Mathematica [A]  time = 0.0416579, size = 41, normalized size = 1. \[ \frac{(A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2} \sqrt{c}}-\frac{A}{b x} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(b*x^2 - c*x^4),x]

[Out]

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

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Maple [A]  time = 0.009, size = 39, normalized size = 1. \[ -{\frac{-Ac-Bb}{b}{\it Artanh} \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{A}{bx}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/(-c*x^4+b*x^2),x)

[Out]

-(-A*c-B*b)/b/(b*c)^(1/2)*arctanh(c*x/(b*c)^(1/2))-A/b/x

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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)/(c*x^4 - b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219043, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (B b + A c\right )} x \log \left (\frac{2 \, b c x +{\left (c x^{2} + b\right )} \sqrt{b c}}{c x^{2} - b}\right ) - 2 \, \sqrt{b c} A}{2 \, \sqrt{b c} b x}, \frac{{\left (B b + A c\right )} x \arctan \left (\frac{\sqrt{-b c} x}{b}\right ) - \sqrt{-b c} A}{\sqrt{-b c} b x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(B*x^2 + A)/(c*x^4 - b*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.893399, size = 75, normalized size = 1.83 \[ - \frac{A}{b x} - \frac{\sqrt{\frac{1}{b^{3} c}} \left (A c + B b\right ) \log{\left (- b^{2} \sqrt{\frac{1}{b^{3} c}} + x \right )}}{2} + \frac{\sqrt{\frac{1}{b^{3} c}} \left (A c + B b\right ) \log{\left (b^{2} \sqrt{\frac{1}{b^{3} c}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/(-c*x**4+b*x**2),x)

[Out]

-A/(b*x) - sqrt(1/(b**3*c))*(A*c + B*b)*log(-b**2*sqrt(1/(b**3*c)) + x)/2 + sqrt
(1/(b**3*c))*(A*c + B*b)*log(b**2*sqrt(1/(b**3*c)) + x)/2

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GIAC/XCAS [A]  time = 0.208797, size = 51, normalized size = 1.24 \[ -\frac{{\left (B b + A c\right )} \arctan \left (\frac{c x}{\sqrt{-b c}}\right )}{\sqrt{-b c} b} - \frac{A}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(B*x^2 + A)/(c*x^4 - b*x^2),x, algorithm="giac")

[Out]

-(B*b + A*c)*arctan(c*x/sqrt(-b*c))/(sqrt(-b*c)*b) - A/(b*x)