3.116 \(\int \frac{A+B x^2}{a-b x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0543235, antiderivative size = 39, 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 \[ \frac{(a B+A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}-\frac{B x}{b} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(a - b*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.5302, size = 34, normalized size = 0.87 \[ - \frac{B x}{b} + \frac{\left (A b + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{a} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0376031, size = 39, normalized size = 1. \[ \frac{(a B+A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}-\frac{B x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 37, normalized size = 1. \[ -{\frac{Bx}{b}}-{\frac{-Ab-Ba}{b}{\it Artanh} \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/(-b*x^2+a),x)

[Out]

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

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235768, size = 1, normalized size = 0.03 \[ \left [-\frac{2 \, \sqrt{a b} B x -{\left (B a + A b\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} + a\right )} \sqrt{a b}}{b x^{2} - a}\right )}{2 \, \sqrt{a b} b}, -\frac{\sqrt{-a b} B x -{\left (B a + A b\right )} \arctan \left (\frac{\sqrt{-a b} x}{a}\right )}{\sqrt{-a b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.63868, size = 75, normalized size = 1.92 \[ - \frac{B x}{b} - \frac{\sqrt{\frac{1}{a b^{3}}} \left (A b + B a\right ) \log{\left (- a b \sqrt{\frac{1}{a b^{3}}} + x \right )}}{2} + \frac{\sqrt{\frac{1}{a b^{3}}} \left (A b + B a\right ) \log{\left (a b \sqrt{\frac{1}{a b^{3}}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.227883, size = 49, normalized size = 1.26 \[ -\frac{B x}{b} - \frac{{\left (B a + A b\right )} \arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{\sqrt{-a b} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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