Optimal. Leaf size=54 \[ \frac{x (A b-a B)}{a b \sqrt{a+b x^2}}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.052163, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{x (A b-a B)}{a b \sqrt{a+b x^2}}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(a + b*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 8.16072, size = 46, normalized size = 0.85 \[ \frac{B \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{b^{\frac{3}{2}}} + \frac{x \left (A b - B a\right )}{a b \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/(b*x**2+a)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.062452, size = 58, normalized size = 1.07 \[ \frac{B \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{b^{3/2}}-\frac{x (a B-A b)}{a b \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(a + b*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.007, size = 54, normalized size = 1. \[{\frac{Ax}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{Bx}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{B\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/(b*x^2+a)^(3/2),x)
[Out]
_______________________________________________________________________________________
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)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.237452, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \, \sqrt{b x^{2} + a}{\left (B a - A b\right )} \sqrt{b} x -{\left (B a b x^{2} + B a^{2}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{2 \,{\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt{b}}, -\frac{\sqrt{b x^{2} + a}{\left (B a - A b\right )} \sqrt{-b} x -{\left (B a b x^{2} + B a^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{{\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt{-b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(b*x^2 + a)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 11.0108, size = 60, normalized size = 1.11 \[ \frac{A x}{a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{2}}{a}}} + B \left (\frac{\operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} - \frac{x}{\sqrt{a} b \sqrt{1 + \frac{b x^{2}}{a}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/(b*x**2+a)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.243235, size = 69, normalized size = 1.28 \[ -\frac{B{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{b^{\frac{3}{2}}} - \frac{{\left (B a - A b\right )} x}{\sqrt{b x^{2} + a} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(b*x^2 + a)^(3/2),x, algorithm="giac")
[Out]