Optimal. Leaf size=34 \[ -\frac{1}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0181075, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{1}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^(-3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 1.98816, size = 32, normalized size = 0.94 \[ - \frac{2 a + 2 b x}{4 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.016426, size = 23, normalized size = 0.68 \[ -\frac{a+b x}{2 b \left ((a+b x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(-3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.004, size = 20, normalized size = 0.6 \[ -{\frac{bx+a}{2\,b} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.724247, size = 22, normalized size = 0.65 \[ -\frac{1}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(-3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.217467, size = 32, normalized size = 0.94 \[ -\frac{1}{2 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(-3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.548649, size = 4, normalized size = 0.12 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(-3/2),x, algorithm="giac")
[Out]