Optimal. Leaf size=33 \[ -\frac{\left (a^2-b^2 x^2\right )^{3/2}}{3 a b (a+b x)^3} \]
[Out]
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Rubi [A] time = 0.0361296, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{\left (a^2-b^2 x^2\right )^{3/2}}{3 a b (a+b x)^3} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 - b^2*x^2]/(a + b*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 4.91411, size = 26, normalized size = 0.79 \[ - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{3 a b \left (a + b x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.0282372, size = 39, normalized size = 1.18 \[ -\frac{(a-b x) \sqrt{a^2-b^2 x^2}}{3 a b (a+b x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^3,x]
[Out]
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Maple [A] time = 0.006, size = 36, normalized size = 1.1 \[ -{\frac{-bx+a}{3\, \left ( bx+a \right ) ^{2}ba}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b^2*x^2+a^2)^(1/2)/(b*x+a)^3,x)
[Out]
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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(sqrt(-b^2*x^2 + a^2)/(b*x + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21491, size = 126, normalized size = 3.82 \[ -\frac{2 \,{\left (b^{2} x^{3} - 3 \, a^{2} x + 3 \, \sqrt{-b^{2} x^{2} + a^{2}} a x\right )}}{3 \,{\left (a b^{3} x^{3} - 3 \, a^{3} b x - 2 \, a^{4} +{\left (a b^{2} x^{2} + 3 \, a^{2} b x + 2 \, a^{3}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b^2*x^2 + a^2)/(b*x + a)^3,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- a + b x\right ) \left (a + b x\right )}}{\left (a + b x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.232948, size = 100, normalized size = 3.03 \[ \frac{2 \,{\left (\frac{3 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + 1\right )}}{3 \, a{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{3}{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b^2*x^2 + a^2)/(b*x + a)^3,x, algorithm="giac")
[Out]