Optimal. Leaf size=95 \[ -\frac{\sin ^{-1}(a x)}{a^3}-\frac{3 \left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^4}+\frac{2 \sqrt{1-a^2 x^2}}{a^3 (1-a x)} \]
[Out]
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Rubi [A] time = 0.250216, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{\sin ^{-1}(a x)}{a^3}-\frac{3 \left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^4}+\frac{2 \sqrt{1-a^2 x^2}}{a^3 (1-a x)} \]
Antiderivative was successfully verified.
[In] Int[(x^2*Sqrt[1 - a^2*x^2])/(1 - a*x)^4,x]
[Out]
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Rubi in Sympy [A] time = 24.4743, size = 78, normalized size = 0.82 \[ - \frac{\operatorname{asin}{\left (a x \right )}}{a^{3}} + \frac{2 \sqrt{- a^{2} x^{2} + 1}}{a^{3} \left (- a x + 1\right )} - \frac{3 \left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{5 a^{3} \left (- a x + 1\right )^{3}} + \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{5 a^{3} \left (- a x + 1\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(-a**2*x**2+1)**(1/2)/(-a*x+1)**4,x)
[Out]
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Mathematica [A] time = 0.084815, size = 50, normalized size = 0.53 \[ \frac{\frac{\left (-13 a^2 x^2+19 a x-8\right ) \sqrt{1-a^2 x^2}}{(a x-1)^3}-5 \sin ^{-1}(a x)}{5 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*Sqrt[1 - a^2*x^2])/(1 - a*x)^4,x]
[Out]
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Maple [B] time = 0.025, size = 200, normalized size = 2.1 \[{\frac{1}{5\,{a}^{7}} \left ( - \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\, \left ( x-{a}^{-1} \right ) a \right ) ^{{\frac{3}{2}}} \left ( x-{a}^{-1} \right ) ^{-4}}+{\frac{3}{5\,{a}^{6}} \left ( - \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\, \left ( x-{a}^{-1} \right ) a \right ) ^{{\frac{3}{2}}} \left ( x-{a}^{-1} \right ) ^{-3}}+{\frac{1}{{a}^{5}} \left ( - \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\, \left ( x-{a}^{-1} \right ) a \right ) ^{{\frac{3}{2}}} \left ( x-{a}^{-1} \right ) ^{-2}}+{\frac{1}{{a}^{3}}\sqrt{- \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\, \left ( x-{a}^{-1} \right ) a}}-{\frac{1}{{a}^{2}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{- \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\, \left ( x-{a}^{-1} \right ) a}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(-a^2*x^2+1)^(1/2)/(-a*x+1)^4,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-a^{2} x^{2} + 1} x^{2}}{{\left (a x - 1\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-a^2*x^2 + 1)*x^2/(a*x - 1)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.289239, size = 352, normalized size = 3.71 \[ \frac{21 \, a^{5} x^{5} - 20 \, a^{4} x^{4} - 35 \, a^{3} x^{3} + 50 \, a^{2} x^{2} - 20 \, a x + 10 \,{\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 5 \, a^{3} x^{3} + 5 \, a^{2} x^{2} - 10 \, a x +{\left (a^{4} x^{4} - 7 \, a^{2} x^{2} + 10 \, a x - 4\right )} \sqrt{-a^{2} x^{2} + 1} + 4\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - 5 \,{\left (a^{4} x^{4} - 9 \, a^{3} x^{3} + 10 \, a^{2} x^{2} - 4 \, a x\right )} \sqrt{-a^{2} x^{2} + 1}}{5 \,{\left (a^{8} x^{5} - 5 \, a^{7} x^{4} + 5 \, a^{6} x^{3} + 5 \, a^{5} x^{2} - 10 \, a^{4} x + 4 \, a^{3} +{\left (a^{7} x^{4} - 7 \, a^{5} x^{2} + 10 \, a^{4} x - 4 \, a^{3}\right )} \sqrt{-a^{2} x^{2} + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-a^2*x^2 + 1)*x^2/(a*x - 1)^4,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{\left (a x - 1\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(-a**2*x**2+1)**(1/2)/(-a*x+1)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.275597, size = 217, normalized size = 2.28 \[ -\frac{\arcsin \left (a x\right ){\rm sign}\left (a\right )}{a^{2}{\left | a \right |}} - \frac{2 \,{\left (\frac{35 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{55 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac{25 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac{5 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} - 8\right )}}{5 \, a^{2}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{5}{\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-a^2*x^2 + 1)*x^2/(a*x - 1)^4,x, algorithm="giac")
[Out]