3.210 \(\int \frac{x^2 \sqrt{1-a^2 x^2}}{(1-a x)^4} \, dx\)

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]

(2*Sqrt[1 - a^2*x^2])/(a^3*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(5*a^3*(1 - a*x)^4)
- (3*(1 - a^2*x^2)^(3/2))/(5*a^3*(1 - a*x)^3) - ArcSin[a*x]/a^3

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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]

(2*Sqrt[1 - a^2*x^2])/(a^3*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(5*a^3*(1 - a*x)^4)
- (3*(1 - a^2*x^2)^(3/2))/(5*a^3*(1 - a*x)^3) - ArcSin[a*x]/a^3

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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]

-asin(a*x)/a**3 + 2*sqrt(-a**2*x**2 + 1)/(a**3*(-a*x + 1)) - 3*(-a**2*x**2 + 1)*
*(3/2)/(5*a**3*(-a*x + 1)**3) + (-a**2*x**2 + 1)**(3/2)/(5*a**3*(-a*x + 1)**4)

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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]

(((-8 + 19*a*x - 13*a^2*x^2)*Sqrt[1 - a^2*x^2])/(-1 + a*x)^3 - 5*ArcSin[a*x])/(5
*a^3)

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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]

1/5/a^7/(x-1/a)^4*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(3/2)+3/5/a^6/(x-1/a)^3*(-(x-1/a)
^2*a^2-2*(x-1/a)*a)^(3/2)+1/a^5/(x-1/a)^2*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(3/2)+1/a
^3*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2)-1/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-(
x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2))

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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]

integrate(sqrt(-a^2*x^2 + 1)*x^2/(a*x - 1)^4, x)

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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]

1/5*(21*a^5*x^5 - 20*a^4*x^4 - 35*a^3*x^3 + 50*a^2*x^2 - 20*a*x + 10*(a^5*x^5 -
5*a^4*x^4 + 5*a^3*x^3 + 5*a^2*x^2 - 10*a*x + (a^4*x^4 - 7*a^2*x^2 + 10*a*x - 4)*
sqrt(-a^2*x^2 + 1) + 4)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 5*(a^4*x^4 - 9*
a^3*x^3 + 10*a^2*x^2 - 4*a*x)*sqrt(-a^2*x^2 + 1))/(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 + (a^7*x^4 - 7*a^5*x^2 + 10*a^4*x - 4*a^3)*sqr
t(-a^2*x^2 + 1))

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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]

Integral(x**2*sqrt(-(a*x - 1)*(a*x + 1))/(a*x - 1)**4, x)

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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]

-arcsin(a*x)*sign(a)/(a^2*abs(a)) - 2/5*(35*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2
*x) - 55*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 25*(sqrt(-a^2*x^2 + 1)*ab
s(a) + a)^3/(a^6*x^3) - 5*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) - 8)/(a^2*
((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^5*abs(a))