3.619 \(\int \frac{(f+g x)^2 \sqrt{1-x^2}}{(1-x)^4} \, dx\)

Optimal. Leaf size=80 \[ \frac{(x+1)^4 (f+g)^2}{5 \left (1-x^2\right )^{5/2}}+\frac{(x+1)^3 (f-9 g) (f+g)}{15 \left (1-x^2\right )^{3/2}}+\frac{2 g^2 (x+1)}{\sqrt{1-x^2}}-g^2 \sin ^{-1}(x) \]

[Out]

((f + g)^2*(1 + x)^4)/(5*(1 - x^2)^(5/2)) + ((f - 9*g)*(f + g)*(1 + x)^3)/(15*(1
- x^2)^(3/2)) + (2*g^2*(1 + x))/Sqrt[1 - x^2] - g^2*ArcSin[x]

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Rubi [A]  time = 0.27267, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{(x+1)^4 (f+g)^2}{5 \left (1-x^2\right )^{5/2}}+\frac{(x+1)^3 (f-9 g) (f+g)}{15 \left (1-x^2\right )^{3/2}}+\frac{2 g^2 (x+1)}{\sqrt{1-x^2}}-g^2 \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)^2*Sqrt[1 - x^2])/(1 - x)^4,x]

[Out]

((f + g)^2*(1 + x)^4)/(5*(1 - x^2)^(5/2)) + ((f - 9*g)*(f + g)*(1 + x)^3)/(15*(1
- x^2)^(3/2)) + (2*g^2*(1 + x))/Sqrt[1 - x^2] - g^2*ArcSin[x]

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Rubi in Sympy [A]  time = 21.7769, size = 85, normalized size = 1.06 \[ - g^{2} \operatorname{asin}{\left (x \right )} + \frac{2 g^{2} \sqrt{- x^{2} + 1}}{- x + 1} - \frac{2 g \left (f + g\right ) \left (- x^{2} + 1\right )^{\frac{3}{2}}}{3 \left (- x + 1\right )^{3}} + \frac{\left (f + g\right )^{2} \left (- x^{2} + 1\right )^{\frac{3}{2}}}{15 \left (- x + 1\right )^{3}} + \frac{\left (f + g\right )^{2} \left (- x^{2} + 1\right )^{\frac{3}{2}}}{5 \left (- x + 1\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**2*(-x**2+1)**(1/2)/(1-x)**4,x)

[Out]

-g**2*asin(x) + 2*g**2*sqrt(-x**2 + 1)/(-x + 1) - 2*g*(f + g)*(-x**2 + 1)**(3/2)
/(3*(-x + 1)**3) + (f + g)**2*(-x**2 + 1)**(3/2)/(15*(-x + 1)**3) + (f + g)**2*(
-x**2 + 1)**(3/2)/(5*(-x + 1)**4)

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Mathematica [A]  time = 0.157479, size = 104, normalized size = 1.3 \[ \frac{\sqrt{1-x^2} \left (\sqrt{x+1} \left (f^2 \left (x^2-3 x-4\right )-2 f g \left (4 x^2+3 x-1\right )-3 g^2 \left (13 x^2-19 x+8\right )\right )+30 g^2 (x-1)^{5/2} \sinh ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right )\right )}{15 (x-1)^3 \sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)^2*Sqrt[1 - x^2])/(1 - x)^4,x]

[Out]

(Sqrt[1 - x^2]*(Sqrt[1 + x]*(f^2*(-4 - 3*x + x^2) - 2*f*g*(-1 + 3*x + 4*x^2) - 3
*g^2*(8 - 19*x + 13*x^2)) + 30*g^2*(-1 + x)^(5/2)*ArcSinh[Sqrt[-1 + x]/Sqrt[2]])
)/(15*(-1 + x)^3*Sqrt[1 + x])

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Maple [A]  time = 0.022, size = 125, normalized size = 1.6 \[{g}^{2} \left ({\frac{1}{ \left ( -1+x \right ) ^{2}} \left ( - \left ( -1+x \right ) ^{2}-2\,x+2 \right ) ^{{\frac{3}{2}}}}+\sqrt{- \left ( -1+x \right ) ^{2}-2\,x+2}-\arcsin \left ( x \right ) \right ) + \left ({f}^{2}+2\,fg+{g}^{2} \right ) \left ({\frac{1}{5\, \left ( -1+x \right ) ^{4}} \left ( - \left ( -1+x \right ) ^{2}-2\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{15\, \left ( -1+x \right ) ^{3}} \left ( - \left ( -1+x \right ) ^{2}-2\,x+2 \right ) ^{{\frac{3}{2}}}} \right ) +{\frac{2\,g \left ( f+g \right ) }{3\, \left ( -1+x \right ) ^{3}} \left ( - \left ( -1+x \right ) ^{2}-2\,x+2 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^2*(-x^2+1)^(1/2)/(1-x)^4,x)

[Out]

g^2*(1/(-1+x)^2*(-(-1+x)^2-2*x+2)^(3/2)+(-(-1+x)^2-2*x+2)^(1/2)-arcsin(x))+(f^2+
2*f*g+g^2)*(1/5/(-1+x)^4*(-(-1+x)^2-2*x+2)^(3/2)-1/15/(-1+x)^3*(-(-1+x)^2-2*x+2)
^(3/2))+2/3*g*(f+g)/(-1+x)^3*(-(-1+x)^2-2*x+2)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (g x + f\right )}^{2} \sqrt{-x^{2} + 1}}{{\left (x - 1\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)^2*sqrt(-x^2 + 1)/(x - 1)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.28936, size = 416, normalized size = 5.2 \[ \frac{3 \,{\left (f^{2} + 2 \, f g + 21 \, g^{2}\right )} x^{5} - 20 \,{\left (f^{2} - 2 \, f g + 3 \, g^{2}\right )} x^{4} + 5 \,{\left (7 \, f^{2} - 2 \, f g - 21 \, g^{2}\right )} x^{3} + 30 \,{\left (f^{2} - 2 \, f g + 5 \, g^{2}\right )} x^{2} - 60 \,{\left (f^{2} + g^{2}\right )} x + 30 \,{\left (g^{2} x^{5} - 5 \, g^{2} x^{4} + 5 \, g^{2} x^{3} + 5 \, g^{2} x^{2} - 10 \, g^{2} x + 4 \, g^{2} +{\left (g^{2} x^{4} - 7 \, g^{2} x^{2} + 10 \, g^{2} x - 4 \, g^{2}\right )} \sqrt{-x^{2} + 1}\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) + 5 \,{\left ({\left (f^{2} - 2 \, f g - 3 \, g^{2}\right )} x^{4} -{\left (f^{2} - 2 \, f g - 27 \, g^{2}\right )} x^{3} - 6 \,{\left (f^{2} - 2 \, f g + 5 \, g^{2}\right )} x^{2} + 12 \,{\left (f^{2} + g^{2}\right )} x\right )} \sqrt{-x^{2} + 1}}{15 \,{\left (x^{5} - 5 \, x^{4} + 5 \, x^{3} + 5 \, x^{2} +{\left (x^{4} - 7 \, x^{2} + 10 \, x - 4\right )} \sqrt{-x^{2} + 1} - 10 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)^2*sqrt(-x^2 + 1)/(x - 1)^4,x, algorithm="fricas")

[Out]

1/15*(3*(f^2 + 2*f*g + 21*g^2)*x^5 - 20*(f^2 - 2*f*g + 3*g^2)*x^4 + 5*(7*f^2 - 2
*f*g - 21*g^2)*x^3 + 30*(f^2 - 2*f*g + 5*g^2)*x^2 - 60*(f^2 + g^2)*x + 30*(g^2*x
^5 - 5*g^2*x^4 + 5*g^2*x^3 + 5*g^2*x^2 - 10*g^2*x + 4*g^2 + (g^2*x^4 - 7*g^2*x^2
+ 10*g^2*x - 4*g^2)*sqrt(-x^2 + 1))*arctan((sqrt(-x^2 + 1) - 1)/x) + 5*((f^2 -
2*f*g - 3*g^2)*x^4 - (f^2 - 2*f*g - 27*g^2)*x^3 - 6*(f^2 - 2*f*g + 5*g^2)*x^2 +
12*(f^2 + g^2)*x)*sqrt(-x^2 + 1))/(x^5 - 5*x^4 + 5*x^3 + 5*x^2 + (x^4 - 7*x^2 +
10*x - 4)*sqrt(-x^2 + 1) - 10*x + 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right )} \left (f + g x\right )^{2}}{\left (x - 1\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)**2*(-x**2+1)**(1/2)/(1-x)**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.296428, size = 359, normalized size = 4.49 \[ -g^{2} \arcsin \left (x\right ) + \frac{2 \,{\left (4 \, f^{2} - 2 \, f g + 24 \, g^{2} + \frac{5 \, f^{2}{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} - \frac{10 \, f g{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} + \frac{105 \, g^{2}{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} + \frac{25 \, f^{2}{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + \frac{10 \, f g{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + \frac{165 \, g^{2}{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + \frac{15 \, f^{2}{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}}{x^{3}} - \frac{30 \, f g{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}}{x^{3}} + \frac{75 \, g^{2}{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}}{x^{3}} + \frac{15 \, f^{2}{\left (\sqrt{-x^{2} + 1} - 1\right )}^{4}}{x^{4}} + \frac{15 \, g^{2}{\left (\sqrt{-x^{2} + 1} - 1\right )}^{4}}{x^{4}}\right )}}{15 \,{\left (\frac{\sqrt{-x^{2} + 1} - 1}{x} + 1\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)^2*sqrt(-x^2 + 1)/(x - 1)^4,x, algorithm="giac")

[Out]

-g^2*arcsin(x) + 2/15*(4*f^2 - 2*f*g + 24*g^2 + 5*f^2*(sqrt(-x^2 + 1) - 1)/x - 1
0*f*g*(sqrt(-x^2 + 1) - 1)/x + 105*g^2*(sqrt(-x^2 + 1) - 1)/x + 25*f^2*(sqrt(-x^
2 + 1) - 1)^2/x^2 + 10*f*g*(sqrt(-x^2 + 1) - 1)^2/x^2 + 165*g^2*(sqrt(-x^2 + 1)
- 1)^2/x^2 + 15*f^2*(sqrt(-x^2 + 1) - 1)^3/x^3 - 30*f*g*(sqrt(-x^2 + 1) - 1)^3/x
^3 + 75*g^2*(sqrt(-x^2 + 1) - 1)^3/x^3 + 15*f^2*(sqrt(-x^2 + 1) - 1)^4/x^4 + 15*
g^2*(sqrt(-x^2 + 1) - 1)^4/x^4)/((sqrt(-x^2 + 1) - 1)/x + 1)^5