3.221 \(\int \frac{x^2 \sqrt{1+x} \sqrt [4]{1-x^2}}{\sqrt{1-x} \left (\sqrt{1-x}-\sqrt{1+x}\right )} \, dx\)

Optimal. Leaf size=304 \[ \frac{1}{6} \sqrt{x+1} \left (1-x^2\right )^{5/4}+\frac{x \left (1-x^2\right )^{5/4}}{6 \sqrt{1-x}}+\frac{7 \left (1-x^2\right )^{5/4}}{24 \sqrt{1-x}}+\frac{1}{24} (x+1)^{3/4} (1-x)^{5/4}+\frac{5}{16} \sqrt [4]{x+1} (1-x)^{3/4}-\frac{1}{16} (x+1)^{3/4} \sqrt [4]{1-x}+\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}}-\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{8 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}} \]

[Out]

(5*(1 - x)^(3/4)*(1 + x)^(1/4))/16 - ((1 - x)^(1/4)*(1 + x)^(3/4))/16 + ((1 - x)
^(5/4)*(1 + x)^(3/4))/24 + (7*(1 - x^2)^(5/4))/(24*Sqrt[1 - x]) + (x*(1 - x^2)^(
5/4))/(6*Sqrt[1 - x]) + (Sqrt[1 + x]*(1 - x^2)^(5/4))/6 - (3*ArcTan[1 - (Sqrt[2]
*(1 - x)^(1/4))/(1 + x)^(1/4)])/(8*Sqrt[2]) + (3*ArcTan[1 + (Sqrt[2]*(1 - x)^(1/
4))/(1 + x)^(1/4)])/(8*Sqrt[2]) + Log[1 + Sqrt[1 - x]/Sqrt[1 + x] - (Sqrt[2]*(1
- x)^(1/4))/(1 + x)^(1/4)]/(8*Sqrt[2]) - Log[1 + Sqrt[1 - x]/Sqrt[1 + x] + (Sqrt
[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(8*Sqrt[2])

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Rubi [A]  time = 1.36897, antiderivative size = 319, normalized size of antiderivative = 1.05, number of steps used = 31, number of rules used = 14, integrand size = 52, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}+\frac{1}{6} \sqrt{x+1} \left (1-x^2\right )^{5/4}+\frac{1}{6} (1-x)^{7/4} (x+1)^{5/4}+\frac{1}{24} (1-x)^{5/4} (x+1)^{3/4}-\frac{1}{16} \sqrt [4]{1-x} (x+1)^{3/4}+\frac{5}{24} (1-x)^{7/4} \sqrt [4]{x+1}-\frac{5}{48} (1-x)^{3/4} \sqrt [4]{x+1}+\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}}-\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{8 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(1 - x)^(3/4)*(1 + x)^(1/4))/48 + (5*(1 - x)^(7/4)*(1 + x)^(1/4))/24 - ((1 -
x)^(1/4)*(1 + x)^(3/4))/16 + ((1 - x)^(5/4)*(1 + x)^(3/4))/24 + ((1 - x)^(7/4)*
(1 + x)^(5/4))/6 + (Sqrt[1 + x]*(1 - x^2)^(5/4))/6 + (1 - x^2)^(9/4)/(3*(1 - x)^
(3/2)) - (3*ArcTan[1 - (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)])/(8*Sqrt[2]) + (3*
ArcTan[1 + (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)])/(8*Sqrt[2]) + Log[1 + Sqrt[1
- x]/Sqrt[1 + x] - (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(8*Sqrt[2]) - Log[1 +
Sqrt[1 - x]/Sqrt[1 + x] + (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(8*Sqrt[2])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\left (- x + 1\right )^{\frac{5}{4}} \left (x + 1\right )^{\frac{3}{4}}}{24} - \frac{\sqrt [4]{- x + 1} \left (x + 1\right )^{\frac{3}{4}}}{16} + \frac{\sqrt{x + 1} \left (- x^{2} + 1\right )^{\frac{5}{4}}}{6} - \frac{\sqrt{2} \log{\left (- \frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} + \frac{\sqrt{- x + 1}}{\sqrt{x + 1}} + 1 \right )}}{64} + \frac{\sqrt{2} \log{\left (\frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} + \frac{\sqrt{- x + 1}}{\sqrt{x + 1}} + 1 \right )}}{64} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} - 1 \right )}}{32} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} + 1 \right )}}{32} + \int ^{\sqrt{- x + 1}} \left (x^{2} - 2\right ) \left (x^{2} - 1\right ) \sqrt [4]{- \left (- x^{2} + 1\right )^{2} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-x + 1)**(5/4)*(x + 1)**(3/4)/24 - (-x + 1)**(1/4)*(x + 1)**(3/4)/16 + sqrt(x +
1)*(-x**2 + 1)**(5/4)/6 - sqrt(2)*log(-sqrt(2)*(-x + 1)**(1/4)/(x + 1)**(1/4) +
sqrt(-x + 1)/sqrt(x + 1) + 1)/64 + sqrt(2)*log(sqrt(2)*(-x + 1)**(1/4)/(x + 1)*
*(1/4) + sqrt(-x + 1)/sqrt(x + 1) + 1)/64 + sqrt(2)*atan(sqrt(2)*(-x + 1)**(1/4)
/(x + 1)**(1/4) - 1)/32 + sqrt(2)*atan(sqrt(2)*(-x + 1)**(1/4)/(x + 1)**(1/4) +
1)/32 + Integral((x**2 - 2)*(x**2 - 1)*(-(-x**2 + 1)**2 + 1)**(1/4), (x, sqrt(-x
+ 1)))

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Mathematica [C]  time = 0.39659, size = 165, normalized size = 0.54 \[ \frac{\sqrt [4]{-(x-1)^2-2 (x-1)} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{1-x}{2}\right )}{8 \sqrt [4]{2} \sqrt [4]{x+1}}+\frac{5 \left (-(x-1)^2-2 (x-1)\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{1-x}{2}\right )}{24\ 2^{3/4} (x+1)^{3/4}}-\frac{1}{48} \sqrt{x+1} \sqrt [4]{1-x^2} \left (8 x^2-\frac{\sqrt{1-x^2} \left (8 x^2+22 x+29\right )}{x+1}+2 x-7\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 + x]*(1 - x^2)^(1/4)*(-7 + 2*x + 8*x^2 - (Sqrt[1 - x^2]*(29 + 22*x + 8*
x^2))/(1 + x)))/48 + ((-2*(-1 + x) - (-1 + x)^2)^(1/4)*Hypergeometric2F1[1/4, 1/
4, 5/4, (1 - x)/2])/(8*2^(1/4)*(1 + x)^(1/4)) + (5*(-2*(-1 + x) - (-1 + x)^2)^(3
/4)*Hypergeometric2F1[3/4, 3/4, 7/4, (1 - x)/2])/(24*2^(3/4)*(1 + x)^(3/4))

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{{x}^{2}\sqrt [4]{-{x}^{2}+1}\sqrt{1+x}{\frac{1}{\sqrt{1-x}}} \left ( \sqrt{1-x}-\sqrt{1+x} \right ) ^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^2*(-x^2+1)^(1/4)*(1+x)^(1/2)/(1-x)^(1/2)/((1-x)^(1/2)-(1+x)^(1/2)),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.259284, size = 774, normalized size = 2.55 \[ -\frac{1}{48} \,{\left (8 \, x^{2} + 2 \, x - 7\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} + \frac{1}{48} \,{\left (8 \, x^{2} + 22 \, x + 29\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{-x + 1} - \frac{1}{16} \, \sqrt{2} \arctan \left (\frac{x + 1}{\sqrt{2}{\left (x + 1\right )} \sqrt{\frac{\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} + x + \sqrt{-x^{2} + 1} + 1}{x + 1}} + \sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} + x + 1}\right ) - \frac{1}{16} \, \sqrt{2} \arctan \left (\frac{x + 1}{\sqrt{2}{\left (x + 1\right )} \sqrt{-\frac{\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} - x - \sqrt{-x^{2} + 1} - 1}{x + 1}} + \sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} - x - 1}\right ) - \frac{5}{16} \, \sqrt{2} \arctan \left (\frac{x - 1}{\sqrt{2}{\left (x - 1\right )} \sqrt{\frac{\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{-x + 1} + x - \sqrt{-x^{2} + 1} - 1}{x - 1}} + \sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{-x + 1} + x - 1}\right ) - \frac{5}{16} \, \sqrt{2} \arctan \left (\frac{x - 1}{\sqrt{2}{\left (x - 1\right )} \sqrt{-\frac{\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{-x + 1} - x + \sqrt{-x^{2} + 1} + 1}{x - 1}} + \sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{-x + 1} - x + 1}\right ) + \frac{1}{64} \, \sqrt{2} \log \left (\frac{2 \,{\left (\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} + x + \sqrt{-x^{2} + 1} + 1\right )}}{x + 1}\right ) - \frac{1}{64} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{x + 1} - x - \sqrt{-x^{2} + 1} - 1\right )}}{x + 1}\right ) + \frac{5}{64} \, \sqrt{2} \log \left (\frac{2 \,{\left (\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{-x + 1} + x - \sqrt{-x^{2} + 1} - 1\right )}}{x - 1}\right ) - \frac{5}{64} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2}{\left (-x^{2} + 1\right )}^{\frac{1}{4}} \sqrt{-x + 1} - x + \sqrt{-x^{2} + 1} + 1\right )}}{x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(8*x^2 + 2*x - 7)*(-x^2 + 1)^(1/4)*sqrt(x + 1) + 1/48*(8*x^2 + 22*x + 29)*
(-x^2 + 1)^(1/4)*sqrt(-x + 1) - 1/16*sqrt(2)*arctan((x + 1)/(sqrt(2)*(x + 1)*sqr
t((sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(x + 1) + x + sqrt(-x^2 + 1) + 1)/(x + 1)) + sqr
t(2)*(-x^2 + 1)^(1/4)*sqrt(x + 1) + x + 1)) - 1/16*sqrt(2)*arctan((x + 1)/(sqrt(
2)*(x + 1)*sqrt(-(sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(x + 1) - x - sqrt(-x^2 + 1) - 1)
/(x + 1)) + sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(x + 1) - x - 1)) - 5/16*sqrt(2)*arctan
((x - 1)/(sqrt(2)*(x - 1)*sqrt((sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(-x + 1) + x - sqrt
(-x^2 + 1) - 1)/(x - 1)) + sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(-x + 1) + x - 1)) - 5/1
6*sqrt(2)*arctan((x - 1)/(sqrt(2)*(x - 1)*sqrt(-(sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(-
x + 1) - x + sqrt(-x^2 + 1) + 1)/(x - 1)) + sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(-x + 1
) - x + 1)) + 1/64*sqrt(2)*log(2*(sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(x + 1) + x + sqr
t(-x^2 + 1) + 1)/(x + 1)) - 1/64*sqrt(2)*log(-2*(sqrt(2)*(-x^2 + 1)^(1/4)*sqrt(x
+ 1) - x - sqrt(-x^2 + 1) - 1)/(x + 1)) + 5/64*sqrt(2)*log(2*(sqrt(2)*(-x^2 + 1
)^(1/4)*sqrt(-x + 1) + x - sqrt(-x^2 + 1) - 1)/(x - 1)) - 5/64*sqrt(2)*log(-2*(s
qrt(2)*(-x^2 + 1)^(1/4)*sqrt(-x + 1) - x + sqrt(-x^2 + 1) + 1)/(x - 1))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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