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

Optimal. Leaf size=234 \[ -\frac{1}{3} (1-x)^{3/4} x (x+1)^{5/4}-\frac{1}{12} (1-x)^{3/4} (x+1)^{5/4}-\frac{3}{8} (1-x)^{3/4} \sqrt [4]{x+1}-\frac{3 \log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{16 \sqrt{2}}+\frac{3 \log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{16 \sqrt{2}}+\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{8 \sqrt{2}}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}} \]

[Out]

(-3*(1 - x)^(3/4)*(1 + x)^(1/4))/8 - ((1 - x)^(3/4)*(1 + x)^(5/4))/12 - ((1 - x)
^(3/4)*x*(1 + x)^(5/4))/3 + (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]
) - (3*Log[1 + Sqrt[1 - x]/Sqrt[1 + x] - (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)])
/(16*Sqrt[2]) + (3*Log[1 + Sqrt[1 - x]/Sqrt[1 + x] + (Sqrt[2]*(1 - x)^(1/4))/(1
+ x)^(1/4)])/(16*Sqrt[2])

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Rubi [A]  time = 0.286204, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{1}{3} (1-x)^{3/4} x (x+1)^{5/4}-\frac{1}{12} (1-x)^{3/4} (x+1)^{5/4}-\frac{3}{8} (1-x)^{3/4} \sqrt [4]{x+1}-\frac{3 \log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{16 \sqrt{2}}+\frac{3 \log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{16 \sqrt{2}}+\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{8 \sqrt{2}}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(1 - x)^(3/4)*(1 + x)^(1/4))/8 - ((1 - x)^(3/4)*(1 + x)^(5/4))/12 - ((1 - x)
^(3/4)*x*(1 + x)^(5/4))/3 + (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]
) - (3*Log[1 + Sqrt[1 - x]/Sqrt[1 + x] - (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)])
/(16*Sqrt[2]) + (3*Log[1 + Sqrt[1 - x]/Sqrt[1 + x] + (Sqrt[2]*(1 - x)^(1/4))/(1
+ x)^(1/4)])/(16*Sqrt[2])

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Rubi in Sympy [A]  time = 22.7873, size = 197, normalized size = 0.84 \[ - \frac{x \left (- x + 1\right )^{\frac{3}{4}} \left (x + 1\right )^{\frac{5}{4}}}{3} - \frac{\left (- x + 1\right )^{\frac{3}{4}} \left (x + 1\right )^{\frac{5}{4}}}{12} - \frac{3 \left (- x + 1\right )^{\frac{3}{4}} \sqrt [4]{x + 1}}{8} - \frac{3 \sqrt{2} \log{\left (1 + \frac{\sqrt{x + 1}}{\sqrt{- x + 1}} - \frac{\sqrt{2} \sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )}}{32} + \frac{3 \sqrt{2} \log{\left (1 + \frac{\sqrt{x + 1}}{\sqrt{- x + 1}} + \frac{\sqrt{2} \sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )}}{32} - \frac{3 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )}}{16} + \frac{3 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(-x + 1)**(3/4)*(x + 1)**(5/4)/3 - (-x + 1)**(3/4)*(x + 1)**(5/4)/12 - 3*(-x
+ 1)**(3/4)*(x + 1)**(1/4)/8 - 3*sqrt(2)*log(1 + sqrt(x + 1)/sqrt(-x + 1) - sqrt
(2)*(x + 1)**(1/4)/(-x + 1)**(1/4))/32 + 3*sqrt(2)*log(1 + sqrt(x + 1)/sqrt(-x +
1) + sqrt(2)*(x + 1)**(1/4)/(-x + 1)**(1/4))/32 - 3*sqrt(2)*atan(1 - sqrt(2)*(x
+ 1)**(1/4)/(-x + 1)**(1/4))/16 + 3*sqrt(2)*atan(1 + sqrt(2)*(x + 1)**(1/4)/(-x
+ 1)**(1/4))/16

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Mathematica [C]  time = 0.0748283, size = 57, normalized size = 0.24 \[ \frac{1}{24} \sqrt [4]{x+1} \left (9\ 2^{3/4} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{x+1}{2}\right )-(1-x)^{3/4} \left (8 x^2+10 x+11\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((1 + x)^(1/4)*(-((1 - x)^(3/4)*(11 + 10*x + 8*x^2)) + 9*2^(3/4)*Hypergeometric2
F1[1/4, 1/4, 5/4, (1 + x)/2]))/24

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{{x}^{2}\sqrt [4]{1+x}{\frac{1}{\sqrt [4]{1-x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.247824, size = 378, normalized size = 1.62 \[ -\frac{1}{24} \,{\left (8 \, x^{2} + 10 \, x + 11\right )}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + \frac{3}{8} \, \sqrt{2} \arctan \left (\frac{x - 1}{\sqrt{2}{\left (x - 1\right )} \sqrt{\frac{\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + x - \sqrt{x + 1} \sqrt{-x + 1} - 1}{x - 1}} + \sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + x - 1}\right ) + \frac{3}{8} \, \sqrt{2} \arctan \left (\frac{x - 1}{\sqrt{2}{\left (x - 1\right )} \sqrt{-\frac{\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - x + \sqrt{x + 1} \sqrt{-x + 1} + 1}{x - 1}} + \sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - x + 1}\right ) - \frac{3}{32} \, \sqrt{2} \log \left (\frac{2 \,{\left (\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + x - \sqrt{x + 1} \sqrt{-x + 1} - 1\right )}}{x - 1}\right ) + \frac{3}{32} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - x + \sqrt{x + 1} \sqrt{-x + 1} + 1\right )}}{x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(8*x^2 + 10*x + 11)*(x + 1)^(1/4)*(-x + 1)^(3/4) + 3/8*sqrt(2)*arctan((x -
1)/(sqrt(2)*(x - 1)*sqrt((sqrt(2)*(x + 1)^(1/4)*(-x + 1)^(3/4) + x - sqrt(x + 1
)*sqrt(-x + 1) - 1)/(x - 1)) + sqrt(2)*(x + 1)^(1/4)*(-x + 1)^(3/4) + x - 1)) +
3/8*sqrt(2)*arctan((x - 1)/(sqrt(2)*(x - 1)*sqrt(-(sqrt(2)*(x + 1)^(1/4)*(-x + 1
)^(3/4) - x + sqrt(x + 1)*sqrt(-x + 1) + 1)/(x - 1)) + sqrt(2)*(x + 1)^(1/4)*(-x
+ 1)^(3/4) - x + 1)) - 3/32*sqrt(2)*log(2*(sqrt(2)*(x + 1)^(1/4)*(-x + 1)^(3/4)
+ x - sqrt(x + 1)*sqrt(-x + 1) - 1)/(x - 1)) + 3/32*sqrt(2)*log(-2*(sqrt(2)*(x
+ 1)^(1/4)*(-x + 1)^(3/4) - x + sqrt(x + 1)*sqrt(-x + 1) + 1)/(x - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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