∖int √ __ 1−xx(1+x)2∕3 ___________________________________________________−(1−x)5∕6 3 √__1+x+(1−x)2∕3√_

3.222 \(\int \frac{\sqrt{1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=292 \[ -\frac{1}{12} (1-x)^{2/3} \sqrt [3]{x+1} (1-3 x)-\frac{1}{4} (1-x) (x+3)+\frac{1}{12} \sqrt [3]{1-x} (x+1)^{2/3} (3 x+1)+\frac{1}{12} \sqrt [6]{1-x} (x+1)^{5/6} (3 x+2)-\frac{1}{12} (1-x)^{5/6} \sqrt [6]{x+1} (3 x+10)+\frac{1}{4} \sqrt{1-x} x \sqrt{x+1}+\frac{1}{6} \tan ^{-1}\left (\frac{\sqrt [6]{x+1}}{\sqrt [6]{1-x}}\right )-\frac{4 \tan ^{-1}\left (\frac{\sqrt [3]{1-x}-2 \sqrt [3]{x+1}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{3 \sqrt{3}}-\frac{5}{6} \tan ^{-1}\left (\frac{\sqrt [3]{1-x}-\sqrt [3]{x+1}}{\sqrt [6]{1-x} \sqrt [6]{x+1}}\right )+\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{1-x} \sqrt [6]{x+1}}{\sqrt [3]{1-x}+\sqrt [3]{x+1}}\right )}{6 \sqrt{3}} \]

[Out]

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

1 - x)*(3 + x))/4 + ((1 - x)^(1/3)*(1 + x)^(2/3)*(1 + 3*x))/12 + ((1 - x)^(1/6)*

(1 + x)^(5/6)*(2 + 3*x))/12 - ((1 - x)^(5/6)*(1 + x)^(1/6)*(10 + 3*x))/12 + ArcT

an[(1 + x)^(1/6)/(1 - x)^(1/6)]/6 - (4*ArcTan[((1 - x)^(1/3) - 2*(1 + x)^(1/3))/

(Sqrt[3]*(1 - x)^(1/3))])/(3*Sqrt[3]) - (5*ArcTan[((1 - x)^(1/3) - (1 + x)^(1/3)

)/((1 - x)^(1/6)*(1 + x)^(1/6))])/6 + ArcTanh[(Sqrt[3]*(1 - x)^(1/6)*(1 + x)^(1/

6))/((1 - x)^(1/3) + (1 + x)^(1/3))]/(6*Sqrt[3])

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Rubi [A] time = 3.29845, antiderivative size = 522, normalized size of antiderivative = 1.79, number of steps used = 44, number of rules used = 19, integrand size = 56, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.339 \[ \frac{x^2}{4}+\frac{1}{4} \sqrt{1-x^2} x+\frac{x}{2}-\frac{1}{4} (1-x)^{5/6} (x+1)^{7/6}-\frac{1}{4} (1-x)^{7/6} (x+1)^{5/6}+\frac{5}{12} \sqrt [6]{1-x} (x+1)^{5/6}-\frac{1}{4} (1-x)^{4/3} (x+1)^{2/3}+\frac{1}{3} \sqrt [3]{1-x} (x+1)^{2/3}-\frac{1}{4} (1-x)^{5/3} \sqrt [3]{x+1}+\frac{1}{6} (1-x)^{2/3} \sqrt [3]{x+1}-\frac{7}{12} (1-x)^{5/6} \sqrt [6]{x+1}-\frac{1}{9} \log (1-x)+\frac{1}{9} \log (x+1)+\frac{1}{3} \log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+1\right )-\frac{\log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{12 \sqrt{3}}+\frac{\log \left (\frac{\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac{\sqrt{3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{12 \sqrt{3}}-\frac{1}{3} \log \left (\frac{\sqrt [3]{x+1}}{\sqrt [3]{1-x}}+1\right )+\frac{1}{4} \sin ^{-1}(x)-\frac{2}{3} \tan ^{-1}\left (\frac{\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1-x}}{\sqrt{3} \sqrt [3]{x+1}}\right )}{3 \sqrt{3}}+\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\frac{1}{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt{3}\right )-\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{x+1}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{3 \sqrt{3}} \]

Warning: Unable to verify antiderivative.

[In] Int[(Sqrt[1 - x]*x*(1 + x)^(2/3))/(-((1 - x)^(5/6)*(1 + x)^(1/3)) + (1 - x)^(2/3)*Sqrt[1 + x]),x]

[Out]

x/2 + x^2/4 - (7*(1 - x)^(5/6)*(1 + x)^(1/6))/12 + ((1 - x)^(2/3)*(1 + x)^(1/3))

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

)^(4/3)*(1 + x)^(2/3))/4 + (5*(1 - x)^(1/6)*(1 + x)^(5/6))/12 - ((1 - x)^(7/6)*(

1 + x)^(5/6))/4 - ((1 - x)^(5/6)*(1 + x)^(7/6))/4 + (x*Sqrt[1 - x^2])/4 + ArcSin

[x]/4 - (2*ArcTan[(1 - x)^(1/6)/(1 + x)^(1/6)])/3 + (2*ArcTan[1/Sqrt[3] - (2*(1

- x)^(1/3))/(Sqrt[3]*(1 + x)^(1/3))])/(3*Sqrt[3]) + ArcTan[Sqrt[3] - (2*(1 - x)^

(1/6))/(1 + x)^(1/6)]/3 - ArcTan[Sqrt[3] + (2*(1 - x)^(1/6))/(1 + x)^(1/6)]/3 -

(2*ArcTan[1/Sqrt[3] - (2*(1 + x)^(1/3))/(Sqrt[3]*(1 - x)^(1/3))])/(3*Sqrt[3]) -

Log[1 - x]/9 + Log[1 + x]/9 + Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3)]/3 - Log[1 + (

1 - x)^(1/3)/(1 + x)^(1/3) - (Sqrt[3]*(1 - x)^(1/6))/(1 + x)^(1/6)]/(12*Sqrt[3])

+ Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3) + (Sqrt[3]*(1 - x)^(1/6))/(1 + x)^(1/6)]/

(12*Sqrt[3]) - Log[1 + (1 + x)^(1/3)/(1 - x)^(1/3)]/3

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate(x*(1+x)**(2/3)*(1-x)**(1/2)/(-(1-x)**(5/6)*(1+x)**(1/3)+(1-x)**(2/3)*(1+x)**(1/2)),x)

[Out]

Timed out

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Mathematica [C] time = 1.23889, size = 391, normalized size = 1.34 \[ -\frac{1}{12} \sqrt [3]{x+1} \left (-4\ 2^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{x+1}{2}\right )+\frac{(3 x+10) \left (1-x^2\right )^{5/6}}{x+1}-\frac{(3 x+2) \sqrt{1-x^2}}{\sqrt [3]{1-x}}-(3 x+1) \sqrt [3]{1-x^2}-3 \sqrt [3]{1-x} x \sqrt [6]{1-x^2}-\frac{3 \sqrt [3]{1-x} x (x+2)}{\sqrt [3]{1-x^2}}+(1-x)^{2/3} (1-3 x)\right )-\frac{2^{2/3} \sqrt [3]{-(x-1)^2-2 (x-1)} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{1-x}{2}\right )}{3 \sqrt [3]{x+1}}-\frac{7 \left (-(x-1)^2-2 (x-1)\right )^{5/6} \, _2F_1\left (\frac{5}{6},\frac{5}{6};\frac{11}{6};\frac{1-x}{2}\right )}{30\ 2^{5/6} (x+1)^{5/6}}+\frac{\sqrt [3]{x+1} \sqrt{2 (x+1)-(x+1)^2} \, _2F_1\left (\frac{5}{6},\frac{5}{6};\frac{11}{6};\frac{x+1}{2}\right )}{6\ 2^{5/6} \sqrt{1-x}}+\frac{\sqrt [3]{1-x} \sqrt{x-1} (x+1)^{5/6} \log \left (\sqrt{x-1}+\sqrt{x+1}\right )}{2 \left (2 (x+1)-(x+1)^2\right )^{5/6}} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[(Sqrt[1 - x]*x*(1 + x)^(2/3))/(-((1 - x)^(5/6)*(1 + x)^(1/3)) + (1 - x)^(2/3)*Sqrt[1 + x]),x]

[Out]

-(2^(2/3)*(-2*(-1 + x) - (-1 + x)^2)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, (1 -

x)/2])/(3*(1 + x)^(1/3)) - ((1 + x)^(1/3)*((1 - 3*x)*(1 - x)^(2/3) - (3*(1 - x)

^(1/3)*x*(2 + x))/(1 - x^2)^(1/3) - 3*(1 - x)^(1/3)*x*(1 - x^2)^(1/6) - (1 + 3*x

)*(1 - x^2)^(1/3) - ((2 + 3*x)*Sqrt[1 - x^2])/(1 - x)^(1/3) + ((10 + 3*x)*(1 - x

^2)^(5/6))/(1 + x) - 4*2^(2/3)*Hypergeometric2F1[1/3, 1/3, 4/3, (1 + x)/2]))/12

- (7*(-2*(-1 + x) - (-1 + x)^2)^(5/6)*Hypergeometric2F1[5/6, 5/6, 11/6, (1 - x)/

2])/(30*2^(5/6)*(1 + x)^(5/6)) + ((1 + x)^(1/3)*Sqrt[2*(1 + x) - (1 + x)^2]*Hype

rgeometric2F1[5/6, 5/6, 11/6, (1 + x)/2])/(6*2^(5/6)*Sqrt[1 - x]) + ((1 - x)^(1/

3)*Sqrt[-1 + x]*(1 + x)^(5/6)*Log[Sqrt[-1 + x] + Sqrt[1 + x]])/(2*(2*(1 + x) - (

1 + x)^2)^(5/6))

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Maple [F] time = 0.059, size = 0, normalized size = 0. \[ \int{x \left ( 1+x \right ) ^{{\frac{2}{3}}}\sqrt{1-x} \left ( - \left ( 1-x \right ) ^{{\frac{5}{6}}}\sqrt [3]{1+x}+ \left ( 1-x \right ) ^{{\frac{2}{3}}}\sqrt{1+x} \right ) ^{-1}}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int(x*(1+x)^(2/3)*(1-x)^(1/2)/(-(1-x)^(5/6)*(1+x)^(1/3)+(1-x)^(2/3)*(1+x)^(1/2)),x)

[Out]

int(x*(1+x)^(2/3)*(1-x)^(1/2)/(-(1-x)^(5/6)*(1+x)^(1/3)+(1-x)^(2/3)*(1+x)^(1/2))

,x)

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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + 1\right )}^{\frac{2}{3}} x \sqrt{-x + 1}}{\sqrt{x + 1}{\left (-x + 1\right )}^{\frac{2}{3}} -{\left (x + 1\right )}^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{5}{6}}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((x + 1)^(2/3)*x*sqrt(-x + 1)/(sqrt(x + 1)*(-x + 1)^(2/3) - (x + 1)^(1/3)*(-x + 1)^(5/6)),x, algorithm="maxima")

[Out]

integrate((x + 1)^(2/3)*x*sqrt(-x + 1)/(sqrt(x + 1)*(-x + 1)^(2/3) - (x + 1)^(1/

3)*(-x + 1)^(5/6)), x)

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Fricas [A] time = 0.431658, size = 1841, normalized size = 6.3 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((x + 1)^(2/3)*x*sqrt(-x + 1)/(sqrt(x + 1)*(-x + 1)^(2/3) - (x + 1)^(1/3)*(-x + 1)^(5/6)),x, algorithm="fricas")

[Out]

1/72*(6*sqrt(3)*(3*x^3 - 4*x^2 + 2*x - 6)*(x + 1)^(5/6)*(-x + 1)^(1/6) + 6*sqrt(

3)*(3*x^3 + 7*x^2 + 8*x - 22)*(x + 1)^(2/3)*(-x + 1)^(1/3) + 18*sqrt(3)*(x^3 + 2

*x^2 + 2*x)*sqrt(x + 1)*sqrt(-x + 1) + 6*sqrt(3)*(3*x^3 + 5*x^2 + 4*x + 6)*(x +

1)^(1/3)*(-x + 1)^(2/3) - 6*sqrt(3)*(3*x^3 + 4*x^2 - 14*x - 22)*(x + 1)^(1/6)*(-

x + 1)^(5/6) - 48*(x^2 + 2*sqrt(x + 1)*sqrt(-x + 1) - 2)*arctan(-1/3*(sqrt(3)*(x

+ 1) - 2*sqrt(3)*(x + 1)^(2/3)*(-x + 1)^(1/3))/(x + 1)) + 20*(sqrt(3)*(x^2 - 2)

+ 2*sqrt(3)*sqrt(x + 1)*sqrt(-x + 1))*arctan((x + 1)/(sqrt(3)*(x + 1) + 2*(x +

1)*sqrt((sqrt(3)*(x + 1)^(5/6)*(-x + 1)^(1/6) + x + (x + 1)^(2/3)*(-x + 1)^(1/3)

+ 1)/(x + 1)) + 2*(x + 1)^(5/6)*(-x + 1)^(1/6))) + 20*(sqrt(3)*(x^2 - 2) + 2*sq

rt(3)*sqrt(x + 1)*sqrt(-x + 1))*arctan(-(x + 1)/(sqrt(3)*(x + 1) - 2*(x + 1)*sqr

t(-(sqrt(3)*(x + 1)^(5/6)*(-x + 1)^(1/6) - x - (x + 1)^(2/3)*(-x + 1)^(1/3) - 1)

/(x + 1)) - 2*(x + 1)^(5/6)*(-x + 1)^(1/6))) - 48*(x^2 + 2*sqrt(x + 1)*sqrt(-x +

1) - 2)*arctan(1/3*(sqrt(3)*(x - 1) + 2*sqrt(3)*(x + 1)^(1/3)*(-x + 1)^(2/3))/(

x - 1)) + 28*(sqrt(3)*(x^2 - 2) + 2*sqrt(3)*sqrt(x + 1)*sqrt(-x + 1))*arctan((x

- 1)/(sqrt(3)*(x - 1) + 2*(x - 1)*sqrt((sqrt(3)*(x + 1)^(1/6)*(-x + 1)^(5/6) + x

- (x + 1)^(1/3)*(-x + 1)^(2/3) - 1)/(x - 1)) + 2*(x + 1)^(1/6)*(-x + 1)^(5/6)))

+ 28*(sqrt(3)*(x^2 - 2) + 2*sqrt(3)*sqrt(x + 1)*sqrt(-x + 1))*arctan(-(x - 1)/(

sqrt(3)*(x - 1) - 2*(x - 1)*sqrt(-(sqrt(3)*(x + 1)^(1/6)*(-x + 1)^(5/6) - x + (x

+ 1)^(1/3)*(-x + 1)^(2/3) + 1)/(x - 1)) - 2*(x + 1)^(1/6)*(-x + 1)^(5/6))) - 20

*(sqrt(3)*(x^2 - 2) + 2*sqrt(3)*sqrt(x + 1)*sqrt(-x + 1))*arctan((-x + 1)^(1/6)/

(x + 1)^(1/6)) - 28*(sqrt(3)*(x^2 - 2) + 2*sqrt(3)*sqrt(x + 1)*sqrt(-x + 1))*arc

tan((x + 1)^(1/6)*(-x + 1)^(5/6)/(x - 1)) - 36*(sqrt(3)*(x^2 - 2) + 2*sqrt(3)*sq

rt(x + 1)*sqrt(-x + 1))*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - 15*(x^2 + 2*s

qrt(x + 1)*sqrt(-x + 1) - 2)*log(25*(sqrt(3)*(x + 1)^(5/6)*(-x + 1)^(1/6) + x +

(x + 1)^(2/3)*(-x + 1)^(1/3) + 1)/(x + 1)) + 15*(x^2 + 2*sqrt(x + 1)*sqrt(-x + 1

) - 2)*log(-25*(sqrt(3)*(x + 1)^(5/6)*(-x + 1)^(1/6) - x - (x + 1)^(2/3)*(-x + 1

)^(1/3) - 1)/(x + 1)) - 16*(sqrt(3)*(x^2 - 2) + 2*sqrt(3)*sqrt(x + 1)*sqrt(-x +

1))*log((x + (x + 1)^(2/3)*(-x + 1)^(1/3) + 1)/(x + 1)) + 8*(sqrt(3)*(x^2 - 2) +

2*sqrt(3)*sqrt(x + 1)*sqrt(-x + 1))*log((x - (x + 1)^(2/3)*(-x + 1)^(1/3) + (x

+ 1)^(1/3)*(-x + 1)^(2/3) + 1)/(x + 1)) - 21*(x^2 + 2*sqrt(x + 1)*sqrt(-x + 1) -

2)*log(49*(sqrt(3)*(x + 1)^(1/6)*(-x + 1)^(5/6) + x - (x + 1)^(1/3)*(-x + 1)^(2

/3) - 1)/(x - 1)) + 21*(x^2 + 2*sqrt(x + 1)*sqrt(-x + 1) - 2)*log(-49*(sqrt(3)*(

x + 1)^(1/6)*(-x + 1)^(5/6) - x + (x + 1)^(1/3)*(-x + 1)^(2/3) + 1)/(x - 1)) - 8

*(sqrt(3)*(x^2 - 2) + 2*sqrt(3)*sqrt(x + 1)*sqrt(-x + 1))*log((x - (x + 1)^(2/3)

*(-x + 1)^(1/3) + (x + 1)^(1/3)*(-x + 1)^(2/3) - 1)/(x - 1)) + 16*(sqrt(3)*(x^2

- 2) + 2*sqrt(3)*sqrt(x + 1)*sqrt(-x + 1))*log(-(x - (x + 1)^(1/3)*(-x + 1)^(2/3

) - 1)/(x - 1)) + 18*sqrt(3)*(x^4 - 2*x^2 - 2*x))/(sqrt(3)*(x^2 - 2) + 2*sqrt(3)

*sqrt(x + 1)*sqrt(-x + 1))

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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \sqrt{- x + 1} \left (x + 1\right )^{\frac{2}{3}}}{- \left (- x + 1\right )^{\frac{5}{6}} \sqrt [3]{x + 1} + \left (- x + 1\right )^{\frac{2}{3}} \sqrt{x + 1}}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(x*(1+x)**(2/3)*(1-x)**(1/2)/(-(1-x)**(5/6)*(1+x)**(1/3)+(1-x)**(2/3)*(1+x)**(1/2)),x)

[Out]

Integral(x*sqrt(-x + 1)*(x + 1)**(2/3)/(-(-x + 1)**(5/6)*(x + 1)**(1/3) + (-x +

1)**(2/3)*sqrt(x + 1)), x)

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GIAC/XCAS [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 + 1)^(2/3)*x*sqrt(-x + 1)/(sqrt(x + 1)*(-x + 1)^(2/3) - (x + 1)^(1/3)*(-x + 1)^(5/6)),x, algorithm="giac")

[Out]

Timed out

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