3.711 \(\int \frac{x^4 (1+x)^{3/2}}{\sqrt{1-x}} \, dx\)

Optimal. Leaf size=118 \[ -\frac{1}{6} \sqrt{1-x} (x+1)^{5/2} x^3-\frac{1}{15} \sqrt{1-x} (x+1)^{5/2} x^2-\frac{11}{48} \sqrt{1-x} (x+1)^{3/2}-\frac{1}{120} \sqrt{1-x} (x+1)^{5/2} (19 x+18)-\frac{11}{16} \sqrt{1-x} \sqrt{x+1}+\frac{11}{16} \sin ^{-1}(x) \]

[Out]

(-11*Sqrt[1 - x]*Sqrt[1 + x])/16 - (11*Sqrt[1 - x]*(1 + x)^(3/2))/48 - (Sqrt[1 -
x]*x^2*(1 + x)^(5/2))/15 - (Sqrt[1 - x]*x^3*(1 + x)^(5/2))/6 - (Sqrt[1 - x]*(1
+ x)^(5/2)*(18 + 19*x))/120 + (11*ArcSin[x])/16

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Rubi [A]  time = 0.157265, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{1}{6} \sqrt{1-x} (x+1)^{5/2} x^3-\frac{1}{15} \sqrt{1-x} (x+1)^{5/2} x^2-\frac{11}{48} \sqrt{1-x} (x+1)^{3/2}-\frac{1}{120} \sqrt{1-x} (x+1)^{5/2} (19 x+18)-\frac{11}{16} \sqrt{1-x} \sqrt{x+1}+\frac{11}{16} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(1 + x)^(3/2))/Sqrt[1 - x],x]

[Out]

(-11*Sqrt[1 - x]*Sqrt[1 + x])/16 - (11*Sqrt[1 - x]*(1 + x)^(3/2))/48 - (Sqrt[1 -
x]*x^2*(1 + x)^(5/2))/15 - (Sqrt[1 - x]*x^3*(1 + x)^(5/2))/6 - (Sqrt[1 - x]*(1
+ x)^(5/2)*(18 + 19*x))/120 + (11*ArcSin[x])/16

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Rubi in Sympy [A]  time = 12.9892, size = 97, normalized size = 0.82 \[ - \frac{x^{3} \sqrt{- x + 1} \left (x + 1\right )^{\frac{5}{2}}}{6} - \frac{x^{2} \sqrt{- x + 1} \left (x + 1\right )^{\frac{5}{2}}}{15} - \frac{\sqrt{- x + 1} \left (x + 1\right )^{\frac{5}{2}} \left (57 x + 54\right )}{360} - \frac{11 \sqrt{- x + 1} \left (x + 1\right )^{\frac{3}{2}}}{48} - \frac{11 \sqrt{- x + 1} \sqrt{x + 1}}{16} + \frac{11 \operatorname{asin}{\left (x \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(1+x)**(3/2)/(1-x)**(1/2),x)

[Out]

-x**3*sqrt(-x + 1)*(x + 1)**(5/2)/6 - x**2*sqrt(-x + 1)*(x + 1)**(5/2)/15 - sqrt
(-x + 1)*(x + 1)**(5/2)*(57*x + 54)/360 - 11*sqrt(-x + 1)*(x + 1)**(3/2)/48 - 11
*sqrt(-x + 1)*sqrt(x + 1)/16 + 11*asin(x)/16

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Mathematica [A]  time = 0.0598163, size = 59, normalized size = 0.5 \[ \frac{11}{8} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-\frac{1}{240} \sqrt{1-x^2} \left (40 x^5+96 x^4+110 x^3+128 x^2+165 x+256\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(1 + x)^(3/2))/Sqrt[1 - x],x]

[Out]

-(Sqrt[1 - x^2]*(256 + 165*x + 128*x^2 + 110*x^3 + 96*x^4 + 40*x^5))/240 + (11*A
rcSin[Sqrt[1 + x]/Sqrt[2]])/8

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Maple [A]  time = 0.021, size = 108, normalized size = 0.9 \[{\frac{1}{240}\sqrt{1-x}\sqrt{1+x} \left ( -40\,{x}^{5}\sqrt{-{x}^{2}+1}-96\,{x}^{4}\sqrt{-{x}^{2}+1}-110\,{x}^{3}\sqrt{-{x}^{2}+1}-128\,{x}^{2}\sqrt{-{x}^{2}+1}-165\,x\sqrt{-{x}^{2}+1}+165\,\arcsin \left ( x \right ) -256\,\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(1+x)^(3/2)/(1-x)^(1/2),x)

[Out]

1/240*(1+x)^(1/2)*(1-x)^(1/2)*(-40*x^5*(-x^2+1)^(1/2)-96*x^4*(-x^2+1)^(1/2)-110*
x^3*(-x^2+1)^(1/2)-128*x^2*(-x^2+1)^(1/2)-165*x*(-x^2+1)^(1/2)+165*arcsin(x)-256
*(-x^2+1)^(1/2))/(-x^2+1)^(1/2)

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Maxima [A]  time = 1.49661, size = 113, normalized size = 0.96 \[ -\frac{1}{6} \, \sqrt{-x^{2} + 1} x^{5} - \frac{2}{5} \, \sqrt{-x^{2} + 1} x^{4} - \frac{11}{24} \, \sqrt{-x^{2} + 1} x^{3} - \frac{8}{15} \, \sqrt{-x^{2} + 1} x^{2} - \frac{11}{16} \, \sqrt{-x^{2} + 1} x - \frac{16}{15} \, \sqrt{-x^{2} + 1} + \frac{11}{16} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)*x^4/sqrt(-x + 1),x, algorithm="maxima")

[Out]

-1/6*sqrt(-x^2 + 1)*x^5 - 2/5*sqrt(-x^2 + 1)*x^4 - 11/24*sqrt(-x^2 + 1)*x^3 - 8/
15*sqrt(-x^2 + 1)*x^2 - 11/16*sqrt(-x^2 + 1)*x - 16/15*sqrt(-x^2 + 1) + 11/16*ar
csin(x)

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Fricas [A]  time = 0.213631, size = 284, normalized size = 2.41 \[ \frac{240 \, x^{11} + 576 \, x^{10} - 860 \, x^{9} - 2880 \, x^{8} - 630 \, x^{7} + 2560 \, x^{6} - 510 \, x^{5} + 7040 \, x^{3} -{\left (40 \, x^{11} + 96 \, x^{10} - 610 \, x^{9} - 1600 \, x^{8} + 105 \, x^{7} + 2560 \, x^{6} + 1030 \, x^{5} + 4400 \, x^{3} - 5280 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 330 \,{\left (x^{6} - 18 \, x^{4} + 48 \, x^{2} + 2 \,{\left (3 \, x^{4} - 16 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} - 32\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 5280 \, x}{240 \,{\left (x^{6} - 18 \, x^{4} + 48 \, x^{2} + 2 \,{\left (3 \, x^{4} - 16 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} - 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)*x^4/sqrt(-x + 1),x, algorithm="fricas")

[Out]

1/240*(240*x^11 + 576*x^10 - 860*x^9 - 2880*x^8 - 630*x^7 + 2560*x^6 - 510*x^5 +
7040*x^3 - (40*x^11 + 96*x^10 - 610*x^9 - 1600*x^8 + 105*x^7 + 2560*x^6 + 1030*
x^5 + 4400*x^3 - 5280*x)*sqrt(x + 1)*sqrt(-x + 1) - 330*(x^6 - 18*x^4 + 48*x^2 +
2*(3*x^4 - 16*x^2 + 16)*sqrt(x + 1)*sqrt(-x + 1) - 32)*arctan((sqrt(x + 1)*sqrt
(-x + 1) - 1)/x) - 5280*x)/(x^6 - 18*x^4 + 48*x^2 + 2*(3*x^4 - 16*x^2 + 16)*sqrt
(x + 1)*sqrt(-x + 1) - 32)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.235303, size = 80, normalized size = 0.68 \[ -\frac{1}{240} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \, x - 8\right )}{\left (x + 1\right )} + 63\right )}{\left (x + 1\right )} - 13\right )}{\left (x + 1\right )} + 55\right )}{\left (x + 1\right )} + 165\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{11}{8} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)*x^4/sqrt(-x + 1),x, algorithm="giac")

[Out]

-1/240*((2*((4*(5*x - 8)*(x + 1) + 63)*(x + 1) - 13)*(x + 1) + 55)*(x + 1) + 165
)*sqrt(x + 1)*sqrt(-x + 1) + 11/8*arcsin(1/2*sqrt(2)*sqrt(x + 1))