3.1106 \(\int \frac{(1-x)^{7/2}}{\sqrt{1+x}} \, dx\)

Optimal. Leaf size=87 \[ \frac{1}{4} \sqrt{x+1} (1-x)^{7/2}+\frac{7}{12} \sqrt{x+1} (1-x)^{5/2}+\frac{35}{24} \sqrt{x+1} (1-x)^{3/2}+\frac{35}{8} \sqrt{x+1} \sqrt{1-x}+\frac{35}{8} \sin ^{-1}(x) \]

[Out]

(35*Sqrt[1 - x]*Sqrt[1 + x])/8 + (35*(1 - x)^(3/2)*Sqrt[1 + x])/24 + (7*(1 - x)^
(5/2)*Sqrt[1 + x])/12 + ((1 - x)^(7/2)*Sqrt[1 + x])/4 + (35*ArcSin[x])/8

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Rubi [A]  time = 0.0661754, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{1}{4} \sqrt{x+1} (1-x)^{7/2}+\frac{7}{12} \sqrt{x+1} (1-x)^{5/2}+\frac{35}{24} \sqrt{x+1} (1-x)^{3/2}+\frac{35}{8} \sqrt{x+1} \sqrt{1-x}+\frac{35}{8} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^(7/2)/Sqrt[1 + x],x]

[Out]

(35*Sqrt[1 - x]*Sqrt[1 + x])/8 + (35*(1 - x)^(3/2)*Sqrt[1 + x])/24 + (7*(1 - x)^
(5/2)*Sqrt[1 + x])/12 + ((1 - x)^(7/2)*Sqrt[1 + x])/4 + (35*ArcSin[x])/8

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Rubi in Sympy [A]  time = 8.40539, size = 71, normalized size = 0.82 \[ \frac{\left (- x + 1\right )^{\frac{7}{2}} \sqrt{x + 1}}{4} + \frac{7 \left (- x + 1\right )^{\frac{5}{2}} \sqrt{x + 1}}{12} + \frac{35 \left (- x + 1\right )^{\frac{3}{2}} \sqrt{x + 1}}{24} + \frac{35 \sqrt{- x + 1} \sqrt{x + 1}}{8} + \frac{35 \operatorname{asin}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**(7/2)/(1+x)**(1/2),x)

[Out]

(-x + 1)**(7/2)*sqrt(x + 1)/4 + 7*(-x + 1)**(5/2)*sqrt(x + 1)/12 + 35*(-x + 1)**
(3/2)*sqrt(x + 1)/24 + 35*sqrt(-x + 1)*sqrt(x + 1)/8 + 35*asin(x)/8

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Mathematica [A]  time = 0.0381746, size = 49, normalized size = 0.56 \[ \frac{1}{24} \sqrt{1-x^2} \left (-6 x^3+32 x^2-81 x+160\right )+\frac{35}{4} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)^(7/2)/Sqrt[1 + x],x]

[Out]

(Sqrt[1 - x^2]*(160 - 81*x + 32*x^2 - 6*x^3))/24 + (35*ArcSin[Sqrt[1 + x]/Sqrt[2
]])/4

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Maple [A]  time = 0.007, size = 85, normalized size = 1. \[{\frac{1}{4} \left ( 1-x \right ) ^{{\frac{7}{2}}}\sqrt{1+x}}+{\frac{7}{12} \left ( 1-x \right ) ^{{\frac{5}{2}}}\sqrt{1+x}}+{\frac{35}{24} \left ( 1-x \right ) ^{{\frac{3}{2}}}\sqrt{1+x}}+{\frac{35}{8}\sqrt{1-x}\sqrt{1+x}}+{\frac{35\,\arcsin \left ( x \right ) }{8}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(7/2)/(1+x)^(1/2),x)

[Out]

1/4*(1-x)^(7/2)*(1+x)^(1/2)+7/12*(1-x)^(5/2)*(1+x)^(1/2)+35/24*(1-x)^(3/2)*(1+x)
^(1/2)+35/8*(1-x)^(1/2)*(1+x)^(1/2)+35/8*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(
1/2)*arcsin(x)

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Maxima [A]  time = 1.50939, size = 76, normalized size = 0.87 \[ -\frac{1}{4} \, \sqrt{-x^{2} + 1} x^{3} + \frac{4}{3} \, \sqrt{-x^{2} + 1} x^{2} - \frac{27}{8} \, \sqrt{-x^{2} + 1} x + \frac{20}{3} \, \sqrt{-x^{2} + 1} + \frac{35}{8} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^(7/2)/sqrt(x + 1),x, algorithm="maxima")

[Out]

-1/4*sqrt(-x^2 + 1)*x^3 + 4/3*sqrt(-x^2 + 1)*x^2 - 27/8*sqrt(-x^2 + 1)*x + 20/3*
sqrt(-x^2 + 1) + 35/8*arcsin(x)

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Fricas [A]  time = 0.208456, size = 224, normalized size = 2.57 \[ \frac{24 \, x^{7} - 128 \, x^{6} + 252 \, x^{5} - 96 \, x^{4} - 924 \, x^{3} + 384 \, x^{2} -{\left (6 \, x^{7} - 32 \, x^{6} + 33 \, x^{5} + 96 \, x^{4} - 600 \, x^{3} + 384 \, x^{2} + 648 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 210 \,{\left (x^{4} - 8 \, x^{2} + 4 \,{\left (x^{2} - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} + 8\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 648 \, x}{24 \,{\left (x^{4} - 8 \, x^{2} + 4 \,{\left (x^{2} - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^(7/2)/sqrt(x + 1),x, algorithm="fricas")

[Out]

1/24*(24*x^7 - 128*x^6 + 252*x^5 - 96*x^4 - 924*x^3 + 384*x^2 - (6*x^7 - 32*x^6
+ 33*x^5 + 96*x^4 - 600*x^3 + 384*x^2 + 648*x)*sqrt(x + 1)*sqrt(-x + 1) - 210*(x
^4 - 8*x^2 + 4*(x^2 - 2)*sqrt(x + 1)*sqrt(-x + 1) + 8)*arctan((sqrt(x + 1)*sqrt(
-x + 1) - 1)/x) + 648*x)/(x^4 - 8*x^2 + 4*(x^2 - 2)*sqrt(x + 1)*sqrt(-x + 1) + 8
)

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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((1-x)**(7/2)/(1+x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226323, size = 136, normalized size = 1.56 \[ -\frac{1}{24} \,{\left ({\left (2 \,{\left (3 \, x - 10\right )}{\left (x + 1\right )} + 43\right )}{\left (x + 1\right )} - 39\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{2} \,{\left ({\left (2 \, x - 5\right )}{\left (x + 1\right )} + 9\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{3}{2} \, \sqrt{x + 1}{\left (x - 2\right )} \sqrt{-x + 1} + \sqrt{x + 1} \sqrt{-x + 1} + \frac{35}{4} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^(7/2)/sqrt(x + 1),x, algorithm="giac")

[Out]

-1/24*((2*(3*x - 10)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt(-x + 1) + 1/2*
((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(-x + 1) - 3/2*sqrt(x + 1)*(x - 2)*sqrt(
-x + 1) + sqrt(x + 1)*sqrt(-x + 1) + 35/4*arcsin(1/2*sqrt(2)*sqrt(x + 1))