3.1135 \(\int \frac{1}{(1-x)^{9/2} (1+x)^{5/2}} \, dx\)

Optimal. Leaf size=83 \[ \frac{8 x}{21 \sqrt{1-x} \sqrt{x+1}}+\frac{4 x}{21 (1-x)^{3/2} (x+1)^{3/2}}+\frac{1}{7 (1-x)^{5/2} (x+1)^{3/2}}+\frac{1}{7 (1-x)^{7/2} (x+1)^{3/2}} \]

[Out]

1/(7*(1 - x)^(7/2)*(1 + x)^(3/2)) + 1/(7*(1 - x)^(5/2)*(1 + x)^(3/2)) + (4*x)/(2
1*(1 - x)^(3/2)*(1 + x)^(3/2)) + (8*x)/(21*Sqrt[1 - x]*Sqrt[1 + x])

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Rubi [A]  time = 0.0509387, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{8 x}{21 \sqrt{1-x} \sqrt{x+1}}+\frac{4 x}{21 (1-x)^{3/2} (x+1)^{3/2}}+\frac{1}{7 (1-x)^{5/2} (x+1)^{3/2}}+\frac{1}{7 (1-x)^{7/2} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - x)^(9/2)*(1 + x)^(5/2)),x]

[Out]

1/(7*(1 - x)^(7/2)*(1 + x)^(3/2)) + 1/(7*(1 - x)^(5/2)*(1 + x)^(3/2)) + (4*x)/(2
1*(1 - x)^(3/2)*(1 + x)^(3/2)) + (8*x)/(21*Sqrt[1 - x]*Sqrt[1 + x])

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Rubi in Sympy [A]  time = 7.1652, size = 70, normalized size = 0.84 \[ \frac{8 x}{21 \sqrt{- x + 1} \sqrt{x + 1}} + \frac{4 x}{21 \left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}} + \frac{1}{7 \left (- x + 1\right )^{\frac{5}{2}} \left (x + 1\right )^{\frac{3}{2}}} + \frac{1}{7 \left (- x + 1\right )^{\frac{7}{2}} \left (x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x)**(9/2)/(1+x)**(5/2),x)

[Out]

8*x/(21*sqrt(-x + 1)*sqrt(x + 1)) + 4*x/(21*(-x + 1)**(3/2)*(x + 1)**(3/2)) + 1/
(7*(-x + 1)**(5/2)*(x + 1)**(3/2)) + 1/(7*(-x + 1)**(7/2)*(x + 1)**(3/2))

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Mathematica [A]  time = 0.0366742, size = 45, normalized size = 0.54 \[ \frac{-8 x^5+16 x^4+4 x^3-24 x^2+9 x+6}{21 (1-x)^{7/2} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - x)^(9/2)*(1 + x)^(5/2)),x]

[Out]

(6 + 9*x - 24*x^2 + 4*x^3 + 16*x^4 - 8*x^5)/(21*(1 - x)^(7/2)*(1 + x)^(3/2))

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Maple [A]  time = 0.006, size = 40, normalized size = 0.5 \[ -{\frac{8\,{x}^{5}-16\,{x}^{4}-4\,{x}^{3}+24\,{x}^{2}-9\,x-6}{21} \left ( 1-x \right ) ^{-{\frac{7}{2}}} \left ( 1+x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-x)^(9/2)/(1+x)^(5/2),x)

[Out]

-1/21*(8*x^5-16*x^4-4*x^3+24*x^2-9*x-6)/(1+x)^(3/2)/(1-x)^(7/2)

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Maxima [A]  time = 1.34598, size = 123, normalized size = 1.48 \[ \frac{8 \, x}{21 \, \sqrt{-x^{2} + 1}} + \frac{4 \, x}{21 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{1}{7 \,{\left ({\left (-x^{2} + 1\right )}^{\frac{3}{2}} x^{2} - 2 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x +{\left (-x^{2} + 1\right )}^{\frac{3}{2}}\right )}} - \frac{1}{7 \,{\left ({\left (-x^{2} + 1\right )}^{\frac{3}{2}} x -{\left (-x^{2} + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 1)^(5/2)*(-x + 1)^(9/2)),x, algorithm="maxima")

[Out]

8/21*x/sqrt(-x^2 + 1) + 4/21*x/(-x^2 + 1)^(3/2) + 1/7/((-x^2 + 1)^(3/2)*x^2 - 2*
(-x^2 + 1)^(3/2)*x + (-x^2 + 1)^(3/2)) - 1/7/((-x^2 + 1)^(3/2)*x - (-x^2 + 1)^(3
/2))

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Fricas [A]  time = 0.208142, size = 286, normalized size = 3.45 \[ \frac{6 \, x^{10} + 28 \, x^{9} - 158 \, x^{8} - 12 \, x^{7} + 602 \, x^{6} - 329 \, x^{5} - 784 \, x^{4} + 644 \, x^{3} + 336 \, x^{2} -{\left (8 \, x^{9} - 46 \, x^{8} - 40 \, x^{7} + 336 \, x^{6} - 133 \, x^{5} - 616 \, x^{4} + 476 \, x^{3} + 336 \, x^{2} - 336 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 336 \, x}{21 \,{\left (x^{10} - 2 \, x^{9} - 13 \, x^{8} + 28 \, x^{7} + 27 \, x^{6} - 82 \, x^{5} - 3 \, x^{4} + 88 \, x^{3} - 28 \, x^{2} +{\left (5 \, x^{8} - 10 \, x^{7} - 20 \, x^{6} + 50 \, x^{5} + 11 \, x^{4} - 72 \, x^{3} + 20 \, x^{2} + 32 \, x - 16\right )} \sqrt{x + 1} \sqrt{-x + 1} - 32 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 1)^(5/2)*(-x + 1)^(9/2)),x, algorithm="fricas")

[Out]

1/21*(6*x^10 + 28*x^9 - 158*x^8 - 12*x^7 + 602*x^6 - 329*x^5 - 784*x^4 + 644*x^3
+ 336*x^2 - (8*x^9 - 46*x^8 - 40*x^7 + 336*x^6 - 133*x^5 - 616*x^4 + 476*x^3 +
336*x^2 - 336*x)*sqrt(x + 1)*sqrt(-x + 1) - 336*x)/(x^10 - 2*x^9 - 13*x^8 + 28*x
^7 + 27*x^6 - 82*x^5 - 3*x^4 + 88*x^3 - 28*x^2 + (5*x^8 - 10*x^7 - 20*x^6 + 50*x
^5 + 11*x^4 - 72*x^3 + 20*x^2 + 32*x - 16)*sqrt(x + 1)*sqrt(-x + 1) - 32*x + 16)

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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/(1-x)**(9/2)/(1+x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214652, size = 169, normalized size = 2.04 \[ \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{768 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \frac{19 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{256 \, \sqrt{x + 1}} - \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{57 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{768 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} - \frac{{\left ({\left ({\left (79 \, x - 432\right )}{\left (x + 1\right )} + 1120\right )}{\left (x + 1\right )} - 840\right )} \sqrt{x + 1} \sqrt{-x + 1}}{336 \,{\left (x - 1\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 1)^(5/2)*(-x + 1)^(9/2)),x, algorithm="giac")

[Out]

1/768*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 19/256*(sqrt(2) - sqrt(-x + 1))
/sqrt(x + 1) - 1/768*(x + 1)^(3/2)*(57*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(
sqrt(2) - sqrt(-x + 1))^3 - 1/336*(((79*x - 432)*(x + 1) + 1120)*(x + 1) - 840)*
sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^4