3.1090 \(\int (1-x)^{7/2} (1+x)^{5/2} \, dx\)

Optimal. Leaf size=90 \[ \frac{1}{7} (1-x)^{7/2} (x+1)^{7/2}+\frac{1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac{5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac{5}{16} \sqrt{1-x} x \sqrt{x+1}+\frac{5}{16} \sin ^{-1}(x) \]

[Out]

(5*Sqrt[1 - x]*x*Sqrt[1 + x])/16 + (5*(1 - x)^(3/2)*x*(1 + x)^(3/2))/24 + ((1 -
x)^(5/2)*x*(1 + x)^(5/2))/6 + ((1 - x)^(7/2)*(1 + x)^(7/2))/7 + (5*ArcSin[x])/16

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Rubi [A]  time = 0.0543882, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{1}{7} (1-x)^{7/2} (x+1)^{7/2}+\frac{1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac{5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac{5}{16} \sqrt{1-x} x \sqrt{x+1}+\frac{5}{16} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^(7/2)*(1 + x)^(5/2),x]

[Out]

(5*Sqrt[1 - x]*x*Sqrt[1 + x])/16 + (5*(1 - x)^(3/2)*x*(1 + x)^(3/2))/24 + ((1 -
x)^(5/2)*x*(1 + x)^(5/2))/6 + ((1 - x)^(7/2)*(1 + x)^(7/2))/7 + (5*ArcSin[x])/16

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Rubi in Sympy [A]  time = 7.96631, size = 75, normalized size = 0.83 \[ \frac{x \left (- x + 1\right )^{\frac{5}{2}} \left (x + 1\right )^{\frac{5}{2}}}{6} + \frac{5 x \left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{24} + \frac{5 x \sqrt{- x + 1} \sqrt{x + 1}}{16} + \frac{\left (- x + 1\right )^{\frac{7}{2}} \left (x + 1\right )^{\frac{7}{2}}}{7} + \frac{5 \operatorname{asin}{\left (x \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**(7/2)*(1+x)**(5/2),x)

[Out]

x*(-x + 1)**(5/2)*(x + 1)**(5/2)/6 + 5*x*(-x + 1)**(3/2)*(x + 1)**(3/2)/24 + 5*x
*sqrt(-x + 1)*sqrt(x + 1)/16 + (-x + 1)**(7/2)*(x + 1)**(7/2)/7 + 5*asin(x)/16

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Mathematica [A]  time = 0.0477767, size = 64, normalized size = 0.71 \[ \frac{1}{336} \sqrt{1-x^2} \left (-48 x^6+56 x^5+144 x^4-182 x^3-144 x^2+231 x+48\right )+\frac{5}{8} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)^(7/2)*(1 + x)^(5/2),x]

[Out]

(Sqrt[1 - x^2]*(48 + 231*x - 144*x^2 - 182*x^3 + 144*x^4 + 56*x^5 - 48*x^6))/336
+ (5*ArcSin[Sqrt[1 + x]/Sqrt[2]])/8

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Maple [A]  time = 0.007, size = 127, normalized size = 1.4 \[{\frac{1}{7} \left ( 1-x \right ) ^{{\frac{7}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{8}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{7}{2}}}}-{\frac{1}{24}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{5}{48}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{5}{16}\sqrt{1-x}\sqrt{1+x}}+{\frac{5\,\arcsin \left ( x \right ) }{16}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(7/2)*(1+x)^(5/2),x)

[Out]

1/7*(1-x)^(7/2)*(1+x)^(7/2)+1/6*(1-x)^(5/2)*(1+x)^(7/2)+1/6*(1-x)^(3/2)*(1+x)^(7
/2)+1/8*(1-x)^(1/2)*(1+x)^(7/2)-1/24*(1-x)^(1/2)*(1+x)^(5/2)-5/48*(1-x)^(1/2)*(1
+x)^(3/2)-5/16*(1-x)^(1/2)*(1+x)^(1/2)+5/16*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x
)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.49761, size = 70, normalized size = 0.78 \[ \frac{1}{7} \,{\left (-x^{2} + 1\right )}^{\frac{7}{2}} + \frac{1}{6} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} x + \frac{5}{24} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{5}{16} \, \sqrt{-x^{2} + 1} x + \frac{5}{16} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)*(-x + 1)^(7/2),x, algorithm="maxima")

[Out]

1/7*(-x^2 + 1)^(7/2) + 1/6*(-x^2 + 1)^(5/2)*x + 5/24*(-x^2 + 1)^(3/2)*x + 5/16*s
qrt(-x^2 + 1)*x + 5/16*arcsin(x)

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Fricas [A]  time = 0.213426, size = 351, normalized size = 3.9 \[ -\frac{48 \, x^{14} - 56 \, x^{13} - 1344 \, x^{12} + 1582 \, x^{11} + 8736 \, x^{10} - 10605 \, x^{9} - 25536 \, x^{8} + 32767 \, x^{7} + 39648 \, x^{6} - 53816 \, x^{5} - 32256 \, x^{4} + 44912 \, x^{3} + 10752 \, x^{2} + 7 \,{\left (48 \, x^{12} - 56 \, x^{11} - 528 \, x^{10} + 630 \, x^{9} + 2064 \, x^{8} - 2583 \, x^{7} - 3936 \, x^{6} + 5272 \, x^{5} + 3840 \, x^{4} - 5360 \, x^{3} - 1536 \, x^{2} + 2112 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 210 \,{\left (7 \, x^{6} - 56 \, x^{4} + 112 \, x^{2} -{\left (x^{6} - 24 \, x^{4} + 80 \, x^{2} - 64\right )} \sqrt{x + 1} \sqrt{-x + 1} - 64\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 14784 \, x}{336 \,{\left (7 \, x^{6} - 56 \, x^{4} + 112 \, x^{2} -{\left (x^{6} - 24 \, x^{4} + 80 \, x^{2} - 64\right )} \sqrt{x + 1} \sqrt{-x + 1} - 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)*(-x + 1)^(7/2),x, algorithm="fricas")

[Out]

-1/336*(48*x^14 - 56*x^13 - 1344*x^12 + 1582*x^11 + 8736*x^10 - 10605*x^9 - 2553
6*x^8 + 32767*x^7 + 39648*x^6 - 53816*x^5 - 32256*x^4 + 44912*x^3 + 10752*x^2 +
7*(48*x^12 - 56*x^11 - 528*x^10 + 630*x^9 + 2064*x^8 - 2583*x^7 - 3936*x^6 + 527
2*x^5 + 3840*x^4 - 5360*x^3 - 1536*x^2 + 2112*x)*sqrt(x + 1)*sqrt(-x + 1) + 210*
(7*x^6 - 56*x^4 + 112*x^2 - (x^6 - 24*x^4 + 80*x^2 - 64)*sqrt(x + 1)*sqrt(-x + 1
) - 64)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - 14784*x)/(7*x^6 - 56*x^4 + 11
2*x^2 - (x^6 - 24*x^4 + 80*x^2 - 64)*sqrt(x + 1)*sqrt(-x + 1) - 64)

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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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.246151, size = 259, normalized size = 2.88 \[ -\frac{1}{105} \,{\left ({\left (3 \,{\left ({\left (5 \,{\left (x + 1\right )}{\left (x - 5\right )} + 74\right )}{\left (x + 1\right )} - 96\right )}{\left (x + 1\right )} + 203\right )}{\left (x + 1\right )} - 70\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} + \frac{2}{15} \,{\left ({\left (3 \,{\left (x + 1\right )}{\left (x - 3\right )} + 17\right )}{\left (x + 1\right )} - 10\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} - \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} + \frac{1}{48} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left (x + 1\right )}{\left (x - 4\right )} + 39\right )}{\left (x + 1\right )} - 37\right )}{\left (x + 1\right )} + 31\right )}{\left (x + 1\right )} - 3\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{1}{4} \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 2\right )} + 5\right )}{\left (x + 1\right )} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \frac{5}{8} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)*(-x + 1)^(7/2),x, algorithm="giac")

[Out]

-1/105*((3*((5*(x + 1)*(x - 5) + 74)*(x + 1) - 96)*(x + 1) + 203)*(x + 1) - 70)*
(x + 1)^(3/2)*sqrt(-x + 1) + 2/15*((3*(x + 1)*(x - 3) + 17)*(x + 1) - 10)*(x + 1
)^(3/2)*sqrt(-x + 1) - 1/3*(x + 1)^(3/2)*(x - 1)*sqrt(-x + 1) + 1/48*((2*((4*(x
+ 1)*(x - 4) + 39)*(x + 1) - 37)*(x + 1) + 31)*(x + 1) - 3)*sqrt(x + 1)*sqrt(-x
+ 1) - 1/4*((2*(x + 1)*(x - 2) + 5)*(x + 1) - 1)*sqrt(x + 1)*sqrt(-x + 1) + 1/2*
sqrt(x + 1)*x*sqrt(-x + 1) + 5/8*arcsin(1/2*sqrt(2)*sqrt(x + 1))