3.1075 \(\int (1-x)^{9/2} (1+x)^{3/2} \, dx\)

Optimal. Leaf size=109 \[ \frac {1}{7} (x+1)^{5/2} (1-x)^{9/2}+\frac {3}{14} (x+1)^{5/2} (1-x)^{7/2}+\frac {3}{10} (x+1)^{5/2} (1-x)^{5/2}+\frac {3}{8} x (x+1)^{3/2} (1-x)^{3/2}+\frac {9}{16} x \sqrt {x+1} \sqrt {1-x}+\frac {9}{16} \sin ^{-1}(x) \]

[Out]

3/8*(1-x)^(3/2)*x*(1+x)^(3/2)+3/10*(1-x)^(5/2)*(1+x)^(5/2)+3/14*(1-x)^(7/2)*(1+x)^(5/2)+1/7*(1-x)^(9/2)*(1+x)^
(5/2)+9/16*arcsin(x)+9/16*x*(1-x)^(1/2)*(1+x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {49, 38, 41, 216} \[ \frac {1}{7} (x+1)^{5/2} (1-x)^{9/2}+\frac {3}{14} (x+1)^{5/2} (1-x)^{7/2}+\frac {3}{10} (x+1)^{5/2} (1-x)^{5/2}+\frac {3}{8} x (x+1)^{3/2} (1-x)^{3/2}+\frac {9}{16} x \sqrt {x+1} \sqrt {1-x}+\frac {9}{16} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(9/2)*(1 + x)^(3/2),x]

[Out]

(9*Sqrt[1 - x]*x*Sqrt[1 + x])/16 + (3*(1 - x)^(3/2)*x*(1 + x)^(3/2))/8 + (3*(1 - x)^(5/2)*(1 + x)^(5/2))/10 +
(3*(1 - x)^(7/2)*(1 + x)^(5/2))/14 + ((1 - x)^(9/2)*(1 + x)^(5/2))/7 + (9*ArcSin[x])/16

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(x*(a + b*x)^m*(c + d*x)^m)/(2*m + 1)
, x] + Dist[(2*a*c*m)/(2*m + 1), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 49

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*(m
+ n + 1)), x] + Dist[(2*c*n)/(m + n + 1), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x]
&& EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0] && IGtQ[n + 1/2, 0] && LtQ[m, n]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int (1-x)^{9/2} (1+x)^{3/2} \, dx &=\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{7} \int (1-x)^{7/2} (1+x)^{3/2} \, dx\\ &=\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {3}{2} \int (1-x)^{5/2} (1+x)^{3/2} \, dx\\ &=\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {3}{2} \int (1-x)^{3/2} (1+x)^{3/2} \, dx\\ &=\frac {3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{8} \int \sqrt {1-x} \sqrt {1+x} \, dx\\ &=\frac {9}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{16} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {9}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{16} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {9}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{16} \sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 66, normalized size = 0.61 \[ \frac {1}{560} \sqrt {1-x^2} \left (80 x^6-280 x^5+208 x^4+350 x^3-656 x^2+245 x+368\right )-\frac {9}{8} \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(9/2)*(1 + x)^(3/2),x]

[Out]

(Sqrt[1 - x^2]*(368 + 245*x - 656*x^2 + 350*x^3 + 208*x^4 - 280*x^5 + 80*x^6))/560 - (9*ArcSin[Sqrt[1 - x]/Sqr
t[2]])/8

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fricas [A]  time = 0.43, size = 67, normalized size = 0.61 \[ \frac {1}{560} \, {\left (80 \, x^{6} - 280 \, x^{5} + 208 \, x^{4} + 350 \, x^{3} - 656 \, x^{2} + 245 \, x + 368\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {9}{8} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(9/2)*(1+x)^(3/2),x, algorithm="fricas")

[Out]

1/560*(80*x^6 - 280*x^5 + 208*x^4 + 350*x^3 - 656*x^2 + 245*x + 368)*sqrt(x + 1)*sqrt(-x + 1) - 9/8*arctan((sq
rt(x + 1)*sqrt(-x + 1) - 1)/x)

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giac [B]  time = 1.26, size = 237, normalized size = 2.17 \[ \frac {1}{1680} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, x - 37\right )} {\left (x + 1\right )} + 661\right )} {\left (x + 1\right )} - 4551\right )} {\left (x + 1\right )} + 4781\right )} {\left (x + 1\right )} - 6335\right )} {\left (x + 1\right )} + 2835\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{120} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - 26\right )} {\left (x + 1\right )} + 321\right )} {\left (x + 1\right )} - 451\right )} {\left (x + 1\right )} + 745\right )} {\left (x + 1\right )} - 405\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{120} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{6} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{6} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {9}{8} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(9/2)*(1+x)^(3/2),x, algorithm="giac")

[Out]

1/1680*((2*((4*(5*(6*x - 37)*(x + 1) + 661)*(x + 1) - 4551)*(x + 1) + 4781)*(x + 1) - 6335)*(x + 1) + 2835)*sq
rt(x + 1)*sqrt(-x + 1) - 1/120*((2*((4*(5*x - 26)*(x + 1) + 321)*(x + 1) - 451)*(x + 1) + 745)*(x + 1) - 405)*
sqrt(x + 1)*sqrt(-x + 1) - 1/120*((2*(3*(4*x - 17)*(x + 1) + 133)*(x + 1) - 295)*(x + 1) + 195)*sqrt(x + 1)*sq
rt(-x + 1) + 1/6*((2*(3*x - 10)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt(-x + 1) - 1/6*((2*x - 5)*(x + 1)
+ 9)*sqrt(x + 1)*sqrt(-x + 1) - sqrt(x + 1)*(x - 2)*sqrt(-x + 1) + sqrt(x + 1)*sqrt(-x + 1) + 9/8*arcsin(1/2*s
qrt(2)*sqrt(x + 1))

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maple [A]  time = 0.00, size = 127, normalized size = 1.17 \[ \frac {9 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{16 \sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (-x +1\right )^{\frac {9}{2}} \left (x +1\right )^{\frac {5}{2}}}{7}+\frac {3 \left (-x +1\right )^{\frac {7}{2}} \left (x +1\right )^{\frac {5}{2}}}{14}+\frac {3 \left (-x +1\right )^{\frac {5}{2}} \left (x +1\right )^{\frac {5}{2}}}{10}+\frac {3 \left (-x +1\right )^{\frac {3}{2}} \left (x +1\right )^{\frac {5}{2}}}{8}+\frac {3 \sqrt {-x +1}\, \left (x +1\right )^{\frac {5}{2}}}{8}-\frac {3 \sqrt {-x +1}\, \left (x +1\right )^{\frac {3}{2}}}{16}-\frac {9 \sqrt {-x +1}\, \sqrt {x +1}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x+1)^(9/2)*(x+1)^(3/2),x)

[Out]

1/7*(-x+1)^(9/2)*(x+1)^(5/2)+3/14*(-x+1)^(7/2)*(x+1)^(5/2)+3/10*(-x+1)^(5/2)*(x+1)^(5/2)+3/8*(-x+1)^(3/2)*(x+1
)^(5/2)+3/8*(-x+1)^(1/2)*(x+1)^(5/2)-3/16*(-x+1)^(1/2)*(x+1)^(3/2)-9/16*(-x+1)^(1/2)*(x+1)^(1/2)+9/16*((x+1)*(
-x+1))^(1/2)/(x+1)^(1/2)/(-x+1)^(1/2)*arcsin(x)

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maxima [A]  time = 3.03, size = 66, normalized size = 0.61 \[ \frac {1}{7} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x^{2} - \frac {1}{2} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {23}{35} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} + \frac {3}{8} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {9}{16} \, \sqrt {-x^{2} + 1} x + \frac {9}{16} \, \arcsin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(9/2)*(1+x)^(3/2),x, algorithm="maxima")

[Out]

1/7*(-x^2 + 1)^(5/2)*x^2 - 1/2*(-x^2 + 1)^(5/2)*x + 23/35*(-x^2 + 1)^(5/2) + 3/8*(-x^2 + 1)^(3/2)*x + 9/16*sqr
t(-x^2 + 1)*x + 9/16*arcsin(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (1-x\right )}^{9/2}\,{\left (x+1\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(9/2)*(x + 1)^(3/2),x)

[Out]

int((1 - x)^(9/2)*(x + 1)^(3/2), x)

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sympy [A]  time = 75.20, size = 325, normalized size = 2.98 \[ \begin {cases} - \frac {9 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} + \frac {i \left (x + 1\right )^{\frac {15}{2}}}{7 \sqrt {x - 1}} - \frac {23 i \left (x + 1\right )^{\frac {13}{2}}}{14 \sqrt {x - 1}} + \frac {541 i \left (x + 1\right )^{\frac {11}{2}}}{70 \sqrt {x - 1}} - \frac {5249 i \left (x + 1\right )^{\frac {9}{2}}}{280 \sqrt {x - 1}} + \frac {6653 i \left (x + 1\right )^{\frac {7}{2}}}{280 \sqrt {x - 1}} - \frac {1027 i \left (x + 1\right )^{\frac {5}{2}}}{80 \sqrt {x - 1}} - \frac {3 i \left (x + 1\right )^{\frac {3}{2}}}{16 \sqrt {x - 1}} + \frac {9 i \sqrt {x + 1}}{8 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\frac {9 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} - \frac {\left (x + 1\right )^{\frac {15}{2}}}{7 \sqrt {1 - x}} + \frac {23 \left (x + 1\right )^{\frac {13}{2}}}{14 \sqrt {1 - x}} - \frac {541 \left (x + 1\right )^{\frac {11}{2}}}{70 \sqrt {1 - x}} + \frac {5249 \left (x + 1\right )^{\frac {9}{2}}}{280 \sqrt {1 - x}} - \frac {6653 \left (x + 1\right )^{\frac {7}{2}}}{280 \sqrt {1 - x}} + \frac {1027 \left (x + 1\right )^{\frac {5}{2}}}{80 \sqrt {1 - x}} + \frac {3 \left (x + 1\right )^{\frac {3}{2}}}{16 \sqrt {1 - x}} - \frac {9 \sqrt {x + 1}}{8 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(9/2)*(1+x)**(3/2),x)

[Out]

Piecewise((-9*I*acosh(sqrt(2)*sqrt(x + 1)/2)/8 + I*(x + 1)**(15/2)/(7*sqrt(x - 1)) - 23*I*(x + 1)**(13/2)/(14*
sqrt(x - 1)) + 541*I*(x + 1)**(11/2)/(70*sqrt(x - 1)) - 5249*I*(x + 1)**(9/2)/(280*sqrt(x - 1)) + 6653*I*(x +
1)**(7/2)/(280*sqrt(x - 1)) - 1027*I*(x + 1)**(5/2)/(80*sqrt(x - 1)) - 3*I*(x + 1)**(3/2)/(16*sqrt(x - 1)) + 9
*I*sqrt(x + 1)/(8*sqrt(x - 1)), Abs(x + 1)/2 > 1), (9*asin(sqrt(2)*sqrt(x + 1)/2)/8 - (x + 1)**(15/2)/(7*sqrt(
1 - x)) + 23*(x + 1)**(13/2)/(14*sqrt(1 - x)) - 541*(x + 1)**(11/2)/(70*sqrt(1 - x)) + 5249*(x + 1)**(9/2)/(28
0*sqrt(1 - x)) - 6653*(x + 1)**(7/2)/(280*sqrt(1 - x)) + 1027*(x + 1)**(5/2)/(80*sqrt(1 - x)) + 3*(x + 1)**(3/
2)/(16*sqrt(1 - x)) - 9*sqrt(x + 1)/(8*sqrt(1 - x)), True))

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