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

Optimal. Leaf size=110 \[ \frac {1}{8} (x+1)^{7/2} (1-x)^{9/2}+\frac {9}{56} (x+1)^{7/2} (1-x)^{7/2}+\frac {3}{16} x (x+1)^{5/2} (1-x)^{5/2}+\frac {15}{64} x (x+1)^{3/2} (1-x)^{3/2}+\frac {45}{128} x \sqrt {x+1} \sqrt {1-x}+\frac {45}{128} \sin ^{-1}(x) \]

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Rubi [A]  time = 0.02, antiderivative size = 110, 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} \begin {gather*} \frac {1}{8} (x+1)^{7/2} (1-x)^{9/2}+\frac {9}{56} (x+1)^{7/2} (1-x)^{7/2}+\frac {3}{16} x (x+1)^{5/2} (1-x)^{5/2}+\frac {15}{64} x (x+1)^{3/2} (1-x)^{3/2}+\frac {45}{128} x \sqrt {x+1} \sqrt {1-x}+\frac {45}{128} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(9/2)*(1 + x)^(5/2),x]

[Out]

(45*Sqrt[1 - x]*x*Sqrt[1 + x])/128 + (15*(1 - x)^(3/2)*x*(1 + x)^(3/2))/64 + (3*(1 - x)^(5/2)*x*(1 + x)^(5/2))
/16 + (9*(1 - x)^(7/2)*(1 + x)^(7/2))/56 + ((1 - x)^(9/2)*(1 + x)^(7/2))/8 + (45*ArcSin[x])/128

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)^{5/2} \, dx &=\frac {1}{8} (1-x)^{9/2} (1+x)^{7/2}+\frac {9}{8} \int (1-x)^{7/2} (1+x)^{5/2} \, dx\\ &=\frac {9}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {1}{8} (1-x)^{9/2} (1+x)^{7/2}+\frac {9}{8} \int (1-x)^{5/2} (1+x)^{5/2} \, dx\\ &=\frac {3}{16} (1-x)^{5/2} x (1+x)^{5/2}+\frac {9}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {1}{8} (1-x)^{9/2} (1+x)^{7/2}+\frac {15}{16} \int (1-x)^{3/2} (1+x)^{3/2} \, dx\\ &=\frac {15}{64} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{16} (1-x)^{5/2} x (1+x)^{5/2}+\frac {9}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {1}{8} (1-x)^{9/2} (1+x)^{7/2}+\frac {45}{64} \int \sqrt {1-x} \sqrt {1+x} \, dx\\ &=\frac {45}{128} \sqrt {1-x} x \sqrt {1+x}+\frac {15}{64} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{16} (1-x)^{5/2} x (1+x)^{5/2}+\frac {9}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {1}{8} (1-x)^{9/2} (1+x)^{7/2}+\frac {45}{128} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {45}{128} \sqrt {1-x} x \sqrt {1+x}+\frac {15}{64} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{16} (1-x)^{5/2} x (1+x)^{5/2}+\frac {9}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {1}{8} (1-x)^{9/2} (1+x)^{7/2}+\frac {45}{128} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {45}{128} \sqrt {1-x} x \sqrt {1+x}+\frac {15}{64} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{16} (1-x)^{5/2} x (1+x)^{5/2}+\frac {9}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {1}{8} (1-x)^{9/2} (1+x)^{7/2}+\frac {45}{128} \sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 70, normalized size = 0.64 \begin {gather*} \frac {1}{896} \left (\sqrt {1-x^2} \left (112 x^7-256 x^6-168 x^5+768 x^4-210 x^3-768 x^2+581 x+256\right )-630 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - x^2]*(256 + 581*x - 768*x^2 - 210*x^3 + 768*x^4 - 168*x^5 - 256*x^6 + 112*x^7) - 630*ArcSin[Sqrt[1 -
x]/Sqrt[2]])/896

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IntegrateAlgebraic [A]  time = 0.16, size = 187, normalized size = 1.70 \begin {gather*} \frac {-\frac {315 (1-x)^{15/2}}{(x+1)^{15/2}}-\frac {2415 (1-x)^{13/2}}{(x+1)^{13/2}}-\frac {8043 (1-x)^{11/2}}{(x+1)^{11/2}}+\frac {17609 (1-x)^{9/2}}{(x+1)^{9/2}}+\frac {15159 (1-x)^{7/2}}{(x+1)^{7/2}}+\frac {8043 (1-x)^{5/2}}{(x+1)^{5/2}}+\frac {2415 (1-x)^{3/2}}{(x+1)^{3/2}}+\frac {315 \sqrt {1-x}}{\sqrt {x+1}}}{448 \left (\frac {1-x}{x+1}+1\right )^8}-\frac {45}{64} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-315*(1 - x)^(15/2))/(1 + x)^(15/2) - (2415*(1 - x)^(13/2))/(1 + x)^(13/2) - (8043*(1 - x)^(11/2))/(1 + x)^(
11/2) + (17609*(1 - x)^(9/2))/(1 + x)^(9/2) + (15159*(1 - x)^(7/2))/(1 + x)^(7/2) + (8043*(1 - x)^(5/2))/(1 +
x)^(5/2) + (2415*(1 - x)^(3/2))/(1 + x)^(3/2) + (315*Sqrt[1 - x])/Sqrt[1 + x])/(448*(1 + (1 - x)/(1 + x))^8) -
(45*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]])/64

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fricas [A]  time = 1.16, size = 72, normalized size = 0.65 \begin {gather*} \frac {1}{896} \, {\left (112 \, x^{7} - 256 \, x^{6} - 168 \, x^{5} + 768 \, x^{4} - 210 \, x^{3} - 768 \, x^{2} + 581 \, x + 256\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {45}{64} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/896*(112*x^7 - 256*x^6 - 168*x^5 + 768*x^4 - 210*x^3 - 768*x^2 + 581*x + 256)*sqrt(x + 1)*sqrt(-x + 1) - 45/
64*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

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giac [B]  time = 1.49, size = 296, normalized size = 2.69 \begin {gather*} \frac {1}{13440} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, {\left (7 \, x - 50\right )} {\left (x + 1\right )} + 1219\right )} {\left (x + 1\right )} - 12463\right )} {\left (x + 1\right )} + 64233\right )} {\left (x + 1\right )} - 53963\right )} {\left (x + 1\right )} + 59465\right )} {\left (x + 1\right )} - 23205\right )} \sqrt {x + 1} \sqrt {-x + 1} - \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}{80} \, {\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}{40} \, {\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}{8} \, {\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 {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {45}{64} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13440*((2*((4*(5*(6*(7*x - 50)*(x + 1) + 1219)*(x + 1) - 12463)*(x + 1) + 64233)*(x + 1) - 53963)*(x + 1) +
59465)*(x + 1) - 23205)*sqrt(x + 1)*sqrt(-x + 1) - 1/1680*((2*((4*(5*(6*x - 37)*(x + 1) + 661)*(x + 1) - 4551)
*(x + 1) + 4781)*(x + 1) - 6335)*(x + 1) + 2835)*sqrt(x + 1)*sqrt(-x + 1) - 1/80*((2*((4*(5*x - 26)*(x + 1) +
321)*(x + 1) - 451)*(x + 1) + 745)*(x + 1) - 405)*sqrt(x + 1)*sqrt(-x + 1) + 1/40*((2*(3*(4*x - 17)*(x + 1) +
133)*(x + 1) - 295)*(x + 1) + 195)*sqrt(x + 1)*sqrt(-x + 1) + 1/8*((2*(3*x - 10)*(x + 1) + 43)*(x + 1) - 39)*s
qrt(x + 1)*sqrt(-x + 1) - 1/2*((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(-x + 1) - 1/2*sqrt(x + 1)*(x - 2)*sqrt(
-x + 1) + sqrt(x + 1)*sqrt(-x + 1) + 45/64*arcsin(1/2*sqrt(2)*sqrt(x + 1))

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maple [A]  time = 0.00, size = 141, normalized size = 1.28 \begin {gather*} \frac {45 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{128 \sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (-x +1\right )^{\frac {9}{2}} \left (x +1\right )^{\frac {7}{2}}}{8}+\frac {9 \left (-x +1\right )^{\frac {7}{2}} \left (x +1\right )^{\frac {7}{2}}}{56}+\frac {3 \left (-x +1\right )^{\frac {5}{2}} \left (x +1\right )^{\frac {7}{2}}}{16}+\frac {3 \left (-x +1\right )^{\frac {3}{2}} \left (x +1\right )^{\frac {7}{2}}}{16}+\frac {9 \sqrt {-x +1}\, \left (x +1\right )^{\frac {7}{2}}}{64}-\frac {3 \sqrt {-x +1}\, \left (x +1\right )^{\frac {5}{2}}}{64}-\frac {15 \sqrt {-x +1}\, \left (x +1\right )^{\frac {3}{2}}}{128}-\frac {45 \sqrt {-x +1}\, \sqrt {x +1}}{128} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(-x+1)^(9/2)*(x+1)^(7/2)+9/56*(-x+1)^(7/2)*(x+1)^(7/2)+3/16*(-x+1)^(5/2)*(x+1)^(7/2)+3/16*(-x+1)^(3/2)*(x+
1)^(7/2)+9/64*(-x+1)^(1/2)*(x+1)^(7/2)-3/64*(-x+1)^(1/2)*(x+1)^(5/2)-15/128*(-x+1)^(1/2)*(x+1)^(3/2)-45/128*(-
x+1)^(1/2)*(x+1)^(1/2)+45/128*((x+1)*(-x+1))^(1/2)/(x+1)^(1/2)/(-x+1)^(1/2)*arcsin(x)

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maxima [A]  time = 2.97, size = 64, normalized size = 0.58 \begin {gather*} -\frac {1}{8} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} x + \frac {2}{7} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} + \frac {3}{16} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {15}{64} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {45}{128} \, \sqrt {-x^{2} + 1} x + \frac {45}{128} \, \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(-x^2 + 1)^(7/2)*x + 2/7*(-x^2 + 1)^(7/2) + 3/16*(-x^2 + 1)^(5/2)*x + 15/64*(-x^2 + 1)^(3/2)*x + 45/128*s
qrt(-x^2 + 1)*x + 45/128*arcsin(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (1-x\right )}^{9/2}\,{\left (x+1\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(9/2)*(x + 1)^(5/2),x)

[Out]

int((1 - x)^(9/2)*(x + 1)^(5/2), x)

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sympy [A]  time = 117.57, size = 360, normalized size = 3.27 \begin {gather*} \begin {cases} - \frac {45 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{64} + \frac {i \left (x + 1\right )^{\frac {17}{2}}}{8 \sqrt {x - 1}} - \frac {79 i \left (x + 1\right )^{\frac {15}{2}}}{56 \sqrt {x - 1}} + \frac {725 i \left (x + 1\right )^{\frac {13}{2}}}{112 \sqrt {x - 1}} - \frac {1699 i \left (x + 1\right )^{\frac {11}{2}}}{112 \sqrt {x - 1}} + \frac {8191 i \left (x + 1\right )^{\frac {9}{2}}}{448 \sqrt {x - 1}} - \frac {4099 i \left (x + 1\right )^{\frac {7}{2}}}{448 \sqrt {x - 1}} - \frac {3 i \left (x + 1\right )^{\frac {5}{2}}}{128 \sqrt {x - 1}} - \frac {15 i \left (x + 1\right )^{\frac {3}{2}}}{128 \sqrt {x - 1}} + \frac {45 i \sqrt {x + 1}}{64 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\frac {45 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{64} - \frac {\left (x + 1\right )^{\frac {17}{2}}}{8 \sqrt {1 - x}} + \frac {79 \left (x + 1\right )^{\frac {15}{2}}}{56 \sqrt {1 - x}} - \frac {725 \left (x + 1\right )^{\frac {13}{2}}}{112 \sqrt {1 - x}} + \frac {1699 \left (x + 1\right )^{\frac {11}{2}}}{112 \sqrt {1 - x}} - \frac {8191 \left (x + 1\right )^{\frac {9}{2}}}{448 \sqrt {1 - x}} + \frac {4099 \left (x + 1\right )^{\frac {7}{2}}}{448 \sqrt {1 - x}} + \frac {3 \left (x + 1\right )^{\frac {5}{2}}}{128 \sqrt {1 - x}} + \frac {15 \left (x + 1\right )^{\frac {3}{2}}}{128 \sqrt {1 - x}} - \frac {45 \sqrt {x + 1}}{64 \sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-45*I*acosh(sqrt(2)*sqrt(x + 1)/2)/64 + I*(x + 1)**(17/2)/(8*sqrt(x - 1)) - 79*I*(x + 1)**(15/2)/(5
6*sqrt(x - 1)) + 725*I*(x + 1)**(13/2)/(112*sqrt(x - 1)) - 1699*I*(x + 1)**(11/2)/(112*sqrt(x - 1)) + 8191*I*(
x + 1)**(9/2)/(448*sqrt(x - 1)) - 4099*I*(x + 1)**(7/2)/(448*sqrt(x - 1)) - 3*I*(x + 1)**(5/2)/(128*sqrt(x - 1
)) - 15*I*(x + 1)**(3/2)/(128*sqrt(x - 1)) + 45*I*sqrt(x + 1)/(64*sqrt(x - 1)), Abs(x + 1)/2 > 1), (45*asin(sq
rt(2)*sqrt(x + 1)/2)/64 - (x + 1)**(17/2)/(8*sqrt(1 - x)) + 79*(x + 1)**(15/2)/(56*sqrt(1 - x)) - 725*(x + 1)*
*(13/2)/(112*sqrt(1 - x)) + 1699*(x + 1)**(11/2)/(112*sqrt(1 - x)) - 8191*(x + 1)**(9/2)/(448*sqrt(1 - x)) + 4
099*(x + 1)**(7/2)/(448*sqrt(1 - x)) + 3*(x + 1)**(5/2)/(128*sqrt(1 - x)) + 15*(x + 1)**(3/2)/(128*sqrt(1 - x)
) - 45*sqrt(x + 1)/(64*sqrt(1 - x)), True))

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