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

Optimal. Leaf size=103 \[ -\frac {2 (1-x)^{9/2}}{3 (x+1)^{3/2}}+\frac {6 (1-x)^{7/2}}{\sqrt {x+1}}+7 \sqrt {x+1} (1-x)^{5/2}+\frac {35}{2} \sqrt {x+1} (1-x)^{3/2}+\frac {105}{2} \sqrt {x+1} \sqrt {1-x}+\frac {105}{2} \sin ^{-1}(x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 103, 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 = {47, 50, 41, 216} \begin {gather*} -\frac {2 (1-x)^{9/2}}{3 (x+1)^{3/2}}+\frac {6 (1-x)^{7/2}}{\sqrt {x+1}}+7 \sqrt {x+1} (1-x)^{5/2}+\frac {35}{2} \sqrt {x+1} (1-x)^{3/2}+\frac {105}{2} \sqrt {x+1} \sqrt {1-x}+\frac {105}{2} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(9/2)/(1 + x)^(5/2),x]

[Out]

(-2*(1 - x)^(9/2))/(3*(1 + x)^(3/2)) + (6*(1 - x)^(7/2))/Sqrt[1 + x] + (105*Sqrt[1 - x]*Sqrt[1 + x])/2 + (35*(
1 - x)^(3/2)*Sqrt[1 + x])/2 + 7*(1 - x)^(5/2)*Sqrt[1 + x] + (105*ArcSin[x])/2

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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

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[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx &=-\frac {2 (1-x)^{9/2}}{3 (1+x)^{3/2}}-3 \int \frac {(1-x)^{7/2}}{(1+x)^{3/2}} \, dx\\ &=-\frac {2 (1-x)^{9/2}}{3 (1+x)^{3/2}}+\frac {6 (1-x)^{7/2}}{\sqrt {1+x}}+21 \int \frac {(1-x)^{5/2}}{\sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{9/2}}{3 (1+x)^{3/2}}+\frac {6 (1-x)^{7/2}}{\sqrt {1+x}}+7 (1-x)^{5/2} \sqrt {1+x}+35 \int \frac {(1-x)^{3/2}}{\sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{9/2}}{3 (1+x)^{3/2}}+\frac {6 (1-x)^{7/2}}{\sqrt {1+x}}+\frac {35}{2} (1-x)^{3/2} \sqrt {1+x}+7 (1-x)^{5/2} \sqrt {1+x}+\frac {105}{2} \int \frac {\sqrt {1-x}}{\sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{9/2}}{3 (1+x)^{3/2}}+\frac {6 (1-x)^{7/2}}{\sqrt {1+x}}+\frac {105}{2} \sqrt {1-x} \sqrt {1+x}+\frac {35}{2} (1-x)^{3/2} \sqrt {1+x}+7 (1-x)^{5/2} \sqrt {1+x}+\frac {105}{2} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{9/2}}{3 (1+x)^{3/2}}+\frac {6 (1-x)^{7/2}}{\sqrt {1+x}}+\frac {105}{2} \sqrt {1-x} \sqrt {1+x}+\frac {35}{2} (1-x)^{3/2} \sqrt {1+x}+7 (1-x)^{5/2} \sqrt {1+x}+\frac {105}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {2 (1-x)^{9/2}}{3 (1+x)^{3/2}}+\frac {6 (1-x)^{7/2}}{\sqrt {1+x}}+\frac {105}{2} \sqrt {1-x} \sqrt {1+x}+\frac {35}{2} (1-x)^{3/2} \sqrt {1+x}+7 (1-x)^{5/2} \sqrt {1+x}+\frac {105}{2} \sin ^{-1}(x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 37, normalized size = 0.36 \begin {gather*} -\frac {(1-x)^{11/2} \, _2F_1\left (\frac {5}{2},\frac {11}{2};\frac {13}{2};\frac {1-x}{2}\right )}{22 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/22*((1 - x)^(11/2)*Hypergeometric2F1[5/2, 11/2, 13/2, (1 - x)/2])/Sqrt[2]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.10, size = 131, normalized size = 1.27 \begin {gather*} \frac {-\frac {16 (1-x)^{9/2}}{(x+1)^{9/2}}+\frac {144 (1-x)^{7/2}}{(x+1)^{7/2}}+\frac {693 (1-x)^{5/2}}{(x+1)^{5/2}}+\frac {840 (1-x)^{3/2}}{(x+1)^{3/2}}+\frac {315 \sqrt {1-x}}{\sqrt {x+1}}}{3 \left (\frac {1-x}{x+1}+1\right )^3}-105 \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]

((-16*(1 - x)^(9/2))/(1 + x)^(9/2) + (144*(1 - x)^(7/2))/(1 + x)^(7/2) + (693*(1 - x)^(5/2))/(1 + x)^(5/2) + (
840*(1 - x)^(3/2))/(1 + x)^(3/2) + (315*Sqrt[1 - x])/Sqrt[1 + x])/(3*(1 + (1 - x)/(1 + x))^3) - 105*ArcTan[Sqr
t[1 - x]/Sqrt[1 + x]]

________________________________________________________________________________________

fricas [A]  time = 1.50, size = 85, normalized size = 0.83 \begin {gather*} \frac {494 \, x^{2} + {\left (2 \, x^{4} - 17 \, x^{3} + 102 \, x^{2} + 679 \, x + 494\right )} \sqrt {x + 1} \sqrt {-x + 1} - 630 \, {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 988 \, x + 494}{6 \, {\left (x^{2} + 2 \, 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="fricas")

[Out]

1/6*(494*x^2 + (2*x^4 - 17*x^3 + 102*x^2 + 679*x + 494)*sqrt(x + 1)*sqrt(-x + 1) - 630*(x^2 + 2*x + 1)*arctan(
(sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 988*x + 494)/(x^2 + 2*x + 1)

________________________________________________________________________________________

giac [A]  time = 0.89, size = 127, normalized size = 1.23 \begin {gather*} \frac {1}{6} \, {\left ({\left (2 \, x - 23\right )} {\left (x + 1\right )} + 165\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {2 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{3 \, {\left (x + 1\right )}^{\frac {3}{2}}} - \frac {34 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} + \frac {2 \, {\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {51 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{3 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} + 105 \, \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]

1/6*((2*x - 23)*(x + 1) + 165)*sqrt(x + 1)*sqrt(-x + 1) + 2/3*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) - 34*(s
qrt(2) - sqrt(-x + 1))/sqrt(x + 1) + 2/3*(x + 1)^(3/2)*(51*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) - 1)/(sqrt(2) -
sqrt(-x + 1))^3 + 105*arcsin(1/2*sqrt(2)*sqrt(x + 1))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 89, normalized size = 0.86 \begin {gather*} \frac {105 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{2 \sqrt {x +1}\, \sqrt {-x +1}}-\frac {\left (2 x^{5}-19 x^{4}+119 x^{3}+577 x^{2}-185 x -494\right ) \sqrt {\left (x +1\right ) \left (-x +1\right )}}{6 \left (x +1\right )^{\frac {3}{2}} \sqrt {-\left (x +1\right ) \left (x -1\right )}\, \sqrt {-x +1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*x^5-19*x^4+119*x^3+577*x^2-185*x-494)/(x+1)^(3/2)/(-(x+1)*(x-1))^(1/2)*((x+1)*(-x+1))^(1/2)/(-x+1)^(1/
2)+105/2*((x+1)*(-x+1))^(1/2)/(x+1)^(1/2)/(-x+1)^(1/2)*arcsin(x)

________________________________________________________________________________________

maxima [A]  time = 3.01, size = 125, normalized size = 1.21 \begin {gather*} \frac {x^{6}}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {7 \, x^{5}}{2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {23 \, x^{4}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {35}{2} \, x {\left (\frac {3 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {2}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}\right )} - \frac {143 \, x}{6 \, \sqrt {-x^{2} + 1}} - \frac {127 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {22 \, x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {247}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {105}{2} \, \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]

1/3*x^6/(-x^2 + 1)^(3/2) - 7/2*x^5/(-x^2 + 1)^(3/2) + 23*x^4/(-x^2 + 1)^(3/2) + 35/2*x*(3*x^2/(-x^2 + 1)^(3/2)
- 2/(-x^2 + 1)^(3/2)) - 143/6*x/sqrt(-x^2 + 1) - 127*x^2/(-x^2 + 1)^(3/2) + 22/3*x/(-x^2 + 1)^(3/2) + 247/3/(
-x^2 + 1)^(3/2) + 105/2*arcsin(x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\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)

________________________________________________________________________________________

sympy [A]  time = 45.20, size = 250, normalized size = 2.43 \begin {gather*} \begin {cases} - 105 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {i \left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {x - 1}} - \frac {29 i \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {x - 1}} + \frac {215 i \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {x - 1}} + \frac {43 i \sqrt {x + 1}}{3 \sqrt {x - 1}} - \frac {448 i}{3 \sqrt {x - 1} \sqrt {x + 1}} + \frac {64 i}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {3}{2}}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\105 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {\left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {1 - x}} + \frac {29 \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {1 - x}} - \frac {215 \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {1 - x}} - \frac {43 \sqrt {x + 1}}{3 \sqrt {1 - x}} + \frac {448}{3 \sqrt {1 - x} \sqrt {x + 1}} - \frac {64}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {3}{2}}} & \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((-105*I*acosh(sqrt(2)*sqrt(x + 1)/2) + I*(x + 1)**(7/2)/(3*sqrt(x - 1)) - 29*I*(x + 1)**(5/2)/(6*sqr
t(x - 1)) + 215*I*(x + 1)**(3/2)/(6*sqrt(x - 1)) + 43*I*sqrt(x + 1)/(3*sqrt(x - 1)) - 448*I/(3*sqrt(x - 1)*sqr
t(x + 1)) + 64*I/(3*sqrt(x - 1)*(x + 1)**(3/2)), Abs(x + 1)/2 > 1), (105*asin(sqrt(2)*sqrt(x + 1)/2) - (x + 1)
**(7/2)/(3*sqrt(1 - x)) + 29*(x + 1)**(5/2)/(6*sqrt(1 - x)) - 215*(x + 1)**(3/2)/(6*sqrt(1 - x)) - 43*sqrt(x +
1)/(3*sqrt(1 - x)) + 448/(3*sqrt(1 - x)*sqrt(x + 1)) - 64/(3*sqrt(1 - x)*(x + 1)**(3/2)), True))

________________________________________________________________________________________