∫ (1−x)9∕2 ______________(1+x)5∕2 dx [1126]

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

-2/3*(1-x)^(9/2)/(1+x)^(3/2)+105/2*arcsin(x)+6*(1-x)^(7/2)/(1+x)^(1/2)+35/2*(1-x)^(3/2)*(1+x)^(1/2)+7*(1-x)^(5

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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 = {49, 52, 41, 222} \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*}

(-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

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]))

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]

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]

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

\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*}

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Mathematica [A]
time = 0.09, size = 61, normalized size = 0.59 \begin {gather*} \frac {\sqrt {1-x} \left (494+679 x+102 x^2-17 x^3+2 x^4\right )}{6 (1+x)^{3/2}}-105 \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}

(Sqrt[1 - x]*(494 + 679*x + 102*x^2 - 17*x^3 + 2*x^4))/(6*(1 + x)^(3/2)) - 105*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]]

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 38.27, size = 193, normalized size = 1.87 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-630 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \left (-1+x\right )^{\frac {3}{2}} \left (1+x\right )^2+128 \left (-1+x\right ) \sqrt {1+x}+896 \left (1-x\right ) \left (1+x\right )^{\frac {3}{2}}+\left (-1+x\right ) \left (301+215 x-29 \left (1+x\right )^2+2 \left (1+x\right )^3\right ) \left (1+x\right )^{\frac {5}{2}}\right )}{6 \left (-1+x\right )^{\frac {3}{2}} \left (1+x\right )^2},\text {Abs}\left [1+x\right ]>2\right \}\right \},\frac {-215 \left (1+x\right )^{\frac {3}{2}}}{6 \sqrt {1-x}}-\frac {64}{3 \left (1+x\right )^{\frac {3}{2}} \sqrt {1-x}}-\frac {43 \sqrt {1+x}}{3 \sqrt {1-x}}-\frac {\left (1+x\right )^{\frac {7}{2}}}{3 \sqrt {1-x}}+\frac {29 \left (1+x\right )^{\frac {5}{2}}}{6 \sqrt {1-x}}+105 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]+\frac {448}{3 \sqrt {1+x} \sqrt {1-x}}\right ] \end {gather*}

Piecewise[{{I / 6 (-630 ArcCosh[Sqrt[2] Sqrt[1 + x] / 2] (-1 + x) ^ (3 / 2) (1 + x) ^ 2 + 128 (-1 + x) Sqrt[1

+ x] + 896 (1 - x) (1 + x) ^ (3 / 2) + (-1 + x) (301 + 215 x - 29 (1 + x) ^ 2 + 2 (1 + x) ^ 3) (1 + x) ^ (5 /

2)) / ((-1 + x) ^ (3 / 2) (1 + x) ^ 2), Abs[1 + x] > 2}}, -215 (1 + x) ^ (3 / 2) / (6 Sqrt[1 - x]) - 64 / (3 (

1 + x) ^ (3 / 2) Sqrt[1 - x]) - 43 Sqrt[1 + x] / (3 Sqrt[1 - x]) - (1 + x) ^ (7 / 2) / (3 Sqrt[1 - x]) + 29 (1

+ x) ^ (5 / 2) / (6 Sqrt[1 - x]) + 105 ArcSin[Sqrt[2] Sqrt[1 + x] / 2] + 448 / (3 Sqrt[1 + x] Sqrt[1 - x])]

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method	result	size

risch	\(-\frac {\left (2 x^{5}-19 x^{4}+119 x^{3}+577 x^{2}-185 x -494\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{6 \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {105 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\)	\(89\)

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

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Maxima [A]
time = 0.34, 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 \left (x\right ) \end {gather*}

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/(

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Fricas [A]
time = 0.30, 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*}

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(

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Sympy [A]
time = 53.13, size = 248, normalized size = 2.41 \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}\: \left |{x + 1}\right | > 2 \\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*}

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), (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))

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Giac [A]
time = 0.01, size = 127, normalized size = 1.23 \begin {gather*} \frac {2 \left (\left (\left (\left (\frac {1}{6} \sqrt {-x+1} \sqrt {-x+1}+\frac {3}{4}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {21}{4}\right ) \sqrt {-x+1} \sqrt {-x+1}-70\right ) \sqrt {-x+1} \sqrt {-x+1}+105\right ) \sqrt {-x+1} \sqrt {x+1}}{\left (x+1\right )^{2}}-105 \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right ) \end {gather*}

1/6*((((2*x - 11)*(x - 1) + 63)*(x - 1) + 840)*(x - 1) + 1260)*sqrt(-x + 1)/(x + 1)^(3/2) - 105*arcsin(1/2*sqr

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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*}

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3.12.26 \(\int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx\) [1126]