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

Optimal result

Integrand size = 17, antiderivative size = 103 \[ \int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx=-\frac {2 (1-x)^{9/2}}{3 (1+x)^{3/2}}+\frac {6 (1-x)^{7/2}}{\sqrt {1+x}}+\frac {105}{2} \sqrt {1-x} \sqrt {1+x}+\frac {35}{2} (1-x)^{3/2} \sqrt {1+x}+7 (1-x)^{5/2} \sqrt {1+x}+\frac {105 \arcsin (x)}{2} \]

[Out]

-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
/2)*(1+x)^(1/2)+105/2*(1-x)^(1/2)*(1+x)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {49, 52, 41, 222} \[ \int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx=\frac {105 \arcsin (x)}{2}-\frac {2 (1-x)^{9/2}}{3 (x+1)^{3/2}}+\frac {6 (1-x)^{7/2}}{\sqrt {x+1}}+7 \sqrt {x+1} (1-x)^{5/2}+\frac {35}{2} \sqrt {x+1} (1-x)^{3/2}+\frac {105}{2} \sqrt {x+1} \sqrt {1-x} \]

[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 49

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 52

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 222

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*} \text {integral}& = -\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 [A] (verified)

Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.59 \[ \int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx=\frac {\sqrt {1-x} \left (494+679 x+102 x^2-17 x^3+2 x^4\right )}{6 (1+x)^{3/2}}-105 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]

[In]

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

[Out]

(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^2]/(-1 + x)]

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86

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

[In]

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

[Out]

-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
)+105/2*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.83 \[ \int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx=\frac {494 \, x^{2} + {\left (2 \, x^{4} - 17 \, x^{3} + 102 \, x^{2} + 679 \, x + 494\right )} \sqrt {x + 1} \sqrt {-x + 1} - 630 \, {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 988 \, x + 494}{6 \, {\left (x^{2} + 2 \, x + 1\right )}} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 57.29 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.41 \[ \int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx=\begin {cases} - 105 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {i \left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {x - 1}} - \frac {29 i \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {x - 1}} + \frac {215 i \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {x - 1}} + \frac {43 i \sqrt {x + 1}}{3 \sqrt {x - 1}} - \frac {448 i}{3 \sqrt {x - 1} \sqrt {x + 1}} + \frac {64 i}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {3}{2}}} & \text {for}\: \left |{x + 1}\right | > 2 \\105 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {\left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {1 - x}} + \frac {29 \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {1 - x}} - \frac {215 \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {1 - x}} - \frac {43 \sqrt {x + 1}}{3 \sqrt {1 - x}} + \frac {448}{3 \sqrt {1 - x} \sqrt {x + 1}} - \frac {64}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[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), (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))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.21 \[ \int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx=\frac {x^{6}}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {7 \, x^{5}}{2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {23 \, x^{4}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {35}{2} \, x {\left (\frac {3 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {2}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}\right )} - \frac {143 \, x}{6 \, \sqrt {-x^{2} + 1}} - \frac {127 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {22 \, x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {247}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {105}{2} \, \arcsin \left (x\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.23 \[ \int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx=\frac {1}{6} \, {\left ({\left (2 \, x - 23\right )} {\left (x + 1\right )} + 165\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {2 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{3 \, {\left (x + 1\right )}^{\frac {3}{2}}} - \frac {34 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} + \frac {2 \, {\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {51 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{3 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} + 105 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]

[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))

Mupad [F(-1)]

Timed out. \[ \int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx=\int \frac {{\left (1-x\right )}^{9/2}}{{\left (x+1\right )}^{5/2}} \,d x \]

[In]

int((1 - x)^(9/2)/(x + 1)^(5/2),x)

[Out]

int((1 - x)^(9/2)/(x + 1)^(5/2), x)