\(\int (1-x)^{9/2} \sqrt {1+x} \, dx\) [1063]

Optimal result

Integrand size = 17, antiderivative size = 108 \[ \int (1-x)^{9/2} \sqrt {1+x} \, dx=\frac {21}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {7}{8} (1-x)^{3/2} (1+x)^{3/2}+\frac {21}{40} (1-x)^{5/2} (1+x)^{3/2}+\frac {3}{10} (1-x)^{7/2} (1+x)^{3/2}+\frac {1}{6} (1-x)^{9/2} (1+x)^{3/2}+\frac {21 \arcsin (x)}{16} \]

[Out]

7/8*(1-x)^(3/2)*(1+x)^(3/2)+21/40*(1-x)^(5/2)*(1+x)^(3/2)+3/10*(1-x)^(7/2)*(1+x)^(3/2)+1/6*(1-x)^(9/2)*(1+x)^(
3/2)+21/16*arcsin(x)+21/16*x*(1-x)^(1/2)*(1+x)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 108, 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 = {51, 38, 41, 222} \[ \int (1-x)^{9/2} \sqrt {1+x} \, dx=\frac {21 \arcsin (x)}{16}+\frac {1}{6} (x+1)^{3/2} (1-x)^{9/2}+\frac {3}{10} (x+1)^{3/2} (1-x)^{7/2}+\frac {21}{40} (x+1)^{3/2} (1-x)^{5/2}+\frac {7}{8} (x+1)^{3/2} (1-x)^{3/2}+\frac {21}{16} x \sqrt {x+1} \sqrt {1-x} \]

[In]

Int[(1 - x)^(9/2)*Sqrt[1 + x],x]

[Out]

(21*Sqrt[1 - x]*x*Sqrt[1 + x])/16 + (7*(1 - x)^(3/2)*(1 + x)^(3/2))/8 + (21*(1 - x)^(5/2)*(1 + x)^(3/2))/40 +
(3*(1 - x)^(7/2)*(1 + x)^(3/2))/10 + ((1 - x)^(9/2)*(1 + x)^(3/2))/6 + (21*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 51

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 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 {1}{6} (1-x)^{9/2} (1+x)^{3/2}+\frac {3}{2} \int (1-x)^{7/2} \sqrt {1+x} \, dx \\ & = \frac {3}{10} (1-x)^{7/2} (1+x)^{3/2}+\frac {1}{6} (1-x)^{9/2} (1+x)^{3/2}+\frac {21}{10} \int (1-x)^{5/2} \sqrt {1+x} \, dx \\ & = \frac {21}{40} (1-x)^{5/2} (1+x)^{3/2}+\frac {3}{10} (1-x)^{7/2} (1+x)^{3/2}+\frac {1}{6} (1-x)^{9/2} (1+x)^{3/2}+\frac {21}{8} \int (1-x)^{3/2} \sqrt {1+x} \, dx \\ & = \frac {7}{8} (1-x)^{3/2} (1+x)^{3/2}+\frac {21}{40} (1-x)^{5/2} (1+x)^{3/2}+\frac {3}{10} (1-x)^{7/2} (1+x)^{3/2}+\frac {1}{6} (1-x)^{9/2} (1+x)^{3/2}+\frac {21}{8} \int \sqrt {1-x} \sqrt {1+x} \, dx \\ & = \frac {21}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {7}{8} (1-x)^{3/2} (1+x)^{3/2}+\frac {21}{40} (1-x)^{5/2} (1+x)^{3/2}+\frac {3}{10} (1-x)^{7/2} (1+x)^{3/2}+\frac {1}{6} (1-x)^{9/2} (1+x)^{3/2}+\frac {21}{16} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = \frac {21}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {7}{8} (1-x)^{3/2} (1+x)^{3/2}+\frac {21}{40} (1-x)^{5/2} (1+x)^{3/2}+\frac {3}{10} (1-x)^{7/2} (1+x)^{3/2}+\frac {1}{6} (1-x)^{9/2} (1+x)^{3/2}+\frac {21}{16} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = \frac {21}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {7}{8} (1-x)^{3/2} (1+x)^{3/2}+\frac {21}{40} (1-x)^{5/2} (1+x)^{3/2}+\frac {3}{10} (1-x)^{7/2} (1+x)^{3/2}+\frac {1}{6} (1-x)^{9/2} (1+x)^{3/2}+\frac {21}{16} \sin ^{-1}(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.58 \[ \int (1-x)^{9/2} \sqrt {1+x} \, dx=\frac {1}{240} \sqrt {1-x^2} \left (448-75 x-256 x^2+350 x^3-192 x^4+40 x^5\right )-\frac {21}{8} \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]

[In]

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

[Out]

(Sqrt[1 - x^2]*(448 - 75*x - 256*x^2 + 350*x^3 - 192*x^4 + 40*x^5))/240 - (21*ArcTan[Sqrt[1 - x^2]/(-1 + x)])/
8

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.85

method	result	size
risch	\(-\frac {\left (40 x^{5}-192 x^{4}+350 x^{3}-256 x^{2}-75 x +448\right ) \left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{240 \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {21 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\)	\(92\)
default	\(\frac {\left (1-x \right )^{\frac {9}{2}} \left (1+x \right )^{\frac {3}{2}}}{6}+\frac {3 \left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {3}{2}}}{10}+\frac {21 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {3}{2}}}{40}+\frac {7 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}{8}+\frac {21 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{16}-\frac {21 \sqrt {1-x}\, \sqrt {1+x}}{16}+\frac {21 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\)	\(113\)

[In]

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

[Out]

-1/240*(40*x^5-192*x^4+350*x^3-256*x^2-75*x+448)*(-1+x)*(1+x)^(1/2)/(-(-1+x)*(1+x))^(1/2)*((1+x)*(1-x))^(1/2)/
(1-x)^(1/2)+21/16*((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 = 62, normalized size of antiderivative = 0.57 \[ \int (1-x)^{9/2} \sqrt {1+x} \, dx=\frac {1}{240} \, {\left (40 \, x^{5} - 192 \, x^{4} + 350 \, x^{3} - 256 \, x^{2} - 75 \, x + 448\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {21}{8} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]

[In]

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

[Out]

1/240*(40*x^5 - 192*x^4 + 350*x^3 - 256*x^2 - 75*x + 448)*sqrt(x + 1)*sqrt(-x + 1) - 21/8*arctan((sqrt(x + 1)*
sqrt(-x + 1) - 1)/x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 125.26 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.66 \[ \int (1-x)^{9/2} \sqrt {1+x} \, dx=\begin {cases} - \frac {21 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} + \frac {i \left (x + 1\right )^{\frac {13}{2}}}{6 \sqrt {x - 1}} - \frac {59 i \left (x + 1\right )^{\frac {11}{2}}}{30 \sqrt {x - 1}} + \frac {1151 i \left (x + 1\right )^{\frac {9}{2}}}{120 \sqrt {x - 1}} - \frac {2947 i \left (x + 1\right )^{\frac {7}{2}}}{120 \sqrt {x - 1}} + \frac {8171 i \left (x + 1\right )^{\frac {5}{2}}}{240 \sqrt {x - 1}} - \frac {1045 i \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {x - 1}} + \frac {21 i \sqrt {x + 1}}{8 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {21 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} - \frac {\left (x + 1\right )^{\frac {13}{2}}}{6 \sqrt {1 - x}} + \frac {59 \left (x + 1\right )^{\frac {11}{2}}}{30 \sqrt {1 - x}} - \frac {1151 \left (x + 1\right )^{\frac {9}{2}}}{120 \sqrt {1 - x}} + \frac {2947 \left (x + 1\right )^{\frac {7}{2}}}{120 \sqrt {1 - x}} - \frac {8171 \left (x + 1\right )^{\frac {5}{2}}}{240 \sqrt {1 - x}} + \frac {1045 \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {1 - x}} - \frac {21 \sqrt {x + 1}}{8 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-21*I*acosh(sqrt(2)*sqrt(x + 1)/2)/8 + I*(x + 1)**(13/2)/(6*sqrt(x - 1)) - 59*I*(x + 1)**(11/2)/(30
*sqrt(x - 1)) + 1151*I*(x + 1)**(9/2)/(120*sqrt(x - 1)) - 2947*I*(x + 1)**(7/2)/(120*sqrt(x - 1)) + 8171*I*(x
+ 1)**(5/2)/(240*sqrt(x - 1)) - 1045*I*(x + 1)**(3/2)/(48*sqrt(x - 1)) + 21*I*sqrt(x + 1)/(8*sqrt(x - 1)), Abs
(x + 1) > 2), (21*asin(sqrt(2)*sqrt(x + 1)/2)/8 - (x + 1)**(13/2)/(6*sqrt(1 - x)) + 59*(x + 1)**(11/2)/(30*sqr
t(1 - x)) - 1151*(x + 1)**(9/2)/(120*sqrt(1 - x)) + 2947*(x + 1)**(7/2)/(120*sqrt(1 - x)) - 8171*(x + 1)**(5/2
)/(240*sqrt(1 - x)) + 1045*(x + 1)**(3/2)/(48*sqrt(1 - x)) - 21*sqrt(x + 1)/(8*sqrt(1 - x)), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.63 \[ \int (1-x)^{9/2} \sqrt {1+x} \, dx=-\frac {1}{6} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{3} + \frac {4}{5} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2} - \frac {13}{8} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {28}{15} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} + \frac {21}{16} \, \sqrt {-x^{2} + 1} x + \frac {21}{16} \, \arcsin \left (x\right ) \]

[In]

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

[Out]

-1/6*(-x^2 + 1)^(3/2)*x^3 + 4/5*(-x^2 + 1)^(3/2)*x^2 - 13/8*(-x^2 + 1)^(3/2)*x + 28/15*(-x^2 + 1)^(3/2) + 21/1
6*sqrt(-x^2 + 1)*x + 21/16*arcsin(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (76) = 152\).

Time = 0.37 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.71 \[ \int (1-x)^{9/2} \sqrt {1+x} \, dx=\frac {1}{240} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - 26\right )} {\left (x + 1\right )} + 321\right )} {\left (x + 1\right )} - 451\right )} {\left (x + 1\right )} + 745\right )} {\left (x + 1\right )} - 405\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{40} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {3}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {21}{8} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]

[In]

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

[Out]

1/240*((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) + 1/12*((2*(3*x
- 10)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt(-x + 1) + 1/3*((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(-x
+ 1) - 3/2*sqrt(x + 1)*(x - 2)*sqrt(-x + 1) + sqrt(x + 1)*sqrt(-x + 1) + 21/8*arcsin(1/2*sqrt(2)*sqrt(x + 1))

Mupad [F(-1)]

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

[In]

int((1 - x)^(9/2)*(x + 1)^(1/2),x)

[Out]

int((1 - x)^(9/2)*(x + 1)^(1/2), x)