\(\int (1-x)^{9/2} (1+x)^{5/2} \, dx\) [1089]

Optimal result

Integrand size = 17, antiderivative size = 110 \[ \int (1-x)^{9/2} (1+x)^{5/2} \, dx=\frac {45}{128} \sqrt {1-x} x \sqrt {1+x}+\frac {15}{64} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{16} (1-x)^{5/2} x (1+x)^{5/2}+\frac {9}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {1}{8} (1-x)^{9/2} (1+x)^{7/2}+\frac {45 \arcsin (x)}{128} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 110, 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} \[ \int (1-x)^{9/2} (1+x)^{5/2} \, dx=\frac {45 \arcsin (x)}{128}+\frac {1}{8} (x+1)^{7/2} (1-x)^{9/2}+\frac {9}{56} (x+1)^{7/2} (1-x)^{7/2}+\frac {3}{16} x (x+1)^{5/2} (1-x)^{5/2}+\frac {15}{64} x (x+1)^{3/2} (1-x)^{3/2}+\frac {45}{128} x \sqrt {x+1} \sqrt {1-x} \]

[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 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}{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*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.65 \[ \int (1-x)^{9/2} (1+x)^{5/2} \, dx=\frac {1}{896} \left (\sqrt {1-x^2} \left (256+581 x-768 x^2-210 x^3+768 x^4-168 x^5-256 x^6+112 x^7\right )-630 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right )\right ) \]

[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*ArcTan[Sqrt[1 -
x^2]/(-1 + x)])/896

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.93

method	result	size
risch	\(-\frac {\left (112 x^{7}-256 x^{6}-168 x^{5}+768 x^{4}-210 x^{3}-768 x^{2}+581 x +256\right ) \left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{896 \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {45 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{128 \sqrt {1+x}\, \sqrt {1-x}}\)	\(102\)
default	\(\frac {\left (1-x \right )^{\frac {9}{2}} \left (1+x \right )^{\frac {7}{2}}}{8}+\frac {9 \left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {7}{2}}}{56}+\frac {3 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {7}{2}}}{16}+\frac {3 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {7}{2}}}{16}+\frac {9 \sqrt {1-x}\, \left (1+x \right )^{\frac {7}{2}}}{64}-\frac {3 \sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{64}-\frac {15 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{128}-\frac {45 \sqrt {1-x}\, \sqrt {1+x}}{128}+\frac {45 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{128 \sqrt {1+x}\, \sqrt {1-x}}\)	\(141\)

[In]

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

[Out]

-1/896*(112*x^7-256*x^6-168*x^5+768*x^4-210*x^3-768*x^2+581*x+256)*(-1+x)*(1+x)^(1/2)/(-(-1+x)*(1+x))^(1/2)*((
1+x)*(1-x))^(1/2)/(1-x)^(1/2)+45/128*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.65 \[ \int (1-x)^{9/2} (1+x)^{5/2} \, dx=\frac {1}{896} \, {\left (112 \, x^{7} - 256 \, x^{6} - 168 \, x^{5} + 768 \, x^{4} - 210 \, x^{3} - 768 \, x^{2} + 581 \, x + 256\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {45}{64} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]

[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)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.58 \[ \int (1-x)^{9/2} (1+x)^{5/2} \, dx=-\frac {1}{8} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} x + \frac {2}{7} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} + \frac {3}{16} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {15}{64} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {45}{128} \, \sqrt {-x^{2} + 1} x + \frac {45}{128} \, \arcsin \left (x\right ) \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (78) = 156\).

Time = 0.37 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.69 \[ \int (1-x)^{9/2} (1+x)^{5/2} \, dx=\frac {1}{13440} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, {\left (7 \, x - 50\right )} {\left (x + 1\right )} + 1219\right )} {\left (x + 1\right )} - 12463\right )} {\left (x + 1\right )} + 64233\right )} {\left (x + 1\right )} - 53963\right )} {\left (x + 1\right )} + 59465\right )} {\left (x + 1\right )} - 23205\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{1680} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, x - 37\right )} {\left (x + 1\right )} + 661\right )} {\left (x + 1\right )} - 4551\right )} {\left (x + 1\right )} + 4781\right )} {\left (x + 1\right )} - 6335\right )} {\left (x + 1\right )} + 2835\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{80} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - 26\right )} {\left (x + 1\right )} + 321\right )} {\left (x + 1\right )} - 451\right )} {\left (x + 1\right )} + 745\right )} {\left (x + 1\right )} - 405\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{40} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{8} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{2} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {45}{64} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]

[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))

Mupad [F(-1)]

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

[In]

int((1 - x)^(9/2)*(x + 1)^(5/2),x)

[Out]

int((1 - x)^(9/2)*(x + 1)^(5/2), x)