\(\int (1-x)^{11/2} (1+x)^{5/2} \, dx\) [1088]

Optimal result

Integrand size = 17, antiderivative size = 130 \[ \int (1-x)^{11/2} (1+x)^{5/2} \, dx=\frac {55}{128} \sqrt {1-x} x \sqrt {1+x}+\frac {55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac {11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac {11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {55 \arcsin (x)}{128} \]

[Out]

55/192*(1-x)^(3/2)*x*(1+x)^(3/2)+11/48*(1-x)^(5/2)*x*(1+x)^(5/2)+11/56*(1-x)^(7/2)*(1+x)^(7/2)+11/72*(1-x)^(9/
2)*(1+x)^(7/2)+1/9*(1-x)^(11/2)*(1+x)^(7/2)+55/128*arcsin(x)+55/128*x*(1-x)^(1/2)*(1+x)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {51, 38, 41, 222} \[ \int (1-x)^{11/2} (1+x)^{5/2} \, dx=\frac {55 \arcsin (x)}{128}+\frac {1}{9} (x+1)^{7/2} (1-x)^{11/2}+\frac {11}{72} (x+1)^{7/2} (1-x)^{9/2}+\frac {11}{56} (x+1)^{7/2} (1-x)^{7/2}+\frac {11}{48} x (x+1)^{5/2} (1-x)^{5/2}+\frac {55}{192} x (x+1)^{3/2} (1-x)^{3/2}+\frac {55}{128} x \sqrt {x+1} \sqrt {1-x} \]

[In]

Int[(1 - x)^(11/2)*(1 + x)^(5/2),x]

[Out]

(55*Sqrt[1 - x]*x*Sqrt[1 + x])/128 + (55*(1 - x)^(3/2)*x*(1 + x)^(3/2))/192 + (11*(1 - x)^(5/2)*x*(1 + x)^(5/2
))/48 + (11*(1 - x)^(7/2)*(1 + x)^(7/2))/56 + (11*(1 - x)^(9/2)*(1 + x)^(7/2))/72 + ((1 - x)^(11/2)*(1 + x)^(7
/2))/9 + (55*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}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {11}{9} \int (1-x)^{9/2} (1+x)^{5/2} \, dx \\ & = \frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {11}{8} \int (1-x)^{7/2} (1+x)^{5/2} \, dx \\ & = \frac {11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {11}{8} \int (1-x)^{5/2} (1+x)^{5/2} \, dx \\ & = \frac {11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac {11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {55}{48} \int (1-x)^{3/2} (1+x)^{3/2} \, dx \\ & = \frac {55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac {11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac {11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {55}{64} \int \sqrt {1-x} \sqrt {1+x} \, dx \\ & = \frac {55}{128} \sqrt {1-x} x \sqrt {1+x}+\frac {55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac {11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac {11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {55}{128} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = \frac {55}{128} \sqrt {1-x} x \sqrt {1+x}+\frac {55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac {11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac {11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {55}{128} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = \frac {55}{128} \sqrt {1-x} x \sqrt {1+x}+\frac {55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac {11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac {11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {55}{128} \sin ^{-1}(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60 \[ \int (1-x)^{11/2} (1+x)^{5/2} \, dx=\frac {\sqrt {1-x^2} \left (3712+4599 x-10240 x^2+3066 x^3+8448 x^4-7224 x^5-1024 x^6+3024 x^7-896 x^8\right )}{8064}-\frac {55}{64} \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]

[In]

Integrate[(1 - x)^(11/2)*(1 + x)^(5/2),x]

[Out]

(Sqrt[1 - x^2]*(3712 + 4599*x - 10240*x^2 + 3066*x^3 + 8448*x^4 - 7224*x^5 - 1024*x^6 + 3024*x^7 - 896*x^8))/8
064 - (55*ArcTan[Sqrt[1 - x^2]/(-1 + x)])/64

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82

method	result	size
risch	\(\frac {\left (896 x^{8}-3024 x^{7}+1024 x^{6}+7224 x^{5}-8448 x^{4}-3066 x^{3}+10240 x^{2}-4599 x -3712\right ) \left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{8064 \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {55 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{128 \sqrt {1+x}\, \sqrt {1-x}}\)	\(107\)
default	\(\frac {\left (1-x \right )^{\frac {11}{2}} \left (1+x \right )^{\frac {7}{2}}}{9}+\frac {11 \left (1-x \right )^{\frac {9}{2}} \left (1+x \right )^{\frac {7}{2}}}{72}+\frac {11 \left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {7}{2}}}{56}+\frac {11 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {7}{2}}}{48}+\frac {11 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {7}{2}}}{48}+\frac {11 \sqrt {1-x}\, \left (1+x \right )^{\frac {7}{2}}}{64}-\frac {11 \sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{192}-\frac {55 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{384}-\frac {55 \sqrt {1-x}\, \sqrt {1+x}}{128}+\frac {55 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{128 \sqrt {1+x}\, \sqrt {1-x}}\)	\(155\)

[In]

int((1-x)^(11/2)*(1+x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/8064*(896*x^8-3024*x^7+1024*x^6+7224*x^5-8448*x^4-3066*x^3+10240*x^2-4599*x-3712)*(-1+x)*(1+x)^(1/2)/(-(-1+x
)*(1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)+55/128*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.59 \[ \int (1-x)^{11/2} (1+x)^{5/2} \, dx=-\frac {1}{8064} \, {\left (896 \, x^{8} - 3024 \, x^{7} + 1024 \, x^{6} + 7224 \, x^{5} - 8448 \, x^{4} - 3066 \, x^{3} + 10240 \, x^{2} - 4599 \, x - 3712\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {55}{64} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]

[In]

integrate((1-x)^(11/2)*(1+x)^(5/2),x, algorithm="fricas")

[Out]

-1/8064*(896*x^8 - 3024*x^7 + 1024*x^6 + 7224*x^5 - 8448*x^4 - 3066*x^3 + 10240*x^2 - 4599*x - 3712)*sqrt(x +
1)*sqrt(-x + 1) - 55/64*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

Sympy [F(-1)]

Timed out. \[ \int (1-x)^{11/2} (1+x)^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((1-x)**(11/2)*(1+x)**(5/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60 \[ \int (1-x)^{11/2} (1+x)^{5/2} \, dx=\frac {1}{9} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} x^{2} - \frac {3}{8} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} x + \frac {29}{63} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} + \frac {11}{48} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {55}{192} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {55}{128} \, \sqrt {-x^{2} + 1} x + \frac {55}{128} \, \arcsin \left (x\right ) \]

[In]

integrate((1-x)^(11/2)*(1+x)^(5/2),x, algorithm="maxima")

[Out]

1/9*(-x^2 + 1)^(7/2)*x^2 - 3/8*(-x^2 + 1)^(7/2)*x + 29/63*(-x^2 + 1)^(7/2) + 11/48*(-x^2 + 1)^(5/2)*x + 55/192
*(-x^2 + 1)^(3/2)*x + 55/128*sqrt(-x^2 + 1)*x + 55/128*arcsin(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (92) = 184\).

Time = 0.38 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.48 \[ \int (1-x)^{11/2} (1+x)^{5/2} \, dx=-\frac {1}{40320} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (2 \, {\left (7 \, {\left (8 \, x - 65\right )} {\left (x + 1\right )} + 2073\right )} {\left (x + 1\right )} - 9833\right )} {\left (x + 1\right )} + 75293\right )} {\left (x + 1\right )} - 310203\right )} {\left (x + 1\right )} + 216993\right )} {\left (x + 1\right )} - 205275\right )} {\left (x + 1\right )} + 69615\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{6720} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, {\left (7 \, x - 50\right )} {\left (x + 1\right )} + 1219\right )} {\left (x + 1\right )} - 12463\right )} {\left (x + 1\right )} + 64233\right )} {\left (x + 1\right )} - 53963\right )} {\left (x + 1\right )} + 59465\right )} {\left (x + 1\right )} - 23205\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{840} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, x - 37\right )} {\left (x + 1\right )} + 661\right )} {\left (x + 1\right )} - 4551\right )} {\left (x + 1\right )} + 4781\right )} {\left (x + 1\right )} - 6335\right )} {\left (x + 1\right )} + 2835\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{40} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - 26\right )} {\left (x + 1\right )} + 321\right )} {\left (x + 1\right )} - 451\right )} {\left (x + 1\right )} + 745\right )} {\left (x + 1\right )} - 405\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{4} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {55}{64} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]

[In]

integrate((1-x)^(11/2)*(1+x)^(5/2),x, algorithm="giac")

[Out]

-1/40320*((2*((4*(5*(2*(7*(8*x - 65)*(x + 1) + 2073)*(x + 1) - 9833)*(x + 1) + 75293)*(x + 1) - 310203)*(x + 1
) + 216993)*(x + 1) - 205275)*(x + 1) + 69615)*sqrt(x + 1)*sqrt(-x + 1) + 1/6720*((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)*s
qrt(-x + 1) + 1/840*((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) - 1/40*((2*((4*(5*x - 26)*(x + 1) + 321)*(x + 1) - 451)*(x + 1) + 745)*(x
+ 1) - 405)*sqrt(x + 1)*sqrt(-x + 1) + 1/4*((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) - sqrt(x + 1)*(x - 2)*sqrt(-x + 1) + sqrt(x + 1)*sqrt(
-x + 1) + 55/64*arcsin(1/2*sqrt(2)*sqrt(x + 1))

Mupad [F(-1)]

Timed out. \[ \int (1-x)^{11/2} (1+x)^{5/2} \, dx=\int {\left (1-x\right )}^{11/2}\,{\left (x+1\right )}^{5/2} \,d x \]

[In]

int((1 - x)^(11/2)*(x + 1)^(5/2),x)

[Out]

int((1 - x)^(11/2)*(x + 1)^(5/2), x)