\(\int (1-x)^{3/2} (1+x)^{5/2} \, dx\) [1092]

Optimal result

Integrand size = 17, antiderivative size = 69 \[ \int (1-x)^{3/2} (1+x)^{5/2} \, dx=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3 \arcsin (x)}{8} \]

[Out]

1/4*(1-x)^(3/2)*x*(1+x)^(3/2)-1/5*(1-x)^(5/2)*(1+x)^(5/2)+3/8*arcsin(x)+3/8*x*(1-x)^(1/2)*(1+x)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {51, 38, 41, 222} \[ \int (1-x)^{3/2} (1+x)^{5/2} \, dx=\frac {3 \arcsin (x)}{8}-\frac {1}{5} (1-x)^{5/2} (x+1)^{5/2}+\frac {1}{4} (1-x)^{3/2} x (x+1)^{3/2}+\frac {3}{8} \sqrt {1-x} x \sqrt {x+1} \]

[In]

Int[(1 - x)^(3/2)*(1 + x)^(5/2),x]

[Out]

(3*Sqrt[1 - x]*x*Sqrt[1 + x])/8 + ((1 - x)^(3/2)*x*(1 + x)^(3/2))/4 - ((1 - x)^(5/2)*(1 + x)^(5/2))/5 + (3*Arc
Sin[x])/8

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}{5} (1-x)^{5/2} (1+x)^{5/2}+\int (1-x)^{3/2} (1+x)^{3/2} \, dx \\ & = \frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{4} \int \sqrt {1-x} \sqrt {1+x} \, dx \\ & = \frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{8} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = \frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = \frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{8} \sin ^{-1}(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int (1-x)^{3/2} (1+x)^{5/2} \, dx=-\frac {1}{40} \sqrt {1-x^2} \left (8-25 x-16 x^2+10 x^3+8 x^4\right )-\frac {3}{4} \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]

[In]

Integrate[(1 - x)^(3/2)*(1 + x)^(5/2),x]

[Out]

-1/40*(Sqrt[1 - x^2]*(8 - 25*x - 16*x^2 + 10*x^3 + 8*x^4)) - (3*ArcTan[Sqrt[1 - x^2]/(-1 + x)])/4

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.26

method	result	size
risch	\(\frac {\left (8 x^{4}+10 x^{3}-16 x^{2}-25 x +8\right ) \left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{40 \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {3 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\)	\(87\)
default	\(\frac {\left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {7}{2}}}{5}+\frac {3 \sqrt {1-x}\, \left (1+x \right )^{\frac {7}{2}}}{20}-\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{20}-\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{8}-\frac {3 \sqrt {1-x}\, \sqrt {1+x}}{8}+\frac {3 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\)	\(99\)

[In]

int((1-x)^(3/2)*(1+x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/40*(8*x^4+10*x^3-16*x^2-25*x+8)*(-1+x)*(1+x)^(1/2)/(-(-1+x)*(1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)+3/8
*((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 = 57, normalized size of antiderivative = 0.83 \[ \int (1-x)^{3/2} (1+x)^{5/2} \, dx=-\frac {1}{40} \, {\left (8 \, x^{4} + 10 \, x^{3} - 16 \, x^{2} - 25 \, x + 8\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {3}{4} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]

[In]

integrate((1-x)^(3/2)*(1+x)^(5/2),x, algorithm="fricas")

[Out]

-1/40*(8*x^4 + 10*x^3 - 16*x^2 - 25*x + 8)*sqrt(x + 1)*sqrt(-x + 1) - 3/4*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1
)/x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 84.97 (sec) , antiderivative size = 245, normalized size of antiderivative = 3.55 \[ \int (1-x)^{3/2} (1+x)^{5/2} \, dx=\begin {cases} - \frac {3 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} - \frac {i \left (x + 1\right )^{\frac {11}{2}}}{5 \sqrt {x - 1}} + \frac {19 i \left (x + 1\right )^{\frac {9}{2}}}{20 \sqrt {x - 1}} - \frac {23 i \left (x + 1\right )^{\frac {7}{2}}}{20 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{40 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {3}{2}}}{8 \sqrt {x - 1}} + \frac {3 i \sqrt {x + 1}}{4 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} + \frac {\left (x + 1\right )^{\frac {11}{2}}}{5 \sqrt {1 - x}} - \frac {19 \left (x + 1\right )^{\frac {9}{2}}}{20 \sqrt {1 - x}} + \frac {23 \left (x + 1\right )^{\frac {7}{2}}}{20 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{40 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {3}{2}}}{8 \sqrt {1 - x}} - \frac {3 \sqrt {x + 1}}{4 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]

[In]

integrate((1-x)**(3/2)*(1+x)**(5/2),x)

[Out]

Piecewise((-3*I*acosh(sqrt(2)*sqrt(x + 1)/2)/4 - I*(x + 1)**(11/2)/(5*sqrt(x - 1)) + 19*I*(x + 1)**(9/2)/(20*s
qrt(x - 1)) - 23*I*(x + 1)**(7/2)/(20*sqrt(x - 1)) - I*(x + 1)**(5/2)/(40*sqrt(x - 1)) - I*(x + 1)**(3/2)/(8*s
qrt(x - 1)) + 3*I*sqrt(x + 1)/(4*sqrt(x - 1)), Abs(x + 1) > 2), (3*asin(sqrt(2)*sqrt(x + 1)/2)/4 + (x + 1)**(1
1/2)/(5*sqrt(1 - x)) - 19*(x + 1)**(9/2)/(20*sqrt(1 - x)) + 23*(x + 1)**(7/2)/(20*sqrt(1 - x)) + (x + 1)**(5/2
)/(40*sqrt(1 - x)) + (x + 1)**(3/2)/(8*sqrt(1 - x)) - 3*sqrt(x + 1)/(4*sqrt(1 - x)), True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.58 \[ \int (1-x)^{3/2} (1+x)^{5/2} \, dx=-\frac {1}{5} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} + \frac {1}{4} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {3}{8} \, \sqrt {-x^{2} + 1} x + \frac {3}{8} \, \arcsin \left (x\right ) \]

[In]

integrate((1-x)^(3/2)*(1+x)^(5/2),x, algorithm="maxima")

[Out]

-1/5*(-x^2 + 1)^(5/2) + 1/4*(-x^2 + 1)^(3/2)*x + 3/8*sqrt(-x^2 + 1)*x + 3/8*arcsin(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (49) = 98\).

Time = 0.32 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.65 \[ \int (1-x)^{3/2} (1+x)^{5/2} \, dx=-\frac {1}{120} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} + \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {3}{4} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]

[In]

integrate((1-x)^(3/2)*(1+x)^(5/2),x, algorithm="giac")

[Out]

-1/120*((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) + sqrt(x + 1)*(x - 2)*sqrt(-x + 1) + sqrt(x + 1)*s
qrt(-x + 1) + 3/4*arcsin(1/2*sqrt(2)*sqrt(x + 1))

Mupad [F(-1)]

Timed out. \[ \int (1-x)^{3/2} (1+x)^{5/2} \, dx=\int {\left (1-x\right )}^{3/2}\,{\left (x+1\right )}^{5/2} \,d x \]

[In]

int((1 - x)^(3/2)*(x + 1)^(5/2),x)

[Out]

int((1 - x)^(3/2)*(x + 1)^(5/2), x)