\(\int \frac {(1+x)^{3/2}}{(1-x)^{9/2}} \, dx\) [1084]

Optimal result

Integrand size = 17, antiderivative size = 41 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{9/2}} \, dx=\frac {(1+x)^{5/2}}{7 (1-x)^{7/2}}+\frac {(1+x)^{5/2}}{35 (1-x)^{5/2}} \]

[Out]

1/7*(1+x)^(5/2)/(1-x)^(7/2)+1/35*(1+x)^(5/2)/(1-x)^(5/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37} \[ \int \frac {(1+x)^{3/2}}{(1-x)^{9/2}} \, dx=\frac {(x+1)^{5/2}}{35 (1-x)^{5/2}}+\frac {(x+1)^{5/2}}{7 (1-x)^{7/2}} \]

[In]

Int[(1 + x)^(3/2)/(1 - x)^(9/2),x]

[Out]

(1 + x)^(5/2)/(7*(1 - x)^(7/2)) + (1 + x)^(5/2)/(35*(1 - x)^(5/2))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = \frac {(1+x)^{5/2}}{7 (1-x)^{7/2}}+\frac {1}{7} \int \frac {(1+x)^{3/2}}{(1-x)^{7/2}} \, dx \\ & = \frac {(1+x)^{5/2}}{7 (1-x)^{7/2}}+\frac {(1+x)^{5/2}}{35 (1-x)^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.61 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{9/2}} \, dx=\frac {(6-x) (1+x)^{5/2}}{35 (1-x)^{7/2}} \]

[In]

Integrate[(1 + x)^(3/2)/(1 - x)^(9/2),x]

[Out]

((6 - x)*(1 + x)^(5/2))/(35*(1 - x)^(7/2))

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.44

method	result	size
gosper	\(-\frac {\left (x -6\right ) \left (1+x \right )^{\frac {5}{2}}}{35 \left (1-x \right )^{\frac {7}{2}}}\)	\(18\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{4}-3 x^{3}-15 x^{2}-17 x -6\right )}{35 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{3} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\)	\(59\)
default	\(\frac {\left (1+x \right )^{\frac {3}{2}}}{2 \left (1-x \right )^{\frac {7}{2}}}-\frac {3 \sqrt {1+x}}{7 \left (1-x \right )^{\frac {7}{2}}}+\frac {3 \sqrt {1+x}}{70 \left (1-x \right )^{\frac {5}{2}}}+\frac {\sqrt {1+x}}{35 \left (1-x \right )^{\frac {3}{2}}}+\frac {\sqrt {1+x}}{35 \sqrt {1-x}}\)	\(72\)

[In]

int((1+x)^(3/2)/(1-x)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-1/35*(x-6)/(1-x)^(7/2)*(1+x)^(5/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (29) = 58\).

Time = 0.23 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.68 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{9/2}} \, dx=\frac {6 \, x^{4} - 24 \, x^{3} + 36 \, x^{2} - {\left (x^{3} - 4 \, x^{2} - 11 \, x - 6\right )} \sqrt {x + 1} \sqrt {-x + 1} - 24 \, x + 6}{35 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} \]

[In]

integrate((1+x)^(3/2)/(1-x)^(9/2),x, algorithm="fricas")

[Out]

1/35*(6*x^4 - 24*x^3 + 36*x^2 - (x^3 - 4*x^2 - 11*x - 6)*sqrt(x + 1)*sqrt(-x + 1) - 24*x + 6)/(x^4 - 4*x^3 + 6
*x^2 - 4*x + 1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 20.20 (sec) , antiderivative size = 226, normalized size of antiderivative = 5.51 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{9/2}} \, dx=\begin {cases} - \frac {i \left (x + 1\right )^{\frac {7}{2}}}{35 \sqrt {x - 1} \left (x + 1\right )^{3} - 210 \sqrt {x - 1} \left (x + 1\right )^{2} + 420 \sqrt {x - 1} \left (x + 1\right ) - 280 \sqrt {x - 1}} + \frac {7 i \left (x + 1\right )^{\frac {5}{2}}}{35 \sqrt {x - 1} \left (x + 1\right )^{3} - 210 \sqrt {x - 1} \left (x + 1\right )^{2} + 420 \sqrt {x - 1} \left (x + 1\right ) - 280 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {\left (x + 1\right )^{\frac {7}{2}}}{35 \sqrt {1 - x} \left (x + 1\right )^{3} - 210 \sqrt {1 - x} \left (x + 1\right )^{2} + 420 \sqrt {1 - x} \left (x + 1\right ) - 280 \sqrt {1 - x}} - \frac {7 \left (x + 1\right )^{\frac {5}{2}}}{35 \sqrt {1 - x} \left (x + 1\right )^{3} - 210 \sqrt {1 - x} \left (x + 1\right )^{2} + 420 \sqrt {1 - x} \left (x + 1\right ) - 280 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]

[In]

integrate((1+x)**(3/2)/(1-x)**(9/2),x)

[Out]

Piecewise((-I*(x + 1)**(7/2)/(35*sqrt(x - 1)*(x + 1)**3 - 210*sqrt(x - 1)*(x + 1)**2 + 420*sqrt(x - 1)*(x + 1)
- 280*sqrt(x - 1)) + 7*I*(x + 1)**(5/2)/(35*sqrt(x - 1)*(x + 1)**3 - 210*sqrt(x - 1)*(x + 1)**2 + 420*sqrt(x
- 1)*(x + 1) - 280*sqrt(x - 1)), Abs(x + 1) > 2), ((x + 1)**(7/2)/(35*sqrt(1 - x)*(x + 1)**3 - 210*sqrt(1 - x)
*(x + 1)**2 + 420*sqrt(1 - x)*(x + 1) - 280*sqrt(1 - x)) - 7*(x + 1)**(5/2)/(35*sqrt(1 - x)*(x + 1)**3 - 210*s
qrt(1 - x)*(x + 1)**2 + 420*sqrt(1 - x)*(x + 1) - 280*sqrt(1 - x)), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (29) = 58\).

Time = 0.20 (sec) , antiderivative size = 131, normalized size of antiderivative = 3.20 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{9/2}} \, dx=-\frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {3 \, \sqrt {-x^{2} + 1}}{7 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac {3 \, \sqrt {-x^{2} + 1}}{70 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{35 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{35 \, {\left (x - 1\right )}} \]

[In]

integrate((1+x)^(3/2)/(1-x)^(9/2),x, algorithm="maxima")

[Out]

-1/2*(-x^2 + 1)^(3/2)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) - 3/7*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*
x + 1) - 3/70*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) + 1/35*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) - 1/35*sqrt(-x^2 +
1)/(x - 1)

Giac [A] (verification not implemented)

none

Time = 0.51 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.54 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{9/2}} \, dx=-\frac {{\left (x + 1\right )}^{\frac {5}{2}} {\left (x - 6\right )} \sqrt {-x + 1}}{35 \, {\left (x - 1\right )}^{4}} \]

[In]

integrate((1+x)^(3/2)/(1-x)^(9/2),x, algorithm="giac")

[Out]

-1/35*(x + 1)^(5/2)*(x - 6)*sqrt(-x + 1)/(x - 1)^4

Mupad [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.56 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{9/2}} \, dx=\frac {\sqrt {1-x}\,\left (\frac {11\,x\,\sqrt {x+1}}{35}+\frac {6\,\sqrt {x+1}}{35}+\frac {4\,x^2\,\sqrt {x+1}}{35}-\frac {x^3\,\sqrt {x+1}}{35}\right )}{x^4-4\,x^3+6\,x^2-4\,x+1} \]

[In]

int((x + 1)^(3/2)/(1 - x)^(9/2),x)

[Out]

((1 - x)^(1/2)*((11*x*(x + 1)^(1/2))/35 + (6*(x + 1)^(1/2))/35 + (4*x^2*(x + 1)^(1/2))/35 - (x^3*(x + 1)^(1/2)
)/35))/(6*x^2 - 4*x - 4*x^3 + x^4 + 1)