\(\int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx\) [1098]

Optimal result

Integrand size = 17, antiderivative size = 20 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx=\frac {(1+x)^{7/2}}{7 (1-x)^{7/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {37} \[ \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx=\frac {(x+1)^{7/2}}{7 (1-x)^{7/2}} \]

[In]

Int[(1 + x)^(5/2)/(1 - x)^(9/2),x]

[Out]

(1 + x)^(7/2)/(7*(1 - x)^(7/2))

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]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx=\frac {(1+x)^{7/2}}{7 (1-x)^{7/2}} \]

[In]

Integrate[(1 + x)^(5/2)/(1 - x)^(9/2),x]

[Out]

(1 + x)^(7/2)/(7*(1 - x)^(7/2))

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75

method	result	size
gosper	\(\frac {\left (1+x \right )^{\frac {7}{2}}}{7 \left (1-x \right )^{\frac {7}{2}}}\)	\(15\)
risch	\(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{4}+4 x^{3}+6 x^{2}+4 x +1\right )}{7 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{3} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\)	\(59\)
default	\(\frac {\left (1+x \right )^{\frac {5}{2}}}{\left (1-x \right )^{\frac {7}{2}}}-\frac {5 \left (1+x \right )^{\frac {3}{2}}}{2 \left (1-x \right )^{\frac {7}{2}}}+\frac {15 \sqrt {1+x}}{7 \left (1-x \right )^{\frac {7}{2}}}-\frac {3 \sqrt {1+x}}{14 \left (1-x \right )^{\frac {5}{2}}}-\frac {\sqrt {1+x}}{7 \left (1-x \right )^{\frac {3}{2}}}-\frac {\sqrt {1+x}}{7 \sqrt {1-x}}\)	\(85\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (14) = 28\).

Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.30 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx=\frac {x^{4} - 4 \, x^{3} + 6 \, x^{2} + {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \sqrt {x + 1} \sqrt {-x + 1} - 4 \, x + 1}{7 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} \]

[In]

integrate((1+x)^(5/2)/(1-x)^(9/2),x, algorithm="fricas")

[Out]

1/7*(x^4 - 4*x^3 + 6*x^2 + (x^3 + 3*x^2 + 3*x + 1)*sqrt(x + 1)*sqrt(-x + 1) - 4*x + 1)/(x^4 - 4*x^3 + 6*x^2 -
4*x + 1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 20.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 5.70 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx=\begin {cases} \frac {i \left (x + 1\right )^{\frac {7}{2}}}{7 \sqrt {x - 1} \left (x + 1\right )^{3} - 42 \sqrt {x - 1} \left (x + 1\right )^{2} + 84 \sqrt {x - 1} \left (x + 1\right ) - 56 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- \frac {\left (x + 1\right )^{\frac {7}{2}}}{7 \sqrt {1 - x} \left (x + 1\right )^{3} - 42 \sqrt {1 - x} \left (x + 1\right )^{2} + 84 \sqrt {1 - x} \left (x + 1\right ) - 56 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((I*(x + 1)**(7/2)/(7*sqrt(x - 1)*(x + 1)**3 - 42*sqrt(x - 1)*(x + 1)**2 + 84*sqrt(x - 1)*(x + 1) - 5
6*sqrt(x - 1)), Abs(x + 1) > 2), (-(x + 1)**(7/2)/(7*sqrt(1 - x)*(x + 1)**3 - 42*sqrt(1 - x)*(x + 1)**2 + 84*s
qrt(1 - x)*(x + 1) - 56*sqrt(1 - x)), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (14) = 28\).

Time = 0.22 (sec) , antiderivative size = 171, normalized size of antiderivative = 8.55 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx=\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1} + \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac {15 \, \sqrt {-x^{2} + 1}}{7 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {3 \, \sqrt {-x^{2} + 1}}{14 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{7 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{7 \, {\left (x - 1\right )}} \]

[In]

integrate((1+x)^(5/2)/(1-x)^(9/2),x, algorithm="maxima")

[Out]

(-x^2 + 1)^(5/2)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1) + 5/2*(-x^2 + 1)^(3/2)/(x^5 - 5*x^4 + 10*x
^3 - 10*x^2 + 5*x - 1) + 15/7*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 3/14*sqrt(-x^2 + 1)/(x^3 - 3*x^
2 + 3*x - 1) - 1/7*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 1/7*sqrt(-x^2 + 1)/(x - 1)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx=\frac {{\left (x + 1\right )}^{\frac {7}{2}} \sqrt {-x + 1}}{7 \, {\left (x - 1\right )}^{4}} \]

[In]

integrate((1+x)^(5/2)/(1-x)^(9/2),x, algorithm="giac")

[Out]

1/7*(x + 1)^(7/2)*sqrt(-x + 1)/(x - 1)^4

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.20 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx=\frac {\sqrt {1-x}\,\left (\frac {3\,x\,\sqrt {x+1}}{7}+\frac {\sqrt {x+1}}{7}+\frac {3\,x^2\,\sqrt {x+1}}{7}+\frac {x^3\,\sqrt {x+1}}{7}\right )}{x^4-4\,x^3+6\,x^2-4\,x+1} \]

[In]

int((x + 1)^(5/2)/(1 - x)^(9/2),x)

[Out]

((1 - x)^(1/2)*((3*x*(x + 1)^(1/2))/7 + (x + 1)^(1/2)/7 + (3*x^2*(x + 1)^(1/2))/7 + (x^3*(x + 1)^(1/2))/7))/(6
*x^2 - 4*x - 4*x^3 + x^4 + 1)