\(\int \frac {(1+x)^{3/2}}{(1-x)^{7/2}} \, dx\) [1083]

Optimal result

Integrand size = 17, antiderivative size = 20 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{7/2}} \, dx=\frac {(1+x)^{5/2}}{5 (1-x)^{5/2}} \]

[Out]

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

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} \[ \int \frac {(1+x)^{3/2}}{(1-x)^{7/2}} \, dx=\frac {(x+1)^{5/2}}{5 (1-x)^{5/2}} \]

[In]

Int[(1 + x)^(3/2)/(1 - x)^(7/2),x]

[Out]

(1 + x)^(5/2)/(5*(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]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{7/2}} \, dx=\frac {(1+x)^{5/2}}{5 (1-x)^{5/2}} \]

[In]

Integrate[(1 + x)^(3/2)/(1 - x)^(7/2),x]

[Out]

(1 + x)^(5/2)/(5*(1 - x)^(5/2))

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75

method	result	size
gosper	\(\frac {\left (1+x \right )^{\frac {5}{2}}}{5 \left (1-x \right )^{\frac {5}{2}}}\)	\(15\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{3}+3 x^{2}+3 x +1\right )}{5 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{2} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\)	\(54\)
default	\(\frac {\left (1+x \right )^{\frac {3}{2}}}{\left (1-x \right )^{\frac {5}{2}}}-\frac {6 \sqrt {1+x}}{5 \left (1-x \right )^{\frac {5}{2}}}+\frac {\sqrt {1+x}}{5 \left (1-x \right )^{\frac {3}{2}}}+\frac {\sqrt {1+x}}{5 \sqrt {1-x}}\)	\(57\)

[In]

int((1+x)^(3/2)/(1-x)^(7/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (14) = 28\).

Time = 0.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.60 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{7/2}} \, dx=\frac {x^{3} - 3 \, x^{2} - {\left (x^{2} + 2 \, x + 1\right )} \sqrt {x + 1} \sqrt {-x + 1} + 3 \, x - 1}{5 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} \]

[In]

integrate((1+x)^(3/2)/(1-x)^(7/2),x, algorithm="fricas")

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.35 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{7/2}} \, dx=\begin {cases} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{5 \sqrt {x - 1} \left (x + 1\right )^{2} - 20 \sqrt {x - 1} \left (x + 1\right ) + 20 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {\left (x + 1\right )^{\frac {5}{2}}}{5 \sqrt {1 - x} \left (x + 1\right )^{2} - 20 \sqrt {1 - x} \left (x + 1\right ) + 20 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]

[In]

integrate((1+x)**(3/2)/(1-x)**(7/2),x)

[Out]

Piecewise((-I*(x + 1)**(5/2)/(5*sqrt(x - 1)*(x + 1)**2 - 20*sqrt(x - 1)*(x + 1) + 20*sqrt(x - 1)), Abs(x + 1)
> 2), ((x + 1)**(5/2)/(5*sqrt(1 - x)*(x + 1)**2 - 20*sqrt(1 - x)*(x + 1) + 20*sqrt(1 - x)), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (14) = 28\).

Time = 0.22 (sec) , antiderivative size = 94, normalized size of antiderivative = 4.70 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{7/2}} \, dx=\frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1} + \frac {6 \, \sqrt {-x^{2} + 1}}{5 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{5 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{5 \, {\left (x - 1\right )}} \]

[In]

integrate((1+x)^(3/2)/(1-x)^(7/2),x, algorithm="maxima")

[Out]

(-x^2 + 1)^(3/2)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 6/5*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) + 1/5*sqrt(-x^2
+ 1)/(x^2 - 2*x + 1) - 1/5*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 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{7/2}} \, dx=-\frac {{\left (x + 1\right )}^{\frac {5}{2}} \sqrt {-x + 1}}{5 \, {\left (x - 1\right )}^{3}} \]

[In]

integrate((1+x)^(3/2)/(1-x)^(7/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.50 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{7/2}} \, dx=-\frac {\sqrt {1-x}\,\left (\frac {2\,x\,\sqrt {x+1}}{5}+\frac {\sqrt {x+1}}{5}+\frac {x^2\,\sqrt {x+1}}{5}\right )}{x^3-3\,x^2+3\,x-1} \]

[In]

int((x + 1)^(3/2)/(1 - x)^(7/2),x)

[Out]

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