\(\int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx\) [1136]

Optimal result

Integrand size = 17, antiderivative size = 103 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx=\frac {1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{7/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{5/2} (1+x)^{3/2}}+\frac {8 x}{63 (1-x)^{3/2} (1+x)^{3/2}}+\frac {16 x}{63 \sqrt {1-x} \sqrt {1+x}} \]

[Out]

1/9/(1-x)^(9/2)/(1+x)^(3/2)+2/21/(1-x)^(7/2)/(1+x)^(3/2)+2/21/(1-x)^(5/2)/(1+x)^(3/2)+8/63*x/(1-x)^(3/2)/(1+x)
^(3/2)+16/63*x/(1-x)^(1/2)/(1+x)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {47, 40, 39} \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx=\frac {16 x}{63 \sqrt {1-x} \sqrt {x+1}}+\frac {8 x}{63 (1-x)^{3/2} (x+1)^{3/2}}+\frac {2}{21 (1-x)^{5/2} (x+1)^{3/2}}+\frac {2}{21 (1-x)^{7/2} (x+1)^{3/2}}+\frac {1}{9 (1-x)^{9/2} (x+1)^{3/2}} \]

[In]

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

[Out]

1/(9*(1 - x)^(9/2)*(1 + x)^(3/2)) + 2/(21*(1 - x)^(7/2)*(1 + x)^(3/2)) + 2/(21*(1 - x)^(5/2)*(1 + x)^(3/2)) +
(8*x)/(63*(1 - x)^(3/2)*(1 + x)^(3/2)) + (16*x)/(63*Sqrt[1 - x]*Sqrt[1 + x])

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rule 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-x)*(a + b*x)^(m + 1)*((c + d*x)^(m
+ 1)/(2*a*c*(m + 1))), x] + Dist[(2*m + 3)/(2*a*c*(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] && ILtQ[m + 3/2, 0]

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}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac {2}{3} \int \frac {1}{(1-x)^{9/2} (1+x)^{5/2}} \, dx \\ & = \frac {1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{7/2} (1+x)^{3/2}}+\frac {10}{21} \int \frac {1}{(1-x)^{7/2} (1+x)^{5/2}} \, dx \\ & = \frac {1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{7/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{5/2} (1+x)^{3/2}}+\frac {8}{21} \int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx \\ & = \frac {1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{7/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{5/2} (1+x)^{3/2}}+\frac {8 x}{63 (1-x)^{3/2} (1+x)^{3/2}}+\frac {16}{63} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx \\ & = \frac {1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{7/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{5/2} (1+x)^{3/2}}+\frac {8 x}{63 (1-x)^{3/2} (1+x)^{3/2}}+\frac {16 x}{63 \sqrt {1-x} \sqrt {1+x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx=\frac {19+6 x-66 x^2+56 x^3+24 x^4-48 x^5+16 x^6}{63 (1-x)^{9/2} (1+x)^{3/2}} \]

[In]

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

[Out]

(19 + 6*x - 66*x^2 + 56*x^3 + 24*x^4 - 48*x^5 + 16*x^6)/(63*(1 - x)^(9/2)*(1 + x)^(3/2))

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.44

method	result	size
gosper	\(\frac {16 x^{6}-48 x^{5}+24 x^{4}+56 x^{3}-66 x^{2}+6 x +19}{63 \left (1+x \right )^{\frac {3}{2}} \left (1-x \right )^{\frac {9}{2}}}\)	\(45\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (16 x^{6}-48 x^{5}+24 x^{4}+56 x^{3}-66 x^{2}+6 x +19\right )}{63 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}} \left (-1+x \right )^{4} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\)	\(71\)
default	\(\frac {1}{9 \left (1-x \right )^{\frac {9}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {2}{21 \left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {2}{21 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {8}{63 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {8}{21 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}-\frac {16 \sqrt {1-x}}{63 \left (1+x \right )^{\frac {3}{2}}}-\frac {16 \sqrt {1-x}}{63 \sqrt {1+x}}\)	\(100\)

[In]

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

[Out]

1/63/(1+x)^(3/2)/(1-x)^(9/2)*(16*x^6-48*x^5+24*x^4+56*x^3-66*x^2+6*x+19)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx=\frac {19 \, x^{7} - 57 \, x^{6} + 19 \, x^{5} + 95 \, x^{4} - 95 \, x^{3} - 19 \, x^{2} - {\left (16 \, x^{6} - 48 \, x^{5} + 24 \, x^{4} + 56 \, x^{3} - 66 \, x^{2} + 6 \, x + 19\right )} \sqrt {x + 1} \sqrt {-x + 1} + 57 \, x - 19}{63 \, {\left (x^{7} - 3 \, x^{6} + x^{5} + 5 \, x^{4} - 5 \, x^{3} - x^{2} + 3 \, x - 1\right )}} \]

[In]

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

[Out]

1/63*(19*x^7 - 57*x^6 + 19*x^5 + 95*x^4 - 95*x^3 - 19*x^2 - (16*x^6 - 48*x^5 + 24*x^4 + 56*x^3 - 66*x^2 + 6*x
+ 19)*sqrt(x + 1)*sqrt(-x + 1) + 57*x - 19)/(x^7 - 3*x^6 + x^5 + 5*x^4 - 5*x^3 - x^2 + 3*x - 1)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.42 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx=\frac {16 \, x}{63 \, \sqrt {-x^{2} + 1}} + \frac {8 \, x}{63 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {1}{9 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{3} - 3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2} + 3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x - {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} + \frac {2}{21 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2} - 2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} - \frac {2}{21 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x - {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} \]

[In]

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

[Out]

16/63*x/sqrt(-x^2 + 1) + 8/63*x/(-x^2 + 1)^(3/2) - 1/9/((-x^2 + 1)^(3/2)*x^3 - 3*(-x^2 + 1)^(3/2)*x^2 + 3*(-x^
2 + 1)^(3/2)*x - (-x^2 + 1)^(3/2)) + 2/21/((-x^2 + 1)^(3/2)*x^2 - 2*(-x^2 + 1)^(3/2)*x + (-x^2 + 1)^(3/2)) - 2
/21/((-x^2 + 1)^(3/2)*x - (-x^2 + 1)^(3/2))

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx=\frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{1536 \, {\left (x + 1\right )}^{\frac {3}{2}}} + \frac {23 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{512 \, \sqrt {x + 1}} - \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {69 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{1536 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} - \frac {{\left ({\left ({\left ({\left (667 \, x - 5021\right )} {\left (x + 1\right )} + 18396\right )} {\left (x + 1\right )} - 26880\right )} {\left (x + 1\right )} + 15120\right )} \sqrt {x + 1} \sqrt {-x + 1}}{4032 \, {\left (x - 1\right )}^{5}} \]

[In]

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

[Out]

1/1536*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 23/512*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/1536*(x + 1)
^(3/2)*(69*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(2) - sqrt(-x + 1))^3 - 1/4032*((((667*x - 5021)*(x +
1) + 18396)*(x + 1) - 26880)*(x + 1) + 15120)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^5

Mupad [B] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx=-\frac {6\,x\,\sqrt {1-x}+19\,\sqrt {1-x}-66\,x^2\,\sqrt {1-x}+56\,x^3\,\sqrt {1-x}+24\,x^4\,\sqrt {1-x}-48\,x^5\,\sqrt {1-x}+16\,x^6\,\sqrt {1-x}}{\left (63\,x+63\right )\,{\left (x-1\right )}^5\,\sqrt {x+1}} \]

[In]

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

[Out]

-(6*x*(1 - x)^(1/2) + 19*(1 - x)^(1/2) - 66*x^2*(1 - x)^(1/2) + 56*x^3*(1 - x)^(1/2) + 24*x^4*(1 - x)^(1/2) -
48*x^5*(1 - x)^(1/2) + 16*x^6*(1 - x)^(1/2))/((63*x + 63)*(x - 1)^5*(x + 1)^(1/2))