Integrand size = 17, antiderivative size = 93 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx=\frac {1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{7/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{5/2} (1+x)^{3/2}}+\frac {8 x}{63 \left (1-x^2\right )^{3/2}}+\frac {16 x}{63 \sqrt {1-x^2}} \] Output:
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/(-x^2+1)^(3/2)+16/63*x/(-x^2+1)^(1/2)
Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.54 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx=\frac {19+6 x-66 x^2+56 x^3+24 x^4-48 x^5+16 x^6}{63 (1-x)^{9/2} (1+x)^{3/2}} \] Input:
Integrate[1/((1 - x)^(11/2)*(1 + x)^(5/2)),x]
Output:
(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))
Time = 0.18 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {55, 55, 55, 39, 209, 208}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{(1-x)^{11/2} (x+1)^{5/2}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {2}{3} \int \frac {1}{(1-x)^{9/2} (x+1)^{5/2}}dx+\frac {1}{9 (1-x)^{9/2} (x+1)^{3/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {2}{3} \left (\frac {5}{7} \int \frac {1}{(1-x)^{7/2} (x+1)^{5/2}}dx+\frac {1}{7 (1-x)^{7/2} (x+1)^{3/2}}\right )+\frac {1}{9 (1-x)^{9/2} (x+1)^{3/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {2}{3} \left (\frac {5}{7} \left (\frac {4}{5} \int \frac {1}{(1-x)^{5/2} (x+1)^{5/2}}dx+\frac {1}{5 (1-x)^{5/2} (x+1)^{3/2}}\right )+\frac {1}{7 (1-x)^{7/2} (x+1)^{3/2}}\right )+\frac {1}{9 (1-x)^{9/2} (x+1)^{3/2}}\) |
\(\Big \downarrow \) 39 |
\(\displaystyle \frac {2}{3} \left (\frac {5}{7} \left (\frac {4}{5} \int \frac {1}{\left (1-x^2\right )^{5/2}}dx+\frac {1}{5 (1-x)^{5/2} (x+1)^{3/2}}\right )+\frac {1}{7 (1-x)^{7/2} (x+1)^{3/2}}\right )+\frac {1}{9 (1-x)^{9/2} (x+1)^{3/2}}\) |
\(\Big \downarrow \) 209 |
\(\displaystyle \frac {2}{3} \left (\frac {5}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {1}{\left (1-x^2\right )^{3/2}}dx+\frac {x}{3 \left (1-x^2\right )^{3/2}}\right )+\frac {1}{5 (1-x)^{5/2} (x+1)^{3/2}}\right )+\frac {1}{7 (1-x)^{7/2} (x+1)^{3/2}}\right )+\frac {1}{9 (1-x)^{9/2} (x+1)^{3/2}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {2}{3} \left (\frac {5}{7} \left (\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-x^2}}+\frac {x}{3 \left (1-x^2\right )^{3/2}}\right )+\frac {1}{5 (1-x)^{5/2} (x+1)^{3/2}}\right )+\frac {1}{7 (1-x)^{7/2} (x+1)^{3/2}}\right )+\frac {1}{9 (1-x)^{9/2} (x+1)^{3/2}}\) |
Input:
Int[1/((1 - x)^(11/2)*(1 + x)^(5/2)),x]
Output:
1/(9*(1 - x)^(9/2)*(1 + x)^(3/2)) + (2*(1/(7*(1 - x)^(7/2)*(1 + x)^(3/2)) + (5*(1/(5*(1 - x)^(5/2)*(1 + x)^(3/2)) + (4*(x/(3*(1 - x^2)^(3/2)) + (2*x )/(3*Sqrt[1 - x^2])))/5))/7))/3
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]))
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] - Simp[d*(S implify[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] && 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] || !SumSimp lerQ[n, 1])
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Time = 0.10 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.48
method | result | size |
gosper | \(\frac {16 x^{6}-48 x^{5}+24 x^{4}+56 x^{3}-66 x^{2}+6 x +19}{63 \left (1+x \right )^{\frac {3}{2}} \left (1-x \right )^{\frac {9}{2}}}\) | \(45\) |
orering | \(-\frac {\left (-1+x \right ) \left (16 x^{6}-48 x^{5}+24 x^{4}+56 x^{3}-66 x^{2}+6 x +19\right )}{63 \left (1+x \right )^{\frac {3}{2}} \left (1-x \right )^{\frac {11}{2}}}\) | \(48\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (16 x^{6}-48 x^{5}+24 x^{4}+56 x^{3}-66 x^{2}+6 x +19\right )}{63 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \left (-1+x \right )^{4}}\) | \(71\) |
default | \(\frac {1}{9 \left (1-x \right )^{\frac {9}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {2}{21 \left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {2}{21 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {8}{63 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {8}{21 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}-\frac {16 \sqrt {1-x}}{63 \left (1+x \right )^{\frac {3}{2}}}-\frac {16 \sqrt {1-x}}{63 \sqrt {1+x}}\) | \(100\) |
Input:
int(1/(1-x)^(11/2)/(1+x)^(5/2),x,method=_RETURNVERBOSE)
Output:
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)
Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx=\frac {19 \, x^{7} - 57 \, x^{6} + 19 \, x^{5} + 95 \, x^{4} - 95 \, x^{3} - 19 \, x^{2} - {\left (16 \, x^{6} - 48 \, x^{5} + 24 \, x^{4} + 56 \, x^{3} - 66 \, x^{2} + 6 \, x + 19\right )} \sqrt {x + 1} \sqrt {-x + 1} + 57 \, x - 19}{63 \, {\left (x^{7} - 3 \, x^{6} + x^{5} + 5 \, x^{4} - 5 \, x^{3} - x^{2} + 3 \, x - 1\right )}} \] Input:
integrate(1/(1-x)^(11/2)/(1+x)^(5/2),x, algorithm="fricas")
Output:
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)
Result contains complex when optimal does not.
Time = 123.70 (sec) , antiderivative size = 789, normalized size of antiderivative = 8.48 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(1-x)**(11/2)/(1+x)**(5/2),x)
Output:
Piecewise((-16*sqrt(-1 + 2/(x + 1))*(x + 1)**6/(-2016*x + 63*(x + 1)**6 - 630*(x + 1)**5 + 2520*(x + 1)**4 - 5040*(x + 1)**3 + 5040*(x + 1)**2 - 201 6) + 144*sqrt(-1 + 2/(x + 1))*(x + 1)**5/(-2016*x + 63*(x + 1)**6 - 630*(x + 1)**5 + 2520*(x + 1)**4 - 5040*(x + 1)**3 + 5040*(x + 1)**2 - 2016) - 5 04*sqrt(-1 + 2/(x + 1))*(x + 1)**4/(-2016*x + 63*(x + 1)**6 - 630*(x + 1)* *5 + 2520*(x + 1)**4 - 5040*(x + 1)**3 + 5040*(x + 1)**2 - 2016) + 840*sqr t(-1 + 2/(x + 1))*(x + 1)**3/(-2016*x + 63*(x + 1)**6 - 630*(x + 1)**5 + 2 520*(x + 1)**4 - 5040*(x + 1)**3 + 5040*(x + 1)**2 - 2016) - 630*sqrt(-1 + 2/(x + 1))*(x + 1)**2/(-2016*x + 63*(x + 1)**6 - 630*(x + 1)**5 + 2520*(x + 1)**4 - 5040*(x + 1)**3 + 5040*(x + 1)**2 - 2016) + 126*sqrt(-1 + 2/(x + 1))*(x + 1)/(-2016*x + 63*(x + 1)**6 - 630*(x + 1)**5 + 2520*(x + 1)**4 - 5040*(x + 1)**3 + 5040*(x + 1)**2 - 2016) + 21*sqrt(-1 + 2/(x + 1))/(-20 16*x + 63*(x + 1)**6 - 630*(x + 1)**5 + 2520*(x + 1)**4 - 5040*(x + 1)**3 + 5040*(x + 1)**2 - 2016), 1/Abs(x + 1) > 1/2), (-16*I*sqrt(1 - 2/(x + 1)) *(x + 1)**6/(-2016*x + 63*(x + 1)**6 - 630*(x + 1)**5 + 2520*(x + 1)**4 - 5040*(x + 1)**3 + 5040*(x + 1)**2 - 2016) + 144*I*sqrt(1 - 2/(x + 1))*(x + 1)**5/(-2016*x + 63*(x + 1)**6 - 630*(x + 1)**5 + 2520*(x + 1)**4 - 5040* (x + 1)**3 + 5040*(x + 1)**2 - 2016) - 504*I*sqrt(1 - 2/(x + 1))*(x + 1)** 4/(-2016*x + 63*(x + 1)**6 - 630*(x + 1)**5 + 2520*(x + 1)**4 - 5040*(x + 1)**3 + 5040*(x + 1)**2 - 2016) + 840*I*sqrt(1 - 2/(x + 1))*(x + 1)**3/...
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (67) = 134\).
Time = 0.03 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.57 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx=\frac {16 \, x}{63 \, \sqrt {-x^{2} + 1}} + \frac {8 \, x}{63 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {1}{9 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{3} - 3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2} + 3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x - {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} + \frac {2}{21 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2} - 2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} - \frac {2}{21 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x - {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} \] Input:
integrate(1/(1-x)^(11/2)/(1+x)^(5/2),x, algorithm="maxima")
Output:
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)) - 2/2 1/((-x^2 + 1)^(3/2)*x - (-x^2 + 1)^(3/2))
Time = 0.13 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx=\frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{1536 \, {\left (x + 1\right )}^{\frac {3}{2}}} + \frac {23 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{512 \, \sqrt {x + 1}} - \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {69 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{1536 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} - \frac {{\left ({\left ({\left ({\left (667 \, x - 5021\right )} {\left (x + 1\right )} + 18396\right )} {\left (x + 1\right )} - 26880\right )} {\left (x + 1\right )} + 15120\right )} \sqrt {x + 1} \sqrt {-x + 1}}{4032 \, {\left (x - 1\right )}^{5}} \] Input:
integrate(1/(1-x)^(11/2)/(1+x)^(5/2),x, algorithm="giac")
Output:
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
Time = 0.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx=-\frac {6\,x\,\sqrt {1-x}+19\,\sqrt {1-x}-66\,x^2\,\sqrt {1-x}+56\,x^3\,\sqrt {1-x}+24\,x^4\,\sqrt {1-x}-48\,x^5\,\sqrt {1-x}+16\,x^6\,\sqrt {1-x}}{\left (63\,x+63\right )\,{\left (x-1\right )}^5\,\sqrt {x+1}} \] Input:
int(1/((1 - x)^(11/2)*(x + 1)^(5/2)),x)
Output:
-(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))
Time = 0.16 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx=\frac {16 x^{6}-48 x^{5}+24 x^{4}+56 x^{3}-66 x^{2}+6 x +19}{63 \sqrt {x +1}\, \sqrt {1-x}\, \left (x^{5}-3 x^{4}+2 x^{3}+2 x^{2}-3 x +1\right )} \] Input:
int(1/(1-x)^(11/2)/(1+x)^(5/2),x)
Output:
(16*x**6 - 48*x**5 + 24*x**4 + 56*x**3 - 66*x**2 + 6*x + 19)/(63*sqrt(x + 1)*sqrt( - x + 1)*(x**5 - 3*x**4 + 2*x**3 + 2*x**2 - 3*x + 1))