\(\int \frac {(1+x)^{5/2}}{(1-x)^{17/2}} \, dx\) [1102]

Optimal result

Integrand size = 17, antiderivative size = 101 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{17/2}} \, dx=\frac {(1+x)^{7/2}}{15 (1-x)^{15/2}}+\frac {4 (1+x)^{7/2}}{195 (1-x)^{13/2}}+\frac {4 (1+x)^{7/2}}{715 (1-x)^{11/2}}+\frac {8 (1+x)^{7/2}}{6435 (1-x)^{9/2}}+\frac {8 (1+x)^{7/2}}{45045 (1-x)^{7/2}} \]

[Out]

1/15*(1+x)^(7/2)/(1-x)^(15/2)+4/195*(1+x)^(7/2)/(1-x)^(13/2)+4/715*(1+x)^(7/2)/(1-x)^(11/2)+8/6435*(1+x)^(7/2)
/(1-x)^(9/2)+8/45045*(1+x)^(7/2)/(1-x)^(7/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37} \[ \int \frac {(1+x)^{5/2}}{(1-x)^{17/2}} \, dx=\frac {8 (x+1)^{7/2}}{45045 (1-x)^{7/2}}+\frac {8 (x+1)^{7/2}}{6435 (1-x)^{9/2}}+\frac {4 (x+1)^{7/2}}{715 (1-x)^{11/2}}+\frac {4 (x+1)^{7/2}}{195 (1-x)^{13/2}}+\frac {(x+1)^{7/2}}{15 (1-x)^{15/2}} \]

[In]

Int[(1 + x)^(5/2)/(1 - x)^(17/2),x]

[Out]

(1 + x)^(7/2)/(15*(1 - x)^(15/2)) + (4*(1 + x)^(7/2))/(195*(1 - x)^(13/2)) + (4*(1 + x)^(7/2))/(715*(1 - x)^(1
1/2)) + (8*(1 + x)^(7/2))/(6435*(1 - x)^(9/2)) + (8*(1 + x)^(7/2))/(45045*(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]

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)^{7/2}}{15 (1-x)^{15/2}}+\frac {4}{15} \int \frac {(1+x)^{5/2}}{(1-x)^{15/2}} \, dx \\ & = \frac {(1+x)^{7/2}}{15 (1-x)^{15/2}}+\frac {4 (1+x)^{7/2}}{195 (1-x)^{13/2}}+\frac {4}{65} \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx \\ & = \frac {(1+x)^{7/2}}{15 (1-x)^{15/2}}+\frac {4 (1+x)^{7/2}}{195 (1-x)^{13/2}}+\frac {4 (1+x)^{7/2}}{715 (1-x)^{11/2}}+\frac {8}{715} \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx \\ & = \frac {(1+x)^{7/2}}{15 (1-x)^{15/2}}+\frac {4 (1+x)^{7/2}}{195 (1-x)^{13/2}}+\frac {4 (1+x)^{7/2}}{715 (1-x)^{11/2}}+\frac {8 (1+x)^{7/2}}{6435 (1-x)^{9/2}}+\frac {8 \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx}{6435} \\ & = \frac {(1+x)^{7/2}}{15 (1-x)^{15/2}}+\frac {4 (1+x)^{7/2}}{195 (1-x)^{13/2}}+\frac {4 (1+x)^{7/2}}{715 (1-x)^{11/2}}+\frac {8 (1+x)^{7/2}}{6435 (1-x)^{9/2}}+\frac {8 (1+x)^{7/2}}{45045 (1-x)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.40 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{17/2}} \, dx=\frac {(1+x)^{7/2} \left (4243-1628 x+468 x^2-88 x^3+8 x^4\right )}{45045 (1-x)^{15/2}} \]

[In]

Integrate[(1 + x)^(5/2)/(1 - x)^(17/2),x]

[Out]

((1 + x)^(7/2)*(4243 - 1628*x + 468*x^2 - 88*x^3 + 8*x^4))/(45045*(1 - x)^(15/2))

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.35

method	result	size
gosper	\(\frac {\left (1+x \right )^{\frac {7}{2}} \left (8 x^{4}-88 x^{3}+468 x^{2}-1628 x +4243\right )}{45045 \left (1-x \right )^{\frac {15}{2}}}\)	\(35\)
risch	\(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{8}-56 x^{7}+164 x^{6}-252 x^{5}+195 x^{4}+8988 x^{3}+19414 x^{2}+15344 x +4243\right )}{45045 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{7} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\)	\(81\)
default	\(\frac {\left (1+x \right )^{\frac {5}{2}}}{5 \left (1-x \right )^{\frac {15}{2}}}-\frac {\left (1+x \right )^{\frac {3}{2}}}{6 \left (1-x \right )^{\frac {15}{2}}}+\frac {\sqrt {1+x}}{15 \left (1-x \right )^{\frac {15}{2}}}-\frac {\sqrt {1+x}}{390 \left (1-x \right )^{\frac {13}{2}}}-\frac {\sqrt {1+x}}{715 \left (1-x \right )^{\frac {11}{2}}}-\frac {\sqrt {1+x}}{1287 \left (1-x \right )^{\frac {9}{2}}}-\frac {4 \sqrt {1+x}}{9009 \left (1-x \right )^{\frac {7}{2}}}-\frac {4 \sqrt {1+x}}{15015 \left (1-x \right )^{\frac {5}{2}}}-\frac {8 \sqrt {1+x}}{45045 \left (1-x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1+x}}{45045 \sqrt {1-x}}\)	\(142\)

[In]

int((1+x)^(5/2)/(1-x)^(17/2),x,method=_RETURNVERBOSE)

[Out]

1/45045*(1+x)^(7/2)/(1-x)^(15/2)*(8*x^4-88*x^3+468*x^2-1628*x+4243)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.29 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{17/2}} \, dx=\frac {4243 \, x^{8} - 33944 \, x^{7} + 118804 \, x^{6} - 237608 \, x^{5} + 297010 \, x^{4} - 237608 \, x^{3} + 118804 \, x^{2} + {\left (8 \, x^{7} - 64 \, x^{6} + 228 \, x^{5} - 480 \, x^{4} + 675 \, x^{3} + 8313 \, x^{2} + 11101 \, x + 4243\right )} \sqrt {x + 1} \sqrt {-x + 1} - 33944 \, x + 4243}{45045 \, {\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} \]

[In]

integrate((1+x)^(5/2)/(1-x)^(17/2),x, algorithm="fricas")

[Out]

1/45045*(4243*x^8 - 33944*x^7 + 118804*x^6 - 237608*x^5 + 297010*x^4 - 237608*x^3 + 118804*x^2 + (8*x^7 - 64*x
^6 + 228*x^5 - 480*x^4 + 675*x^3 + 8313*x^2 + 11101*x + 4243)*sqrt(x + 1)*sqrt(-x + 1) - 33944*x + 4243)/(x^8
- 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1)

Sympy [F(-1)]

Timed out. \[ \int \frac {(1+x)^{5/2}}{(1-x)^{17/2}} \, dx=\text {Timed out} \]

[In]

integrate((1+x)**(5/2)/(1-x)**(17/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (71) = 142\).

Time = 0.21 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.82 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{17/2}} \, dx=\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{5 \, {\left (x^{10} - 10 \, x^{9} + 45 \, x^{8} - 120 \, x^{7} + 210 \, x^{6} - 252 \, x^{5} + 210 \, x^{4} - 120 \, x^{3} + 45 \, x^{2} - 10 \, x + 1\right )}} + \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{6 \, {\left (x^{9} - 9 \, x^{8} + 36 \, x^{7} - 84 \, x^{6} + 126 \, x^{5} - 126 \, x^{4} + 84 \, x^{3} - 36 \, x^{2} + 9 \, x - 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{15 \, {\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{390 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{715 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{1287 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {4 \, \sqrt {-x^{2} + 1}}{9009 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {4 \, \sqrt {-x^{2} + 1}}{15015 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {8 \, \sqrt {-x^{2} + 1}}{45045 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {8 \, \sqrt {-x^{2} + 1}}{45045 \, {\left (x - 1\right )}} \]

[In]

integrate((1+x)^(5/2)/(1-x)^(17/2),x, algorithm="maxima")

[Out]

1/5*(-x^2 + 1)^(5/2)/(x^10 - 10*x^9 + 45*x^8 - 120*x^7 + 210*x^6 - 252*x^5 + 210*x^4 - 120*x^3 + 45*x^2 - 10*x
+ 1) + 1/6*(-x^2 + 1)^(3/2)/(x^9 - 9*x^8 + 36*x^7 - 84*x^6 + 126*x^5 - 126*x^4 + 84*x^3 - 36*x^2 + 9*x - 1) +
1/15*sqrt(-x^2 + 1)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) + 1/390*sqrt(-x^2 +
1)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x - 1) - 1/715*sqrt(-x^2 + 1)/(x^6 - 6*x^5 + 15*x^4 -
20*x^3 + 15*x^2 - 6*x + 1) + 1/1287*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) - 4/9009*sqrt(-x^
2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 4/15015*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 8/45045*sqrt(-x^2 +
1)/(x^2 - 2*x + 1) + 8/45045*sqrt(-x^2 + 1)/(x - 1)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.42 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{17/2}} \, dx=\frac {{\left (4 \, {\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 14\right )} + 195\right )} {\left (x + 1\right )} - 715\right )} {\left (x + 1\right )} + 6435\right )} {\left (x + 1\right )}^{\frac {7}{2}} \sqrt {-x + 1}}{45045 \, {\left (x - 1\right )}^{8}} \]

[In]

integrate((1+x)^(5/2)/(1-x)^(17/2),x, algorithm="giac")

[Out]

1/45045*(4*((2*(x + 1)*(x - 14) + 195)*(x + 1) - 715)*(x + 1) + 6435)*(x + 1)^(7/2)*sqrt(-x + 1)/(x - 1)^8

Mupad [B] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.23 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{17/2}} \, dx=\frac {\sqrt {1-x}\,\left (\frac {11101\,x\,\sqrt {x+1}}{45045}+\frac {4243\,\sqrt {x+1}}{45045}+\frac {2771\,x^2\,\sqrt {x+1}}{15015}+\frac {15\,x^3\,\sqrt {x+1}}{1001}-\frac {32\,x^4\,\sqrt {x+1}}{3003}+\frac {76\,x^5\,\sqrt {x+1}}{15015}-\frac {64\,x^6\,\sqrt {x+1}}{45045}+\frac {8\,x^7\,\sqrt {x+1}}{45045}\right )}{x^8-8\,x^7+28\,x^6-56\,x^5+70\,x^4-56\,x^3+28\,x^2-8\,x+1} \]

[In]

int((x + 1)^(5/2)/(1 - x)^(17/2),x)

[Out]

((1 - x)^(1/2)*((11101*x*(x + 1)^(1/2))/45045 + (4243*(x + 1)^(1/2))/45045 + (2771*x^2*(x + 1)^(1/2))/15015 +
(15*x^3*(x + 1)^(1/2))/1001 - (32*x^4*(x + 1)^(1/2))/3003 + (76*x^5*(x + 1)^(1/2))/15015 - (64*x^6*(x + 1)^(1/
2))/45045 + (8*x^7*(x + 1)^(1/2))/45045))/(28*x^2 - 8*x - 56*x^3 + 70*x^4 - 56*x^5 + 28*x^6 - 8*x^7 + x^8 + 1)