\(\int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx\) [1103]

Optimal result

Integrand size = 17, antiderivative size = 121 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx=\frac {(1+x)^{7/2}}{17 (1-x)^{17/2}}+\frac {(1+x)^{7/2}}{51 (1-x)^{15/2}}+\frac {4 (1+x)^{7/2}}{663 (1-x)^{13/2}}+\frac {4 (1+x)^{7/2}}{2431 (1-x)^{11/2}}+\frac {8 (1+x)^{7/2}}{21879 (1-x)^{9/2}}+\frac {8 (1+x)^{7/2}}{153153 (1-x)^{7/2}} \]

[Out]

1/17*(1+x)^(7/2)/(1-x)^(17/2)+1/51*(1+x)^(7/2)/(1-x)^(15/2)+4/663*(1+x)^(7/2)/(1-x)^(13/2)+4/2431*(1+x)^(7/2)/
(1-x)^(11/2)+8/21879*(1+x)^(7/2)/(1-x)^(9/2)+8/153153*(1+x)^(7/2)/(1-x)^(7/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37} \[ \int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx=\frac {8 (x+1)^{7/2}}{153153 (1-x)^{7/2}}+\frac {8 (x+1)^{7/2}}{21879 (1-x)^{9/2}}+\frac {4 (x+1)^{7/2}}{2431 (1-x)^{11/2}}+\frac {4 (x+1)^{7/2}}{663 (1-x)^{13/2}}+\frac {(x+1)^{7/2}}{51 (1-x)^{15/2}}+\frac {(x+1)^{7/2}}{17 (1-x)^{17/2}} \]

[In]

Int[(1 + x)^(5/2)/(1 - x)^(19/2),x]

[Out]

(1 + x)^(7/2)/(17*(1 - x)^(17/2)) + (1 + x)^(7/2)/(51*(1 - x)^(15/2)) + (4*(1 + x)^(7/2))/(663*(1 - x)^(13/2))
+ (4*(1 + x)^(7/2))/(2431*(1 - x)^(11/2)) + (8*(1 + x)^(7/2))/(21879*(1 - x)^(9/2)) + (8*(1 + x)^(7/2))/(1531
53*(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}}{17 (1-x)^{17/2}}+\frac {5}{17} \int \frac {(1+x)^{5/2}}{(1-x)^{17/2}} \, dx \\ & = \frac {(1+x)^{7/2}}{17 (1-x)^{17/2}}+\frac {(1+x)^{7/2}}{51 (1-x)^{15/2}}+\frac {4}{51} \int \frac {(1+x)^{5/2}}{(1-x)^{15/2}} \, dx \\ & = \frac {(1+x)^{7/2}}{17 (1-x)^{17/2}}+\frac {(1+x)^{7/2}}{51 (1-x)^{15/2}}+\frac {4 (1+x)^{7/2}}{663 (1-x)^{13/2}}+\frac {4}{221} \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx \\ & = \frac {(1+x)^{7/2}}{17 (1-x)^{17/2}}+\frac {(1+x)^{7/2}}{51 (1-x)^{15/2}}+\frac {4 (1+x)^{7/2}}{663 (1-x)^{13/2}}+\frac {4 (1+x)^{7/2}}{2431 (1-x)^{11/2}}+\frac {8 \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx}{2431} \\ & = \frac {(1+x)^{7/2}}{17 (1-x)^{17/2}}+\frac {(1+x)^{7/2}}{51 (1-x)^{15/2}}+\frac {4 (1+x)^{7/2}}{663 (1-x)^{13/2}}+\frac {4 (1+x)^{7/2}}{2431 (1-x)^{11/2}}+\frac {8 (1+x)^{7/2}}{21879 (1-x)^{9/2}}+\frac {8 \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx}{21879} \\ & = \frac {(1+x)^{7/2}}{17 (1-x)^{17/2}}+\frac {(1+x)^{7/2}}{51 (1-x)^{15/2}}+\frac {4 (1+x)^{7/2}}{663 (1-x)^{13/2}}+\frac {4 (1+x)^{7/2}}{2431 (1-x)^{11/2}}+\frac {8 (1+x)^{7/2}}{21879 (1-x)^{9/2}}+\frac {8 (1+x)^{7/2}}{153153 (1-x)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.37 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx=\frac {(1+x)^{7/2} \left (13252-5871 x+2096 x^2-556 x^3+96 x^4-8 x^5\right )}{153153 (1-x)^{17/2}} \]

[In]

Integrate[(1 + x)^(5/2)/(1 - x)^(19/2),x]

[Out]

((1 + x)^(7/2)*(13252 - 5871*x + 2096*x^2 - 556*x^3 + 96*x^4 - 8*x^5))/(153153*(1 - x)^(17/2))

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.33

method	result	size
gosper	\(-\frac {\left (1+x \right )^{\frac {7}{2}} \left (8 x^{5}-96 x^{4}+556 x^{3}-2096 x^{2}+5871 x -13252\right )}{153153 \left (1-x \right )^{\frac {17}{2}}}\)	\(40\)
risch	\(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{9}-64 x^{8}+220 x^{7}-416 x^{6}+447 x^{5}-216 x^{4}-25610 x^{3}-58124 x^{2}-47137 x -13252\right )}{153153 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{8} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\)	\(86\)
default	\(\frac {\left (1+x \right )^{\frac {5}{2}}}{6 \left (1-x \right )^{\frac {17}{2}}}-\frac {5 \left (1+x \right )^{\frac {3}{2}}}{42 \left (1-x \right )^{\frac {17}{2}}}+\frac {5 \sqrt {1+x}}{119 \left (1-x \right )^{\frac {17}{2}}}-\frac {\sqrt {1+x}}{714 \left (1-x \right )^{\frac {15}{2}}}-\frac {\sqrt {1+x}}{1326 \left (1-x \right )^{\frac {13}{2}}}-\frac {\sqrt {1+x}}{2431 \left (1-x \right )^{\frac {11}{2}}}-\frac {5 \sqrt {1+x}}{21879 \left (1-x \right )^{\frac {9}{2}}}-\frac {20 \sqrt {1+x}}{153153 \left (1-x \right )^{\frac {7}{2}}}-\frac {4 \sqrt {1+x}}{51051 \left (1-x \right )^{\frac {5}{2}}}-\frac {8 \sqrt {1+x}}{153153 \left (1-x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1+x}}{153153 \sqrt {1-x}}\)	\(156\)

[In]

int((1+x)^(5/2)/(1-x)^(19/2),x,method=_RETURNVERBOSE)

[Out]

-1/153153*(1+x)^(7/2)/(1-x)^(17/2)*(8*x^5-96*x^4+556*x^3-2096*x^2+5871*x-13252)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.20 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx=\frac {13252 \, x^{9} - 119268 \, x^{8} + 477072 \, x^{7} - 1113168 \, x^{6} + 1669752 \, x^{5} - 1669752 \, x^{4} + 1113168 \, x^{3} - 477072 \, x^{2} + {\left (8 \, x^{8} - 72 \, x^{7} + 292 \, x^{6} - 708 \, x^{5} + 1155 \, x^{4} - 1371 \, x^{3} - 24239 \, x^{2} - 33885 \, x - 13252\right )} \sqrt {x + 1} \sqrt {-x + 1} + 119268 \, x - 13252}{153153 \, {\left (x^{9} - 9 \, x^{8} + 36 \, x^{7} - 84 \, x^{6} + 126 \, x^{5} - 126 \, x^{4} + 84 \, x^{3} - 36 \, x^{2} + 9 \, x - 1\right )}} \]

[In]

integrate((1+x)^(5/2)/(1-x)^(19/2),x, algorithm="fricas")

[Out]

1/153153*(13252*x^9 - 119268*x^8 + 477072*x^7 - 1113168*x^6 + 1669752*x^5 - 1669752*x^4 + 1113168*x^3 - 477072
*x^2 + (8*x^8 - 72*x^7 + 292*x^6 - 708*x^5 + 1155*x^4 - 1371*x^3 - 24239*x^2 - 33885*x - 13252)*sqrt(x + 1)*sq
rt(-x + 1) + 119268*x - 13252)/(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)

Sympy [F(-1)]

Timed out. \[ \int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx=\text {Timed out} \]

[In]

integrate((1+x)**(5/2)/(1-x)**(19/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (85) = 170\).

Time = 0.21 (sec) , antiderivative size = 452, normalized size of antiderivative = 3.74 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx=-\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{6 \, {\left (x^{11} - 11 \, x^{10} + 55 \, x^{9} - 165 \, x^{8} + 330 \, x^{7} - 462 \, x^{6} + 462 \, x^{5} - 330 \, x^{4} + 165 \, x^{3} - 55 \, x^{2} + 11 \, x - 1\right )}} - \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{42 \, {\left (x^{10} - 10 \, x^{9} + 45 \, x^{8} - 120 \, x^{7} + 210 \, x^{6} - 252 \, x^{5} + 210 \, x^{4} - 120 \, x^{3} + 45 \, x^{2} - 10 \, x + 1\right )}} - \frac {5 \, \sqrt {-x^{2} + 1}}{119 \, {\left (x^{9} - 9 \, x^{8} + 36 \, x^{7} - 84 \, x^{6} + 126 \, x^{5} - 126 \, x^{4} + 84 \, x^{3} - 36 \, x^{2} + 9 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{714 \, {\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{1326 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{2431 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac {5 \, \sqrt {-x^{2} + 1}}{21879 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {20 \, \sqrt {-x^{2} + 1}}{153153 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {4 \, \sqrt {-x^{2} + 1}}{51051 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {8 \, \sqrt {-x^{2} + 1}}{153153 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {8 \, \sqrt {-x^{2} + 1}}{153153 \, {\left (x - 1\right )}} \]

[In]

integrate((1+x)^(5/2)/(1-x)^(19/2),x, algorithm="maxima")

[Out]

-1/6*(-x^2 + 1)^(5/2)/(x^11 - 11*x^10 + 55*x^9 - 165*x^8 + 330*x^7 - 462*x^6 + 462*x^5 - 330*x^4 + 165*x^3 - 5
5*x^2 + 11*x - 1) - 5/42*(-x^2 + 1)^(3/2)/(x^10 - 10*x^9 + 45*x^8 - 120*x^7 + 210*x^6 - 252*x^5 + 210*x^4 - 12
0*x^3 + 45*x^2 - 10*x + 1) - 5/119*sqrt(-x^2 + 1)/(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) - 1/714*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
) + 1/1326*sqrt(-x^2 + 1)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x - 1) - 1/2431*sqrt(-x^2 + 1)/
(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1) + 5/21879*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5
*x - 1) - 20/153153*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 4/51051*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x
- 1) - 8/153153*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 8/153153*sqrt(-x^2 + 1)/(x - 1)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.40 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx=\frac {{\left ({\left (4 \, {\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 16\right )} + 255\right )} {\left (x + 1\right )} - 1105\right )} {\left (x + 1\right )} + 12155\right )} {\left (x + 1\right )} - 21879\right )} {\left (x + 1\right )}^{\frac {7}{2}} \sqrt {-x + 1}}{153153 \, {\left (x - 1\right )}^{9}} \]

[In]

integrate((1+x)^(5/2)/(1-x)^(19/2),x, algorithm="giac")

[Out]

1/153153*((4*((2*(x + 1)*(x - 16) + 255)*(x + 1) - 1105)*(x + 1) + 12155)*(x + 1) - 21879)*(x + 1)^(7/2)*sqrt(
-x + 1)/(x - 1)^9

Mupad [B] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.16 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx=-\frac {\sqrt {1-x}\,\left (\frac {3765\,x\,\sqrt {x+1}}{17017}+\frac {13252\,\sqrt {x+1}}{153153}+\frac {24239\,x^2\,\sqrt {x+1}}{153153}+\frac {457\,x^3\,\sqrt {x+1}}{51051}-\frac {5\,x^4\,\sqrt {x+1}}{663}+\frac {236\,x^5\,\sqrt {x+1}}{51051}-\frac {292\,x^6\,\sqrt {x+1}}{153153}+\frac {8\,x^7\,\sqrt {x+1}}{17017}-\frac {8\,x^8\,\sqrt {x+1}}{153153}\right )}{x^9-9\,x^8+36\,x^7-84\,x^6+126\,x^5-126\,x^4+84\,x^3-36\,x^2+9\,x-1} \]

[In]

int((x + 1)^(5/2)/(1 - x)^(19/2),x)

[Out]

-((1 - x)^(1/2)*((3765*x*(x + 1)^(1/2))/17017 + (13252*(x + 1)^(1/2))/153153 + (24239*x^2*(x + 1)^(1/2))/15315
3 + (457*x^3*(x + 1)^(1/2))/51051 - (5*x^4*(x + 1)^(1/2))/663 + (236*x^5*(x + 1)^(1/2))/51051 - (292*x^6*(x +
1)^(1/2))/153153 + (8*x^7*(x + 1)^(1/2))/17017 - (8*x^8*(x + 1)^(1/2))/153153))/(9*x - 36*x^2 + 84*x^3 - 126*x
^4 + 126*x^5 - 84*x^6 + 36*x^7 - 9*x^8 + x^9 - 1)