3.11.100 \(\int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx\) [1100]

Optimal. Leaf size=61 \[ \frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{99 (1-x)^{9/2}}+\frac {2 (1+x)^{7/2}}{693 (1-x)^{7/2}} \]

[Out]

1/11*(1+x)^(7/2)/(1-x)^(11/2)+2/99*(1+x)^(7/2)/(1-x)^(9/2)+2/693*(1+x)^(7/2)/(1-x)^(7/2)

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Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37} \begin {gather*} \frac {2 (x+1)^{7/2}}{693 (1-x)^{7/2}}+\frac {2 (x+1)^{7/2}}{99 (1-x)^{9/2}}+\frac {(x+1)^{7/2}}{11 (1-x)^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(5/2)/(1 - x)^(13/2),x]

[Out]

(1 + x)^(7/2)/(11*(1 - x)^(11/2)) + (2*(1 + x)^(7/2))/(99*(1 - x)^(9/2)) + (2*(1 + x)^(7/2))/(693*(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*} \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx &=\frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2}{11} \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx\\ &=\frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{99 (1-x)^{9/2}}+\frac {2}{99} \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx\\ &=\frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{99 (1-x)^{9/2}}+\frac {2 (1+x)^{7/2}}{693 (1-x)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 30, normalized size = 0.49 \begin {gather*} \frac {(1+x)^{7/2} \left (79-18 x+2 x^2\right )}{693 (1-x)^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(5/2)/(1 - x)^(13/2),x]

[Out]

((1 + x)^(7/2)*(79 - 18*x + 2*x^2))/(693*(1 - x)^(11/2))

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 103.28, size = 475, normalized size = 7.79 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-55+143 x-26 \left (1+x\right )^2+2 \left (1+x\right )^3\right ) \left (1+x\right )^{\frac {7}{2}}}{693 \sqrt {-1+x} \left (1-6 x+15 x^2-20 x^3+15 x^4-6 x^5+x^6\right )},\text {Abs}\left [1+x\right ]>2\right \}\right \},\frac {-143 \left (1+x\right )^{\frac {9}{2}}}{-133056 \left (1+x\right ) \sqrt {1-x}-110880 \left (1+x\right )^3 \sqrt {1-x}-8316 \left (1+x\right )^5 \sqrt {1-x}+693 \left (1+x\right )^6 \sqrt {1-x}+41580 \left (1+x\right )^4 \sqrt {1-x}+44352 \sqrt {1-x}+166320 \left (1+x\right )^2 \sqrt {1-x}}-\frac {2 \left (1+x\right )^{\frac {13}{2}}}{-133056 \left (1+x\right ) \sqrt {1-x}-110880 \left (1+x\right )^3 \sqrt {1-x}-8316 \left (1+x\right )^5 \sqrt {1-x}+693 \left (1+x\right )^6 \sqrt {1-x}+41580 \left (1+x\right )^4 \sqrt {1-x}+44352 \sqrt {1-x}+166320 \left (1+x\right )^2 \sqrt {1-x}}+\frac {26 \left (1+x\right )^{\frac {11}{2}}}{-133056 \left (1+x\right ) \sqrt {1-x}-110880 \left (1+x\right )^3 \sqrt {1-x}-8316 \left (1+x\right )^5 \sqrt {1-x}+693 \left (1+x\right )^6 \sqrt {1-x}+41580 \left (1+x\right )^4 \sqrt {1-x}+44352 \sqrt {1-x}+166320 \left (1+x\right )^2 \sqrt {1-x}}+\frac {198 \left (1+x\right )^{\frac {7}{2}}}{-133056 \left (1+x\right ) \sqrt {1-x}-110880 \left (1+x\right )^3 \sqrt {1-x}-8316 \left (1+x\right )^5 \sqrt {1-x}+693 \left (1+x\right )^6 \sqrt {1-x}+41580 \left (1+x\right )^4 \sqrt {1-x}+44352 \sqrt {1-x}+166320 \left (1+x\right )^2 \sqrt {1-x}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(1 + x)^(5/2)/(1 - x)^(13/2),x]')

[Out]

Piecewise[{{I / 693 (-55 + 143 x - 26 (1 + x) ^ 2 + 2 (1 + x) ^ 3) (1 + x) ^ (7 / 2) / (Sqrt[-1 + x] (1 - 6 x
+ 15 x ^ 2 - 20 x ^ 3 + 15 x ^ 4 - 6 x ^ 5 + x ^ 6)), Abs[1 + x] > 2}}, -143 (1 + x) ^ (9 / 2) / (-133056 (1 +
 x) Sqrt[1 - x] - 110880 (1 + x) ^ 3 Sqrt[1 - x] - 8316 (1 + x) ^ 5 Sqrt[1 - x] + 693 (1 + x) ^ 6 Sqrt[1 - x]
+ 41580 (1 + x) ^ 4 Sqrt[1 - x] + 44352 Sqrt[1 - x] + 166320 (1 + x) ^ 2 Sqrt[1 - x]) - 2 (1 + x) ^ (13 / 2) /
 (-133056 (1 + x) Sqrt[1 - x] - 110880 (1 + x) ^ 3 Sqrt[1 - x] - 8316 (1 + x) ^ 5 Sqrt[1 - x] + 693 (1 + x) ^
6 Sqrt[1 - x] + 41580 (1 + x) ^ 4 Sqrt[1 - x] + 44352 Sqrt[1 - x] + 166320 (1 + x) ^ 2 Sqrt[1 - x]) + 26 (1 +
x) ^ (11 / 2) / (-133056 (1 + x) Sqrt[1 - x] - 110880 (1 + x) ^ 3 Sqrt[1 - x] - 8316 (1 + x) ^ 5 Sqrt[1 - x] +
 693 (1 + x) ^ 6 Sqrt[1 - x] + 41580 (1 + x) ^ 4 Sqrt[1 - x] + 44352 Sqrt[1 - x] + 166320 (1 + x) ^ 2 Sqrt[1 -
 x]) + 198 (1 + x) ^ (7 / 2) / (-133056 (1 + x) Sqrt[1 - x] - 110880 (1 + x) ^ 3 Sqrt[1 - x] - 8316 (1 + x) ^
5 Sqrt[1 - x] + 693 (1 + x) ^ 6 Sqrt[1 - x] + 41580 (1 + x) ^ 4 Sqrt[1 - x] + 44352 Sqrt[1 - x] + 166320 (1 +
x) ^ 2 Sqrt[1 - x])]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(113\) vs. \(2(43)=86\).
time = 0.14, size = 114, normalized size = 1.87

method result size
gosper \(\frac {\left (1+x \right )^{\frac {7}{2}} \left (2 x^{2}-18 x +79\right )}{693 \left (1-x \right )^{\frac {11}{2}}}\) \(25\)
risch \(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{6}-10 x^{5}+19 x^{4}+216 x^{3}+404 x^{2}+298 x +79\right )}{693 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{5} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) \(71\)
default \(\frac {\left (1+x \right )^{\frac {5}{2}}}{3 \left (1-x \right )^{\frac {11}{2}}}-\frac {5 \left (1+x \right )^{\frac {3}{2}}}{12 \left (1-x \right )^{\frac {11}{2}}}+\frac {5 \sqrt {1+x}}{22 \left (1-x \right )^{\frac {11}{2}}}-\frac {5 \sqrt {1+x}}{396 \left (1-x \right )^{\frac {9}{2}}}-\frac {5 \sqrt {1+x}}{693 \left (1-x \right )^{\frac {7}{2}}}-\frac {\sqrt {1+x}}{231 \left (1-x \right )^{\frac {5}{2}}}-\frac {2 \sqrt {1+x}}{693 \left (1-x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1+x}}{693 \sqrt {1-x}}\) \(114\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(5/2)/(1-x)^(13/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(1+x)^(5/2)/(1-x)^(11/2)-5/12*(1+x)^(3/2)/(1-x)^(11/2)+5/22*(1+x)^(1/2)/(1-x)^(11/2)-5/396*(1+x)^(1/2)/(1-
x)^(9/2)-5/693*(1+x)^(1/2)/(1-x)^(7/2)-1/231*(1+x)^(1/2)/(1-x)^(5/2)-2/693*(1+x)^(1/2)/(1-x)^(3/2)-2/693*(1+x)
^(1/2)/(1-x)^(1/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (43) = 86\).
time = 0.26, size = 269, normalized size = 4.41 \begin {gather*} \frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{3 \, {\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 {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{12 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} + \frac {5 \, \sqrt {-x^{2} + 1}}{22 \, {\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}}{396 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {5 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{231 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(13/2),x, algorithm="maxima")

[Out]

1/3*(-x^2 + 1)^(5/2)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) + 5/12*(-x^2 + 1)^(3
/2)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x - 1) + 5/22*sqrt(-x^2 + 1)/(x^6 - 6*x^5 + 15*x^4 -
20*x^3 + 15*x^2 - 6*x + 1) + 5/396*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) - 5/693*sqrt(-x^2
+ 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 1/231*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 2/693*sqrt(-x^2 + 1)/(x^
2 - 2*x + 1) + 2/693*sqrt(-x^2 + 1)/(x - 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (43) = 86\).
time = 0.30, size = 100, normalized size = 1.64 \begin {gather*} \frac {79 \, x^{6} - 474 \, x^{5} + 1185 \, x^{4} - 1580 \, x^{3} + 1185 \, x^{2} + {\left (2 \, x^{5} - 12 \, x^{4} + 31 \, x^{3} + 185 \, x^{2} + 219 \, x + 79\right )} \sqrt {x + 1} \sqrt {-x + 1} - 474 \, x + 79}{693 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(13/2),x, algorithm="fricas")

[Out]

1/693*(79*x^6 - 474*x^5 + 1185*x^4 - 1580*x^3 + 1185*x^2 + (2*x^5 - 12*x^4 + 31*x^3 + 185*x^2 + 219*x + 79)*sq
rt(x + 1)*sqrt(-x + 1) - 474*x + 79)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1)

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Sympy [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(5/2)/(1-x)**(13/2),x)

[Out]

Timed out

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Giac [A]
time = 0.02, size = 105, normalized size = 1.72 \begin {gather*} \frac {2 \left (\left (\frac {1}{693} \sqrt {x+1} \sqrt {x+1}-\frac 1{63}\right ) \sqrt {x+1} \sqrt {x+1}+\frac 1{14}\right ) \sqrt {x+1} \sqrt {x+1} \sqrt {x+1} \sqrt {x+1} \sqrt {x+1} \sqrt {x+1} \sqrt {x+1} \sqrt {-x+1}}{\left (-x+1\right )^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(13/2),x)

[Out]

1/693*(2*(x + 1)*(x - 10) + 99)*(x + 1)^(7/2)*sqrt(-x + 1)/(x - 1)^6

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Mupad [B]
time = 0.31, size = 94, normalized size = 1.54 \begin {gather*} \frac {\sqrt {1-x}\,\left (\frac {73\,x\,\sqrt {x+1}}{231}+\frac {79\,\sqrt {x+1}}{693}+\frac {185\,x^2\,\sqrt {x+1}}{693}+\frac {31\,x^3\,\sqrt {x+1}}{693}-\frac {4\,x^4\,\sqrt {x+1}}{231}+\frac {2\,x^5\,\sqrt {x+1}}{693}\right )}{x^6-6\,x^5+15\,x^4-20\,x^3+15\,x^2-6\,x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(5/2)/(1 - x)^(13/2),x)

[Out]

((1 - x)^(1/2)*((73*x*(x + 1)^(1/2))/231 + (79*(x + 1)^(1/2))/693 + (185*x^2*(x + 1)^(1/2))/693 + (31*x^3*(x +
 1)^(1/2))/693 - (4*x^4*(x + 1)^(1/2))/231 + (2*x^5*(x + 1)^(1/2))/693))/(15*x^2 - 6*x - 20*x^3 + 15*x^4 - 6*x
^5 + x^6 + 1)

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