3.12.24 \(\int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx\) [1124]

Optimal. Leaf size=82 \[ \frac {1}{7 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{3/2} \sqrt {1+x}}+\frac {8 x}{35 \sqrt {1-x} \sqrt {1+x}} \]

[Out]

1/7/(1-x)^(7/2)/(1+x)^(1/2)+4/35/(1-x)^(5/2)/(1+x)^(1/2)+4/35/(1-x)^(3/2)/(1+x)^(1/2)+8/35*x/(1-x)^(1/2)/(1+x)
^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 39} \begin {gather*} \frac {8 x}{35 \sqrt {1-x} \sqrt {x+1}}+\frac {4}{35 (1-x)^{3/2} \sqrt {x+1}}+\frac {4}{35 (1-x)^{5/2} \sqrt {x+1}}+\frac {1}{7 (1-x)^{7/2} \sqrt {x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(9/2)*(1 + x)^(3/2)),x]

[Out]

1/(7*(1 - x)^(7/2)*Sqrt[1 + x]) + 4/(35*(1 - x)^(5/2)*Sqrt[1 + x]) + 4/(35*(1 - x)^(3/2)*Sqrt[1 + x]) + (8*x)/
(35*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 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}{(1-x)^{9/2} (1+x)^{3/2}} \, dx &=\frac {1}{7 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{7} \int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{7 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{5/2} \sqrt {1+x}}+\frac {12}{35} \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{7 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{3/2} \sqrt {1+x}}+\frac {8}{35} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{7 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{3/2} \sqrt {1+x}}+\frac {8 x}{35 \sqrt {1-x} \sqrt {1+x}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 40, normalized size = 0.49 \begin {gather*} \frac {-13+4 x+20 x^2-24 x^3+8 x^4}{35 (-1+x)^3 \sqrt {1-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(9/2)*(1 + x)^(3/2)),x]

[Out]

(-13 + 4*x + 20*x^2 - 24*x^3 + 8*x^4)/(35*(-1 + x)^3*Sqrt[1 - x^2])

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 34.25, size = 290, normalized size = 3.54 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\left (13-4 x-20 x^2+24 x^3-8 x^4\right ) \sqrt {\frac {1-x}{1+x}}}{35 \left (1-4 x+6 x^2-4 x^3+x^4\right )},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},\frac {-140 I \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}}{-560-1120 x-280 \left (1+x\right )^3+35 \left (1+x\right )^4+840 \left (1+x\right )^2}-\frac {35 I \sqrt {1-\frac {2}{1+x}}}{-560-1120 x-280 \left (1+x\right )^3+35 \left (1+x\right )^4+840 \left (1+x\right )^2}-\frac {8 I \left (1+x\right )^4 \sqrt {1-\frac {2}{1+x}}}{-560-1120 x-280 \left (1+x\right )^3+35 \left (1+x\right )^4+840 \left (1+x\right )^2}+\frac {I 56 \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}}{-560-1120 x-280 \left (1+x\right )^3+35 \left (1+x\right )^4+840 \left (1+x\right )^2}+\frac {I 140 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}{-560-1120 x-280 \left (1+x\right )^3+35 \left (1+x\right )^4+840 \left (1+x\right )^2}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/((1 - x)^(9/2)*(1 + x)^(3/2)),x]')

[Out]

Piecewise[{{(13 - 4 x - 20 x ^ 2 + 24 x ^ 3 - 8 x ^ 4) Sqrt[(1 - x) / (1 + x)] / (35 (1 - 4 x + 6 x ^ 2 - 4 x
^ 3 + x ^ 4)), 1 / Abs[1 + x] > 1 / 2}}, -140 I (1 + x) ^ 2 Sqrt[1 - 2 / (1 + x)] / (-560 - 1120 x - 280 (1 +
x) ^ 3 + 35 (1 + x) ^ 4 + 840 (1 + x) ^ 2) - 35 I Sqrt[1 - 2 / (1 + x)] / (-560 - 1120 x - 280 (1 + x) ^ 3 + 3
5 (1 + x) ^ 4 + 840 (1 + x) ^ 2) - 8 I (1 + x) ^ 4 Sqrt[1 - 2 / (1 + x)] / (-560 - 1120 x - 280 (1 + x) ^ 3 +
35 (1 + x) ^ 4 + 840 (1 + x) ^ 2) + I 56 (1 + x) ^ 3 Sqrt[1 - 2 / (1 + x)] / (-560 - 1120 x - 280 (1 + x) ^ 3
+ 35 (1 + x) ^ 4 + 840 (1 + x) ^ 2) + I 140 (1 + x) Sqrt[1 - 2 / (1 + x)] / (-560 - 1120 x - 280 (1 + x) ^ 3 +
35 (1 + x) ^ 4 + 840 (1 + x) ^ 2)]

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Maple [A]
time = 0.14, size = 72, normalized size = 0.88

method	result	size
gosper	\(-\frac {8 x^{4}-24 x^{3}+20 x^{2}+4 x -13}{35 \sqrt {1+x}\, \left (1-x \right )^{\frac {7}{2}}}\)	\(35\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{4}-24 x^{3}+20 x^{2}+4 x -13\right )}{35 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{3} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(61\)
default	\(\frac {1}{7 \left (1-x \right )^{\frac {7}{2}} \sqrt {1+x}}+\frac {4}{35 \left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}+\frac {4}{35 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}+\frac {8}{35 \sqrt {1-x}\, \sqrt {1+x}}-\frac {8 \sqrt {1-x}}{35 \sqrt {1+x}}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(9/2)/(1+x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/7/(1-x)^(7/2)/(1+x)^(1/2)+4/35/(1-x)^(5/2)/(1+x)^(1/2)+4/35/(1-x)^(3/2)/(1+x)^(1/2)+8/35/(1-x)^(1/2)/(1+x)^(
1/2)-8/35*(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. 134 vs. \(2 (58) = 116\).
time = 0.27, size = 134, normalized size = 1.63 \begin {gather*} \frac {8 \, x}{35 \, \sqrt {-x^{2} + 1}} - \frac {1}{7 \, {\left (\sqrt {-x^{2} + 1} x^{3} - 3 \, \sqrt {-x^{2} + 1} x^{2} + 3 \, \sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} + \frac {4}{35 \, {\left (\sqrt {-x^{2} + 1} x^{2} - 2 \, \sqrt {-x^{2} + 1} x + \sqrt {-x^{2} + 1}\right )}} - \frac {4}{35 \, {\left (\sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(9/2)/(1+x)^(3/2),x, algorithm="maxima")

[Out]

8/35*x/sqrt(-x^2 + 1) - 1/7/(sqrt(-x^2 + 1)*x^3 - 3*sqrt(-x^2 + 1)*x^2 + 3*sqrt(-x^2 + 1)*x - sqrt(-x^2 + 1))
+ 4/35/(sqrt(-x^2 + 1)*x^2 - 2*sqrt(-x^2 + 1)*x + sqrt(-x^2 + 1)) - 4/35/(sqrt(-x^2 + 1)*x - sqrt(-x^2 + 1))

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Fricas [A]
time = 0.29, size = 86, normalized size = 1.05 \begin {gather*} \frac {13 \, x^{5} - 39 \, x^{4} + 26 \, x^{3} + 26 \, x^{2} - {\left (8 \, x^{4} - 24 \, x^{3} + 20 \, x^{2} + 4 \, x - 13\right )} \sqrt {x + 1} \sqrt {-x + 1} - 39 \, x + 13}{35 \, {\left (x^{5} - 3 \, x^{4} + 2 \, x^{3} + 2 \, x^{2} - 3 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(9/2)/(1+x)^(3/2),x, algorithm="fricas")

[Out]

1/35*(13*x^5 - 39*x^4 + 26*x^3 + 26*x^2 - (8*x^4 - 24*x^3 + 20*x^2 + 4*x - 13)*sqrt(x + 1)*sqrt(-x + 1) - 39*x
+ 13)/(x^5 - 3*x^4 + 2*x^3 + 2*x^2 - 3*x + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 42.08, size = 425, normalized size = 5.18 \begin {gather*} \begin {cases} - \frac {8 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {56 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {35 \sqrt {-1 + \frac {2}{x + 1}}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {8 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {56 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {35 i \sqrt {1 - \frac {2}{x + 1}}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)**(9/2)/(1+x)**(3/2),x)

[Out]

Piecewise((-8*sqrt(-1 + 2/(x + 1))*(x + 1)**4/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560
) + 56*sqrt(-1 + 2/(x + 1))*(x + 1)**3/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) - 140
*sqrt(-1 + 2/(x + 1))*(x + 1)**2/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) + 140*sqrt(
-1 + 2/(x + 1))*(x + 1)/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) - 35*sqrt(-1 + 2/(x
+ 1))/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560), 1/Abs(x + 1) > 1/2), (-8*I*sqrt(1 - 2
/(x + 1))*(x + 1)**4/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) + 56*I*sqrt(1 - 2/(x +
1))*(x + 1)**3/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) - 140*I*sqrt(1 - 2/(x + 1))*(
x + 1)**2/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) + 140*I*sqrt(1 - 2/(x + 1))*(x + 1
)/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) - 35*I*sqrt(1 - 2/(x + 1))/(-1120*x + 35*(
x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (58) = 116\).
time = 0.02, size = 315, normalized size = 3.84 \begin {gather*} -2 \left (\frac {-\frac {1}{7}\cdot 4722366482869645213696 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{7}-\frac {1}{5}\cdot 51946031311566097350656 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{5}-89724963174523259060224 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{3}+\frac {441541266148311827480576 \left (-2 \sqrt {x+1}+2 \sqrt {2}\right )}{\sqrt {-x+1}}}{19342813113834066795298816}+\frac {6545 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{6}+665 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{4}+77 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{2}+5}{143360 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{7}}+\frac {\sqrt {-x+1} \sqrt {x+1}}{32 \left (x+1\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(9/2)/(1+x)^(3/2),x)

[Out]

-1/14336*(sqrt(2) - sqrt(x + 1))^7/(-x + 1)^(7/2) - 11/10240*(sqrt(2) - sqrt(x + 1))^5/(-x + 1)^(5/2) - 19/204
8*(sqrt(2) - sqrt(x + 1))^3/(-x + 1)^(3/2) - 187/2048*(sqrt(2) - sqrt(x + 1))/sqrt(-x + 1) - 1/16*sqrt(-x + 1)
/sqrt(x + 1) - 1/71680*(6545*(sqrt(2) - sqrt(x + 1))^6/(x - 1)^3 - 665*(sqrt(2) - sqrt(x + 1))^4/(x - 1)^2 + 7
7*(sqrt(2) - sqrt(x + 1))^2/(x - 1) - 5)*(-x + 1)^(7/2)/(sqrt(2) - sqrt(x + 1))^7

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Mupad [B]
time = 0.36, size = 68, normalized size = 0.83 \begin {gather*} -\frac {4\,x\,\sqrt {1-x}-13\,\sqrt {1-x}+20\,x^2\,\sqrt {1-x}-24\,x^3\,\sqrt {1-x}+8\,x^4\,\sqrt {1-x}}{35\,{\left (x-1\right )}^4\,\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x)^(9/2)*(x + 1)^(3/2)),x)

[Out]

-(4*x*(1 - x)^(1/2) - 13*(1 - x)^(1/2) + 20*x^2*(1 - x)^(1/2) - 24*x^3*(1 - x)^(1/2) + 8*x^4*(1 - x)^(1/2))/(3
5*(x - 1)^4*(x + 1)^(1/2))

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