3.11.27 \(\int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx\)

Optimal. Leaf size=63 \[ \frac {2 (x+1)^{5/2}}{3 (1-x)^{3/2}}-\frac {10 (x+1)^{3/2}}{3 \sqrt {1-x}}-5 \sqrt {1-x} \sqrt {x+1}+5 \sin ^{-1}(x) \]

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Rubi [A]  time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {47, 50, 41, 216} \begin {gather*} \frac {2 (x+1)^{5/2}}{3 (1-x)^{3/2}}-\frac {10 (x+1)^{3/2}}{3 \sqrt {1-x}}-5 \sqrt {1-x} \sqrt {x+1}+5 \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(5/2)/(1 - x)^(5/2),x]

[Out]

-5*Sqrt[1 - x]*Sqrt[1 + x] - (10*(1 + x)^(3/2))/(3*Sqrt[1 - x]) + (2*(1 + x)^(5/2))/(3*(1 - x)^(3/2)) + 5*ArcS
in[x]

Rule 41

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]))

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx &=\frac {2 (1+x)^{5/2}}{3 (1-x)^{3/2}}-\frac {5}{3} \int \frac {(1+x)^{3/2}}{(1-x)^{3/2}} \, dx\\ &=-\frac {10 (1+x)^{3/2}}{3 \sqrt {1-x}}+\frac {2 (1+x)^{5/2}}{3 (1-x)^{3/2}}+5 \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=-5 \sqrt {1-x} \sqrt {1+x}-\frac {10 (1+x)^{3/2}}{3 \sqrt {1-x}}+\frac {2 (1+x)^{5/2}}{3 (1-x)^{3/2}}+5 \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-5 \sqrt {1-x} \sqrt {1+x}-\frac {10 (1+x)^{3/2}}{3 \sqrt {1-x}}+\frac {2 (1+x)^{5/2}}{3 (1-x)^{3/2}}+5 \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-5 \sqrt {1-x} \sqrt {1+x}-\frac {10 (1+x)^{3/2}}{3 \sqrt {1-x}}+\frac {2 (1+x)^{5/2}}{3 (1-x)^{3/2}}+5 \sin ^{-1}(x)\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 37, normalized size = 0.59 \begin {gather*} \frac {8 \sqrt {2} \, _2F_1\left (-\frac {5}{2},-\frac {3}{2};-\frac {1}{2};\frac {1-x}{2}\right )}{3 (1-x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*Sqrt[2]*Hypergeometric2F1[-5/2, -3/2, -1/2, (1 - x)/2])/(3*(1 - x)^(3/2))

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IntegrateAlgebraic [C]  time = 0.18, size = 73, normalized size = 1.16 \begin {gather*} \frac {\sqrt {1-x} \left (-3 (x+1)^{5/2}+40 (x+1)^{3/2}-60 \sqrt {x+1}\right )}{3 (x-1)^2}+10 i \log \left (\sqrt {1-x}-i \sqrt {x+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - x]*(-60*Sqrt[1 + x] + 40*(1 + x)^(3/2) - 3*(1 + x)^(5/2)))/(3*(-1 + x)^2) + (10*I)*Log[Sqrt[1 - x] -
I*Sqrt[1 + x]]

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fricas [A]  time = 0.71, size = 75, normalized size = 1.19 \begin {gather*} -\frac {23 \, x^{2} + {\left (3 \, x^{2} - 34 \, x + 23\right )} \sqrt {x + 1} \sqrt {-x + 1} + 30 \, {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - 46 \, x + 23}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(23*x^2 + (3*x^2 - 34*x + 23)*sqrt(x + 1)*sqrt(-x + 1) + 30*(x^2 - 2*x + 1)*arctan((sqrt(x + 1)*sqrt(-x +
1) - 1)/x) - 46*x + 23)/(x^2 - 2*x + 1)

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giac [A]  time = 0.98, size = 44, normalized size = 0.70 \begin {gather*} -\frac {{\left ({\left (3 \, x - 37\right )} {\left (x + 1\right )} + 60\right )} \sqrt {x + 1} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} + 10 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*((3*x - 37)*(x + 1) + 60)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^2 + 10*arcsin(1/2*sqrt(2)*sqrt(x + 1))

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maple [A]  time = 0.02, size = 84, normalized size = 1.33 \begin {gather*} \frac {5 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{\sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (3 x^{3}-31 x^{2}-11 x +23\right ) \sqrt {\left (x +1\right ) \left (-x +1\right )}}{3 \left (x -1\right ) \sqrt {-\left (x +1\right ) \left (x -1\right )}\, \sqrt {-x +1}\, \sqrt {x +1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)^(5/2)/(-x+1)^(5/2),x)

[Out]

1/3*(3*x^3-31*x^2-11*x+23)/(x-1)/(-(x+1)*(x-1))^(1/2)*((x+1)*(-x+1))^(1/2)/(-x+1)^(1/2)/(x+1)^(1/2)+5*((x+1)*(
-x+1))^(1/2)/(x+1)^(1/2)/(-x+1)^(1/2)*arcsin(x)

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maxima [B]  time = 2.97, size = 99, normalized size = 1.57 \begin {gather*} -\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1} - \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {10 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {35 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x - 1\right )}} + 5 \, \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-x^2 + 1)^(5/2)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) - 5/3*(-x^2 + 1)^(3/2)/(x^3 - 3*x^2 + 3*x - 1) + 10/3*sqrt(-
x^2 + 1)/(x^2 - 2*x + 1) + 35/3*sqrt(-x^2 + 1)/(x - 1) + 5*arcsin(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (x+1\right )}^{5/2}}{{\left (1-x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(5/2)/(1 - x)^(5/2),x)

[Out]

int((x + 1)^(5/2)/(1 - x)^(5/2), x)

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sympy [B]  time = 7.47, size = 576, normalized size = 9.14 \begin {gather*} \begin {cases} \frac {30 i \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} - \frac {15 \pi \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} - \frac {60 i \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} + \frac {30 \pi \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} + \frac {3 i \left (x + 1\right )^{15}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} - \frac {40 i \left (x + 1\right )^{14}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} + \frac {60 i \left (x + 1\right )^{13}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\frac {30 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} - \frac {60 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} + \frac {3 \left (x + 1\right )^{15}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} - \frac {40 \left (x + 1\right )^{14}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} + \frac {60 \left (x + 1\right )^{13}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((30*I*sqrt(x - 1)*(x + 1)**(27/2)*acosh(sqrt(2)*sqrt(x + 1)/2)/(-3*sqrt(x - 1)*(x + 1)**(27/2) + 6*s
qrt(x - 1)*(x + 1)**(25/2)) - 15*pi*sqrt(x - 1)*(x + 1)**(27/2)/(-3*sqrt(x - 1)*(x + 1)**(27/2) + 6*sqrt(x - 1
)*(x + 1)**(25/2)) - 60*I*sqrt(x - 1)*(x + 1)**(25/2)*acosh(sqrt(2)*sqrt(x + 1)/2)/(-3*sqrt(x - 1)*(x + 1)**(2
7/2) + 6*sqrt(x - 1)*(x + 1)**(25/2)) + 30*pi*sqrt(x - 1)*(x + 1)**(25/2)/(-3*sqrt(x - 1)*(x + 1)**(27/2) + 6*
sqrt(x - 1)*(x + 1)**(25/2)) + 3*I*(x + 1)**15/(-3*sqrt(x - 1)*(x + 1)**(27/2) + 6*sqrt(x - 1)*(x + 1)**(25/2)
) - 40*I*(x + 1)**14/(-3*sqrt(x - 1)*(x + 1)**(27/2) + 6*sqrt(x - 1)*(x + 1)**(25/2)) + 60*I*(x + 1)**13/(-3*s
qrt(x - 1)*(x + 1)**(27/2) + 6*sqrt(x - 1)*(x + 1)**(25/2)), Abs(x + 1)/2 > 1), (30*sqrt(1 - x)*(x + 1)**(27/2
)*asin(sqrt(2)*sqrt(x + 1)/2)/(3*sqrt(1 - x)*(x + 1)**(27/2) - 6*sqrt(1 - x)*(x + 1)**(25/2)) - 60*sqrt(1 - x)
*(x + 1)**(25/2)*asin(sqrt(2)*sqrt(x + 1)/2)/(3*sqrt(1 - x)*(x + 1)**(27/2) - 6*sqrt(1 - x)*(x + 1)**(25/2)) +
3*(x + 1)**15/(3*sqrt(1 - x)*(x + 1)**(27/2) - 6*sqrt(1 - x)*(x + 1)**(25/2)) - 40*(x + 1)**14/(3*sqrt(1 - x)
*(x + 1)**(27/2) - 6*sqrt(1 - x)*(x + 1)**(25/2)) + 60*(x + 1)**13/(3*sqrt(1 - x)*(x + 1)**(27/2) - 6*sqrt(1 -
x)*(x + 1)**(25/2)), True))

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