3.11.96 \(\int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx\) [1096]

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

[Out]

2/3*(1+x)^(5/2)/(1-x)^(3/2)+5*arcsin(x)-10/3*(1+x)^(3/2)/(1-x)^(1/2)-5*(1-x)^(1/2)*(1+x)^(1/2)

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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 = {49, 52, 41, 222} \begin {gather*} 5 \text {ArcSin}(x)+\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} \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 49

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 52

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 222

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 [A]
time = 0.14, size = 51, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {1-x^2} \left (23-34 x+3 x^2\right )}{3 (-1+x)^2}+10 \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(Sqrt[1 - x^2]*(23 - 34*x + 3*x^2))/(-1 + x)^2 + 10*ArcTan[Sqrt[1 + x]/Sqrt[1 - x]]

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Maple [A]
time = 0.16, size = 84, normalized size = 1.33

method	result	size
risch	\(\frac {\left (3 x^{3}-31 x^{2}-11 x +23\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{3 \left (-1+x \right ) \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (47) = 94\).
time = 0.50, 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 \left (x\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="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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Fricas [A]
time = 0.59, 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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Sympy [C] Result contains complex when optimal does not.
time = 4.53, size = 575, normalized size = 9.13 \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}\: \left |{x + 1}\right | > 2 \\\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)**(27/
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*sqr
t(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*sqrt(x
- 1)*(x + 1)**(27/2) - 6*sqrt(x - 1)*(x + 1)**(25/2)), Abs(x + 1) > 2), (30*sqrt(1 - x)*(x + 1)**(27/2)*asin(s
qrt(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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Giac [A]
time = 1.32, 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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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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