3.1128 \(\int \frac{(1-x)^{5/2}}{(1+x)^{5/2}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.010729, antiderivative size = 63, normalized size of antiderivative = 1., 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} \[ -\frac{2 (1-x)^{5/2}}{3 (x+1)^{3/2}}+\frac{10 (1-x)^{3/2}}{3 \sqrt{x+1}}+5 \sqrt{x+1} \sqrt{1-x}+5 \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(5/2)/(1 + x)^(5/2),x]

[Out]

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

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

Mathematica [C]  time = 0.0102119, size = 37, normalized size = 0.59 \[ -\frac{(1-x)^{7/2} \, _2F_1\left (\frac{5}{2},\frac{7}{2};\frac{9}{2};\frac{1-x}{2}\right )}{14 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - x)^(7/2)*Hypergeometric2F1[5/2, 7/2, 9/2, (1 - x)/2])/(14*Sqrt[2])

________________________________________________________________________________________

Maple [A]  time = 0.019, size = 79, normalized size = 1.3 \begin{align*} -{\frac{3\,{x}^{3}+31\,{x}^{2}-11\,x-23}{3}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) } \left ( 1+x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}}+5\,{\frac{\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }\arcsin \left ( x \right ) }{\sqrt{1-x}\sqrt{1+x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.50109, size = 132, normalized size = 2.1 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.85971, size = 204, normalized size = 3.24 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Sympy [C]  time = 12.9692, size = 160, normalized size = 2.54 \begin{align*} \begin{cases} \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) + \frac{28 \sqrt{-1 + \frac{2}{x + 1}}}{3} - \frac{8 \sqrt{-1 + \frac{2}{x + 1}}}{3 \left (x + 1\right )} + 5 i \log{\left (\frac{1}{x + 1} \right )} + 5 i \log{\left (x + 1 \right )} + 10 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) + \frac{28 i \sqrt{1 - \frac{2}{x + 1}}}{3} - \frac{8 i \sqrt{1 - \frac{2}{x + 1}}}{3 \left (x + 1\right )} + 5 i \log{\left (\frac{1}{x + 1} \right )} - 10 i \log{\left (\sqrt{1 - \frac{2}{x + 1}} + 1 \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sqrt(-1 + 2/(x + 1))*(x + 1) + 28*sqrt(-1 + 2/(x + 1))/3 - 8*sqrt(-1 + 2/(x + 1))/(3*(x + 1)) + 5*I
*log(1/(x + 1)) + 5*I*log(x + 1) + 10*asin(sqrt(2)*sqrt(x + 1)/2), 2/Abs(x + 1) > 1), (I*sqrt(1 - 2/(x + 1))*(
x + 1) + 28*I*sqrt(1 - 2/(x + 1))/3 - 8*I*sqrt(1 - 2/(x + 1))/(3*(x + 1)) + 5*I*log(1/(x + 1)) - 10*I*log(sqrt
(1 - 2/(x + 1)) + 1), True))

________________________________________________________________________________________

Giac [B]  time = 1.11284, size = 155, normalized size = 2.46 \begin{align*} \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{6 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \sqrt{x + 1} \sqrt{-x + 1} - \frac{9 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{2 \, \sqrt{x + 1}} + \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{27 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{6 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} + 10 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + sqrt(x + 1)*sqrt(-x + 1) - 9/2*(sqrt(2) - sqrt(-x + 1))/sqrt(x
+ 1) + 1/6*(x + 1)^(3/2)*(27*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) - 1)/(sqrt(2) - sqrt(-x + 1))^3 + 10*arcsin(1/
2*sqrt(2)*sqrt(x + 1))