3.1097 \(\int \frac{(1+x)^{5/2}}{(1-x)^{7/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0073775, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {47, 41, 216} \[ \frac{2 (x+1)^{5/2}}{5 (1-x)^{5/2}}-\frac{2 (x+1)^{3/2}}{3 (1-x)^{3/2}}+\frac{2 \sqrt{x+1}}{\sqrt{1-x}}-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(5/2)/(1 - x)^(7/2),x]

[Out]

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

Mathematica [C]  time = 0.0083435, size = 37, normalized size = 0.59 \[ \frac{8 \sqrt{2} \, _2F_1\left (-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};\frac{1-x}{2}\right )}{5 (1-x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(5/2)/(1 - x)^(7/2),x]

[Out]

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

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Maple [A]  time = 0.017, size = 84, normalized size = 1.3 \begin{align*}{\frac{46\,{x}^{3}-2\,{x}^{2}-22\,x+26}{15\, \left ( -1+x \right ) ^{2}}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}}-{\arcsin \left ( x \right ) \sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(5/2)/(1-x)^(7/2),x)

[Out]

2/15*(23*x^3-x^2-11*x+13)/(-1+x)^2/(-(1+x)*(-1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)/(1+x)^(1/2)-((1+x)*(1
-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [B]  time = 1.55084, size = 216, normalized size = 3.43 \begin{align*} -\frac{{\left (-x^{2} + 1\right )}^{\frac{5}{2}}}{5 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1} + \frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac{6 \, \sqrt{-x^{2} + 1}}{5 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{7 \, \sqrt{-x^{2} + 1}}{15 \,{\left (x^{2} - 2 \, x + 1\right )}} - \frac{38 \, \sqrt{-x^{2} + 1}}{15 \,{\left (x - 1\right )}} - \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(7/2),x, algorithm="maxima")

[Out]

-1/5*(-x^2 + 1)^(5/2)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) + (-x^2 + 1)^(3/2)/(x^4 - 4*x^3 + 6*x^2 - 4*x
+ 1) + 1/3*(-x^2 + 1)^(3/2)/(x^3 - 3*x^2 + 3*x - 1) + 6/5*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 7/15*sqrt(-
x^2 + 1)/(x^2 - 2*x + 1) - 38/15*sqrt(-x^2 + 1)/(x - 1) - arcsin(x)

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Fricas [A]  time = 1.60495, size = 240, normalized size = 3.81 \begin{align*} \frac{2 \,{\left (13 \, x^{3} - 39 \, x^{2} -{\left (23 \, x^{2} - 24 \, x + 13\right )} \sqrt{x + 1} \sqrt{-x + 1} + 15 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 39 \, x - 13\right )}}{15 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(7/2),x, algorithm="fricas")

[Out]

2/15*(13*x^3 - 39*x^2 - (23*x^2 - 24*x + 13)*sqrt(x + 1)*sqrt(-x + 1) + 15*(x^3 - 3*x^2 + 3*x - 1)*arctan((sqr
t(x + 1)*sqrt(-x + 1) - 1)/x) + 39*x - 13)/(x^3 - 3*x^2 + 3*x - 1)

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Sympy [B]  time = 37.5888, size = 1608, normalized size = 25.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(5/2)/(1-x)**(7/2),x)

[Out]

Piecewise((30*I*sqrt(x - 1)*(x + 1)**(35/2)*acosh(sqrt(2)*sqrt(x + 1)/2)/(15*sqrt(x - 1)*(x + 1)**(35/2) - 90*
sqrt(x - 1)*(x + 1)**(33/2) + 180*sqrt(x - 1)*(x + 1)**(31/2) - 120*sqrt(x - 1)*(x + 1)**(29/2)) - 15*pi*sqrt(
x - 1)*(x + 1)**(35/2)/(15*sqrt(x - 1)*(x + 1)**(35/2) - 90*sqrt(x - 1)*(x + 1)**(33/2) + 180*sqrt(x - 1)*(x +
1)**(31/2) - 120*sqrt(x - 1)*(x + 1)**(29/2)) - 180*I*sqrt(x - 1)*(x + 1)**(33/2)*acosh(sqrt(2)*sqrt(x + 1)/2
)/(15*sqrt(x - 1)*(x + 1)**(35/2) - 90*sqrt(x - 1)*(x + 1)**(33/2) + 180*sqrt(x - 1)*(x + 1)**(31/2) - 120*sqr
t(x - 1)*(x + 1)**(29/2)) + 90*pi*sqrt(x - 1)*(x + 1)**(33/2)/(15*sqrt(x - 1)*(x + 1)**(35/2) - 90*sqrt(x - 1)
*(x + 1)**(33/2) + 180*sqrt(x - 1)*(x + 1)**(31/2) - 120*sqrt(x - 1)*(x + 1)**(29/2)) + 360*I*sqrt(x - 1)*(x +
1)**(31/2)*acosh(sqrt(2)*sqrt(x + 1)/2)/(15*sqrt(x - 1)*(x + 1)**(35/2) - 90*sqrt(x - 1)*(x + 1)**(33/2) + 18
0*sqrt(x - 1)*(x + 1)**(31/2) - 120*sqrt(x - 1)*(x + 1)**(29/2)) - 180*pi*sqrt(x - 1)*(x + 1)**(31/2)/(15*sqrt
(x - 1)*(x + 1)**(35/2) - 90*sqrt(x - 1)*(x + 1)**(33/2) + 180*sqrt(x - 1)*(x + 1)**(31/2) - 120*sqrt(x - 1)*(
x + 1)**(29/2)) - 240*I*sqrt(x - 1)*(x + 1)**(29/2)*acosh(sqrt(2)*sqrt(x + 1)/2)/(15*sqrt(x - 1)*(x + 1)**(35/
2) - 90*sqrt(x - 1)*(x + 1)**(33/2) + 180*sqrt(x - 1)*(x + 1)**(31/2) - 120*sqrt(x - 1)*(x + 1)**(29/2)) + 120
*pi*sqrt(x - 1)*(x + 1)**(29/2)/(15*sqrt(x - 1)*(x + 1)**(35/2) - 90*sqrt(x - 1)*(x + 1)**(33/2) + 180*sqrt(x
- 1)*(x + 1)**(31/2) - 120*sqrt(x - 1)*(x + 1)**(29/2)) - 46*I*(x + 1)**18/(15*sqrt(x - 1)*(x + 1)**(35/2) - 9
0*sqrt(x - 1)*(x + 1)**(33/2) + 180*sqrt(x - 1)*(x + 1)**(31/2) - 120*sqrt(x - 1)*(x + 1)**(29/2)) + 232*I*(x
+ 1)**17/(15*sqrt(x - 1)*(x + 1)**(35/2) - 90*sqrt(x - 1)*(x + 1)**(33/2) + 180*sqrt(x - 1)*(x + 1)**(31/2) -
120*sqrt(x - 1)*(x + 1)**(29/2)) - 400*I*(x + 1)**16/(15*sqrt(x - 1)*(x + 1)**(35/2) - 90*sqrt(x - 1)*(x + 1)*
*(33/2) + 180*sqrt(x - 1)*(x + 1)**(31/2) - 120*sqrt(x - 1)*(x + 1)**(29/2)) + 240*I*(x + 1)**15/(15*sqrt(x -
1)*(x + 1)**(35/2) - 90*sqrt(x - 1)*(x + 1)**(33/2) + 180*sqrt(x - 1)*(x + 1)**(31/2) - 120*sqrt(x - 1)*(x + 1
)**(29/2)), Abs(x + 1)/2 > 1), (-30*sqrt(1 - x)*(x + 1)**(35/2)*asin(sqrt(2)*sqrt(x + 1)/2)/(15*sqrt(1 - x)*(x
+ 1)**(35/2) - 90*sqrt(1 - x)*(x + 1)**(33/2) + 180*sqrt(1 - x)*(x + 1)**(31/2) - 120*sqrt(1 - x)*(x + 1)**(2
9/2)) + 180*sqrt(1 - x)*(x + 1)**(33/2)*asin(sqrt(2)*sqrt(x + 1)/2)/(15*sqrt(1 - x)*(x + 1)**(35/2) - 90*sqrt(
1 - x)*(x + 1)**(33/2) + 180*sqrt(1 - x)*(x + 1)**(31/2) - 120*sqrt(1 - x)*(x + 1)**(29/2)) - 360*sqrt(1 - x)*
(x + 1)**(31/2)*asin(sqrt(2)*sqrt(x + 1)/2)/(15*sqrt(1 - x)*(x + 1)**(35/2) - 90*sqrt(1 - x)*(x + 1)**(33/2) +
180*sqrt(1 - x)*(x + 1)**(31/2) - 120*sqrt(1 - x)*(x + 1)**(29/2)) + 240*sqrt(1 - x)*(x + 1)**(29/2)*asin(sqr
t(2)*sqrt(x + 1)/2)/(15*sqrt(1 - x)*(x + 1)**(35/2) - 90*sqrt(1 - x)*(x + 1)**(33/2) + 180*sqrt(1 - x)*(x + 1)
**(31/2) - 120*sqrt(1 - x)*(x + 1)**(29/2)) + 46*(x + 1)**18/(15*sqrt(1 - x)*(x + 1)**(35/2) - 90*sqrt(1 - x)*
(x + 1)**(33/2) + 180*sqrt(1 - x)*(x + 1)**(31/2) - 120*sqrt(1 - x)*(x + 1)**(29/2)) - 232*(x + 1)**17/(15*sqr
t(1 - x)*(x + 1)**(35/2) - 90*sqrt(1 - x)*(x + 1)**(33/2) + 180*sqrt(1 - x)*(x + 1)**(31/2) - 120*sqrt(1 - x)*
(x + 1)**(29/2)) + 400*(x + 1)**16/(15*sqrt(1 - x)*(x + 1)**(35/2) - 90*sqrt(1 - x)*(x + 1)**(33/2) + 180*sqrt
(1 - x)*(x + 1)**(31/2) - 120*sqrt(1 - x)*(x + 1)**(29/2)) - 240*(x + 1)**15/(15*sqrt(1 - x)*(x + 1)**(35/2) -
90*sqrt(1 - x)*(x + 1)**(33/2) + 180*sqrt(1 - x)*(x + 1)**(31/2) - 120*sqrt(1 - x)*(x + 1)**(29/2)), True))

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Giac [A]  time = 1.06599, size = 59, normalized size = 0.94 \begin{align*} -\frac{2 \,{\left ({\left (23 \, x - 47\right )}{\left (x + 1\right )} + 60\right )} \sqrt{x + 1} \sqrt{-x + 1}}{15 \,{\left (x - 1\right )}^{3}} - 2 \, \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)^(7/2),x, algorithm="giac")

[Out]

-2/15*((23*x - 47)*(x + 1) + 60)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^3 - 2*arcsin(1/2*sqrt(2)*sqrt(x + 1))