∫ (1+x)5∕2 ______________(1−x)7∕2 dx [1097]

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

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

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Rubi [A]
time = 0.00, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 = {49, 41, 222} \begin {gather*} -\text {ArcSin}(x)+\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}} \end {gather*}

(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

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]))

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]

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

\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*}

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Mathematica [A]
time = 0.06, size = 51, normalized size = 0.81 \begin {gather*} \frac {2 \sqrt {1+x} \left (13-24 x+23 x^2\right )}{15 (1-x)^{5/2}}+2 \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}

(2*Sqrt[1 + x]*(13 - 24*x + 23*x^2))/(15*(1 - x)^(5/2)) + 2*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]]

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method	result	size

risch	\(\frac {2 \left (23 x^{3}-x^{2}-11 x +13\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{15 \left (-1+x \right )^{2} \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\)	\(84\)

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (47) = 94\).
time = 0.54, size = 160, normalized size = 2.54 \begin {gather*} -\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 {gather*}

-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(-

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Fricas [A]
time = 0.94, size = 91, normalized size = 1.44 \begin {gather*} \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 {gather*}

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

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Sympy [C] Result contains complex when optimal does not.
time = 7.28, size = 1606, normalized size = 25.49 \begin {gather*} \text {Too large to display} \end {gather*}

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), (-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)**(29/

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) + 1

80*sqrt(1 - x)*(x + 1)**(31/2) - 120*sqrt(1 - x)*(x + 1)**(29/2)) + 240*sqrt(1 - x)*(x + 1)**(29/2)*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)) + 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*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)) + 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) - 9

0*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 = 1.24, size = 44, normalized size = 0.70 \begin {gather*} -\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 {gather*}

-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))

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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 )}^{7/2}} \,d x \end {gather*}

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3.11.97 \(\int \frac {(1+x)^{5/2}}{(1-x)^{7/2}} \, dx\) [1097]