∫ (1+x)5∕2 ______________(1−x)3∕2 dx [1095]

Optimal. Leaf size=65 \[ \frac {15}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-\frac {15}{2} \sin ^{-1}(x) \]

-15/2*arcsin(x)+2*(1+x)^(5/2)/(1-x)^(1/2)+5/2*(1-x)^(1/2)*(1+x)^(3/2)+15/2*(1-x)^(1/2)*(1+x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 65, 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*} -\frac {15 \text {ArcSin}(x)}{2}+\frac {2 (x+1)^{5/2}}{\sqrt {1-x}}+\frac {5}{2} \sqrt {1-x} (x+1)^{3/2}+\frac {15}{2} \sqrt {1-x} \sqrt {x+1} \end {gather*}

(15*Sqrt[1 - x]*Sqrt[1 + x])/2 + (5*Sqrt[1 - x]*(1 + x)^(3/2))/2 + (2*(1 + x)^(5/2))/Sqrt[1 - x] - (15*ArcSin[

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[((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]

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)^{3/2}} \, dx &=\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-5 \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=\frac {5}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-\frac {15}{2} \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=\frac {15}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-\frac {15}{2} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {15}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-\frac {15}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {15}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-\frac {15}{2} \sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 49, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {1+x} \left (-24+7 x+x^2\right )}{2 \sqrt {1-x}}+15 \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}

-1/2*(Sqrt[1 + x]*(-24 + 7*x + x^2))/Sqrt[1 - x] + 15*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]]

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method	result	size

risch	\(-\frac {\left (x^{3}+8 x^{2}-17 x -24\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}-\frac {15 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\)	\(77\)

-1/2*(x^3+8*x^2-17*x-24)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)/(1+x)^(1/2)-15/2*((1+x)*(1-x))^

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Maxima [A]
time = 0.54, size = 56, normalized size = 0.86 \begin {gather*} -\frac {x^{3}}{2 \, \sqrt {-x^{2} + 1}} - \frac {4 \, x^{2}}{\sqrt {-x^{2} + 1}} + \frac {17 \, x}{2 \, \sqrt {-x^{2} + 1}} + \frac {12}{\sqrt {-x^{2} + 1}} - \frac {15}{2} \, \arcsin \left (x\right ) \end {gather*}

-1/2*x^3/sqrt(-x^2 + 1) - 4*x^2/sqrt(-x^2 + 1) + 17/2*x/sqrt(-x^2 + 1) + 12/sqrt(-x^2 + 1) - 15/2*arcsin(x)

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Fricas [A]
time = 0.62, size = 58, normalized size = 0.89 \begin {gather*} \frac {{\left (x^{2} + 7 \, x - 24\right )} \sqrt {x + 1} \sqrt {-x + 1} + 30 \, {\left (x - 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 24 \, x - 24}{2 \, {\left (x - 1\right )}} \end {gather*}

1/2*((x^2 + 7*x - 24)*sqrt(x + 1)*sqrt(-x + 1) + 30*(x - 1)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 24*x -

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Sympy [C] Result contains complex when optimal does not.
time = 5.57, size = 138, normalized size = 2.12 \begin {gather*} \begin {cases} 15 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {i \left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {x - 1}} + \frac {5 i \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {x - 1}} - \frac {15 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- 15 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {\left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {1 - x}} - \frac {5 \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {1 - x}} + \frac {15 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}

Piecewise((15*I*acosh(sqrt(2)*sqrt(x + 1)/2) + I*(x + 1)**(5/2)/(2*sqrt(x - 1)) + 5*I*(x + 1)**(3/2)/(2*sqrt(x

- 1)) - 15*I*sqrt(x + 1)/sqrt(x - 1), Abs(x + 1) > 2), (-15*asin(sqrt(2)*sqrt(x + 1)/2) - (x + 1)**(5/2)/(2*s

qrt(1 - x)) - 5*(x + 1)**(3/2)/(2*sqrt(1 - x)) + 15*sqrt(x + 1)/sqrt(1 - x), True))

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Giac [A]
time = 1.22, size = 42, normalized size = 0.65 \begin {gather*} \frac {{\left ({\left (x + 6\right )} {\left (x + 1\right )} - 30\right )} \sqrt {x + 1} \sqrt {-x + 1}}{2 \, {\left (x - 1\right )}} - 15 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \end {gather*}

1/2*((x + 6)*(x + 1) - 30)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1) - 15*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 )}^{3/2}} \,d x \end {gather*}

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3.11.95 \(\int \frac {(1+x)^{5/2}}{(1-x)^{3/2}} \, dx\) [1095]