∫ (1−x)5∕2 ______________(1+x)3∕2 dx [1117]

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

-15/2*arcsin(x)-2*(1-x)^(5/2)/(1+x)^(1/2)-5/2*(1-x)^(3/2)*(1+x)^(1/2)-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 {2 (1-x)^{5/2}}{\sqrt {x+1}}-\frac {5}{2} \sqrt {x+1} (1-x)^{3/2}-\frac {15}{2} \sqrt {x+1} \sqrt {1-x}-\frac {15}{2} \sin ^{-1}(x) \end {gather*}

(-2*(1 - x)^(5/2))/Sqrt[1 + x] - (15*Sqrt[1 - x]*Sqrt[1 + x])/2 - (5*(1 - x)^(3/2)*Sqrt[1 + x])/2 - (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 {2 (1-x)^{5/2}}{\sqrt {1+x}}-\frac {5}{2} (1-x)^{3/2} \sqrt {1+x}-\frac {15}{2} \int \frac {\sqrt {1-x}}{\sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{5/2}}{\sqrt {1+x}}-\frac {15}{2} \sqrt {1-x} \sqrt {1+x}-\frac {5}{2} (1-x)^{3/2} \sqrt {1+x}-\frac {15}{2} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{5/2}}{\sqrt {1+x}}-\frac {15}{2} \sqrt {1-x} \sqrt {1+x}-\frac {5}{2} (1-x)^{3/2} \sqrt {1+x}-\frac {15}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {2 (1-x)^{5/2}}{\sqrt {1+x}}-\frac {15}{2} \sqrt {1-x} \sqrt {1+x}-\frac {5}{2} (1-x)^{3/2} \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*}

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 6.24, size = 129, normalized size = 1.98 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-7 x \sqrt {-1+x}+x^2 \sqrt {-1+x}-24 \sqrt {-1+x}+30 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \sqrt {1+x}\right )}{2 \sqrt {1+x}},\text {Abs}\left [1+x\right ]>2\right \}\right \},\frac {-16}{\sqrt {1+x} \sqrt {1-x}}-15 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]-\frac {\sqrt {1+x}}{\sqrt {1-x}}-\frac {\left (1+x\right )^{\frac {5}{2}}}{2 \sqrt {1-x}}+\frac {11 \left (1+x\right )^{\frac {3}{2}}}{2 \sqrt {1-x}}\right ] \end {gather*}

Piecewise[{{I / 2 (-7 x Sqrt[-1 + x] + x ^ 2 Sqrt[-1 + x] - 24 Sqrt[-1 + x] + 30 ArcCosh[Sqrt[2] Sqrt[1 + x] /

2] Sqrt[1 + x]) / Sqrt[1 + x], Abs[1 + x] > 2}}, -16 / (Sqrt[1 + x] Sqrt[1 - x]) - 15 ArcSin[Sqrt[2] Sqrt[1 +

x] / 2] - Sqrt[1 + x] / Sqrt[1 - x] - (1 + x) ^ (5 / 2) / (2 Sqrt[1 - x]) + 11 (1 + x) ^ (3 / 2) / (2 Sqrt[1

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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 = 0.35, 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 = 0.29, 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 [A]
time = 4.89, size = 167, normalized size = 2.57 \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 {11 i \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {x - 1}} + \frac {i \sqrt {x + 1}}{\sqrt {x - 1}} + \frac {16 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 {11 \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {1 - x}} - \frac {\sqrt {x + 1}}{\sqrt {1 - x}} - \frac {16}{\sqrt {1 - x} \sqrt {x + 1}} & \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)) - 11*I*(x + 1)**(3/2)/(2*sqrt(

x - 1)) + I*sqrt(x + 1)/sqrt(x - 1) + 16*I/(sqrt(x - 1)*sqrt(x + 1)), Abs(x + 1) > 2), (-15*asin(sqrt(2)*sqrt(

x + 1)/2) - (x + 1)**(5/2)/(2*sqrt(1 - x)) + 11*(x + 1)**(3/2)/(2*sqrt(1 - x)) - sqrt(x + 1)/sqrt(1 - x) - 16/

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Giac [A]
time = 0.01, size = 86, normalized size = 1.32 \begin {gather*} \frac {2 \left (\left (\frac {1}{4} \sqrt {-x+1} \sqrt {-x+1}+\frac {5}{4}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {15}{2}\right ) \sqrt {-x+1} \sqrt {x+1}}{x+1}+15 \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right ) \end {gather*}

1/2*((x - 1)*(x - 6) - 30)*sqrt(-x + 1)/sqrt(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 (1-x\right )}^{5/2}}{{\left (x+1\right )}^{3/2}} \,d x \end {gather*}

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3.12.17 \(\int \frac {(1-x)^{5/2}}{(1+x)^{3/2}} \, dx\) [1117]