3.11.77 \(\int (1-x)^{5/2} (1+x)^{3/2} \, dx\) [1077]

Optimal. Leaf size=69 \[ \frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{8} \sin ^{-1}(x) \]

[Out]

1/4*(1-x)^(3/2)*x*(1+x)^(3/2)+1/5*(1-x)^(5/2)*(1+x)^(5/2)+3/8*arcsin(x)+3/8*x*(1-x)^(1/2)*(1+x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 69, 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 = {51, 38, 41, 222} \begin {gather*} \frac {1}{5} (1-x)^{5/2} (x+1)^{5/2}+\frac {1}{4} (1-x)^{3/2} x (x+1)^{3/2}+\frac {3}{8} \sqrt {1-x} x \sqrt {x+1}+\frac {3}{8} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(5/2)*(1 + x)^(3/2),x]

[Out]

(3*Sqrt[1 - x]*x*Sqrt[1 + x])/8 + ((1 - x)^(3/2)*x*(1 + x)^(3/2))/4 + ((1 - x)^(5/2)*(1 + x)^(5/2))/5 + (3*Arc
Sin[x])/8

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x*(a + b*x)^m*((c + d*x)^m/(2*m + 1))
, x] + Dist[2*a*c*(m/(2*m + 1)), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

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 51

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[2*c*(n/(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x]
&& EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0] && IGtQ[n + 1/2, 0] && LtQ[m, n]

Rule 222

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 (1-x)^{5/2} (1+x)^{3/2} \, dx &=\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\int (1-x)^{3/2} (1+x)^{3/2} \, dx\\ &=\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{4} \int \sqrt {1-x} \sqrt {1+x} \, dx\\ &=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{8} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{8} \sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 68, normalized size = 0.99 \begin {gather*} \frac {\sqrt {1-x} \left (8+33 x+9 x^2-26 x^3-2 x^4+8 x^5\right )}{40 \sqrt {1+x}}-\frac {3}{4} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(5/2)*(1 + x)^(3/2),x]

[Out]

(Sqrt[1 - x]*(8 + 33*x + 9*x^2 - 26*x^3 - 2*x^4 + 8*x^5))/(40*Sqrt[1 + x]) - (3*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]
])/4

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 35.72, size = 175, normalized size = 2.54 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-129 \left (1+x\right )^{\frac {5}{2}}-58 \left (1+x\right )^{\frac {9}{2}}-30 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \sqrt {-1+x}-5 \left (1+x\right )^{\frac {3}{2}}+8 \left (1+x\right )^{\frac {11}{2}}+30 \sqrt {1+x}+146 \left (1+x\right )^{\frac {7}{2}}\right )}{40 \sqrt {-1+x}},\text {Abs}\left [1+x\right ]>2\right \}\right \},\frac {-73 \left (1+x\right )^{\frac {7}{2}}}{20 \sqrt {1-x}}-\frac {3 \sqrt {1+x}}{4 \sqrt {1-x}}-\frac {\left (1+x\right )^{\frac {11}{2}}}{5 \sqrt {1-x}}+\frac {\left (1+x\right )^{\frac {3}{2}}}{8 \sqrt {1-x}}+\frac {3 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]}{4}+\frac {29 \left (1+x\right )^{\frac {9}{2}}}{20 \sqrt {1-x}}+\frac {129 \left (1+x\right )^{\frac {5}{2}}}{40 \sqrt {1-x}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(1 - x)^(5/2)*(1 + x)^(3/2),x]')

[Out]

Piecewise[{{I / 40 (-129 (1 + x) ^ (5 / 2) - 58 (1 + x) ^ (9 / 2) - 30 ArcCosh[Sqrt[2] Sqrt[1 + x] / 2] Sqrt[-
1 + x] - 5 (1 + x) ^ (3 / 2) + 8 (1 + x) ^ (11 / 2) + 30 Sqrt[1 + x] + 146 (1 + x) ^ (7 / 2)) / Sqrt[-1 + x],
Abs[1 + x] > 2}}, -73 (1 + x) ^ (7 / 2) / (20 Sqrt[1 - x]) - 3 Sqrt[1 + x] / (4 Sqrt[1 - x]) - (1 + x) ^ (11 /
2) / (5 Sqrt[1 - x]) + (1 + x) ^ (3 / 2) / (8 Sqrt[1 - x]) + 3 ArcSin[Sqrt[2] Sqrt[1 + x] / 2] / 4 + 29 (1 +
x) ^ (9 / 2) / (20 Sqrt[1 - x]) + 129 (1 + x) ^ (5 / 2) / (40 Sqrt[1 - x])]

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Maple [A]
time = 0.14, size = 99, normalized size = 1.43

method	result	size
risch	\(-\frac {\left (8 x^{4}-10 x^{3}-16 x^{2}+25 x +8\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{40 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {3 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\)	\(87\)
default	\(\frac {\left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {5}{2}}}{5}+\frac {\left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {5}{2}}}{4}+\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{4}-\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{8}-\frac {3 \sqrt {1-x}\, \sqrt {1+x}}{8}+\frac {3 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)^(5/2)*(1+x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/5*(1-x)^(5/2)*(1+x)^(5/2)+1/4*(1-x)^(3/2)*(1+x)^(5/2)+1/4*(1-x)^(1/2)*(1+x)^(5/2)-1/8*(1-x)^(1/2)*(1+x)^(3/2
)-3/8*(1-x)^(1/2)*(1+x)^(1/2)+3/8*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]
time = 0.35, size = 40, normalized size = 0.58 \begin {gather*} \frac {1}{5} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} + \frac {1}{4} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {3}{8} \, \sqrt {-x^{2} + 1} x + \frac {3}{8} \, \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(5/2)*(1+x)^(3/2),x, algorithm="maxima")

[Out]

1/5*(-x^2 + 1)^(5/2) + 1/4*(-x^2 + 1)^(3/2)*x + 3/8*sqrt(-x^2 + 1)*x + 3/8*arcsin(x)

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Fricas [A]
time = 0.30, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{40} \, {\left (8 \, x^{4} - 10 \, x^{3} - 16 \, x^{2} + 25 \, x + 8\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {3}{4} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(5/2)*(1+x)^(3/2),x, algorithm="fricas")

[Out]

1/40*(8*x^4 - 10*x^3 - 16*x^2 + 25*x + 8)*sqrt(x + 1)*sqrt(-x + 1) - 3/4*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)
/x)

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Sympy [C] Result contains complex when optimal does not.
time = 35.95, size = 248, normalized size = 3.59 \begin {gather*} \begin {cases} - \frac {3 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} + \frac {i \left (x + 1\right )^{\frac {11}{2}}}{5 \sqrt {x - 1}} - \frac {29 i \left (x + 1\right )^{\frac {9}{2}}}{20 \sqrt {x - 1}} + \frac {73 i \left (x + 1\right )^{\frac {7}{2}}}{20 \sqrt {x - 1}} - \frac {129 i \left (x + 1\right )^{\frac {5}{2}}}{40 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {3}{2}}}{8 \sqrt {x - 1}} + \frac {3 i \sqrt {x + 1}}{4 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} - \frac {\left (x + 1\right )^{\frac {11}{2}}}{5 \sqrt {1 - x}} + \frac {29 \left (x + 1\right )^{\frac {9}{2}}}{20 \sqrt {1 - x}} - \frac {73 \left (x + 1\right )^{\frac {7}{2}}}{20 \sqrt {1 - x}} + \frac {129 \left (x + 1\right )^{\frac {5}{2}}}{40 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {3}{2}}}{8 \sqrt {1 - x}} - \frac {3 \sqrt {x + 1}}{4 \sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(5/2)*(1+x)**(3/2),x)

[Out]

Piecewise((-3*I*acosh(sqrt(2)*sqrt(x + 1)/2)/4 + I*(x + 1)**(11/2)/(5*sqrt(x - 1)) - 29*I*(x + 1)**(9/2)/(20*s
qrt(x - 1)) + 73*I*(x + 1)**(7/2)/(20*sqrt(x - 1)) - 129*I*(x + 1)**(5/2)/(40*sqrt(x - 1)) - I*(x + 1)**(3/2)/
(8*sqrt(x - 1)) + 3*I*sqrt(x + 1)/(4*sqrt(x - 1)), Abs(x + 1) > 2), (3*asin(sqrt(2)*sqrt(x + 1)/2)/4 - (x + 1)
**(11/2)/(5*sqrt(1 - x)) + 29*(x + 1)**(9/2)/(20*sqrt(1 - x)) - 73*(x + 1)**(7/2)/(20*sqrt(1 - x)) + 129*(x +
1)**(5/2)/(40*sqrt(1 - x)) + (x + 1)**(3/2)/(8*sqrt(1 - x)) - 3*sqrt(x + 1)/(4*sqrt(1 - x)), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (49) = 98\).
time = 0.02, size = 343, normalized size = 4.97 \begin {gather*} 2 \left (2 \left (\left (\left (\left (\frac {1}{20} \sqrt {-x+1} \sqrt {-x+1}-\frac {21}{80}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {133}{240}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {59}{96}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {13}{32}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {3}{8} \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right )-4 \left (2 \left (\left (\left (\frac {13}{48}-\frac {1}{16} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {43}{96}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {13}{32}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {3}{8} \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right )+4 \left (2 \left (\frac {3}{8}-\frac {1}{8} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )}{2}\right )-2 \left (\frac {1}{2} \sqrt {-x+1} \sqrt {x+1}+\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*((2*(3*(4*x + 17)*(x - 1) + 133)*(x - 1) + 295)*(x - 1) + 195)*sqrt(x + 1)*sqrt(-x + 1) - 1/12*((2*(3*x
+ 10)*(x - 1) + 43)*(x - 1) + 39)*sqrt(x + 1)*sqrt(-x + 1) + (x + 2)*sqrt(x + 1)*sqrt(-x + 1) - sqrt(x + 1)*sq
rt(-x + 1) - 3/4*arcsin(1/2*sqrt(2)*sqrt(-x + 1))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (1-x\right )}^{5/2}\,{\left (x+1\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(5/2)*(x + 1)^(3/2),x)

[Out]

int((1 - x)^(5/2)*(x + 1)^(3/2), x)

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