3.11.65 \(\int (1-x)^{5/2} \sqrt {1+x} \, dx\) [1065]

Optimal. Leaf size=68 \[ \frac {5}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac {5}{8} \sin ^{-1}(x) \]

[Out]

5/12*(1-x)^(3/2)*(1+x)^(3/2)+1/4*(1-x)^(5/2)*(1+x)^(3/2)+5/8*arcsin(x)+5/8*x*(1-x)^(1/2)*(1+x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 68, 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 {5 \text {ArcSin}(x)}{8}+\frac {1}{4} (x+1)^{3/2} (1-x)^{5/2}+\frac {5}{12} (x+1)^{3/2} (1-x)^{3/2}+\frac {5}{8} x \sqrt {x+1} \sqrt {1-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(5/2)*Sqrt[1 + x],x]

[Out]

(5*Sqrt[1 - x]*x*Sqrt[1 + x])/8 + (5*(1 - x)^(3/2)*(1 + x)^(3/2))/12 + ((1 - x)^(5/2)*(1 + x)^(3/2))/4 + (5*Ar
cSin[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} \sqrt {1+x} \, dx &=\frac {1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac {5}{4} \int (1-x)^{3/2} \sqrt {1+x} \, dx\\ &=\frac {5}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac {5}{4} \int \sqrt {1-x} \sqrt {1+x} \, dx\\ &=\frac {5}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac {5}{8} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {5}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac {5}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {5}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {1}{4} (1-x)^{5/2} (1+x)^{3/2}+\frac {5}{8} \sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 63, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {1+x} \left (-16+7 x+25 x^2-22 x^3+6 x^4\right )}{24 \sqrt {1-x}}+\frac {5}{4} \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*(Sqrt[1 + x]*(-16 + 7*x + 25*x^2 - 22*x^3 + 6*x^4))/Sqrt[1 - x] + (5*ArcTan[Sqrt[1 + x]/Sqrt[1 - x]])/4

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Maple [A]
time = 0.16, size = 85, normalized size = 1.25

method	result	size
risch	\(-\frac {\left (6 x^{3}-16 x^{2}+9 x +16\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{24 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\)	\(82\)
default	\(\frac {\left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {3}{2}}}{4}+\frac {5 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}{12}+\frac {5 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{8}-\frac {5 \sqrt {1-x}\, \sqrt {1+x}}{8}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(1-x)^(5/2)*(1+x)^(3/2)+5/12*(1-x)^(3/2)*(1+x)^(3/2)+5/8*(1-x)^(1/2)*(1+x)^(3/2)-5/8*(1-x)^(1/2)*(1+x)^(1/
2)+5/8*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]
time = 0.50, size = 40, normalized size = 0.59 \begin {gather*} -\frac {1}{4} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {2}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} + \frac {5}{8} \, \sqrt {-x^{2} + 1} x + \frac {5}{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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.57, size = 52, normalized size = 0.76 \begin {gather*} \frac {1}{24} \, {\left (6 \, x^{3} - 16 \, x^{2} + 9 \, x + 16\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {5}{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)^(1/2),x, algorithm="fricas")

[Out]

1/24*(6*x^3 - 16*x^2 + 9*x + 16)*sqrt(x + 1)*sqrt(-x + 1) - 5/4*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

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Sympy [C] Result contains complex when optimal does not.
time = 9.19, size = 216, normalized size = 3.18 \begin {gather*} \begin {cases} - \frac {5 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} + \frac {i \left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {x - 1}} - \frac {23 i \left (x + 1\right )^{\frac {7}{2}}}{12 \sqrt {x - 1}} + \frac {127 i \left (x + 1\right )^{\frac {5}{2}}}{24 \sqrt {x - 1}} - \frac {133 i \left (x + 1\right )^{\frac {3}{2}}}{24 \sqrt {x - 1}} + \frac {5 i \sqrt {x + 1}}{4 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} - \frac {\left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {1 - x}} + \frac {23 \left (x + 1\right )^{\frac {7}{2}}}{12 \sqrt {1 - x}} - \frac {127 \left (x + 1\right )^{\frac {5}{2}}}{24 \sqrt {1 - x}} + \frac {133 \left (x + 1\right )^{\frac {3}{2}}}{24 \sqrt {1 - x}} - \frac {5 \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)**(1/2),x)

[Out]

Piecewise((-5*I*acosh(sqrt(2)*sqrt(x + 1)/2)/4 + I*(x + 1)**(9/2)/(4*sqrt(x - 1)) - 23*I*(x + 1)**(7/2)/(12*sq
rt(x - 1)) + 127*I*(x + 1)**(5/2)/(24*sqrt(x - 1)) - 133*I*(x + 1)**(3/2)/(24*sqrt(x - 1)) + 5*I*sqrt(x + 1)/(
4*sqrt(x - 1)), Abs(x + 1) > 2), (5*asin(sqrt(2)*sqrt(x + 1)/2)/4 - (x + 1)**(9/2)/(4*sqrt(1 - x)) + 23*(x + 1
)**(7/2)/(12*sqrt(1 - x)) - 127*(x + 1)**(5/2)/(24*sqrt(1 - x)) + 133*(x + 1)**(3/2)/(24*sqrt(1 - x)) - 5*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. 101 vs. \(2 (48) = 96\).
time = 1.21, size = 101, normalized size = 1.49 \begin {gather*} \frac {1}{24} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{6} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {5}{4} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((2*(3*x - 10)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt(-x + 1) - 1/6*((2*x - 5)*(x + 1) + 9)*sqrt(x
+ 1)*sqrt(-x + 1) - 1/2*sqrt(x + 1)*(x - 2)*sqrt(-x + 1) + sqrt(x + 1)*sqrt(-x + 1) + 5/4*arcsin(1/2*sqrt(2)*s
qrt(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}\,\sqrt {x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(5/2)*(x + 1)^(1/2),x)

[Out]

int((1 - x)^(5/2)*(x + 1)^(1/2), x)

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