3.1106 \(\int \frac{(1-x)^{7/2}}{\sqrt{1+x}} \, dx\)

Optimal. Leaf size=87 \[ \frac{1}{4} \sqrt{x+1} (1-x)^{7/2}+\frac{7}{12} \sqrt{x+1} (1-x)^{5/2}+\frac{35}{24} \sqrt{x+1} (1-x)^{3/2}+\frac{35}{8} \sqrt{x+1} \sqrt{1-x}+\frac{35}{8} \sin ^{-1}(x) \]

[Out]

(35*Sqrt[1 - x]*Sqrt[1 + x])/8 + (35*(1 - x)^(3/2)*Sqrt[1 + x])/24 + (7*(1 - x)^(5/2)*Sqrt[1 + x])/12 + ((1 -
x)^(7/2)*Sqrt[1 + x])/4 + (35*ArcSin[x])/8

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Rubi [A]  time = 0.0176468, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {50, 41, 216} \[ \frac{1}{4} \sqrt{x+1} (1-x)^{7/2}+\frac{7}{12} \sqrt{x+1} (1-x)^{5/2}+\frac{35}{24} \sqrt{x+1} (1-x)^{3/2}+\frac{35}{8} \sqrt{x+1} \sqrt{1-x}+\frac{35}{8} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(7/2)/Sqrt[1 + x],x]

[Out]

(35*Sqrt[1 - x]*Sqrt[1 + x])/8 + (35*(1 - x)^(3/2)*Sqrt[1 + x])/24 + (7*(1 - x)^(5/2)*Sqrt[1 + x])/12 + ((1 -
x)^(7/2)*Sqrt[1 + x])/4 + (35*ArcSin[x])/8

Rule 50

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]

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 216

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 \frac{(1-x)^{7/2}}{\sqrt{1+x}} \, dx &=\frac{1}{4} (1-x)^{7/2} \sqrt{1+x}+\frac{7}{4} \int \frac{(1-x)^{5/2}}{\sqrt{1+x}} \, dx\\ &=\frac{7}{12} (1-x)^{5/2} \sqrt{1+x}+\frac{1}{4} (1-x)^{7/2} \sqrt{1+x}+\frac{35}{12} \int \frac{(1-x)^{3/2}}{\sqrt{1+x}} \, dx\\ &=\frac{35}{24} (1-x)^{3/2} \sqrt{1+x}+\frac{7}{12} (1-x)^{5/2} \sqrt{1+x}+\frac{1}{4} (1-x)^{7/2} \sqrt{1+x}+\frac{35}{8} \int \frac{\sqrt{1-x}}{\sqrt{1+x}} \, dx\\ &=\frac{35}{8} \sqrt{1-x} \sqrt{1+x}+\frac{35}{24} (1-x)^{3/2} \sqrt{1+x}+\frac{7}{12} (1-x)^{5/2} \sqrt{1+x}+\frac{1}{4} (1-x)^{7/2} \sqrt{1+x}+\frac{35}{8} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=\frac{35}{8} \sqrt{1-x} \sqrt{1+x}+\frac{35}{24} (1-x)^{3/2} \sqrt{1+x}+\frac{7}{12} (1-x)^{5/2} \sqrt{1+x}+\frac{1}{4} (1-x)^{7/2} \sqrt{1+x}+\frac{35}{8} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{35}{8} \sqrt{1-x} \sqrt{1+x}+\frac{35}{24} (1-x)^{3/2} \sqrt{1+x}+\frac{7}{12} (1-x)^{5/2} \sqrt{1+x}+\frac{1}{4} (1-x)^{7/2} \sqrt{1+x}+\frac{35}{8} \sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.026514, size = 61, normalized size = 0.7 \[ \frac{\sqrt{x+1} \left (6 x^4-38 x^3+113 x^2-241 x+160\right )}{24 \sqrt{1-x}}-\frac{35}{4} \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(7/2)/Sqrt[1 + x],x]

[Out]

(Sqrt[1 + x]*(160 - 241*x + 113*x^2 - 38*x^3 + 6*x^4))/(24*Sqrt[1 - x]) - (35*ArcSin[Sqrt[1 - x]/Sqrt[2]])/4

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Maple [A]  time = 0.001, size = 85, normalized size = 1. \begin{align*}{\frac{1}{4} \left ( 1-x \right ) ^{{\frac{7}{2}}}\sqrt{1+x}}+{\frac{7}{12} \left ( 1-x \right ) ^{{\frac{5}{2}}}\sqrt{1+x}}+{\frac{35}{24} \left ( 1-x \right ) ^{{\frac{3}{2}}}\sqrt{1+x}}+{\frac{35}{8}\sqrt{1-x}\sqrt{1+x}}+{\frac{35\,\arcsin \left ( x \right ) }{8}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)^(7/2)/(1+x)^(1/2),x)

[Out]

1/4*(1-x)^(7/2)*(1+x)^(1/2)+7/12*(1-x)^(5/2)*(1+x)^(1/2)+35/24*(1-x)^(3/2)*(1+x)^(1/2)+35/8*(1-x)^(1/2)*(1+x)^
(1/2)+35/8*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.51879, size = 76, normalized size = 0.87 \begin{align*} -\frac{1}{4} \, \sqrt{-x^{2} + 1} x^{3} + \frac{4}{3} \, \sqrt{-x^{2} + 1} x^{2} - \frac{27}{8} \, \sqrt{-x^{2} + 1} x + \frac{20}{3} \, \sqrt{-x^{2} + 1} + \frac{35}{8} \, \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(7/2)/(1+x)^(1/2),x, algorithm="maxima")

[Out]

-1/4*sqrt(-x^2 + 1)*x^3 + 4/3*sqrt(-x^2 + 1)*x^2 - 27/8*sqrt(-x^2 + 1)*x + 20/3*sqrt(-x^2 + 1) + 35/8*arcsin(x
)

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Fricas [A]  time = 1.67869, size = 149, normalized size = 1.71 \begin{align*} -\frac{1}{24} \,{\left (6 \, x^{3} - 32 \, x^{2} + 81 \, x - 160\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{35}{4} \, \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(7/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

-1/24*(6*x^3 - 32*x^2 + 81*x - 160)*sqrt(x + 1)*sqrt(-x + 1) - 35/4*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

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Sympy [A]  time = 50.5507, size = 201, normalized size = 2.31 \begin{align*} \begin{cases} - \frac{35 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{31 i \left (x + 1\right )^{\frac{7}{2}}}{12 \sqrt{x - 1}} - \frac{263 i \left (x + 1\right )^{\frac{5}{2}}}{24 \sqrt{x - 1}} + \frac{605 i \left (x + 1\right )^{\frac{3}{2}}}{24 \sqrt{x - 1}} - \frac{93 i \sqrt{x + 1}}{4 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- \frac{\sqrt{1 - x} \left (x + 1\right )^{\frac{7}{2}}}{4} + \frac{25 \sqrt{1 - x} \left (x + 1\right )^{\frac{5}{2}}}{12} - \frac{163 \sqrt{1 - x} \left (x + 1\right )^{\frac{3}{2}}}{24} + \frac{93 \sqrt{1 - x} \sqrt{x + 1}}{8} + \frac{35 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(7/2)/(1+x)**(1/2),x)

[Out]

Piecewise((-35*I*acosh(sqrt(2)*sqrt(x + 1)/2)/4 - I*(x + 1)**(9/2)/(4*sqrt(x - 1)) + 31*I*(x + 1)**(7/2)/(12*s
qrt(x - 1)) - 263*I*(x + 1)**(5/2)/(24*sqrt(x - 1)) + 605*I*(x + 1)**(3/2)/(24*sqrt(x - 1)) - 93*I*sqrt(x + 1)
/(4*sqrt(x - 1)), Abs(x + 1)/2 > 1), (-sqrt(1 - x)*(x + 1)**(7/2)/4 + 25*sqrt(1 - x)*(x + 1)**(5/2)/12 - 163*s
qrt(1 - x)*(x + 1)**(3/2)/24 + 93*sqrt(1 - x)*sqrt(x + 1)/8 + 35*asin(sqrt(2)*sqrt(x + 1)/2)/4, True))

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Giac [A]  time = 1.12798, size = 136, normalized size = 1.56 \begin{align*} -\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}{2} \,{\left ({\left (2 \, x - 5\right )}{\left (x + 1\right )} + 9\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{3}{2} \, \sqrt{x + 1}{\left (x - 2\right )} \sqrt{-x + 1} + \sqrt{x + 1} \sqrt{-x + 1} + \frac{35}{4} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(7/2)/(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) + 1/2*((2*x - 5)*(x + 1) + 9)*sqrt(x
+ 1)*sqrt(-x + 1) - 3/2*sqrt(x + 1)*(x - 2)*sqrt(-x + 1) + sqrt(x + 1)*sqrt(-x + 1) + 35/4*arcsin(1/2*sqrt(2)
*sqrt(x + 1))