3.1127 \(\int \frac{(1-x)^{7/2}}{(1+x)^{5/2}} \, dx\)

Optimal. Leaf size=87 \[ -\frac{2 (1-x)^{7/2}}{3 (x+1)^{3/2}}+\frac{14 (1-x)^{5/2}}{3 \sqrt{x+1}}+\frac{35}{6} \sqrt{x+1} (1-x)^{3/2}+\frac{35}{2} \sqrt{x+1} \sqrt{1-x}+\frac{35}{2} \sin ^{-1}(x) \]

[Out]

(-2*(1 - x)^(7/2))/(3*(1 + x)^(3/2)) + (14*(1 - x)^(5/2))/(3*Sqrt[1 + x]) + (35*Sqrt[1 - x]*Sqrt[1 + x])/2 + (
35*(1 - x)^(3/2)*Sqrt[1 + x])/6 + (35*ArcSin[x])/2

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Rubi [A]  time = 0.0152415, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {47, 50, 41, 216} \[ -\frac{2 (1-x)^{7/2}}{3 (x+1)^{3/2}}+\frac{14 (1-x)^{5/2}}{3 \sqrt{x+1}}+\frac{35}{6} \sqrt{x+1} (1-x)^{3/2}+\frac{35}{2} \sqrt{x+1} \sqrt{1-x}+\frac{35}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(7/2)/(1 + x)^(5/2),x]

[Out]

(-2*(1 - x)^(7/2))/(3*(1 + x)^(3/2)) + (14*(1 - x)^(5/2))/(3*Sqrt[1 + x]) + (35*Sqrt[1 - x]*Sqrt[1 + x])/2 + (
35*(1 - x)^(3/2)*Sqrt[1 + x])/6 + (35*ArcSin[x])/2

Rule 47

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]

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}}{(1+x)^{5/2}} \, dx &=-\frac{2 (1-x)^{7/2}}{3 (1+x)^{3/2}}-\frac{7}{3} \int \frac{(1-x)^{5/2}}{(1+x)^{3/2}} \, dx\\ &=-\frac{2 (1-x)^{7/2}}{3 (1+x)^{3/2}}+\frac{14 (1-x)^{5/2}}{3 \sqrt{1+x}}+\frac{35}{3} \int \frac{(1-x)^{3/2}}{\sqrt{1+x}} \, dx\\ &=-\frac{2 (1-x)^{7/2}}{3 (1+x)^{3/2}}+\frac{14 (1-x)^{5/2}}{3 \sqrt{1+x}}+\frac{35}{6} (1-x)^{3/2} \sqrt{1+x}+\frac{35}{2} \int \frac{\sqrt{1-x}}{\sqrt{1+x}} \, dx\\ &=-\frac{2 (1-x)^{7/2}}{3 (1+x)^{3/2}}+\frac{14 (1-x)^{5/2}}{3 \sqrt{1+x}}+\frac{35}{2} \sqrt{1-x} \sqrt{1+x}+\frac{35}{6} (1-x)^{3/2} \sqrt{1+x}+\frac{35}{2} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{2 (1-x)^{7/2}}{3 (1+x)^{3/2}}+\frac{14 (1-x)^{5/2}}{3 \sqrt{1+x}}+\frac{35}{2} \sqrt{1-x} \sqrt{1+x}+\frac{35}{6} (1-x)^{3/2} \sqrt{1+x}+\frac{35}{2} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{2 (1-x)^{7/2}}{3 (1+x)^{3/2}}+\frac{14 (1-x)^{5/2}}{3 \sqrt{1+x}}+\frac{35}{2} \sqrt{1-x} \sqrt{1+x}+\frac{35}{6} (1-x)^{3/2} \sqrt{1+x}+\frac{35}{2} \sin ^{-1}(x)\\ \end{align*}

Mathematica [C]  time = 0.0118406, size = 37, normalized size = 0.43 \[ -\frac{(1-x)^{9/2} \, _2F_1\left (\frac{5}{2},\frac{9}{2};\frac{11}{2};\frac{1-x}{2}\right )}{18 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(7/2)/(1 + x)^(5/2),x]

[Out]

-((1 - x)^(9/2)*Hypergeometric2F1[5/2, 9/2, 11/2, (1 - x)/2])/(18*Sqrt[2])

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Maple [A]  time = 0.017, size = 84, normalized size = 1. \begin{align*}{\frac{3\,{x}^{4}-33\,{x}^{3}-199\,{x}^{2}+65\,x+164}{6}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) } \left ( 1+x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}}+{\frac{35\,\arcsin \left ( x \right ) }{2}\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)^(5/2),x)

[Out]

1/6*(3*x^4-33*x^3-199*x^2+65*x+164)/(1+x)^(3/2)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)+35/2*((1
+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.52355, size = 150, normalized size = 1.72 \begin{align*} -\frac{x^{5}}{2 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{6 \, x^{4}}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{35}{6} \, x{\left (\frac{3 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{2}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}\right )} - \frac{61 \, x}{6 \, \sqrt{-x^{2} + 1}} - \frac{44 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{16 \, x}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{82}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{35}{2} \, \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(7/2)/(1+x)^(5/2),x, algorithm="maxima")

[Out]

-1/2*x^5/(-x^2 + 1)^(3/2) + 6*x^4/(-x^2 + 1)^(3/2) + 35/6*x*(3*x^2/(-x^2 + 1)^(3/2) - 2/(-x^2 + 1)^(3/2)) - 61
/6*x/sqrt(-x^2 + 1) - 44*x^2/(-x^2 + 1)^(3/2) + 16/3*x/(-x^2 + 1)^(3/2) + 82/3/(-x^2 + 1)^(3/2) + 35/2*arcsin(
x)

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Fricas [A]  time = 1.82426, size = 224, normalized size = 2.57 \begin{align*} \frac{164 \, x^{2} -{\left (3 \, x^{3} - 30 \, x^{2} - 229 \, x - 164\right )} \sqrt{x + 1} \sqrt{-x + 1} - 210 \,{\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 328 \, x + 164}{6 \,{\left (x^{2} + 2 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(7/2)/(1+x)^(5/2),x, algorithm="fricas")

[Out]

1/6*(164*x^2 - (3*x^3 - 30*x^2 - 229*x - 164)*sqrt(x + 1)*sqrt(-x + 1) - 210*(x^2 + 2*x + 1)*arctan((sqrt(x +
1)*sqrt(-x + 1) - 1)/x) + 328*x + 164)/(x^2 + 2*x + 1)

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Sympy [C]  time = 60.2526, size = 207, normalized size = 2.38 \begin{align*} \begin{cases} - \frac{\sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2}}{2} + \frac{13 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )}{2} + \frac{80 \sqrt{-1 + \frac{2}{x + 1}}}{3} - \frac{16 \sqrt{-1 + \frac{2}{x + 1}}}{3 \left (x + 1\right )} + \frac{35 i \log{\left (\frac{1}{x + 1} \right )}}{2} + \frac{35 i \log{\left (x + 1 \right )}}{2} + 35 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- \frac{i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2}}{2} + \frac{13 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )}{2} + \frac{80 i \sqrt{1 - \frac{2}{x + 1}}}{3} - \frac{16 i \sqrt{1 - \frac{2}{x + 1}}}{3 \left (x + 1\right )} + \frac{35 i \log{\left (\frac{1}{x + 1} \right )}}{2} - 35 i \log{\left (\sqrt{1 - \frac{2}{x + 1}} + 1 \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(7/2)/(1+x)**(5/2),x)

[Out]

Piecewise((-sqrt(-1 + 2/(x + 1))*(x + 1)**2/2 + 13*sqrt(-1 + 2/(x + 1))*(x + 1)/2 + 80*sqrt(-1 + 2/(x + 1))/3
- 16*sqrt(-1 + 2/(x + 1))/(3*(x + 1)) + 35*I*log(1/(x + 1))/2 + 35*I*log(x + 1)/2 + 35*asin(sqrt(2)*sqrt(x + 1
)/2), 2/Abs(x + 1) > 1), (-I*sqrt(1 - 2/(x + 1))*(x + 1)**2/2 + 13*I*sqrt(1 - 2/(x + 1))*(x + 1)/2 + 80*I*sqrt
(1 - 2/(x + 1))/3 - 16*I*sqrt(1 - 2/(x + 1))/(3*(x + 1)) + 35*I*log(1/(x + 1))/2 - 35*I*log(sqrt(1 - 2/(x + 1)
) + 1), True))

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Giac [A]  time = 1.13977, size = 161, normalized size = 1.85 \begin{align*} -\frac{1}{2} \, \sqrt{x + 1}{\left (x - 12\right )} \sqrt{-x + 1} + \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{3 \,{\left (x + 1\right )}^{\frac{3}{2}}} - \frac{13 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{\sqrt{x + 1}} + \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{39 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{3 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} + 35 \, \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)^(5/2),x, algorithm="giac")

[Out]

-1/2*sqrt(x + 1)*(x - 12)*sqrt(-x + 1) + 1/3*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) - 13*(sqrt(2) - sqrt(-x
+ 1))/sqrt(x + 1) + 1/3*(x + 1)^(3/2)*(39*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) - 1)/(sqrt(2) - sqrt(-x + 1))^3 +
35*arcsin(1/2*sqrt(2)*sqrt(x + 1))