3.1126 \(\int \frac{(1-x)^{9/2}}{(1+x)^{5/2}} \, dx\)

Optimal. Leaf size=103 \[ -\frac{2 (1-x)^{9/2}}{3 (x+1)^{3/2}}+\frac{6 (1-x)^{7/2}}{\sqrt{x+1}}+7 \sqrt{x+1} (1-x)^{5/2}+\frac{35}{2} \sqrt{x+1} (1-x)^{3/2}+\frac{105}{2} \sqrt{x+1} \sqrt{1-x}+\frac{105}{2} \sin ^{-1}(x) \]

[Out]

(-2*(1 - x)^(9/2))/(3*(1 + x)^(3/2)) + (6*(1 - x)^(7/2))/Sqrt[1 + x] + (105*Sqrt[1 - x]*Sqrt[1 + x])/2 + (35*(
1 - x)^(3/2)*Sqrt[1 + x])/2 + 7*(1 - x)^(5/2)*Sqrt[1 + x] + (105*ArcSin[x])/2

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Rubi [A]  time = 0.0214789, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {47, 50, 41, 216} \[ -\frac{2 (1-x)^{9/2}}{3 (x+1)^{3/2}}+\frac{6 (1-x)^{7/2}}{\sqrt{x+1}}+7 \sqrt{x+1} (1-x)^{5/2}+\frac{35}{2} \sqrt{x+1} (1-x)^{3/2}+\frac{105}{2} \sqrt{x+1} \sqrt{1-x}+\frac{105}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(9/2)/(1 + x)^(5/2),x]

[Out]

(-2*(1 - x)^(9/2))/(3*(1 + x)^(3/2)) + (6*(1 - x)^(7/2))/Sqrt[1 + x] + (105*Sqrt[1 - x]*Sqrt[1 + x])/2 + (35*(
1 - x)^(3/2)*Sqrt[1 + x])/2 + 7*(1 - x)^(5/2)*Sqrt[1 + x] + (105*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)^{9/2}}{(1+x)^{5/2}} \, dx &=-\frac{2 (1-x)^{9/2}}{3 (1+x)^{3/2}}-3 \int \frac{(1-x)^{7/2}}{(1+x)^{3/2}} \, dx\\ &=-\frac{2 (1-x)^{9/2}}{3 (1+x)^{3/2}}+\frac{6 (1-x)^{7/2}}{\sqrt{1+x}}+21 \int \frac{(1-x)^{5/2}}{\sqrt{1+x}} \, dx\\ &=-\frac{2 (1-x)^{9/2}}{3 (1+x)^{3/2}}+\frac{6 (1-x)^{7/2}}{\sqrt{1+x}}+7 (1-x)^{5/2} \sqrt{1+x}+35 \int \frac{(1-x)^{3/2}}{\sqrt{1+x}} \, dx\\ &=-\frac{2 (1-x)^{9/2}}{3 (1+x)^{3/2}}+\frac{6 (1-x)^{7/2}}{\sqrt{1+x}}+\frac{35}{2} (1-x)^{3/2} \sqrt{1+x}+7 (1-x)^{5/2} \sqrt{1+x}+\frac{105}{2} \int \frac{\sqrt{1-x}}{\sqrt{1+x}} \, dx\\ &=-\frac{2 (1-x)^{9/2}}{3 (1+x)^{3/2}}+\frac{6 (1-x)^{7/2}}{\sqrt{1+x}}+\frac{105}{2} \sqrt{1-x} \sqrt{1+x}+\frac{35}{2} (1-x)^{3/2} \sqrt{1+x}+7 (1-x)^{5/2} \sqrt{1+x}+\frac{105}{2} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{2 (1-x)^{9/2}}{3 (1+x)^{3/2}}+\frac{6 (1-x)^{7/2}}{\sqrt{1+x}}+\frac{105}{2} \sqrt{1-x} \sqrt{1+x}+\frac{35}{2} (1-x)^{3/2} \sqrt{1+x}+7 (1-x)^{5/2} \sqrt{1+x}+\frac{105}{2} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{2 (1-x)^{9/2}}{3 (1+x)^{3/2}}+\frac{6 (1-x)^{7/2}}{\sqrt{1+x}}+\frac{105}{2} \sqrt{1-x} \sqrt{1+x}+\frac{35}{2} (1-x)^{3/2} \sqrt{1+x}+7 (1-x)^{5/2} \sqrt{1+x}+\frac{105}{2} \sin ^{-1}(x)\\ \end{align*}

Mathematica [C]  time = 0.0138066, size = 37, normalized size = 0.36 \[ -\frac{(1-x)^{11/2} \, _2F_1\left (\frac{5}{2},\frac{11}{2};\frac{13}{2};\frac{1-x}{2}\right )}{22 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(9/2)/(1 + x)^(5/2),x]

[Out]

-((1 - x)^(11/2)*Hypergeometric2F1[5/2, 11/2, 13/2, (1 - x)/2])/(22*Sqrt[2])

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Maple [A]  time = 0.018, size = 89, normalized size = 0.9 \begin{align*} -{\frac{2\,{x}^{5}-19\,{x}^{4}+119\,{x}^{3}+577\,{x}^{2}-185\,x-494}{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{105\,\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)^(9/2)/(1+x)^(5/2),x)

[Out]

-1/6*(2*x^5-19*x^4+119*x^3+577*x^2-185*x-494)/(1+x)^(3/2)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2
)+105/2*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.48302, size = 169, normalized size = 1.64 \begin{align*} \frac{x^{6}}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{7 \, x^{5}}{2 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{23 \, x^{4}}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{35}{2} \, 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{143 \, x}{6 \, \sqrt{-x^{2} + 1}} - \frac{127 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{22 \, x}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{247}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{105}{2} \, \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(9/2)/(1+x)^(5/2),x, algorithm="maxima")

[Out]

1/3*x^6/(-x^2 + 1)^(3/2) - 7/2*x^5/(-x^2 + 1)^(3/2) + 23*x^4/(-x^2 + 1)^(3/2) + 35/2*x*(3*x^2/(-x^2 + 1)^(3/2)
- 2/(-x^2 + 1)^(3/2)) - 143/6*x/sqrt(-x^2 + 1) - 127*x^2/(-x^2 + 1)^(3/2) + 22/3*x/(-x^2 + 1)^(3/2) + 247/3/(
-x^2 + 1)^(3/2) + 105/2*arcsin(x)

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Fricas [A]  time = 1.91111, size = 238, normalized size = 2.31 \begin{align*} \frac{494 \, x^{2} +{\left (2 \, x^{4} - 17 \, x^{3} + 102 \, x^{2} + 679 \, x + 494\right )} \sqrt{x + 1} \sqrt{-x + 1} - 630 \,{\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 988 \, x + 494}{6 \,{\left (x^{2} + 2 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(9/2)/(1+x)^(5/2),x, algorithm="fricas")

[Out]

1/6*(494*x^2 + (2*x^4 - 17*x^3 + 102*x^2 + 679*x + 494)*sqrt(x + 1)*sqrt(-x + 1) - 630*(x^2 + 2*x + 1)*arctan(
(sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 988*x + 494)/(x^2 + 2*x + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(9/2)/(1+x)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.19299, size = 171, normalized size = 1.66 \begin{align*} \frac{1}{6} \,{\left ({\left (2 \, x - 23\right )}{\left (x + 1\right )} + 165\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{2 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{3 \,{\left (x + 1\right )}^{\frac{3}{2}}} - \frac{34 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{\sqrt{x + 1}} + \frac{2 \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{51 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{3 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} + 105 \, \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)^(9/2)/(1+x)^(5/2),x, algorithm="giac")

[Out]

1/6*((2*x - 23)*(x + 1) + 165)*sqrt(x + 1)*sqrt(-x + 1) + 2/3*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) - 34*(s
qrt(2) - sqrt(-x + 1))/sqrt(x + 1) + 2/3*(x + 1)^(3/2)*(51*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) - 1)/(sqrt(2) -
sqrt(-x + 1))^3 + 105*arcsin(1/2*sqrt(2)*sqrt(x + 1))