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]

-2/3*(1-x)^(7/2)/(1+x)^(3/2)+35/2*arcsin(x)+14/3*(1-x)^(5/2)/(1+x)^(1/2)+35/6*(1-x)^(3/2)*(1+x)^(1/2)+35/2*(1-
x)^(1/2)*(1+x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 87, normalized size of antiderivative = 1.00, 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 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 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 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*}

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Mathematica [C]  time = 0.01, 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]

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

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fricas [A]  time = 0.43, size = 81, normalized size = 0.93 \[ \frac {164 \, x^{2} - {\left (3 \, x^{3} - 30 \, x^{2} - 229 \, x - 164\right )} \sqrt {x + 1} \sqrt {-x + 1} - 210 \, {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 328 \, x + 164}{6 \, {\left (x^{2} + 2 \, x + 1\right )}} \]

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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giac [A]  time = 0.80, size = 119, normalized size = 1.37 \[ -\frac {1}{2} \, \sqrt {x + 1} {\left (x - 12\right )} \sqrt {-x + 1} + \frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{3 \, {\left (x + 1\right )}^{\frac {3}{2}}} - \frac {13 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} + \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {39 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{3 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} + 35 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]

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))

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maple [A]  time = 0.02, size = 84, normalized size = 0.97 \[ \frac {35 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{2 \sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (3 x^{4}-33 x^{3}-199 x^{2}+65 x +164\right ) \sqrt {\left (x +1\right ) \left (-x +1\right )}}{6 \left (x +1\right )^{\frac {3}{2}} \sqrt {-\left (x +1\right ) \left (x -1\right )}\, \sqrt {-x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x+1)^(7/2)/(x+1)^(5/2),x)

[Out]

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

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maxima [A]  time = 3.00, size = 111, normalized size = 1.28 \[ -\frac {x^{5}}{2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {6 \, x^{4}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {35}{6} \, x {\left (\frac {3 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {2}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}\right )} - \frac {61 \, x}{6 \, \sqrt {-x^{2} + 1}} - \frac {44 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {16 \, x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {82}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {35}{2} \, \arcsin \relax (x) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-x\right )}^{7/2}}{{\left (x+1\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(7/2)/(x + 1)^(5/2),x)

[Out]

int((1 - x)^(7/2)/(x + 1)^(5/2), x)

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sympy [C]  time = 17.50, size = 207, normalized size = 2.38 \[ \begin {cases} - \frac {\sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{2} + \frac {13 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{2} + \frac {80 \sqrt {-1 + \frac {2}{x + 1}}}{3} - \frac {16 \sqrt {-1 + \frac {2}{x + 1}}}{3 \left (x + 1\right )} + \frac {35 i \log {\left (\frac {1}{x + 1} \right )}}{2} + \frac {35 i \log {\left (x + 1 \right )}}{2} + 35 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\- \frac {i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{2} + \frac {13 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{2} + \frac {80 i \sqrt {1 - \frac {2}{x + 1}}}{3} - \frac {16 i \sqrt {1 - \frac {2}{x + 1}}}{3 \left (x + 1\right )} + \frac {35 i \log {\left (\frac {1}{x + 1} \right )}}{2} - 35 i \log {\left (\sqrt {1 - \frac {2}{x + 1}} + 1 \right )} & \text {otherwise} \end {cases} \]

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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