3.11.47 \(\int \frac {(1-x)^{7/2}}{(1+x)^{3/2}} \, dx\)

Optimal. Leaf size=85 \[ -\frac {2 (1-x)^{7/2}}{\sqrt {x+1}}-\frac {7}{3} \sqrt {x+1} (1-x)^{5/2}-\frac {35}{6} \sqrt {x+1} (1-x)^{3/2}-\frac {35}{2} \sqrt {x+1} \sqrt {1-x}-\frac {35}{2} \sin ^{-1}(x) \]

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Rubi [A]  time = 0.02, antiderivative size = 85, 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} \begin {gather*} -\frac {2 (1-x)^{7/2}}{\sqrt {x+1}}-\frac {7}{3} \sqrt {x+1} (1-x)^{5/2}-\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) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(7/2)/(1 + x)^(3/2),x]

[Out]

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

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Mathematica [C]  time = 0.01, size = 37, normalized size = 0.44 \begin {gather*} -\frac {(1-x)^{9/2} \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {1-x}{2}\right )}{9 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(7/2)/(1 + x)^(3/2),x]

[Out]

-1/9*((1 - x)^(9/2)*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - x)/2])/Sqrt[2]

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IntegrateAlgebraic [A]  time = 0.10, size = 98, normalized size = 1.15 \begin {gather*} 35 \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right )-\frac {\sqrt {1-x} \left (\frac {48 (1-x)^3}{(x+1)^3}+\frac {231 (1-x)^2}{(x+1)^2}+\frac {280 (1-x)}{x+1}+105\right )}{3 \sqrt {x+1} \left (\frac {1-x}{x+1}+1\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - x)^(7/2)/(1 + x)^(3/2),x]

[Out]

-1/3*(Sqrt[1 - x]*(105 + (48*(1 - x)^3)/(1 + x)^3 + (231*(1 - x)^2)/(1 + x)^2 + (280*(1 - x))/(1 + x)))/(Sqrt[
1 + x]*(1 + (1 - x)/(1 + x))^3) + 35*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]]

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fricas [A]  time = 1.04, size = 65, normalized size = 0.76 \begin {gather*} -\frac {{\left (2 \, x^{3} - 13 \, x^{2} + 55 \, x + 166\right )} \sqrt {x + 1} \sqrt {-x + 1} - 210 \, {\left (x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 166 \, x + 166}{6 \, {\left (x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(7/2)/(1+x)^(3/2),x, algorithm="fricas")

[Out]

-1/6*((2*x^3 - 13*x^2 + 55*x + 166)*sqrt(x + 1)*sqrt(-x + 1) - 210*(x + 1)*arctan((sqrt(x + 1)*sqrt(-x + 1) -
1)/x) + 166*x + 166)/(x + 1)

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giac [A]  time = 0.75, size = 81, normalized size = 0.95 \begin {gather*} -\frac {1}{6} \, {\left ({\left (2 \, x - 17\right )} {\left (x + 1\right )} + 87\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {8 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} - \frac {8 \, \sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} - 35 \, \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)^(7/2)/(1+x)^(3/2),x, algorithm="giac")

[Out]

-1/6*((2*x - 17)*(x + 1) + 87)*sqrt(x + 1)*sqrt(-x + 1) + 8*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 8*sqrt(x +
1)/(sqrt(2) - sqrt(-x + 1)) - 35*arcsin(1/2*sqrt(2)*sqrt(x + 1))

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maple [A]  time = 0.02, size = 84, normalized size = 0.99 \begin {gather*} -\frac {35 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{2 \sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (2 x^{4}-15 x^{3}+68 x^{2}+111 x -166\right ) \sqrt {\left (x +1\right ) \left (-x +1\right )}}{6 \sqrt {-\left (x +1\right ) \left (x -1\right )}\, \sqrt {-x +1}\, \sqrt {x +1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x+1)^(7/2)/(x+1)^(3/2),x)

[Out]

1/6*(2*x^4-15*x^3+68*x^2+111*x-166)/(-(x+1)*(x-1))^(1/2)*((x+1)*(-x+1))^(1/2)/(-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 = 2.84, size = 70, normalized size = 0.82 \begin {gather*} \frac {x^{4}}{3 \, \sqrt {-x^{2} + 1}} - \frac {5 \, x^{3}}{2 \, \sqrt {-x^{2} + 1}} + \frac {34 \, x^{2}}{3 \, \sqrt {-x^{2} + 1}} + \frac {37 \, x}{2 \, \sqrt {-x^{2} + 1}} - \frac {83}{3 \, \sqrt {-x^{2} + 1}} - \frac {35}{2} \, \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(7/2)/(1+x)^(3/2),x, algorithm="maxima")

[Out]

1/3*x^4/sqrt(-x^2 + 1) - 5/2*x^3/sqrt(-x^2 + 1) + 34/3*x^2/sqrt(-x^2 + 1) + 37/2*x/sqrt(-x^2 + 1) - 83/3/sqrt(
-x^2 + 1) - 35/2*arcsin(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-x\right )}^{7/2}}{{\left (x+1\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(7/2)/(x + 1)^(3/2),x)

[Out]

int((1 - x)^(7/2)/(x + 1)^(3/2), x)

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sympy [A]  time = 17.48, size = 207, normalized size = 2.44 \begin {gather*} \begin {cases} 35 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {x - 1}} + \frac {23 i \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {x - 1}} - \frac {125 i \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {x - 1}} + \frac {13 i \sqrt {x + 1}}{\sqrt {x - 1}} + \frac {32 i}{\sqrt {x - 1} \sqrt {x + 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- 35 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {1 - x}} - \frac {23 \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {1 - x}} + \frac {125 \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {1 - x}} - \frac {13 \sqrt {x + 1}}{\sqrt {1 - x}} - \frac {32}{\sqrt {1 - x} \sqrt {x + 1}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(7/2)/(1+x)**(3/2),x)

[Out]

Piecewise((35*I*acosh(sqrt(2)*sqrt(x + 1)/2) - I*(x + 1)**(7/2)/(3*sqrt(x - 1)) + 23*I*(x + 1)**(5/2)/(6*sqrt(
x - 1)) - 125*I*(x + 1)**(3/2)/(6*sqrt(x - 1)) + 13*I*sqrt(x + 1)/sqrt(x - 1) + 32*I/(sqrt(x - 1)*sqrt(x + 1))
, Abs(x + 1)/2 > 1), (-35*asin(sqrt(2)*sqrt(x + 1)/2) + (x + 1)**(7/2)/(3*sqrt(1 - x)) - 23*(x + 1)**(5/2)/(6*
sqrt(1 - x)) + 125*(x + 1)**(3/2)/(6*sqrt(1 - x)) - 13*sqrt(x + 1)/sqrt(1 - x) - 32/(sqrt(1 - x)*sqrt(x + 1)),
True))

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