3.12.27 \(\int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx\) [1127]

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

[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 = 0.01, 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 = {49, 52, 41, 222} \begin {gather*} \frac {35 \text {ArcSin}(x)}{2}-\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} \end {gather*}

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 49

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 52

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 222

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 [A]
time = 0.08, size = 56, normalized size = 0.64 \begin {gather*} \frac {\sqrt {1-x} \left (164+229 x+30 x^2-3 x^3\right )}{6 (1+x)^{3/2}}-35 \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - x]*(164 + 229*x + 30*x^2 - 3*x^3))/(6*(1 + x)^(3/2)) - 35*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]]

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Maple [A]
time = 0.16, size = 84, normalized size = 0.97

method	result	size
risch	\(\frac {\left (3 x^{4}-33 x^{3}-199 x^{2}+65 x +164\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{6 \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {35 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.49, size = 111, normalized size = 1.28 \begin {gather*} -\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 {gather*}

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 = 1.11, size = 81, normalized size = 0.93 \begin {gather*} \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 {gather*}

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] Result contains complex when optimal does not.
time = 14.61, size = 212, 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 {5}{2}}}{2 \sqrt {x - 1}} + \frac {15 i \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {x - 1}} + \frac {41 i \sqrt {x + 1}}{3 \sqrt {x - 1}} - \frac {176 i}{3 \sqrt {x - 1} \sqrt {x + 1}} + \frac {32 i}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {3}{2}}} & \text {for}\: \left |{x + 1}\right | > 2 \\35 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {1 - x}} - \frac {15 \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {1 - x}} - \frac {41 \sqrt {x + 1}}{3 \sqrt {1 - x}} + \frac {176}{3 \sqrt {1 - x} \sqrt {x + 1}} - \frac {32}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-35*I*acosh(sqrt(2)*sqrt(x + 1)/2) - I*(x + 1)**(5/2)/(2*sqrt(x - 1)) + 15*I*(x + 1)**(3/2)/(2*sqrt
(x - 1)) + 41*I*sqrt(x + 1)/(3*sqrt(x - 1)) - 176*I/(3*sqrt(x - 1)*sqrt(x + 1)) + 32*I/(3*sqrt(x - 1)*(x + 1)*
*(3/2)), Abs(x + 1) > 2), (35*asin(sqrt(2)*sqrt(x + 1)/2) + (x + 1)**(5/2)/(2*sqrt(1 - x)) - 15*(x + 1)**(3/2)
/(2*sqrt(1 - x)) - 41*sqrt(x + 1)/(3*sqrt(1 - x)) + 176/(3*sqrt(1 - x)*sqrt(x + 1)) - 32/(3*sqrt(1 - x)*(x + 1
)**(3/2)), True))

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Giac [A]
time = 1.37, size = 119, normalized size = 1.37 \begin {gather*} -\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 {gather*}

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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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 )}^{5/2}} \,d x \end {gather*}

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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