3.28.87 \(\int \frac {\sqrt {1-x}}{8 (1+x)^{7/2}} \, dx\)

Optimal. Leaf size=269 \[ \frac {\left (\frac {1}{960}-\frac {\sqrt {1-x}}{960}\right ) \left (\sqrt {x+1}-1\right )^9 \left (\frac {15 \left (\sqrt {1-x}-1\right )^8}{\left (\sqrt {x+1}-1\right )^8}-\frac {30 \left (\sqrt {1-x}-1\right )^7}{\left (\sqrt {x+1}-1\right )^7}+\frac {40 \left (\sqrt {1-x}-1\right )^6}{\left (\sqrt {x+1}-1\right )^6}+\frac {50 \left (\sqrt {1-x}-1\right )^5}{\left (\sqrt {x+1}-1\right )^5}-\frac {46 \left (\sqrt {1-x}-1\right )^4}{\left (\sqrt {x+1}-1\right )^4}-\frac {50 \left (\sqrt {1-x}-1\right )^3}{\left (\sqrt {x+1}-1\right )^3}+\frac {40 \left (\sqrt {1-x}-1\right )^2}{\left (\sqrt {x+1}-1\right )^2}+\frac {30 \sqrt {1-x}-30}{\sqrt {x+1}-1}+15\right )}{\left (x+\sqrt {1-x} \sqrt {x+1}-2 \sqrt {x+1}+1\right )^5} \]

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Rubi [A]  time = 0.01, antiderivative size = 41, normalized size of antiderivative = 0.15, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {12, 45, 37} \begin {gather*} -\frac {(1-x)^{3/2}}{120 (x+1)^{3/2}}-\frac {(1-x)^{3/2}}{40 (x+1)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x]/(8*(1 + x)^(7/2)),x]

[Out]

-1/40*(1 - x)^(3/2)/(1 + x)^(5/2) - (1 - x)^(3/2)/(120*(1 + x)^(3/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {\sqrt {1-x}}{8 (1+x)^{7/2}} \, dx &=\frac {1}{8} \int \frac {\sqrt {1-x}}{(1+x)^{7/2}} \, dx\\ &=-\frac {(1-x)^{3/2}}{40 (1+x)^{5/2}}+\frac {1}{40} \int \frac {\sqrt {1-x}}{(1+x)^{5/2}} \, dx\\ &=-\frac {(1-x)^{3/2}}{40 (1+x)^{5/2}}-\frac {(1-x)^{3/2}}{120 (1+x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.09 \begin {gather*} -\frac {(1-x)^{3/2} (x+4)}{120 (x+1)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x]/(8*(1 + x)^(7/2)),x]

[Out]

-1/120*((1 - x)^(3/2)*(4 + x))/(1 + x)^(5/2)

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IntegrateAlgebraic [A]  time = 0.06, size = 34, normalized size = 0.13 \begin {gather*} -\frac {(1-x)^{3/2} \left (5+\frac {3 (1-x)}{1+x}\right )}{240 (1+x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 - x]/(8*(1 + x)^(7/2)),x]

[Out]

-1/240*((1 - x)^(3/2)*(5 + (3*(1 - x))/(1 + x)))/(1 + x)^(3/2)

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fricas [A]  time = 0.46, size = 54, normalized size = 0.20 \begin {gather*} -\frac {4 \, x^{3} + 12 \, x^{2} - {\left (x^{2} + 3 \, x - 4\right )} \sqrt {x + 1} \sqrt {-x + 1} + 12 \, x + 4}{120 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(1-x)^(1/2)/(1+x)^(7/2),x, algorithm="fricas")

[Out]

-1/120*(4*x^3 + 12*x^2 - (x^2 + 3*x - 4)*sqrt(x + 1)*sqrt(-x + 1) + 12*x + 4)/(x^3 + 3*x^2 + 3*x + 1)

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giac [A]  time = 0.19, size = 133, normalized size = 0.49 \begin {gather*} \frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{5}}{2560 \, {\left (x + 1\right )}^{\frac {5}{2}}} + \frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{1536 \, {\left (x + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {2} - \sqrt {-x + 1}}{256 \, \sqrt {x + 1}} + \frac {{\left (\frac {30 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{4}}{{\left (x + 1\right )}^{2}} - \frac {5 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} - 3\right )} {\left (x + 1\right )}^{\frac {5}{2}}}{7680 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(1-x)^(1/2)/(1+x)^(7/2),x, algorithm="giac")

[Out]

1/2560*(sqrt(2) - sqrt(-x + 1))^5/(x + 1)^(5/2) + 1/1536*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) - 1/256*(sqr
t(2) - sqrt(-x + 1))/sqrt(x + 1) + 1/7680*(30*(sqrt(2) - sqrt(-x + 1))^4/(x + 1)^2 - 5*(sqrt(2) - sqrt(-x + 1)
)^2/(x + 1) - 3)*(x + 1)^(5/2)/(sqrt(2) - sqrt(-x + 1))^5

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maple [A]  time = 0.19, size = 18, normalized size = 0.07

method	result	size
gosper	\(-\frac {\left (x +4\right ) \left (1-x \right )^{\frac {3}{2}}}{120 \left (1+x \right )^{\frac {5}{2}}}\)	\(18\)
default	\(-\frac {\sqrt {1-x}}{20 \left (1+x \right )^{\frac {5}{2}}}+\frac {\sqrt {1-x}}{120 \left (1+x \right )^{\frac {3}{2}}}+\frac {\sqrt {1-x}}{120 \sqrt {1+x}}\)	\(44\)
risch	\(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{3}+2 x^{2}-7 x +4\right )}{120 \sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/8*(1-x)^(1/2)/(1+x)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/120*(x+4)*(1-x)^(3/2)/(1+x)^(5/2)

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maxima [A]  time = 0.31, size = 64, normalized size = 0.24 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{20 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{120 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{120 \, {\left (x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(1-x)^(1/2)/(1+x)^(7/2),x, algorithm="maxima")

[Out]

-1/20*sqrt(-x^2 + 1)/(x^3 + 3*x^2 + 3*x + 1) + 1/120*sqrt(-x^2 + 1)/(x^2 + 2*x + 1) + 1/120*sqrt(-x^2 + 1)/(x
+ 1)

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mupad [B]  time = 2.13, size = 49, normalized size = 0.18 \begin {gather*} \frac {3\,x\,\sqrt {1-x}-4\,\sqrt {1-x}+x^2\,\sqrt {1-x}}{\sqrt {x+1}\,\left (120\,x^2+240\,x+120\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(1/2)/(8*(x + 1)^(7/2)),x)

[Out]

(3*x*(1 - x)^(1/2) - 4*(1 - x)^(1/2) + x^2*(1 - x)^(1/2))/((x + 1)^(1/2)*(240*x + 120*x^2 + 120))

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sympy [A]  time = 5.16, size = 162, normalized size = 0.60 \begin {gather*} \frac {\begin {cases} \frac {\sqrt {-1 + \frac {2}{x + 1}}}{15} + \frac {\sqrt {-1 + \frac {2}{x + 1}}}{15 \left (x + 1\right )} - \frac {2 \sqrt {-1 + \frac {2}{x + 1}}}{5 \left (x + 1\right )^{2}} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\\frac {i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 30 x + 15 \left (x + 1\right )^{2} - 30} - \frac {i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 30 x + 15 \left (x + 1\right )^{2} - 30} - \frac {8 i \sqrt {1 - \frac {2}{x + 1}}}{- 30 x + 15 \left (x + 1\right )^{2} - 30} + \frac {12 i \sqrt {1 - \frac {2}{x + 1}}}{\left (x + 1\right ) \left (- 30 x + 15 \left (x + 1\right )^{2} - 30\right )} & \text {otherwise} \end {cases}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(1-x)**(1/2)/(1+x)**(7/2),x)

[Out]

Piecewise((sqrt(-1 + 2/(x + 1))/15 + sqrt(-1 + 2/(x + 1))/(15*(x + 1)) - 2*sqrt(-1 + 2/(x + 1))/(5*(x + 1)**2)
, 2/Abs(x + 1) > 1), (I*sqrt(1 - 2/(x + 1))*(x + 1)**2/(-30*x + 15*(x + 1)**2 - 30) - I*sqrt(1 - 2/(x + 1))*(x
+ 1)/(-30*x + 15*(x + 1)**2 - 30) - 8*I*sqrt(1 - 2/(x + 1))/(-30*x + 15*(x + 1)**2 - 30) + 12*I*sqrt(1 - 2/(x
+ 1))/((x + 1)*(-30*x + 15*(x + 1)**2 - 30)), True))/8

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