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

Optimal. Leaf size=87 \[ \frac {1}{4} \sqrt {x+1} (1-x)^{7/2}+\frac {7}{12} \sqrt {x+1} (1-x)^{5/2}+\frac {35}{24} \sqrt {x+1} (1-x)^{3/2}+\frac {35}{8} \sqrt {x+1} \sqrt {1-x}+\frac {35}{8} \sin ^{-1}(x) \]

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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 = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {50, 41, 216} \begin {gather*} \frac {1}{4} \sqrt {x+1} (1-x)^{7/2}+\frac {7}{12} \sqrt {x+1} (1-x)^{5/2}+\frac {35}{24} \sqrt {x+1} (1-x)^{3/2}+\frac {35}{8} \sqrt {x+1} \sqrt {1-x}+\frac {35}{8} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(7/2)/Sqrt[1 + x],x]

[Out]

(35*Sqrt[1 - x]*Sqrt[1 + x])/8 + (35*(1 - x)^(3/2)*Sqrt[1 + x])/24 + (7*(1 - x)^(5/2)*Sqrt[1 + x])/12 + ((1 -
x)^(7/2)*Sqrt[1 + x])/4 + (35*ArcSin[x])/8

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 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}}{\sqrt {1+x}} \, dx &=\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {7}{4} \int \frac {(1-x)^{5/2}}{\sqrt {1+x}} \, dx\\ &=\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{12} \int \frac {(1-x)^{3/2}}{\sqrt {1+x}} \, dx\\ &=\frac {35}{24} (1-x)^{3/2} \sqrt {1+x}+\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{8} \int \frac {\sqrt {1-x}}{\sqrt {1+x}} \, dx\\ &=\frac {35}{8} \sqrt {1-x} \sqrt {1+x}+\frac {35}{24} (1-x)^{3/2} \sqrt {1+x}+\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{8} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {35}{8} \sqrt {1-x} \sqrt {1+x}+\frac {35}{24} (1-x)^{3/2} \sqrt {1+x}+\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {35}{8} \sqrt {1-x} \sqrt {1+x}+\frac {35}{24} (1-x)^{3/2} \sqrt {1+x}+\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{8} \sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 61, normalized size = 0.70 \begin {gather*} \frac {\sqrt {x+1} \left (6 x^4-38 x^3+113 x^2-241 x+160\right )}{24 \sqrt {1-x}}-\frac {35}{4} \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + x]*(160 - 241*x + 113*x^2 - 38*x^3 + 6*x^4))/(24*Sqrt[1 - x]) - (35*ArcSin[Sqrt[1 - x]/Sqrt[2]])/4

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IntegrateAlgebraic [A]  time = 0.07, size = 100, normalized size = 1.15 \begin {gather*} \frac {\sqrt {x+1} \left (\frac {105 (x+1)^3}{(1-x)^3}+\frac {385 (x+1)^2}{(1-x)^2}+\frac {511 (x+1)}{1-x}+279\right )}{12 \sqrt {1-x} \left (\frac {x+1}{1-x}+1\right )^4}+\frac {35}{4} \tan ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {1-x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + x]*(279 + (511*(1 + x))/(1 - x) + (385*(1 + x)^2)/(1 - x)^2 + (105*(1 + x)^3)/(1 - x)^3))/(12*Sqrt[1
- x]*(1 + (1 + x)/(1 - x))^4) + (35*ArcTan[Sqrt[1 + x]/Sqrt[1 - x]])/4

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fricas [A]  time = 1.29, size = 52, normalized size = 0.60 \begin {gather*} -\frac {1}{24} \, {\left (6 \, x^{3} - 32 \, x^{2} + 81 \, x - 160\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {35}{4} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(6*x^3 - 32*x^2 + 81*x - 160)*sqrt(x + 1)*sqrt(-x + 1) - 35/4*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

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giac [A]  time = 0.76, size = 101, normalized size = 1.16 \begin {gather*} -\frac {1}{24} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{2} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {3}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {35}{4} \, \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)^(1/2),x, algorithm="giac")

[Out]

-1/24*((2*(3*x - 10)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt(-x + 1) + 1/2*((2*x - 5)*(x + 1) + 9)*sqrt(x
+ 1)*sqrt(-x + 1) - 3/2*sqrt(x + 1)*(x - 2)*sqrt(-x + 1) + sqrt(x + 1)*sqrt(-x + 1) + 35/4*arcsin(1/2*sqrt(2)
*sqrt(x + 1))

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maple [A]  time = 0.01, size = 85, normalized size = 0.98 \begin {gather*} \frac {35 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{8 \sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (-x +1\right )^{\frac {7}{2}} \sqrt {x +1}}{4}+\frac {7 \left (-x +1\right )^{\frac {5}{2}} \sqrt {x +1}}{12}+\frac {35 \left (-x +1\right )^{\frac {3}{2}} \sqrt {x +1}}{24}+\frac {35 \sqrt {-x +1}\, \sqrt {x +1}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(-x+1)^(7/2)*(x+1)^(1/2)+7/12*(-x+1)^(5/2)*(x+1)^(1/2)+35/24*(-x+1)^(3/2)*(x+1)^(1/2)+35/8*(-x+1)^(1/2)*(x
+1)^(1/2)+35/8*((x+1)*(-x+1))^(1/2)/(x+1)^(1/2)/(-x+1)^(1/2)*arcsin(x)

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maxima [A]  time = 2.99, size = 56, normalized size = 0.64 \begin {gather*} -\frac {1}{4} \, \sqrt {-x^{2} + 1} x^{3} + \frac {4}{3} \, \sqrt {-x^{2} + 1} x^{2} - \frac {27}{8} \, \sqrt {-x^{2} + 1} x + \frac {20}{3} \, \sqrt {-x^{2} + 1} + \frac {35}{8} \, \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(-x^2 + 1)*x^3 + 4/3*sqrt(-x^2 + 1)*x^2 - 27/8*sqrt(-x^2 + 1)*x + 20/3*sqrt(-x^2 + 1) + 35/8*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}}{\sqrt {x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(7/2)/(x + 1)^(1/2),x)

[Out]

int((1 - x)^(7/2)/(x + 1)^(1/2), x)

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sympy [A]  time = 14.68, size = 201, normalized size = 2.31 \begin {gather*} \begin {cases} - \frac {35 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} - \frac {i \left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {x - 1}} + \frac {31 i \left (x + 1\right )^{\frac {7}{2}}}{12 \sqrt {x - 1}} - \frac {263 i \left (x + 1\right )^{\frac {5}{2}}}{24 \sqrt {x - 1}} + \frac {605 i \left (x + 1\right )^{\frac {3}{2}}}{24 \sqrt {x - 1}} - \frac {93 i \sqrt {x + 1}}{4 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- \frac {\sqrt {1 - x} \left (x + 1\right )^{\frac {7}{2}}}{4} + \frac {25 \sqrt {1 - x} \left (x + 1\right )^{\frac {5}{2}}}{12} - \frac {163 \sqrt {1 - x} \left (x + 1\right )^{\frac {3}{2}}}{24} + \frac {93 \sqrt {1 - x} \sqrt {x + 1}}{8} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-35*I*acosh(sqrt(2)*sqrt(x + 1)/2)/4 - I*(x + 1)**(9/2)/(4*sqrt(x - 1)) + 31*I*(x + 1)**(7/2)/(12*s
qrt(x - 1)) - 263*I*(x + 1)**(5/2)/(24*sqrt(x - 1)) + 605*I*(x + 1)**(3/2)/(24*sqrt(x - 1)) - 93*I*sqrt(x + 1)
/(4*sqrt(x - 1)), Abs(x + 1)/2 > 1), (-sqrt(1 - x)*(x + 1)**(7/2)/4 + 25*sqrt(1 - x)*(x + 1)**(5/2)/12 - 163*s
qrt(1 - x)*(x + 1)**(3/2)/24 + 93*sqrt(1 - x)*sqrt(x + 1)/8 + 35*asin(sqrt(2)*sqrt(x + 1)/2)/4, True))

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