3.12.6 \(\int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx\) [1106]

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

[Out]

35/8*arcsin(x)+35/24*(1-x)^(3/2)*(1+x)^(1/2)+7/12*(1-x)^(5/2)*(1+x)^(1/2)+1/4*(1-x)^(7/2)*(1+x)^(1/2)+35/8*(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 = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {52, 41, 222} \begin {gather*} \frac {35 \text {ArcSin}(x)}{8}+\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} \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 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}}{\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 = 0.07, size = 63, normalized size = 0.72 \begin {gather*} \frac {\sqrt {1+x} \left (160-241 x+113 x^2-38 x^3+6 x^4\right )}{24 \sqrt {1-x}}+\frac {35}{4} \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\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*ArcTan[Sqrt[1 + x]/Sqrt[1 - x]])
/4

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Maple [A]
time = 0.14, size = 85, normalized size = 0.98

method	result	size
risch	\(\frac {\left (6 x^{3}-32 x^{2}+81 x -160\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{24 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {35 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\)	\(82\)
default	\(\frac {\left (1-x \right )^{\frac {7}{2}} \sqrt {1+x}}{4}+\frac {7 \left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}{12}+\frac {35 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}{24}+\frac {35 \sqrt {1-x}\, \sqrt {1+x}}{8}+\frac {35 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, 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 \left (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="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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Fricas [A]
time = 1.04, 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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Sympy [C] Result contains complex when optimal does not.
time = 14.70, size = 197, normalized size = 2.26 \begin {gather*} \begin {cases} - \frac {i \sqrt {x - 1} \left (x + 1\right )^{\frac {7}{2}}}{4} + \frac {25 i \sqrt {x - 1} \left (x + 1\right )^{\frac {5}{2}}}{12} - \frac {163 i \sqrt {x - 1} \left (x + 1\right )^{\frac {3}{2}}}{24} + \frac {93 i \sqrt {x - 1} \sqrt {x + 1}}{8} - \frac {35 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {35 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} + \frac {\left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {1 - x}} - \frac {31 \left (x + 1\right )^{\frac {7}{2}}}{12 \sqrt {1 - x}} + \frac {263 \left (x + 1\right )^{\frac {5}{2}}}{24 \sqrt {1 - x}} - \frac {605 \left (x + 1\right )^{\frac {3}{2}}}{24 \sqrt {1 - x}} + \frac {93 \sqrt {x + 1}}{4 \sqrt {1 - x}} & \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((-I*sqrt(x - 1)*(x + 1)**(7/2)/4 + 25*I*sqrt(x - 1)*(x + 1)**(5/2)/12 - 163*I*sqrt(x - 1)*(x + 1)**(
3/2)/24 + 93*I*sqrt(x - 1)*sqrt(x + 1)/8 - 35*I*acosh(sqrt(2)*sqrt(x + 1)/2)/4, Abs(x + 1) > 2), (35*asin(sqrt
(2)*sqrt(x + 1)/2)/4 + (x + 1)**(9/2)/(4*sqrt(1 - x)) - 31*(x + 1)**(7/2)/(12*sqrt(1 - x)) + 263*(x + 1)**(5/2
)/(24*sqrt(1 - x)) - 605*(x + 1)**(3/2)/(24*sqrt(1 - x)) + 93*sqrt(x + 1)/(4*sqrt(1 - x)), True))

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Giac [A]
time = 1.27, 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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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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