3.11.45 \(\int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx\)

Optimal. Leaf size=81 \[ \frac {2 \sqrt {x+1}}{35 \sqrt {1-x}}+\frac {2 \sqrt {x+1}}{35 (1-x)^{3/2}}+\frac {3 \sqrt {x+1}}{35 (1-x)^{5/2}}+\frac {\sqrt {x+1}}{7 (1-x)^{7/2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {45, 37} \begin {gather*} \frac {2 \sqrt {x+1}}{35 \sqrt {1-x}}+\frac {2 \sqrt {x+1}}{35 (1-x)^{3/2}}+\frac {3 \sqrt {x+1}}{35 (1-x)^{5/2}}+\frac {\sqrt {x+1}}{7 (1-x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(9/2)*Sqrt[1 + x]),x]

[Out]

Sqrt[1 + x]/(7*(1 - x)^(7/2)) + (3*Sqrt[1 + x])/(35*(1 - x)^(5/2)) + (2*Sqrt[1 + x])/(35*(1 - x)^(3/2)) + (2*S
qrt[1 + x])/(35*Sqrt[1 - 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 {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx &=\frac {\sqrt {1+x}}{7 (1-x)^{7/2}}+\frac {3}{7} \int \frac {1}{(1-x)^{7/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{7 (1-x)^{7/2}}+\frac {3 \sqrt {1+x}}{35 (1-x)^{5/2}}+\frac {6}{35} \int \frac {1}{(1-x)^{5/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{7 (1-x)^{7/2}}+\frac {3 \sqrt {1+x}}{35 (1-x)^{5/2}}+\frac {2 \sqrt {1+x}}{35 (1-x)^{3/2}}+\frac {2}{35} \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{7 (1-x)^{7/2}}+\frac {3 \sqrt {1+x}}{35 (1-x)^{5/2}}+\frac {2 \sqrt {1+x}}{35 (1-x)^{3/2}}+\frac {2 \sqrt {1+x}}{35 \sqrt {1-x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 35, normalized size = 0.43 \begin {gather*} \frac {\sqrt {x+1} \left (-2 x^3+8 x^2-13 x+12\right )}{35 (1-x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(9/2)*Sqrt[1 + x]),x]

[Out]

(Sqrt[1 + x]*(12 - 13*x + 8*x^2 - 2*x^3))/(35*(1 - x)^(7/2))

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IntegrateAlgebraic [A]  time = 0.07, size = 62, normalized size = 0.77 \begin {gather*} \frac {\sqrt {x+1} \left (\frac {5 (x+1)^3}{(1-x)^3}+\frac {21 (x+1)^2}{(1-x)^2}+\frac {35 (x+1)}{1-x}+35\right )}{280 \sqrt {1-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((1 - x)^(9/2)*Sqrt[1 + x]),x]

[Out]

(Sqrt[1 + x]*(35 + (35*(1 + x))/(1 - x) + (21*(1 + x)^2)/(1 - x)^2 + (5*(1 + x)^3)/(1 - x)^3))/(280*Sqrt[1 - x
])

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fricas [A]  time = 1.28, size = 71, normalized size = 0.88 \begin {gather*} \frac {12 \, x^{4} - 48 \, x^{3} + 72 \, x^{2} - {\left (2 \, x^{3} - 8 \, x^{2} + 13 \, x - 12\right )} \sqrt {x + 1} \sqrt {-x + 1} - 48 \, x + 12}{35 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(9/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

1/35*(12*x^4 - 48*x^3 + 72*x^2 - (2*x^3 - 8*x^2 + 13*x - 12)*sqrt(x + 1)*sqrt(-x + 1) - 48*x + 12)/(x^4 - 4*x^
3 + 6*x^2 - 4*x + 1)

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giac [A]  time = 0.66, size = 35, normalized size = 0.43 \begin {gather*} -\frac {{\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 6\right )} + 35\right )} {\left (x + 1\right )} - 35\right )} \sqrt {x + 1} \sqrt {-x + 1}}{35 \, {\left (x - 1\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(9/2)/(1+x)^(1/2),x, algorithm="giac")

[Out]

-1/35*((2*(x + 1)*(x - 6) + 35)*(x + 1) - 35)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^4

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maple [A]  time = 0.00, size = 30, normalized size = 0.37 \begin {gather*} -\frac {\sqrt {x +1}\, \left (2 x^{3}-8 x^{2}+13 x -12\right )}{35 \left (-x +1\right )^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x+1)^(9/2)/(x+1)^(1/2),x)

[Out]

-1/35*(x+1)^(1/2)*(2*x^3-8*x^2+13*x-12)/(-x+1)^(7/2)

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maxima [A]  time = 3.03, size = 95, normalized size = 1.17 \begin {gather*} \frac {\sqrt {-x^{2} + 1}}{7 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac {3 \, \sqrt {-x^{2} + 1}}{35 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{35 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{35 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(9/2)/(1+x)^(1/2),x, algorithm="maxima")

[Out]

1/7*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) - 3/35*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) + 2/35*sqrt(-
x^2 + 1)/(x^2 - 2*x + 1) - 2/35*sqrt(-x^2 + 1)/(x - 1)

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mupad [B]  time = 0.34, size = 67, normalized size = 0.83 \begin {gather*} -\frac {x\,\sqrt {1-x}-12\,\sqrt {1-x}+5\,x^2\,\sqrt {1-x}-6\,x^3\,\sqrt {1-x}+2\,x^4\,\sqrt {1-x}}{35\,{\left (x-1\right )}^4\,\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x)^(9/2)*(x + 1)^(1/2)),x)

[Out]

-(x*(1 - x)^(1/2) - 12*(1 - x)^(1/2) + 5*x^2*(1 - x)^(1/2) - 6*x^3*(1 - x)^(1/2) + 2*x^4*(1 - x)^(1/2))/(35*(x
- 1)^4*(x + 1)^(1/2))

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sympy [C]  time = 22.13, size = 595, normalized size = 7.35 \begin {gather*} \begin {cases} \frac {2 i \left (x + 1\right )^{3}}{35 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 i \sqrt {-1 + \frac {2}{x + 1}}} - \frac {14 i \left (x + 1\right )^{2}}{35 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 i \sqrt {-1 + \frac {2}{x + 1}}} + \frac {35 i \left (x + 1\right )}{35 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 i \sqrt {-1 + \frac {2}{x + 1}}} - \frac {35 i}{35 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 i \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 i \sqrt {-1 + \frac {2}{x + 1}}} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\- \frac {2 \left (x + 1\right )^{3}}{- 35 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} + 210 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} - 420 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) + 280 i \sqrt {1 - \frac {2}{x + 1}}} + \frac {14 \left (x + 1\right )^{2}}{- 35 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} + 210 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} - 420 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) + 280 i \sqrt {1 - \frac {2}{x + 1}}} - \frac {35 \left (x + 1\right )}{- 35 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} + 210 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} - 420 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) + 280 i \sqrt {1 - \frac {2}{x + 1}}} + \frac {35}{- 35 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} + 210 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} - 420 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) + 280 i \sqrt {1 - \frac {2}{x + 1}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)**(9/2)/(1+x)**(1/2),x)

[Out]

Piecewise((2*I*(x + 1)**3/(35*I*sqrt(-1 + 2/(x + 1))*(x + 1)**3 - 210*I*sqrt(-1 + 2/(x + 1))*(x + 1)**2 + 420*
I*sqrt(-1 + 2/(x + 1))*(x + 1) - 280*I*sqrt(-1 + 2/(x + 1))) - 14*I*(x + 1)**2/(35*I*sqrt(-1 + 2/(x + 1))*(x +
1)**3 - 210*I*sqrt(-1 + 2/(x + 1))*(x + 1)**2 + 420*I*sqrt(-1 + 2/(x + 1))*(x + 1) - 280*I*sqrt(-1 + 2/(x + 1
))) + 35*I*(x + 1)/(35*I*sqrt(-1 + 2/(x + 1))*(x + 1)**3 - 210*I*sqrt(-1 + 2/(x + 1))*(x + 1)**2 + 420*I*sqrt(
-1 + 2/(x + 1))*(x + 1) - 280*I*sqrt(-1 + 2/(x + 1))) - 35*I/(35*I*sqrt(-1 + 2/(x + 1))*(x + 1)**3 - 210*I*sqr
t(-1 + 2/(x + 1))*(x + 1)**2 + 420*I*sqrt(-1 + 2/(x + 1))*(x + 1) - 280*I*sqrt(-1 + 2/(x + 1))), 2/Abs(x + 1)
> 1), (-2*(x + 1)**3/(-35*I*sqrt(1 - 2/(x + 1))*(x + 1)**3 + 210*I*sqrt(1 - 2/(x + 1))*(x + 1)**2 - 420*I*sqrt
(1 - 2/(x + 1))*(x + 1) + 280*I*sqrt(1 - 2/(x + 1))) + 14*(x + 1)**2/(-35*I*sqrt(1 - 2/(x + 1))*(x + 1)**3 + 2
10*I*sqrt(1 - 2/(x + 1))*(x + 1)**2 - 420*I*sqrt(1 - 2/(x + 1))*(x + 1) + 280*I*sqrt(1 - 2/(x + 1))) - 35*(x +
1)/(-35*I*sqrt(1 - 2/(x + 1))*(x + 1)**3 + 210*I*sqrt(1 - 2/(x + 1))*(x + 1)**2 - 420*I*sqrt(1 - 2/(x + 1))*(
x + 1) + 280*I*sqrt(1 - 2/(x + 1))) + 35/(-35*I*sqrt(1 - 2/(x + 1))*(x + 1)**3 + 210*I*sqrt(1 - 2/(x + 1))*(x
+ 1)**2 - 420*I*sqrt(1 - 2/(x + 1))*(x + 1) + 280*I*sqrt(1 - 2/(x + 1))), True))

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