3.11.4 \(\int \frac {\sqrt {1+x}}{(1-x)^{11/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac {2 (x+1)^{3/2}}{315 (1-x)^{3/2}}+\frac {2 (x+1)^{3/2}}{105 (1-x)^{5/2}}+\frac {(x+1)^{3/2}}{21 (1-x)^{7/2}}+\frac {(x+1)^{3/2}}{9 (1-x)^{9/2}} \]

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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 (x+1)^{3/2}}{315 (1-x)^{3/2}}+\frac {2 (x+1)^{3/2}}{105 (1-x)^{5/2}}+\frac {(x+1)^{3/2}}{21 (1-x)^{7/2}}+\frac {(x+1)^{3/2}}{9 (1-x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x]/(1 - x)^(11/2),x]

[Out]

(1 + x)^(3/2)/(9*(1 - x)^(9/2)) + (1 + x)^(3/2)/(21*(1 - x)^(7/2)) + (2*(1 + x)^(3/2))/(105*(1 - x)^(5/2)) + (
2*(1 + x)^(3/2))/(315*(1 - x)^(3/2))

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}}{(1-x)^{11/2}} \, dx &=\frac {(1+x)^{3/2}}{9 (1-x)^{9/2}}+\frac {1}{3} \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx\\ &=\frac {(1+x)^{3/2}}{9 (1-x)^{9/2}}+\frac {(1+x)^{3/2}}{21 (1-x)^{7/2}}+\frac {2}{21} \int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx\\ &=\frac {(1+x)^{3/2}}{9 (1-x)^{9/2}}+\frac {(1+x)^{3/2}}{21 (1-x)^{7/2}}+\frac {2 (1+x)^{3/2}}{105 (1-x)^{5/2}}+\frac {2}{105} \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx\\ &=\frac {(1+x)^{3/2}}{9 (1-x)^{9/2}}+\frac {(1+x)^{3/2}}{21 (1-x)^{7/2}}+\frac {2 (1+x)^{3/2}}{105 (1-x)^{5/2}}+\frac {2 (1+x)^{3/2}}{315 (1-x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 35, normalized size = 0.43 \begin {gather*} \frac {(x+1)^{3/2} \left (-2 x^3+12 x^2-33 x+58\right )}{315 (1-x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x]/(1 - x)^(11/2),x]

[Out]

((1 + x)^(3/2)*(58 - 33*x + 12*x^2 - 2*x^3))/(315*(1 - x)^(9/2))

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IntegrateAlgebraic [A]  time = 0.07, size = 77, normalized size = 0.95 \begin {gather*} \frac {\frac {35 (x+1)^{9/2}}{(1-x)^{9/2}}+\frac {135 (x+1)^{7/2}}{(1-x)^{7/2}}+\frac {189 (x+1)^{5/2}}{(1-x)^{5/2}}+\frac {105 (x+1)^{3/2}}{(1-x)^{3/2}}}{2520} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 + x]/(1 - x)^(11/2),x]

[Out]

((105*(1 + x)^(3/2))/(1 - x)^(3/2) + (189*(1 + x)^(5/2))/(1 - x)^(5/2) + (135*(1 + x)^(7/2))/(1 - x)^(7/2) + (
35*(1 + x)^(9/2))/(1 - x)^(9/2))/2520

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fricas [A]  time = 1.31, size = 85, normalized size = 1.05 \begin {gather*} \frac {58 \, x^{5} - 290 \, x^{4} + 580 \, x^{3} - 580 \, x^{2} + {\left (2 \, x^{4} - 10 \, x^{3} + 21 \, x^{2} - 25 \, x - 58\right )} \sqrt {x + 1} \sqrt {-x + 1} + 290 \, x - 58}{315 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(1/2)/(1-x)^(11/2),x, algorithm="fricas")

[Out]

1/315*(58*x^5 - 290*x^4 + 580*x^3 - 580*x^2 + (2*x^4 - 10*x^3 + 21*x^2 - 25*x - 58)*sqrt(x + 1)*sqrt(-x + 1) +
290*x - 58)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1)

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giac [A]  time = 1.12, size = 35, normalized size = 0.43 \begin {gather*} \frac {{\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 8\right )} + 63\right )} {\left (x + 1\right )} - 105\right )} {\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1}}{315 \, {\left (x - 1\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(1/2)/(1-x)^(11/2),x, algorithm="giac")

[Out]

1/315*((2*(x + 1)*(x - 8) + 63)*(x + 1) - 105)*(x + 1)^(3/2)*sqrt(-x + 1)/(x - 1)^5

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maple [A]  time = 0.00, size = 30, normalized size = 0.37 \begin {gather*} -\frac {\left (x +1\right )^{\frac {3}{2}} \left (2 x^{3}-12 x^{2}+33 x -58\right )}{315 \left (-x +1\right )^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)^(1/2)/(-x+1)^(11/2),x)

[Out]

-1/315*(x+1)^(3/2)*(2*x^3-12*x^2+33*x-58)/(-x+1)^(9/2)

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maxima [B]  time = 1.35, size = 131, normalized size = 1.62 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} + 1}}{9 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{63 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{105 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{315 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{315 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(1/2)/(1-x)^(11/2),x, algorithm="maxima")

[Out]

-2/9*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) - 1/63*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x
+ 1) + 1/105*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 2/315*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 2/315*sqrt(-x^2
+ 1)/(x - 1)

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mupad [B]  time = 0.28, size = 80, normalized size = 0.99 \begin {gather*} -\frac {\sqrt {1-x}\,\left (\frac {5\,x\,\sqrt {x+1}}{63}+\frac {58\,\sqrt {x+1}}{315}-\frac {x^2\,\sqrt {x+1}}{15}+\frac {2\,x^3\,\sqrt {x+1}}{63}-\frac {2\,x^4\,\sqrt {x+1}}{315}\right )}{x^5-5\,x^4+10\,x^3-10\,x^2+5\,x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(1/2)/(1 - x)^(11/2),x)

[Out]

-((1 - x)^(1/2)*((5*x*(x + 1)^(1/2))/63 + (58*(x + 1)^(1/2))/315 - (x^2*(x + 1)^(1/2))/15 + (2*x^3*(x + 1)^(1/
2))/63 - (2*x^4*(x + 1)^(1/2))/315))/(5*x - 10*x^2 + 10*x^3 - 5*x^4 + x^5 - 1)

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sympy [B]  time = 53.78, size = 1562, normalized size = 19.28

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(1/2)/(1-x)**(11/2),x)

[Out]

Piecewise((2*I*(x + 1)**(15/2)/(315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x - 1)*(x + 1)**6 + 26460*sqrt(x - 1)*(
x + 1)**5 - 88200*sqrt(x - 1)*(x + 1)**4 + 176400*sqrt(x - 1)*(x + 1)**3 - 211680*sqrt(x - 1)*(x + 1)**2 + 141
120*sqrt(x - 1)*(x + 1) - 40320*sqrt(x - 1)) - 30*I*(x + 1)**(13/2)/(315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x
- 1)*(x + 1)**6 + 26460*sqrt(x - 1)*(x + 1)**5 - 88200*sqrt(x - 1)*(x + 1)**4 + 176400*sqrt(x - 1)*(x + 1)**3
- 211680*sqrt(x - 1)*(x + 1)**2 + 141120*sqrt(x - 1)*(x + 1) - 40320*sqrt(x - 1)) + 195*I*(x + 1)**(11/2)/(315
*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x - 1)*(x + 1)**6 + 26460*sqrt(x - 1)*(x + 1)**5 - 88200*sqrt(x - 1)*(x +
1)**4 + 176400*sqrt(x - 1)*(x + 1)**3 - 211680*sqrt(x - 1)*(x + 1)**2 + 141120*sqrt(x - 1)*(x + 1) - 40320*sqr
t(x - 1)) - 715*I*(x + 1)**(9/2)/(315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x - 1)*(x + 1)**6 + 26460*sqrt(x - 1)
*(x + 1)**5 - 88200*sqrt(x - 1)*(x + 1)**4 + 176400*sqrt(x - 1)*(x + 1)**3 - 211680*sqrt(x - 1)*(x + 1)**2 + 1
41120*sqrt(x - 1)*(x + 1) - 40320*sqrt(x - 1)) + 1530*I*(x + 1)**(7/2)/(315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt
(x - 1)*(x + 1)**6 + 26460*sqrt(x - 1)*(x + 1)**5 - 88200*sqrt(x - 1)*(x + 1)**4 + 176400*sqrt(x - 1)*(x + 1)*
*3 - 211680*sqrt(x - 1)*(x + 1)**2 + 141120*sqrt(x - 1)*(x + 1) - 40320*sqrt(x - 1)) - 1764*I*(x + 1)**(5/2)/(
315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x - 1)*(x + 1)**6 + 26460*sqrt(x - 1)*(x + 1)**5 - 88200*sqrt(x - 1)*(x
+ 1)**4 + 176400*sqrt(x - 1)*(x + 1)**3 - 211680*sqrt(x - 1)*(x + 1)**2 + 141120*sqrt(x - 1)*(x + 1) - 40320*
sqrt(x - 1)) + 840*I*(x + 1)**(3/2)/(315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x - 1)*(x + 1)**6 + 26460*sqrt(x -
1)*(x + 1)**5 - 88200*sqrt(x - 1)*(x + 1)**4 + 176400*sqrt(x - 1)*(x + 1)**3 - 211680*sqrt(x - 1)*(x + 1)**2
+ 141120*sqrt(x - 1)*(x + 1) - 40320*sqrt(x - 1)), Abs(x + 1)/2 > 1), (-2*(x + 1)**(15/2)/(315*sqrt(1 - x)*(x
+ 1)**7 - 4410*sqrt(1 - x)*(x + 1)**6 + 26460*sqrt(1 - x)*(x + 1)**5 - 88200*sqrt(1 - x)*(x + 1)**4 + 176400*s
qrt(1 - x)*(x + 1)**3 - 211680*sqrt(1 - x)*(x + 1)**2 + 141120*sqrt(1 - x)*(x + 1) - 40320*sqrt(1 - x)) + 30*(
x + 1)**(13/2)/(315*sqrt(1 - x)*(x + 1)**7 - 4410*sqrt(1 - x)*(x + 1)**6 + 26460*sqrt(1 - x)*(x + 1)**5 - 8820
0*sqrt(1 - x)*(x + 1)**4 + 176400*sqrt(1 - x)*(x + 1)**3 - 211680*sqrt(1 - x)*(x + 1)**2 + 141120*sqrt(1 - x)*
(x + 1) - 40320*sqrt(1 - x)) - 195*(x + 1)**(11/2)/(315*sqrt(1 - x)*(x + 1)**7 - 4410*sqrt(1 - x)*(x + 1)**6 +
26460*sqrt(1 - x)*(x + 1)**5 - 88200*sqrt(1 - x)*(x + 1)**4 + 176400*sqrt(1 - x)*(x + 1)**3 - 211680*sqrt(1 -
x)*(x + 1)**2 + 141120*sqrt(1 - x)*(x + 1) - 40320*sqrt(1 - x)) + 715*(x + 1)**(9/2)/(315*sqrt(1 - x)*(x + 1)
**7 - 4410*sqrt(1 - x)*(x + 1)**6 + 26460*sqrt(1 - x)*(x + 1)**5 - 88200*sqrt(1 - x)*(x + 1)**4 + 176400*sqrt(
1 - x)*(x + 1)**3 - 211680*sqrt(1 - x)*(x + 1)**2 + 141120*sqrt(1 - x)*(x + 1) - 40320*sqrt(1 - x)) - 1530*(x
+ 1)**(7/2)/(315*sqrt(1 - x)*(x + 1)**7 - 4410*sqrt(1 - x)*(x + 1)**6 + 26460*sqrt(1 - x)*(x + 1)**5 - 88200*s
qrt(1 - x)*(x + 1)**4 + 176400*sqrt(1 - x)*(x + 1)**3 - 211680*sqrt(1 - x)*(x + 1)**2 + 141120*sqrt(1 - x)*(x
+ 1) - 40320*sqrt(1 - x)) + 1764*(x + 1)**(5/2)/(315*sqrt(1 - x)*(x + 1)**7 - 4410*sqrt(1 - x)*(x + 1)**6 + 26
460*sqrt(1 - x)*(x + 1)**5 - 88200*sqrt(1 - x)*(x + 1)**4 + 176400*sqrt(1 - x)*(x + 1)**3 - 211680*sqrt(1 - x)
*(x + 1)**2 + 141120*sqrt(1 - x)*(x + 1) - 40320*sqrt(1 - x)) - 840*(x + 1)**(3/2)/(315*sqrt(1 - x)*(x + 1)**7
- 4410*sqrt(1 - x)*(x + 1)**6 + 26460*sqrt(1 - x)*(x + 1)**5 - 88200*sqrt(1 - x)*(x + 1)**4 + 176400*sqrt(1 -
x)*(x + 1)**3 - 211680*sqrt(1 - x)*(x + 1)**2 + 141120*sqrt(1 - x)*(x + 1) - 40320*sqrt(1 - x)), True))

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