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

Optimal. Leaf size=61 \[ \frac {2 (x+1)^{3/2}}{105 (1-x)^{3/2}}+\frac {2 (x+1)^{3/2}}{35 (1-x)^{5/2}}+\frac {(x+1)^{3/2}}{7 (1-x)^{7/2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, 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}}{105 (1-x)^{3/2}}+\frac {2 (x+1)^{3/2}}{35 (1-x)^{5/2}}+\frac {(x+1)^{3/2}}{7 (1-x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x]/(1 - x)^(9/2),x]

[Out]

(1 + x)^(3/2)/(7*(1 - x)^(7/2)) + (2*(1 + x)^(3/2))/(35*(1 - x)^(5/2)) + (2*(1 + x)^(3/2))/(105*(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)^{9/2}} \, dx &=\frac {(1+x)^{3/2}}{7 (1-x)^{7/2}}+\frac {2}{7} \int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx\\ &=\frac {(1+x)^{3/2}}{7 (1-x)^{7/2}}+\frac {2 (1+x)^{3/2}}{35 (1-x)^{5/2}}+\frac {2}{35} \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx\\ &=\frac {(1+x)^{3/2}}{7 (1-x)^{7/2}}+\frac {2 (1+x)^{3/2}}{35 (1-x)^{5/2}}+\frac {2 (1+x)^{3/2}}{105 (1-x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.49 \begin {gather*} \frac {(x+1)^{3/2} \left (2 x^2-10 x+23\right )}{105 (1-x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x)^(3/2)*(23 - 10*x + 2*x^2))/(105*(1 - x)^(7/2))

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IntegrateAlgebraic [A]  time = 0.07, size = 48, normalized size = 0.79 \begin {gather*} \frac {(x+1)^{3/2} \left (\frac {15 (x+1)^2}{(1-x)^2}+\frac {42 (x+1)}{1-x}+35\right )}{420 (1-x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x)^(3/2)*(35 + (42*(1 + x))/(1 - x) + (15*(1 + x)^2)/(1 - x)^2))/(420*(1 - x)^(3/2))

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fricas [A]  time = 1.17, size = 70, normalized size = 1.15 \begin {gather*} \frac {23 \, x^{4} - 92 \, x^{3} + 138 \, x^{2} + {\left (2 \, x^{3} - 8 \, x^{2} + 13 \, x + 23\right )} \sqrt {x + 1} \sqrt {-x + 1} - 92 \, x + 23}{105 \, {\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+x)^(1/2)/(1-x)^(9/2),x, algorithm="fricas")

[Out]

1/105*(23*x^4 - 92*x^3 + 138*x^2 + (2*x^3 - 8*x^2 + 13*x + 23)*sqrt(x + 1)*sqrt(-x + 1) - 92*x + 23)/(x^4 - 4*
x^3 + 6*x^2 - 4*x + 1)

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giac [A]  time = 1.22, size = 29, normalized size = 0.48 \begin {gather*} \frac {{\left (2 \, {\left (x + 1\right )} {\left (x - 6\right )} + 35\right )} {\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1}}{105 \, {\left (x - 1\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(2*(x + 1)*(x - 6) + 35)*(x + 1)^(3/2)*sqrt(-x + 1)/(x - 1)^4

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maple [A]  time = 0.00, size = 25, normalized size = 0.41 \begin {gather*} \frac {\left (x +1\right )^{\frac {3}{2}} \left (2 x^{2}-10 x +23\right )}{105 \left (-x +1\right )^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(x+1)^(3/2)*(2*x^2-10*x+23)/(-x+1)^(7/2)

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maxima [B]  time = 1.27, size = 95, normalized size = 1.56 \begin {gather*} \frac {2 \, \sqrt {-x^{2} + 1}}{7 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{35 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{105 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{105 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 1/35*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 2/105*sqrt(
-x^2 + 1)/(x^2 - 2*x + 1) + 2/105*sqrt(-x^2 + 1)/(x - 1)

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mupad [B]  time = 0.27, size = 64, normalized size = 1.05 \begin {gather*} \frac {\sqrt {1-x}\,\left (\frac {13\,x\,\sqrt {x+1}}{105}+\frac {23\,\sqrt {x+1}}{105}-\frac {8\,x^2\,\sqrt {x+1}}{105}+\frac {2\,x^3\,\sqrt {x+1}}{105}\right )}{x^4-4\,x^3+6\,x^2-4\,x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(1/2)/(1 - x)^(9/2),x)

[Out]

((1 - x)^(1/2)*((13*x*(x + 1)^(1/2))/105 + (23*(x + 1)^(1/2))/105 - (8*x^2*(x + 1)^(1/2))/105 + (2*x^3*(x + 1)
^(1/2))/105))/(6*x^2 - 4*x - 4*x^3 + x^4 + 1)

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sympy [B]  time = 19.92, size = 568, normalized size = 9.31 \begin {gather*} \begin {cases} \frac {2 i \left (x + 1\right )^{\frac {9}{2}}}{105 \sqrt {x - 1} \left (x + 1\right )^{4} - 840 \sqrt {x - 1} \left (x + 1\right )^{3} + 2520 \sqrt {x - 1} \left (x + 1\right )^{2} - 3360 \sqrt {x - 1} \left (x + 1\right ) + 1680 \sqrt {x - 1}} - \frac {18 i \left (x + 1\right )^{\frac {7}{2}}}{105 \sqrt {x - 1} \left (x + 1\right )^{4} - 840 \sqrt {x - 1} \left (x + 1\right )^{3} + 2520 \sqrt {x - 1} \left (x + 1\right )^{2} - 3360 \sqrt {x - 1} \left (x + 1\right ) + 1680 \sqrt {x - 1}} + \frac {63 i \left (x + 1\right )^{\frac {5}{2}}}{105 \sqrt {x - 1} \left (x + 1\right )^{4} - 840 \sqrt {x - 1} \left (x + 1\right )^{3} + 2520 \sqrt {x - 1} \left (x + 1\right )^{2} - 3360 \sqrt {x - 1} \left (x + 1\right ) + 1680 \sqrt {x - 1}} - \frac {70 i \left (x + 1\right )^{\frac {3}{2}}}{105 \sqrt {x - 1} \left (x + 1\right )^{4} - 840 \sqrt {x - 1} \left (x + 1\right )^{3} + 2520 \sqrt {x - 1} \left (x + 1\right )^{2} - 3360 \sqrt {x - 1} \left (x + 1\right ) + 1680 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- \frac {2 \left (x + 1\right )^{\frac {9}{2}}}{105 \sqrt {1 - x} \left (x + 1\right )^{4} - 840 \sqrt {1 - x} \left (x + 1\right )^{3} + 2520 \sqrt {1 - x} \left (x + 1\right )^{2} - 3360 \sqrt {1 - x} \left (x + 1\right ) + 1680 \sqrt {1 - x}} + \frac {18 \left (x + 1\right )^{\frac {7}{2}}}{105 \sqrt {1 - x} \left (x + 1\right )^{4} - 840 \sqrt {1 - x} \left (x + 1\right )^{3} + 2520 \sqrt {1 - x} \left (x + 1\right )^{2} - 3360 \sqrt {1 - x} \left (x + 1\right ) + 1680 \sqrt {1 - x}} - \frac {63 \left (x + 1\right )^{\frac {5}{2}}}{105 \sqrt {1 - x} \left (x + 1\right )^{4} - 840 \sqrt {1 - x} \left (x + 1\right )^{3} + 2520 \sqrt {1 - x} \left (x + 1\right )^{2} - 3360 \sqrt {1 - x} \left (x + 1\right ) + 1680 \sqrt {1 - x}} + \frac {70 \left (x + 1\right )^{\frac {3}{2}}}{105 \sqrt {1 - x} \left (x + 1\right )^{4} - 840 \sqrt {1 - x} \left (x + 1\right )^{3} + 2520 \sqrt {1 - x} \left (x + 1\right )^{2} - 3360 \sqrt {1 - x} \left (x + 1\right ) + 1680 \sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*I*(x + 1)**(9/2)/(105*sqrt(x - 1)*(x + 1)**4 - 840*sqrt(x - 1)*(x + 1)**3 + 2520*sqrt(x - 1)*(x +
1)**2 - 3360*sqrt(x - 1)*(x + 1) + 1680*sqrt(x - 1)) - 18*I*(x + 1)**(7/2)/(105*sqrt(x - 1)*(x + 1)**4 - 840*
sqrt(x - 1)*(x + 1)**3 + 2520*sqrt(x - 1)*(x + 1)**2 - 3360*sqrt(x - 1)*(x + 1) + 1680*sqrt(x - 1)) + 63*I*(x
+ 1)**(5/2)/(105*sqrt(x - 1)*(x + 1)**4 - 840*sqrt(x - 1)*(x + 1)**3 + 2520*sqrt(x - 1)*(x + 1)**2 - 3360*sqrt
(x - 1)*(x + 1) + 1680*sqrt(x - 1)) - 70*I*(x + 1)**(3/2)/(105*sqrt(x - 1)*(x + 1)**4 - 840*sqrt(x - 1)*(x + 1
)**3 + 2520*sqrt(x - 1)*(x + 1)**2 - 3360*sqrt(x - 1)*(x + 1) + 1680*sqrt(x - 1)), Abs(x + 1)/2 > 1), (-2*(x +
1)**(9/2)/(105*sqrt(1 - x)*(x + 1)**4 - 840*sqrt(1 - x)*(x + 1)**3 + 2520*sqrt(1 - x)*(x + 1)**2 - 3360*sqrt(
1 - x)*(x + 1) + 1680*sqrt(1 - x)) + 18*(x + 1)**(7/2)/(105*sqrt(1 - x)*(x + 1)**4 - 840*sqrt(1 - x)*(x + 1)**
3 + 2520*sqrt(1 - x)*(x + 1)**2 - 3360*sqrt(1 - x)*(x + 1) + 1680*sqrt(1 - x)) - 63*(x + 1)**(5/2)/(105*sqrt(1
- x)*(x + 1)**4 - 840*sqrt(1 - x)*(x + 1)**3 + 2520*sqrt(1 - x)*(x + 1)**2 - 3360*sqrt(1 - x)*(x + 1) + 1680*
sqrt(1 - x)) + 70*(x + 1)**(3/2)/(105*sqrt(1 - x)*(x + 1)**4 - 840*sqrt(1 - x)*(x + 1)**3 + 2520*sqrt(1 - x)*(
x + 1)**2 - 3360*sqrt(1 - x)*(x + 1) + 1680*sqrt(1 - x)), True))

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