3.1115 \(\int \frac{1}{(1-x)^{11/2} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=101 \[ \frac{8 \sqrt{x+1}}{315 \sqrt{1-x}}+\frac{8 \sqrt{x+1}}{315 (1-x)^{3/2}}+\frac{4 \sqrt{x+1}}{105 (1-x)^{5/2}}+\frac{4 \sqrt{x+1}}{63 (1-x)^{7/2}}+\frac{\sqrt{x+1}}{9 (1-x)^{9/2}} \]

[Out]

Sqrt[1 + x]/(9*(1 - x)^(9/2)) + (4*Sqrt[1 + x])/(63*(1 - x)^(7/2)) + (4*Sqrt[1 + x])/(105*(1 - x)^(5/2)) + (8*
Sqrt[1 + x])/(315*(1 - x)^(3/2)) + (8*Sqrt[1 + x])/(315*Sqrt[1 - x])

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Rubi [A]  time = 0.0182229, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {45, 37} \[ \frac{8 \sqrt{x+1}}{315 \sqrt{1-x}}+\frac{8 \sqrt{x+1}}{315 (1-x)^{3/2}}+\frac{4 \sqrt{x+1}}{105 (1-x)^{5/2}}+\frac{4 \sqrt{x+1}}{63 (1-x)^{7/2}}+\frac{\sqrt{x+1}}{9 (1-x)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(11/2)*Sqrt[1 + x]),x]

[Out]

Sqrt[1 + x]/(9*(1 - x)^(9/2)) + (4*Sqrt[1 + x])/(63*(1 - x)^(7/2)) + (4*Sqrt[1 + x])/(105*(1 - x)^(5/2)) + (8*
Sqrt[1 + x])/(315*(1 - x)^(3/2)) + (8*Sqrt[1 + x])/(315*Sqrt[1 - x])

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])

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]

Rubi steps

\begin{align*} \int \frac{1}{(1-x)^{11/2} \sqrt{1+x}} \, dx &=\frac{\sqrt{1+x}}{9 (1-x)^{9/2}}+\frac{4}{9} \int \frac{1}{(1-x)^{9/2} \sqrt{1+x}} \, dx\\ &=\frac{\sqrt{1+x}}{9 (1-x)^{9/2}}+\frac{4 \sqrt{1+x}}{63 (1-x)^{7/2}}+\frac{4}{21} \int \frac{1}{(1-x)^{7/2} \sqrt{1+x}} \, dx\\ &=\frac{\sqrt{1+x}}{9 (1-x)^{9/2}}+\frac{4 \sqrt{1+x}}{63 (1-x)^{7/2}}+\frac{4 \sqrt{1+x}}{105 (1-x)^{5/2}}+\frac{8}{105} \int \frac{1}{(1-x)^{5/2} \sqrt{1+x}} \, dx\\ &=\frac{\sqrt{1+x}}{9 (1-x)^{9/2}}+\frac{4 \sqrt{1+x}}{63 (1-x)^{7/2}}+\frac{4 \sqrt{1+x}}{105 (1-x)^{5/2}}+\frac{8 \sqrt{1+x}}{315 (1-x)^{3/2}}+\frac{8}{315} \int \frac{1}{(1-x)^{3/2} \sqrt{1+x}} \, dx\\ &=\frac{\sqrt{1+x}}{9 (1-x)^{9/2}}+\frac{4 \sqrt{1+x}}{63 (1-x)^{7/2}}+\frac{4 \sqrt{1+x}}{105 (1-x)^{5/2}}+\frac{8 \sqrt{1+x}}{315 (1-x)^{3/2}}+\frac{8 \sqrt{1+x}}{315 \sqrt{1-x}}\\ \end{align*}

Mathematica [A]  time = 0.0112795, size = 40, normalized size = 0.4 \[ \frac{\sqrt{x+1} \left (8 x^4-40 x^3+84 x^2-100 x+83\right )}{315 (1-x)^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + x]*(83 - 100*x + 84*x^2 - 40*x^3 + 8*x^4))/(315*(1 - x)^(9/2))

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Maple [A]  time = 0.003, size = 35, normalized size = 0.4 \begin{align*}{\frac{8\,{x}^{4}-40\,{x}^{3}+84\,{x}^{2}-100\,x+83}{315}\sqrt{1+x} \left ( 1-x \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*(1+x)^(1/2)*(8*x^4-40*x^3+84*x^2-100*x+83)/(1-x)^(9/2)

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Maxima [A]  time = 1.50946, size = 177, normalized size = 1.75 \begin{align*} -\frac{\sqrt{-x^{2} + 1}}{9 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac{4 \, \sqrt{-x^{2} + 1}}{63 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac{4 \, \sqrt{-x^{2} + 1}}{105 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac{8 \, \sqrt{-x^{2} + 1}}{315 \,{\left (x^{2} - 2 \, x + 1\right )}} - \frac{8 \, \sqrt{-x^{2} + 1}}{315 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) + 4/63*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x
+ 1) - 4/105*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) + 8/315*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) - 8/315*sqrt(-x^2
+ 1)/(x - 1)

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Fricas [A]  time = 1.84438, size = 225, normalized size = 2.23 \begin{align*} \frac{83 \, x^{5} - 415 \, x^{4} + 830 \, x^{3} - 830 \, x^{2} -{\left (8 \, x^{4} - 40 \, x^{3} + 84 \, x^{2} - 100 \, x + 83\right )} \sqrt{x + 1} \sqrt{-x + 1} + 415 \, x - 83}{315 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*(83*x^5 - 415*x^4 + 830*x^3 - 830*x^2 - (8*x^4 - 40*x^3 + 84*x^2 - 100*x + 83)*sqrt(x + 1)*sqrt(-x + 1)
+ 415*x - 83)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.08185, size = 57, normalized size = 0.56 \begin{align*} -\frac{{\left (4 \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 8\right )} + 63\right )}{\left (x + 1\right )} - 105\right )}{\left (x + 1\right )} + 315\right )} \sqrt{x + 1} \sqrt{-x + 1}}{315 \,{\left (x - 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(4*((2*(x + 1)*(x - 8) + 63)*(x + 1) - 105)*(x + 1) + 315)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^5