3.11.54 \(\int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=62 \[ \frac {2 x}{5 \sqrt {1-x} \sqrt {x+1}}+\frac {1}{5 (1-x)^{3/2} \sqrt {x+1}}+\frac {1}{5 (1-x)^{5/2} \sqrt {x+1}} \]

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Rubi [A]  time = 0.01, antiderivative size = 62, 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, 39} \begin {gather*} \frac {2 x}{5 \sqrt {1-x} \sqrt {x+1}}+\frac {1}{5 (1-x)^{3/2} \sqrt {x+1}}+\frac {1}{5 (1-x)^{5/2} \sqrt {x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(7/2)*(1 + x)^(3/2)),x]

[Out]

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

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

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

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Mathematica [A]  time = 0.01, size = 33, normalized size = 0.53 \begin {gather*} \frac {2 x^3-4 x^2+x+2}{5 (x-1)^2 \sqrt {1-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(7/2)*(1 + x)^(3/2)),x]

[Out]

(2 + x - 4*x^2 + 2*x^3)/(5*(-1 + x)^2*Sqrt[1 - x^2])

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IntegrateAlgebraic [A]  time = 0.08, size = 62, normalized size = 1.00 \begin {gather*} \frac {(x+1)^{5/2} \left (-\frac {5 (1-x)^3}{(x+1)^3}+\frac {15 (1-x)^2}{(x+1)^2}+\frac {5 (1-x)}{x+1}+1\right )}{40 (1-x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((1 - x)^(7/2)*(1 + x)^(3/2)),x]

[Out]

((1 + x)^(5/2)*(1 - (5*(1 - x)^3)/(1 + x)^3 + (15*(1 - x)^2)/(1 + x)^2 + (5*(1 - x))/(1 + x)))/(40*(1 - x)^(5/
2))

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fricas [A]  time = 1.04, size = 59, normalized size = 0.95 \begin {gather*} \frac {2 \, x^{4} - 4 \, x^{3} - {\left (2 \, x^{3} - 4 \, x^{2} + x + 2\right )} \sqrt {x + 1} \sqrt {-x + 1} + 4 \, x - 2}{5 \, {\left (x^{4} - 2 \, x^{3} + 2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(7/2)/(1+x)^(3/2),x, algorithm="fricas")

[Out]

1/5*(2*x^4 - 4*x^3 - (2*x^3 - 4*x^2 + x + 2)*sqrt(x + 1)*sqrt(-x + 1) + 4*x - 2)/(x^4 - 2*x^3 + 2*x - 1)

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giac [A]  time = 0.66, size = 73, normalized size = 1.18 \begin {gather*} \frac {\sqrt {2} - \sqrt {-x + 1}}{16 \, \sqrt {x + 1}} - \frac {\sqrt {x + 1}}{16 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}} - \frac {{\left ({\left (11 \, x - 39\right )} {\left (x + 1\right )} + 60\right )} \sqrt {x + 1} \sqrt {-x + 1}}{40 \, {\left (x - 1\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(7/2)/(1+x)^(3/2),x, algorithm="giac")

[Out]

1/16*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/16*sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) - 1/40*((11*x - 39)*(x +
1) + 60)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^3

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maple [A]  time = 0.00, size = 28, normalized size = 0.45 \begin {gather*} \frac {2 x^{3}-4 x^{2}+x +2}{5 \sqrt {x +1}\, \left (-x +1\right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x+1)^(7/2)/(x+1)^(3/2),x)

[Out]

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

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maxima [A]  time = 1.35, size = 79, normalized size = 1.27 \begin {gather*} \frac {2 \, x}{5 \, \sqrt {-x^{2} + 1}} + \frac {1}{5 \, {\left (\sqrt {-x^{2} + 1} x^{2} - 2 \, \sqrt {-x^{2} + 1} x + \sqrt {-x^{2} + 1}\right )}} - \frac {1}{5 \, {\left (\sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(7/2)/(1+x)^(3/2),x, algorithm="maxima")

[Out]

2/5*x/sqrt(-x^2 + 1) + 1/5/(sqrt(-x^2 + 1)*x^2 - 2*sqrt(-x^2 + 1)*x + sqrt(-x^2 + 1)) - 1/5/(sqrt(-x^2 + 1)*x
- sqrt(-x^2 + 1))

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mupad [B]  time = 0.34, size = 55, normalized size = 0.89 \begin {gather*} -\frac {x\,\sqrt {1-x}+2\,\sqrt {1-x}-4\,x^2\,\sqrt {1-x}+2\,x^3\,\sqrt {1-x}}{5\,{\left (x-1\right )}^3\,\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x)^(7/2)*(x + 1)^(3/2)),x)

[Out]

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

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sympy [B]  time = 16.83, size = 282, normalized size = 4.55 \begin {gather*} \begin {cases} \frac {2 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 60 x - 5 \left (x + 1\right )^{3} + 30 \left (x + 1\right )^{2} - 20} - \frac {10 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 60 x - 5 \left (x + 1\right )^{3} + 30 \left (x + 1\right )^{2} - 20} + \frac {15 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{- 60 x - 5 \left (x + 1\right )^{3} + 30 \left (x + 1\right )^{2} - 20} - \frac {5 \sqrt {-1 + \frac {2}{x + 1}}}{- 60 x - 5 \left (x + 1\right )^{3} + 30 \left (x + 1\right )^{2} - 20} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\\frac {2 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 60 x - 5 \left (x + 1\right )^{3} + 30 \left (x + 1\right )^{2} - 20} - \frac {10 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 60 x - 5 \left (x + 1\right )^{3} + 30 \left (x + 1\right )^{2} - 20} + \frac {15 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 60 x - 5 \left (x + 1\right )^{3} + 30 \left (x + 1\right )^{2} - 20} - \frac {5 i \sqrt {1 - \frac {2}{x + 1}}}{- 60 x - 5 \left (x + 1\right )^{3} + 30 \left (x + 1\right )^{2} - 20} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)**(7/2)/(1+x)**(3/2),x)

[Out]

Piecewise((2*sqrt(-1 + 2/(x + 1))*(x + 1)**3/(-60*x - 5*(x + 1)**3 + 30*(x + 1)**2 - 20) - 10*sqrt(-1 + 2/(x +
1))*(x + 1)**2/(-60*x - 5*(x + 1)**3 + 30*(x + 1)**2 - 20) + 15*sqrt(-1 + 2/(x + 1))*(x + 1)/(-60*x - 5*(x +
1)**3 + 30*(x + 1)**2 - 20) - 5*sqrt(-1 + 2/(x + 1))/(-60*x - 5*(x + 1)**3 + 30*(x + 1)**2 - 20), 2/Abs(x + 1)
> 1), (2*I*sqrt(1 - 2/(x + 1))*(x + 1)**3/(-60*x - 5*(x + 1)**3 + 30*(x + 1)**2 - 20) - 10*I*sqrt(1 - 2/(x +
1))*(x + 1)**2/(-60*x - 5*(x + 1)**3 + 30*(x + 1)**2 - 20) + 15*I*sqrt(1 - 2/(x + 1))*(x + 1)/(-60*x - 5*(x +
1)**3 + 30*(x + 1)**2 - 20) - 5*I*sqrt(1 - 2/(x + 1))/(-60*x - 5*(x + 1)**3 + 30*(x + 1)**2 - 20), True))

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