3.12.35 \(\int \frac {1}{(1-x)^{9/2} (1+x)^{5/2}} \, dx\) [1135]

Optimal. Leaf size=83 \[ \frac {1}{7 (1-x)^{7/2} (1+x)^{3/2}}+\frac {1}{7 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4 x}{21 (1-x)^{3/2} (1+x)^{3/2}}+\frac {8 x}{21 \sqrt {1-x} \sqrt {1+x}} \]

[Out]

1/7/(1-x)^(7/2)/(1+x)^(3/2)+1/7/(1-x)^(5/2)/(1+x)^(3/2)+4/21*x/(1-x)^(3/2)/(1+x)^(3/2)+8/21*x/(1-x)^(1/2)/(1+x
)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {47, 40, 39} \begin {gather*} \frac {8 x}{21 \sqrt {1-x} \sqrt {x+1}}+\frac {4 x}{21 (1-x)^{3/2} (x+1)^{3/2}}+\frac {1}{7 (1-x)^{5/2} (x+1)^{3/2}}+\frac {1}{7 (1-x)^{7/2} (x+1)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(9/2)*(1 + x)^(5/2)),x]

[Out]

1/(7*(1 - x)^(7/2)*(1 + x)^(3/2)) + 1/(7*(1 - x)^(5/2)*(1 + x)^(3/2)) + (4*x)/(21*(1 - x)^(3/2)*(1 + x)^(3/2))
+ (8*x)/(21*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 40

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

Rule 47

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

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Mathematica [A]
time = 0.07, size = 45, normalized size = 0.54 \begin {gather*} \frac {6+9 x-24 x^2+4 x^3+16 x^4-8 x^5}{21 (1-x)^{7/2} (1+x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(9/2)*(1 + x)^(5/2)),x]

[Out]

(6 + 9*x - 24*x^2 + 4*x^3 + 16*x^4 - 8*x^5)/(21*(1 - x)^(7/2)*(1 + x)^(3/2))

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Maple [A]
time = 0.16, size = 86, normalized size = 1.04

method	result	size
gosper	\(-\frac {8 x^{5}-16 x^{4}-4 x^{3}+24 x^{2}-9 x -6}{21 \left (1+x \right )^{\frac {3}{2}} \left (1-x \right )^{\frac {7}{2}}}\)	\(40\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{5}-16 x^{4}-4 x^{3}+24 x^{2}-9 x -6\right )}{21 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}} \left (-1+x \right )^{3} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(66\)
default	\(\frac {1}{7 \left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {1}{7 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {4}{21 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {4}{7 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1-x}}{21 \left (1+x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1-x}}{21 \sqrt {1+x}}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(9/2)/(1+x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/7/(1-x)^(7/2)/(1+x)^(3/2)+1/7/(1-x)^(5/2)/(1+x)^(3/2)+4/21/(1-x)^(3/2)/(1+x)^(3/2)+4/7/(1-x)^(1/2)/(1+x)^(3/
2)-8/21*(1-x)^(1/2)/(1+x)^(3/2)-8/21*(1-x)^(1/2)/(1+x)^(1/2)

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Maxima [A]
time = 0.55, size = 91, normalized size = 1.10 \begin {gather*} \frac {8 \, x}{21 \, \sqrt {-x^{2} + 1}} + \frac {4 \, x}{21 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {1}{7 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2} - 2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} - \frac {1}{7 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x - {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(9/2)/(1+x)^(5/2),x, algorithm="maxima")

[Out]

8/21*x/sqrt(-x^2 + 1) + 4/21*x/(-x^2 + 1)^(3/2) + 1/7/((-x^2 + 1)^(3/2)*x^2 - 2*(-x^2 + 1)^(3/2)*x + (-x^2 + 1
)^(3/2)) - 1/7/((-x^2 + 1)^(3/2)*x - (-x^2 + 1)^(3/2))

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Fricas [A]
time = 1.45, size = 101, normalized size = 1.22 \begin {gather*} \frac {6 \, x^{6} - 12 \, x^{5} - 6 \, x^{4} + 24 \, x^{3} - 6 \, x^{2} - {\left (8 \, x^{5} - 16 \, x^{4} - 4 \, x^{3} + 24 \, x^{2} - 9 \, x - 6\right )} \sqrt {x + 1} \sqrt {-x + 1} - 12 \, x + 6}{21 \, {\left (x^{6} - 2 \, x^{5} - x^{4} + 4 \, x^{3} - x^{2} - 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(9/2)/(1+x)^(5/2),x, algorithm="fricas")

[Out]

1/21*(6*x^6 - 12*x^5 - 6*x^4 + 24*x^3 - 6*x^2 - (8*x^5 - 16*x^4 - 4*x^3 + 24*x^2 - 9*x - 6)*sqrt(x + 1)*sqrt(-
x + 1) - 12*x + 6)/(x^6 - 2*x^5 - x^4 + 4*x^3 - x^2 - 2*x + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 51.38, size = 593, normalized size = 7.14 \begin {gather*} \begin {cases} - \frac {8 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{5}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} + \frac {56 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{4}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} + \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {7 \sqrt {-1 + \frac {2}{x + 1}}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {8 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{5}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} + \frac {56 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{4}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} + \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {35 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {7 i \sqrt {1 - \frac {2}{x + 1}}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} & \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)**(5/2),x)

[Out]

Piecewise((-8*sqrt(-1 + 2/(x + 1))*(x + 1)**5/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672*(
x + 1)**2 + 336) + 56*sqrt(-1 + 2/(x + 1))*(x + 1)**4/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3
- 672*(x + 1)**2 + 336) - 140*sqrt(-1 + 2/(x + 1))*(x + 1)**3/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(
x + 1)**3 - 672*(x + 1)**2 + 336) + 140*sqrt(-1 + 2/(x + 1))*(x + 1)**2/(336*x + 21*(x + 1)**5 - 168*(x + 1)**
4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) - 35*sqrt(-1 + 2/(x + 1))*(x + 1)/(336*x + 21*(x + 1)**5 - 168*(x +
1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) - 7*sqrt(-1 + 2/(x + 1))/(336*x + 21*(x + 1)**5 - 168*(x + 1)*
*4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336), 1/Abs(x + 1) > 1/2), (-8*I*sqrt(1 - 2/(x + 1))*(x + 1)**5/(336*x
+ 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) + 56*I*sqrt(1 - 2/(x + 1))*(x + 1)**
4/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) - 140*I*sqrt(1 - 2/(x + 1))
*(x + 1)**3/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) + 140*I*sqrt(1 -
2/(x + 1))*(x + 1)**2/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) - 35*I*
sqrt(1 - 2/(x + 1))*(x + 1)/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) -
7*I*sqrt(1 - 2/(x + 1))/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336), Tru
e))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (59) = 118\).
time = 1.12, size = 125, normalized size = 1.51 \begin {gather*} \frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{768 \, {\left (x + 1\right )}^{\frac {3}{2}}} + \frac {19 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{256 \, \sqrt {x + 1}} - \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {57 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{768 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} - \frac {{\left ({\left ({\left (79 \, x - 432\right )} {\left (x + 1\right )} + 1120\right )} {\left (x + 1\right )} - 840\right )} \sqrt {x + 1} \sqrt {-x + 1}}{336 \, {\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)^(5/2),x, algorithm="giac")

[Out]

1/768*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 19/256*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/768*(x + 1)^(
3/2)*(57*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(2) - sqrt(-x + 1))^3 - 1/336*(((79*x - 432)*(x + 1) + 1
120)*(x + 1) - 840)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^4

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Mupad [B]
time = 0.41, size = 86, normalized size = 1.04 \begin {gather*} \frac {9\,x\,\sqrt {1-x}+6\,\sqrt {1-x}-24\,x^2\,\sqrt {1-x}+4\,x^3\,\sqrt {1-x}+16\,x^4\,\sqrt {1-x}-8\,x^5\,\sqrt {1-x}}{\left (21\,x+21\right )\,{\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)^(5/2)),x)

[Out]

(9*x*(1 - x)^(1/2) + 6*(1 - x)^(1/2) - 24*x^2*(1 - x)^(1/2) + 4*x^3*(1 - x)^(1/2) + 16*x^4*(1 - x)^(1/2) - 8*x
^5*(1 - x)^(1/2))/((21*x + 21)*(x - 1)^4*(x + 1)^(1/2))

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