3.11.85 \(\int \frac {(1+x)^{3/2}}{(1-x)^{11/2}} \, dx\) [1085]

Optimal. Leaf size=61 \[ \frac {(1+x)^{5/2}}{9 (1-x)^{9/2}}+\frac {2 (1+x)^{5/2}}{63 (1-x)^{7/2}}+\frac {2 (1+x)^{5/2}}{315 (1-x)^{5/2}} \]

[Out]

1/9*(1+x)^(5/2)/(1-x)^(9/2)+2/63*(1+x)^(5/2)/(1-x)^(7/2)+2/315*(1+x)^(5/2)/(1-x)^(5/2)

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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 = {47, 37} \begin {gather*} \frac {2 (x+1)^{5/2}}{315 (1-x)^{5/2}}+\frac {2 (x+1)^{5/2}}{63 (1-x)^{7/2}}+\frac {(x+1)^{5/2}}{9 (1-x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(3/2)/(1 - x)^(11/2),x]

[Out]

(1 + x)^(5/2)/(9*(1 - x)^(9/2)) + (2*(1 + x)^(5/2))/(63*(1 - x)^(7/2)) + (2*(1 + x)^(5/2))/(315*(1 - x)^(5/2))

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

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Mathematica [A]
time = 0.06, size = 30, normalized size = 0.49 \begin {gather*} \frac {(1+x)^{5/2} \left (47-14 x+2 x^2\right )}{315 (1-x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(3/2)/(1 - x)^(11/2),x]

[Out]

((1 + x)^(5/2)*(47 - 14*x + 2*x^2))/(315*(1 - x)^(9/2))

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Maple [A]
time = 0.18, size = 86, normalized size = 1.41

method	result	size
gosper	\(\frac {\left (1+x \right )^{\frac {5}{2}} \left (2 x^{2}-14 x +47\right )}{315 \left (1-x \right )^{\frac {9}{2}}}\)	\(25\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{5}-8 x^{4}+11 x^{3}+101 x^{2}+127 x +47\right )}{315 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{4} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(66\)
default	\(\frac {\left (1+x \right )^{\frac {3}{2}}}{3 \left (1-x \right )^{\frac {9}{2}}}-\frac {2 \sqrt {1+x}}{9 \left (1-x \right )^{\frac {9}{2}}}+\frac {\sqrt {1+x}}{63 \left (1-x \right )^{\frac {7}{2}}}+\frac {\sqrt {1+x}}{105 \left (1-x \right )^{\frac {5}{2}}}+\frac {2 \sqrt {1+x}}{315 \left (1-x \right )^{\frac {3}{2}}}+\frac {2 \sqrt {1+x}}{315 \sqrt {1-x}}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(3/2)/(1-x)^(11/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(1+x)^(3/2)/(1-x)^(9/2)-2/9*(1+x)^(1/2)/(1-x)^(9/2)+1/63*(1+x)^(1/2)/(1-x)^(7/2)+1/105*(1+x)^(1/2)/(1-x)^(
5/2)+2/315*(1+x)^(1/2)/(1-x)^(3/2)+2/315*(1+x)^(1/2)/(1-x)^(1/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (43) = 86\).
time = 0.28, size = 172, normalized size = 2.82 \begin {gather*} \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \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)^(3/2)/(1-x)^(11/2),x, algorithm="maxima")

[Out]

1/3*(-x^2 + 1)^(3/2)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1) + 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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Fricas [A]
time = 0.80, size = 86, normalized size = 1.41 \begin {gather*} \frac {47 \, x^{5} - 235 \, x^{4} + 470 \, x^{3} - 470 \, x^{2} - {\left (2 \, x^{4} - 10 \, x^{3} + 21 \, x^{2} + 80 \, x + 47\right )} \sqrt {x + 1} \sqrt {-x + 1} + 235 \, x - 47}{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)^(3/2)/(1-x)^(11/2),x, algorithm="fricas")

[Out]

1/315*(47*x^5 - 235*x^4 + 470*x^3 - 470*x^2 - (2*x^4 - 10*x^3 + 21*x^2 + 80*x + 47)*sqrt(x + 1)*sqrt(-x + 1) +
235*x - 47)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1)

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Sympy [C] Result contains complex when optimal does not.
time = 46.30, size = 675, normalized size = 11.07 \begin {gather*} \begin {cases} - \frac {2 i \left (x + 1\right )^{\frac {11}{2}}}{315 \sqrt {x - 1} \left (x + 1\right )^{5} - 3150 \sqrt {x - 1} \left (x + 1\right )^{4} + 12600 \sqrt {x - 1} \left (x + 1\right )^{3} - 25200 \sqrt {x - 1} \left (x + 1\right )^{2} + 25200 \sqrt {x - 1} \left (x + 1\right ) - 10080 \sqrt {x - 1}} + \frac {22 i \left (x + 1\right )^{\frac {9}{2}}}{315 \sqrt {x - 1} \left (x + 1\right )^{5} - 3150 \sqrt {x - 1} \left (x + 1\right )^{4} + 12600 \sqrt {x - 1} \left (x + 1\right )^{3} - 25200 \sqrt {x - 1} \left (x + 1\right )^{2} + 25200 \sqrt {x - 1} \left (x + 1\right ) - 10080 \sqrt {x - 1}} - \frac {99 i \left (x + 1\right )^{\frac {7}{2}}}{315 \sqrt {x - 1} \left (x + 1\right )^{5} - 3150 \sqrt {x - 1} \left (x + 1\right )^{4} + 12600 \sqrt {x - 1} \left (x + 1\right )^{3} - 25200 \sqrt {x - 1} \left (x + 1\right )^{2} + 25200 \sqrt {x - 1} \left (x + 1\right ) - 10080 \sqrt {x - 1}} + \frac {126 i \left (x + 1\right )^{\frac {5}{2}}}{315 \sqrt {x - 1} \left (x + 1\right )^{5} - 3150 \sqrt {x - 1} \left (x + 1\right )^{4} + 12600 \sqrt {x - 1} \left (x + 1\right )^{3} - 25200 \sqrt {x - 1} \left (x + 1\right )^{2} + 25200 \sqrt {x - 1} \left (x + 1\right ) - 10080 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {2 \left (x + 1\right )^{\frac {11}{2}}}{315 \sqrt {1 - x} \left (x + 1\right )^{5} - 3150 \sqrt {1 - x} \left (x + 1\right )^{4} + 12600 \sqrt {1 - x} \left (x + 1\right )^{3} - 25200 \sqrt {1 - x} \left (x + 1\right )^{2} + 25200 \sqrt {1 - x} \left (x + 1\right ) - 10080 \sqrt {1 - x}} - \frac {22 \left (x + 1\right )^{\frac {9}{2}}}{315 \sqrt {1 - x} \left (x + 1\right )^{5} - 3150 \sqrt {1 - x} \left (x + 1\right )^{4} + 12600 \sqrt {1 - x} \left (x + 1\right )^{3} - 25200 \sqrt {1 - x} \left (x + 1\right )^{2} + 25200 \sqrt {1 - x} \left (x + 1\right ) - 10080 \sqrt {1 - x}} + \frac {99 \left (x + 1\right )^{\frac {7}{2}}}{315 \sqrt {1 - x} \left (x + 1\right )^{5} - 3150 \sqrt {1 - x} \left (x + 1\right )^{4} + 12600 \sqrt {1 - x} \left (x + 1\right )^{3} - 25200 \sqrt {1 - x} \left (x + 1\right )^{2} + 25200 \sqrt {1 - x} \left (x + 1\right ) - 10080 \sqrt {1 - x}} - \frac {126 \left (x + 1\right )^{\frac {5}{2}}}{315 \sqrt {1 - x} \left (x + 1\right )^{5} - 3150 \sqrt {1 - x} \left (x + 1\right )^{4} + 12600 \sqrt {1 - x} \left (x + 1\right )^{3} - 25200 \sqrt {1 - x} \left (x + 1\right )^{2} + 25200 \sqrt {1 - x} \left (x + 1\right ) - 10080 \sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(3/2)/(1-x)**(11/2),x)

[Out]

Piecewise((-2*I*(x + 1)**(11/2)/(315*sqrt(x - 1)*(x + 1)**5 - 3150*sqrt(x - 1)*(x + 1)**4 + 12600*sqrt(x - 1)*
(x + 1)**3 - 25200*sqrt(x - 1)*(x + 1)**2 + 25200*sqrt(x - 1)*(x + 1) - 10080*sqrt(x - 1)) + 22*I*(x + 1)**(9/
2)/(315*sqrt(x - 1)*(x + 1)**5 - 3150*sqrt(x - 1)*(x + 1)**4 + 12600*sqrt(x - 1)*(x + 1)**3 - 25200*sqrt(x - 1
)*(x + 1)**2 + 25200*sqrt(x - 1)*(x + 1) - 10080*sqrt(x - 1)) - 99*I*(x + 1)**(7/2)/(315*sqrt(x - 1)*(x + 1)**
5 - 3150*sqrt(x - 1)*(x + 1)**4 + 12600*sqrt(x - 1)*(x + 1)**3 - 25200*sqrt(x - 1)*(x + 1)**2 + 25200*sqrt(x -
1)*(x + 1) - 10080*sqrt(x - 1)) + 126*I*(x + 1)**(5/2)/(315*sqrt(x - 1)*(x + 1)**5 - 3150*sqrt(x - 1)*(x + 1)
**4 + 12600*sqrt(x - 1)*(x + 1)**3 - 25200*sqrt(x - 1)*(x + 1)**2 + 25200*sqrt(x - 1)*(x + 1) - 10080*sqrt(x -
1)), Abs(x + 1) > 2), (2*(x + 1)**(11/2)/(315*sqrt(1 - x)*(x + 1)**5 - 3150*sqrt(1 - x)*(x + 1)**4 + 12600*sq
rt(1 - x)*(x + 1)**3 - 25200*sqrt(1 - x)*(x + 1)**2 + 25200*sqrt(1 - x)*(x + 1) - 10080*sqrt(1 - x)) - 22*(x +
1)**(9/2)/(315*sqrt(1 - x)*(x + 1)**5 - 3150*sqrt(1 - x)*(x + 1)**4 + 12600*sqrt(1 - x)*(x + 1)**3 - 25200*sq
rt(1 - x)*(x + 1)**2 + 25200*sqrt(1 - x)*(x + 1) - 10080*sqrt(1 - x)) + 99*(x + 1)**(7/2)/(315*sqrt(1 - x)*(x
+ 1)**5 - 3150*sqrt(1 - x)*(x + 1)**4 + 12600*sqrt(1 - x)*(x + 1)**3 - 25200*sqrt(1 - x)*(x + 1)**2 + 25200*sq
rt(1 - x)*(x + 1) - 10080*sqrt(1 - x)) - 126*(x + 1)**(5/2)/(315*sqrt(1 - x)*(x + 1)**5 - 3150*sqrt(1 - x)*(x
+ 1)**4 + 12600*sqrt(1 - x)*(x + 1)**3 - 25200*sqrt(1 - x)*(x + 1)**2 + 25200*sqrt(1 - x)*(x + 1) - 10080*sqrt
(1 - x)), True))

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Giac [A]
time = 1.37, size = 29, normalized size = 0.48 \begin {gather*} -\frac {{\left (2 \, {\left (x + 1\right )} {\left (x - 8\right )} + 63\right )} {\left (x + 1\right )}^{\frac {5}{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)^(3/2)/(1-x)^(11/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.32, size = 80, normalized size = 1.31 \begin {gather*} -\frac {\sqrt {1-x}\,\left (\frac {16\,x\,\sqrt {x+1}}{63}+\frac {47\,\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)^(3/2)/(1 - x)^(11/2),x)

[Out]

-((1 - x)^(1/2)*((16*x*(x + 1)^(1/2))/63 + (47*(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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