3.11.16 \(\int \frac {(1+x)^{3/2}}{(1-x)^{11/2}} \, dx\)

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

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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)^{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 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+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 = 0.01, size = 30, normalized size = 0.49 \begin {gather*} \frac {(x+1)^{5/2} \left (2 x^2-14 x+47\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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IntegrateAlgebraic [A]  time = 0.07, size = 48, normalized size = 0.79 \begin {gather*} \frac {(x+1)^{9/2} \left (\frac {63 (1-x)^2}{(x+1)^2}+\frac {90 (1-x)}{x+1}+35\right )}{1260 (1-x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x)^(9/2)*(35 + (63*(1 - x)^2)/(1 + x)^2 + (90*(1 - x))/(1 + x)))/(1260*(1 - x)^(9/2))

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fricas [A]  time = 0.67, 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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giac [A]  time = 1.11, 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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maple [A]  time = 0.00, size = 25, normalized size = 0.41 \begin {gather*} \frac {\left (x +1\right )^{\frac {5}{2}} \left (2 x^{2}-14 x +47\right )}{315 \left (-x +1\right )^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.37, 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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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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sympy [B]  time = 51.65, size = 677, normalized size = 11.10 \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}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\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 > 1), (2*(x + 1)**(11/2)/(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)) - 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*
sqrt(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*
sqrt(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*sq
rt(1 - x)), True))

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