3.11.30 \(\int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx\)

Optimal. Leaf size=41 \[ \frac {(x+1)^{7/2}}{63 (1-x)^{7/2}}+\frac {(x+1)^{7/2}}{9 (1-x)^{9/2}} \]

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Rubi [A]  time = 0.00, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, 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 {(x+1)^{7/2}}{63 (1-x)^{7/2}}+\frac {(x+1)^{7/2}}{9 (1-x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(5/2)/(1 - x)^(11/2),x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.56 \begin {gather*} -\frac {(x-8) (x+1)^{7/2}}{63 (1-x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(5/2)/(1 - x)^(11/2),x]

[Out]

-1/63*((-8 + x)*(1 + x)^(7/2))/(1 - x)^(9/2)

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IntegrateAlgebraic [A]  time = 0.07, size = 34, normalized size = 0.83 \begin {gather*} \frac {(x+1)^{9/2} \left (\frac {9 (1-x)}{x+1}+7\right )}{126 (1-x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x)^(5/2)/(1 - x)^(11/2),x]

[Out]

((1 + x)^(9/2)*(7 + (9*(1 - x))/(1 + x)))/(126*(1 - x)^(9/2))

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fricas [B]  time = 1.06, size = 83, normalized size = 2.02 \begin {gather*} \frac {8 \, x^{5} - 40 \, x^{4} + 80 \, x^{3} - 80 \, x^{2} + {\left (x^{4} - 5 \, x^{3} - 21 \, x^{2} - 23 \, x - 8\right )} \sqrt {x + 1} \sqrt {-x + 1} + 40 \, x - 8}{63 \, {\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)^(5/2)/(1-x)^(11/2),x, algorithm="fricas")

[Out]

1/63*(8*x^5 - 40*x^4 + 80*x^3 - 80*x^2 + (x^4 - 5*x^3 - 21*x^2 - 23*x - 8)*sqrt(x + 1)*sqrt(-x + 1) + 40*x - 8
)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1)

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giac [A]  time = 1.23, size = 22, normalized size = 0.54 \begin {gather*} \frac {{\left (x + 1\right )}^{\frac {7}{2}} {\left (x - 8\right )} \sqrt {-x + 1}}{63 \, {\left (x - 1\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(11/2),x, algorithm="giac")

[Out]

1/63*(x + 1)^(7/2)*(x - 8)*sqrt(-x + 1)/(x - 1)^5

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maple [A]  time = 0.00, size = 18, normalized size = 0.44 \begin {gather*} -\frac {\left (x +1\right )^{\frac {7}{2}} \left (x -8\right )}{63 \left (-x +1\right )^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)^(5/2)/(-x+1)^(11/2),x)

[Out]

-1/63*(x+1)^(7/2)*(x-8)/(-x+1)^(9/2)

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maxima [B]  time = 1.40, size = 218, normalized size = 5.32 \begin {gather*} -\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{2 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} - \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{6 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} - \frac {5 \, \sqrt {-x^{2} + 1}}{9 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {5 \, \sqrt {-x^{2} + 1}}{126 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{42 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{63 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{63 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(11/2),x, algorithm="maxima")

[Out]

-1/2*(-x^2 + 1)^(5/2)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x - 1) - 5/6*(-x^2 + 1)^(3/2)/(x^6
- 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1) - 5/9*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) -
5/126*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 1/42*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 1/63*sqr
t(-x^2 + 1)/(x^2 - 2*x + 1) + 1/63*sqrt(-x^2 + 1)/(x - 1)

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mupad [B]  time = 0.30, size = 80, normalized size = 1.95 \begin {gather*} -\frac {\sqrt {1-x}\,\left (\frac {23\,x\,\sqrt {x+1}}{63}+\frac {8\,\sqrt {x+1}}{63}+\frac {x^2\,\sqrt {x+1}}{3}+\frac {5\,x^3\,\sqrt {x+1}}{63}-\frac {x^4\,\sqrt {x+1}}{63}\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)^(5/2)/(1 - x)^(11/2),x)

[Out]

-((1 - x)^(1/2)*((23*x*(x + 1)^(1/2))/63 + (8*(x + 1)^(1/2))/63 + (x^2*(x + 1)^(1/2))/3 + (5*x^3*(x + 1)^(1/2)
)/63 - (x^4*(x + 1)^(1/2))/63))/(5*x - 10*x^2 + 10*x^3 - 5*x^4 + x^5 - 1)

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sympy [B]  time = 53.14, size = 282, normalized size = 6.88 \begin {gather*} \begin {cases} \frac {i \left (x + 1\right )^{\frac {9}{2}}}{63 \sqrt {x - 1} \left (x + 1\right )^{4} - 504 \sqrt {x - 1} \left (x + 1\right )^{3} + 1512 \sqrt {x - 1} \left (x + 1\right )^{2} - 2016 \sqrt {x - 1} \left (x + 1\right ) + 1008 \sqrt {x - 1}} - \frac {9 i \left (x + 1\right )^{\frac {7}{2}}}{63 \sqrt {x - 1} \left (x + 1\right )^{4} - 504 \sqrt {x - 1} \left (x + 1\right )^{3} + 1512 \sqrt {x - 1} \left (x + 1\right )^{2} - 2016 \sqrt {x - 1} \left (x + 1\right ) + 1008 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- \frac {\left (x + 1\right )^{\frac {9}{2}}}{63 \sqrt {1 - x} \left (x + 1\right )^{4} - 504 \sqrt {1 - x} \left (x + 1\right )^{3} + 1512 \sqrt {1 - x} \left (x + 1\right )^{2} - 2016 \sqrt {1 - x} \left (x + 1\right ) + 1008 \sqrt {1 - x}} + \frac {9 \left (x + 1\right )^{\frac {7}{2}}}{63 \sqrt {1 - x} \left (x + 1\right )^{4} - 504 \sqrt {1 - x} \left (x + 1\right )^{3} + 1512 \sqrt {1 - x} \left (x + 1\right )^{2} - 2016 \sqrt {1 - x} \left (x + 1\right ) + 1008 \sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(5/2)/(1-x)**(11/2),x)

[Out]

Piecewise((I*(x + 1)**(9/2)/(63*sqrt(x - 1)*(x + 1)**4 - 504*sqrt(x - 1)*(x + 1)**3 + 1512*sqrt(x - 1)*(x + 1)
**2 - 2016*sqrt(x - 1)*(x + 1) + 1008*sqrt(x - 1)) - 9*I*(x + 1)**(7/2)/(63*sqrt(x - 1)*(x + 1)**4 - 504*sqrt(
x - 1)*(x + 1)**3 + 1512*sqrt(x - 1)*(x + 1)**2 - 2016*sqrt(x - 1)*(x + 1) + 1008*sqrt(x - 1)), Abs(x + 1)/2 >
1), (-(x + 1)**(9/2)/(63*sqrt(1 - x)*(x + 1)**4 - 504*sqrt(1 - x)*(x + 1)**3 + 1512*sqrt(1 - x)*(x + 1)**2 -
2016*sqrt(1 - x)*(x + 1) + 1008*sqrt(1 - x)) + 9*(x + 1)**(7/2)/(63*sqrt(1 - x)*(x + 1)**4 - 504*sqrt(1 - x)*(
x + 1)**3 + 1512*sqrt(1 - x)*(x + 1)**2 - 2016*sqrt(1 - x)*(x + 1) + 1008*sqrt(1 - x)), True))

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