3.11.31 \(\int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx\)

Optimal. Leaf size=61 \[ \frac {2 (x+1)^{7/2}}{693 (1-x)^{7/2}}+\frac {2 (x+1)^{7/2}}{99 (1-x)^{9/2}}+\frac {(x+1)^{7/2}}{11 (1-x)^{11/2}} \]

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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)^{7/2}}{693 (1-x)^{7/2}}+\frac {2 (x+1)^{7/2}}{99 (1-x)^{9/2}}+\frac {(x+1)^{7/2}}{11 (1-x)^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(5/2)/(1 - x)^(13/2),x]

[Out]

(1 + x)^(7/2)/(11*(1 - x)^(11/2)) + (2*(1 + x)^(7/2))/(99*(1 - x)^(9/2)) + (2*(1 + x)^(7/2))/(693*(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)^{13/2}} \, dx &=\frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2}{11} \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx\\ &=\frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{99 (1-x)^{9/2}}+\frac {2}{99} \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx\\ &=\frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{99 (1-x)^{9/2}}+\frac {2 (1+x)^{7/2}}{693 (1-x)^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 30, normalized size = 0.49 \begin {gather*} \frac {(x+1)^{7/2} \left (2 x^2-18 x+79\right )}{693 (1-x)^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(5/2)/(1 - x)^(13/2),x]

[Out]

((1 + x)^(7/2)*(79 - 18*x + 2*x^2))/(693*(1 - x)^(11/2))

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IntegrateAlgebraic [A]  time = 0.08, size = 48, normalized size = 0.79 \begin {gather*} \frac {(x+1)^{11/2} \left (\frac {99 (1-x)^2}{(x+1)^2}+\frac {154 (1-x)}{x+1}+63\right )}{2772 (1-x)^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x)^(5/2)/(1 - x)^(13/2),x]

[Out]

((1 + x)^(11/2)*(63 + (99*(1 - x)^2)/(1 + x)^2 + (154*(1 - x))/(1 + x)))/(2772*(1 - x)^(11/2))

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fricas [B]  time = 0.97, size = 100, normalized size = 1.64 \begin {gather*} \frac {79 \, x^{6} - 474 \, x^{5} + 1185 \, x^{4} - 1580 \, x^{3} + 1185 \, x^{2} + {\left (2 \, x^{5} - 12 \, x^{4} + 31 \, x^{3} + 185 \, x^{2} + 219 \, x + 79\right )} \sqrt {x + 1} \sqrt {-x + 1} - 474 \, x + 79}{693 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(13/2),x, algorithm="fricas")

[Out]

1/693*(79*x^6 - 474*x^5 + 1185*x^4 - 1580*x^3 + 1185*x^2 + (2*x^5 - 12*x^4 + 31*x^3 + 185*x^2 + 219*x + 79)*sq
rt(x + 1)*sqrt(-x + 1) - 474*x + 79)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1)

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giac [A]  time = 1.27, size = 29, normalized size = 0.48 \begin {gather*} \frac {{\left (2 \, {\left (x + 1\right )} {\left (x - 10\right )} + 99\right )} {\left (x + 1\right )}^{\frac {7}{2}} \sqrt {-x + 1}}{693 \, {\left (x - 1\right )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(13/2),x, algorithm="giac")

[Out]

1/693*(2*(x + 1)*(x - 10) + 99)*(x + 1)^(7/2)*sqrt(-x + 1)/(x - 1)^6

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maple [A]  time = 0.00, size = 25, normalized size = 0.41 \begin {gather*} \frac {\left (x +1\right )^{\frac {7}{2}} \left (2 x^{2}-18 x +79\right )}{693 \left (-x +1\right )^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)^(5/2)/(-x+1)^(13/2),x)

[Out]

1/693*(x+1)^(7/2)*(2*x^2-18*x+79)/(-x+1)^(11/2)

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maxima [B]  time = 1.42, size = 269, normalized size = 4.41 \begin {gather*} \frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{3 \, {\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} + \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{12 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} + \frac {5 \, \sqrt {-x^{2} + 1}}{22 \, {\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}}{396 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {5 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{231 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(13/2),x, algorithm="maxima")

[Out]

1/3*(-x^2 + 1)^(5/2)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) + 5/12*(-x^2 + 1)^(3
/2)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x - 1) + 5/22*sqrt(-x^2 + 1)/(x^6 - 6*x^5 + 15*x^4 -
20*x^3 + 15*x^2 - 6*x + 1) + 5/396*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) - 5/693*sqrt(-x^2
+ 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 1/231*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 2/693*sqrt(-x^2 + 1)/(x^
2 - 2*x + 1) + 2/693*sqrt(-x^2 + 1)/(x - 1)

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mupad [B]  time = 0.31, size = 94, normalized size = 1.54 \begin {gather*} \frac {\sqrt {1-x}\,\left (\frac {73\,x\,\sqrt {x+1}}{231}+\frac {79\,\sqrt {x+1}}{693}+\frac {185\,x^2\,\sqrt {x+1}}{693}+\frac {31\,x^3\,\sqrt {x+1}}{693}-\frac {4\,x^4\,\sqrt {x+1}}{231}+\frac {2\,x^5\,\sqrt {x+1}}{693}\right )}{x^6-6\,x^5+15\,x^4-20\,x^3+15\,x^2-6\,x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(5/2)/(1 - x)^(13/2),x)

[Out]

((1 - x)^(1/2)*((73*x*(x + 1)^(1/2))/231 + (79*(x + 1)^(1/2))/693 + (185*x^2*(x + 1)^(1/2))/693 + (31*x^3*(x +
1)^(1/2))/693 - (4*x^4*(x + 1)^(1/2))/231 + (2*x^5*(x + 1)^(1/2))/693))/(15*x^2 - 6*x - 20*x^3 + 15*x^4 - 6*x
^5 + x^6 + 1)

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sympy [B]  time = 133.94, size = 785, normalized size = 12.87 \begin {gather*} \begin {cases} \frac {2 i \left (x + 1\right )^{\frac {13}{2}}}{693 \sqrt {x - 1} \left (x + 1\right )^{6} - 8316 \sqrt {x - 1} \left (x + 1\right )^{5} + 41580 \sqrt {x - 1} \left (x + 1\right )^{4} - 110880 \sqrt {x - 1} \left (x + 1\right )^{3} + 166320 \sqrt {x - 1} \left (x + 1\right )^{2} - 133056 \sqrt {x - 1} \left (x + 1\right ) + 44352 \sqrt {x - 1}} - \frac {26 i \left (x + 1\right )^{\frac {11}{2}}}{693 \sqrt {x - 1} \left (x + 1\right )^{6} - 8316 \sqrt {x - 1} \left (x + 1\right )^{5} + 41580 \sqrt {x - 1} \left (x + 1\right )^{4} - 110880 \sqrt {x - 1} \left (x + 1\right )^{3} + 166320 \sqrt {x - 1} \left (x + 1\right )^{2} - 133056 \sqrt {x - 1} \left (x + 1\right ) + 44352 \sqrt {x - 1}} + \frac {143 i \left (x + 1\right )^{\frac {9}{2}}}{693 \sqrt {x - 1} \left (x + 1\right )^{6} - 8316 \sqrt {x - 1} \left (x + 1\right )^{5} + 41580 \sqrt {x - 1} \left (x + 1\right )^{4} - 110880 \sqrt {x - 1} \left (x + 1\right )^{3} + 166320 \sqrt {x - 1} \left (x + 1\right )^{2} - 133056 \sqrt {x - 1} \left (x + 1\right ) + 44352 \sqrt {x - 1}} - \frac {198 i \left (x + 1\right )^{\frac {7}{2}}}{693 \sqrt {x - 1} \left (x + 1\right )^{6} - 8316 \sqrt {x - 1} \left (x + 1\right )^{5} + 41580 \sqrt {x - 1} \left (x + 1\right )^{4} - 110880 \sqrt {x - 1} \left (x + 1\right )^{3} + 166320 \sqrt {x - 1} \left (x + 1\right )^{2} - 133056 \sqrt {x - 1} \left (x + 1\right ) + 44352 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- \frac {2 \left (x + 1\right )^{\frac {13}{2}}}{693 \sqrt {1 - x} \left (x + 1\right )^{6} - 8316 \sqrt {1 - x} \left (x + 1\right )^{5} + 41580 \sqrt {1 - x} \left (x + 1\right )^{4} - 110880 \sqrt {1 - x} \left (x + 1\right )^{3} + 166320 \sqrt {1 - x} \left (x + 1\right )^{2} - 133056 \sqrt {1 - x} \left (x + 1\right ) + 44352 \sqrt {1 - x}} + \frac {26 \left (x + 1\right )^{\frac {11}{2}}}{693 \sqrt {1 - x} \left (x + 1\right )^{6} - 8316 \sqrt {1 - x} \left (x + 1\right )^{5} + 41580 \sqrt {1 - x} \left (x + 1\right )^{4} - 110880 \sqrt {1 - x} \left (x + 1\right )^{3} + 166320 \sqrt {1 - x} \left (x + 1\right )^{2} - 133056 \sqrt {1 - x} \left (x + 1\right ) + 44352 \sqrt {1 - x}} - \frac {143 \left (x + 1\right )^{\frac {9}{2}}}{693 \sqrt {1 - x} \left (x + 1\right )^{6} - 8316 \sqrt {1 - x} \left (x + 1\right )^{5} + 41580 \sqrt {1 - x} \left (x + 1\right )^{4} - 110880 \sqrt {1 - x} \left (x + 1\right )^{3} + 166320 \sqrt {1 - x} \left (x + 1\right )^{2} - 133056 \sqrt {1 - x} \left (x + 1\right ) + 44352 \sqrt {1 - x}} + \frac {198 \left (x + 1\right )^{\frac {7}{2}}}{693 \sqrt {1 - x} \left (x + 1\right )^{6} - 8316 \sqrt {1 - x} \left (x + 1\right )^{5} + 41580 \sqrt {1 - x} \left (x + 1\right )^{4} - 110880 \sqrt {1 - x} \left (x + 1\right )^{3} + 166320 \sqrt {1 - x} \left (x + 1\right )^{2} - 133056 \sqrt {1 - x} \left (x + 1\right ) + 44352 \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)**(13/2),x)

[Out]

Piecewise((2*I*(x + 1)**(13/2)/(693*sqrt(x - 1)*(x + 1)**6 - 8316*sqrt(x - 1)*(x + 1)**5 + 41580*sqrt(x - 1)*(
x + 1)**4 - 110880*sqrt(x - 1)*(x + 1)**3 + 166320*sqrt(x - 1)*(x + 1)**2 - 133056*sqrt(x - 1)*(x + 1) + 44352
*sqrt(x - 1)) - 26*I*(x + 1)**(11/2)/(693*sqrt(x - 1)*(x + 1)**6 - 8316*sqrt(x - 1)*(x + 1)**5 + 41580*sqrt(x
- 1)*(x + 1)**4 - 110880*sqrt(x - 1)*(x + 1)**3 + 166320*sqrt(x - 1)*(x + 1)**2 - 133056*sqrt(x - 1)*(x + 1) +
44352*sqrt(x - 1)) + 143*I*(x + 1)**(9/2)/(693*sqrt(x - 1)*(x + 1)**6 - 8316*sqrt(x - 1)*(x + 1)**5 + 41580*s
qrt(x - 1)*(x + 1)**4 - 110880*sqrt(x - 1)*(x + 1)**3 + 166320*sqrt(x - 1)*(x + 1)**2 - 133056*sqrt(x - 1)*(x
+ 1) + 44352*sqrt(x - 1)) - 198*I*(x + 1)**(7/2)/(693*sqrt(x - 1)*(x + 1)**6 - 8316*sqrt(x - 1)*(x + 1)**5 + 4
1580*sqrt(x - 1)*(x + 1)**4 - 110880*sqrt(x - 1)*(x + 1)**3 + 166320*sqrt(x - 1)*(x + 1)**2 - 133056*sqrt(x -
1)*(x + 1) + 44352*sqrt(x - 1)), Abs(x + 1)/2 > 1), (-2*(x + 1)**(13/2)/(693*sqrt(1 - x)*(x + 1)**6 - 8316*sqr
t(1 - x)*(x + 1)**5 + 41580*sqrt(1 - x)*(x + 1)**4 - 110880*sqrt(1 - x)*(x + 1)**3 + 166320*sqrt(1 - x)*(x + 1
)**2 - 133056*sqrt(1 - x)*(x + 1) + 44352*sqrt(1 - x)) + 26*(x + 1)**(11/2)/(693*sqrt(1 - x)*(x + 1)**6 - 8316
*sqrt(1 - x)*(x + 1)**5 + 41580*sqrt(1 - x)*(x + 1)**4 - 110880*sqrt(1 - x)*(x + 1)**3 + 166320*sqrt(1 - x)*(x
+ 1)**2 - 133056*sqrt(1 - x)*(x + 1) + 44352*sqrt(1 - x)) - 143*(x + 1)**(9/2)/(693*sqrt(1 - x)*(x + 1)**6 -
8316*sqrt(1 - x)*(x + 1)**5 + 41580*sqrt(1 - x)*(x + 1)**4 - 110880*sqrt(1 - x)*(x + 1)**3 + 166320*sqrt(1 - x
)*(x + 1)**2 - 133056*sqrt(1 - x)*(x + 1) + 44352*sqrt(1 - x)) + 198*(x + 1)**(7/2)/(693*sqrt(1 - x)*(x + 1)**
6 - 8316*sqrt(1 - x)*(x + 1)**5 + 41580*sqrt(1 - x)*(x + 1)**4 - 110880*sqrt(1 - x)*(x + 1)**3 + 166320*sqrt(1
- x)*(x + 1)**2 - 133056*sqrt(1 - x)*(x + 1) + 44352*sqrt(1 - x)), True))

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