3.1102 \(\int \frac{(1+x)^{5/2}}{(1-x)^{17/2}} \, dx\)

Optimal. Leaf size=101 \[ \frac{8 (x+1)^{7/2}}{45045 (1-x)^{7/2}}+\frac{8 (x+1)^{7/2}}{6435 (1-x)^{9/2}}+\frac{4 (x+1)^{7/2}}{715 (1-x)^{11/2}}+\frac{4 (x+1)^{7/2}}{195 (1-x)^{13/2}}+\frac{(x+1)^{7/2}}{15 (1-x)^{15/2}} \]

[Out]

(1 + x)^(7/2)/(15*(1 - x)^(15/2)) + (4*(1 + x)^(7/2))/(195*(1 - x)^(13/2)) + (4*(1 + x)^(7/2))/(715*(1 - x)^(1
1/2)) + (8*(1 + x)^(7/2))/(6435*(1 - x)^(9/2)) + (8*(1 + x)^(7/2))/(45045*(1 - x)^(7/2))

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Rubi [A]  time = 0.0192351, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {45, 37} \[ \frac{8 (x+1)^{7/2}}{45045 (1-x)^{7/2}}+\frac{8 (x+1)^{7/2}}{6435 (1-x)^{9/2}}+\frac{4 (x+1)^{7/2}}{715 (1-x)^{11/2}}+\frac{4 (x+1)^{7/2}}{195 (1-x)^{13/2}}+\frac{(x+1)^{7/2}}{15 (1-x)^{15/2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(5/2)/(1 - x)^(17/2),x]

[Out]

(1 + x)^(7/2)/(15*(1 - x)^(15/2)) + (4*(1 + x)^(7/2))/(195*(1 - x)^(13/2)) + (4*(1 + x)^(7/2))/(715*(1 - x)^(1
1/2)) + (8*(1 + x)^(7/2))/(6435*(1 - x)^(9/2)) + (8*(1 + x)^(7/2))/(45045*(1 - x)^(7/2))

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])

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]

Rubi steps

\begin{align*} \int \frac{(1+x)^{5/2}}{(1-x)^{17/2}} \, dx &=\frac{(1+x)^{7/2}}{15 (1-x)^{15/2}}+\frac{4}{15} \int \frac{(1+x)^{5/2}}{(1-x)^{15/2}} \, dx\\ &=\frac{(1+x)^{7/2}}{15 (1-x)^{15/2}}+\frac{4 (1+x)^{7/2}}{195 (1-x)^{13/2}}+\frac{4}{65} \int \frac{(1+x)^{5/2}}{(1-x)^{13/2}} \, dx\\ &=\frac{(1+x)^{7/2}}{15 (1-x)^{15/2}}+\frac{4 (1+x)^{7/2}}{195 (1-x)^{13/2}}+\frac{4 (1+x)^{7/2}}{715 (1-x)^{11/2}}+\frac{8}{715} \int \frac{(1+x)^{5/2}}{(1-x)^{11/2}} \, dx\\ &=\frac{(1+x)^{7/2}}{15 (1-x)^{15/2}}+\frac{4 (1+x)^{7/2}}{195 (1-x)^{13/2}}+\frac{4 (1+x)^{7/2}}{715 (1-x)^{11/2}}+\frac{8 (1+x)^{7/2}}{6435 (1-x)^{9/2}}+\frac{8 \int \frac{(1+x)^{5/2}}{(1-x)^{9/2}} \, dx}{6435}\\ &=\frac{(1+x)^{7/2}}{15 (1-x)^{15/2}}+\frac{4 (1+x)^{7/2}}{195 (1-x)^{13/2}}+\frac{4 (1+x)^{7/2}}{715 (1-x)^{11/2}}+\frac{8 (1+x)^{7/2}}{6435 (1-x)^{9/2}}+\frac{8 (1+x)^{7/2}}{45045 (1-x)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.020974, size = 40, normalized size = 0.4 \[ \frac{(x+1)^{7/2} \left (8 x^4-88 x^3+468 x^2-1628 x+4243\right )}{45045 (1-x)^{15/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(5/2)/(1 - x)^(17/2),x]

[Out]

((1 + x)^(7/2)*(4243 - 1628*x + 468*x^2 - 88*x^3 + 8*x^4))/(45045*(1 - x)^(15/2))

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Maple [A]  time = 0.003, size = 35, normalized size = 0.4 \begin{align*}{\frac{8\,{x}^{4}-88\,{x}^{3}+468\,{x}^{2}-1628\,x+4243}{45045} \left ( 1+x \right ) ^{{\frac{7}{2}}} \left ( 1-x \right ) ^{-{\frac{15}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(5/2)/(1-x)^(17/2),x)

[Out]

1/45045*(1+x)^(7/2)*(8*x^4-88*x^3+468*x^2-1628*x+4243)/(1-x)^(15/2)

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Maxima [B]  time = 1.01575, size = 521, normalized size = 5.16 \begin{align*} \frac{{\left (-x^{2} + 1\right )}^{\frac{5}{2}}}{5 \,{\left (x^{10} - 10 \, x^{9} + 45 \, x^{8} - 120 \, x^{7} + 210 \, x^{6} - 252 \, x^{5} + 210 \, x^{4} - 120 \, x^{3} + 45 \, x^{2} - 10 \, x + 1\right )}} + \frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{6 \,{\left (x^{9} - 9 \, x^{8} + 36 \, x^{7} - 84 \, x^{6} + 126 \, x^{5} - 126 \, x^{4} + 84 \, x^{3} - 36 \, x^{2} + 9 \, x - 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{15 \,{\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{\sqrt{-x^{2} + 1}}{390 \,{\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{715 \,{\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{1287 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac{4 \, \sqrt{-x^{2} + 1}}{9009 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac{4 \, \sqrt{-x^{2} + 1}}{15015 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{8 \, \sqrt{-x^{2} + 1}}{45045 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{8 \, \sqrt{-x^{2} + 1}}{45045 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(17/2),x, algorithm="maxima")

[Out]

1/5*(-x^2 + 1)^(5/2)/(x^10 - 10*x^9 + 45*x^8 - 120*x^7 + 210*x^6 - 252*x^5 + 210*x^4 - 120*x^3 + 45*x^2 - 10*x
+ 1) + 1/6*(-x^2 + 1)^(3/2)/(x^9 - 9*x^8 + 36*x^7 - 84*x^6 + 126*x^5 - 126*x^4 + 84*x^3 - 36*x^2 + 9*x - 1) +
1/15*sqrt(-x^2 + 1)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) + 1/390*sqrt(-x^2 +
1)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x - 1) - 1/715*sqrt(-x^2 + 1)/(x^6 - 6*x^5 + 15*x^4 -
20*x^3 + 15*x^2 - 6*x + 1) + 1/1287*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) - 4/9009*sqrt(-x^
2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 4/15015*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 8/45045*sqrt(-x^2 +
1)/(x^2 - 2*x + 1) + 8/45045*sqrt(-x^2 + 1)/(x - 1)

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Fricas [A]  time = 1.65709, size = 385, normalized size = 3.81 \begin{align*} \frac{4243 \, x^{8} - 33944 \, x^{7} + 118804 \, x^{6} - 237608 \, x^{5} + 297010 \, x^{4} - 237608 \, x^{3} + 118804 \, x^{2} +{\left (8 \, x^{7} - 64 \, x^{6} + 228 \, x^{5} - 480 \, x^{4} + 675 \, x^{3} + 8313 \, x^{2} + 11101 \, x + 4243\right )} \sqrt{x + 1} \sqrt{-x + 1} - 33944 \, x + 4243}{45045 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(17/2),x, algorithm="fricas")

[Out]

1/45045*(4243*x^8 - 33944*x^7 + 118804*x^6 - 237608*x^5 + 297010*x^4 - 237608*x^3 + 118804*x^2 + (8*x^7 - 64*x
^6 + 228*x^5 - 480*x^4 + 675*x^3 + 8313*x^2 + 11101*x + 4243)*sqrt(x + 1)*sqrt(-x + 1) - 33944*x + 4243)/(x^8
- 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(5/2)/(1-x)**(17/2),x)

[Out]

Timed out

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Giac [A]  time = 1.12352, size = 57, normalized size = 0.56 \begin{align*} \frac{{\left (4 \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 14\right )} + 195\right )}{\left (x + 1\right )} - 715\right )}{\left (x + 1\right )} + 6435\right )}{\left (x + 1\right )}^{\frac{7}{2}} \sqrt{-x + 1}}{45045 \,{\left (x - 1\right )}^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(17/2),x, algorithm="giac")

[Out]

1/45045*(4*((2*(x + 1)*(x - 14) + 195)*(x + 1) - 715)*(x + 1) + 6435)*(x + 1)^(7/2)*sqrt(-x + 1)/(x - 1)^8