3.1087 \(\int \frac{(1+x)^{3/2}}{(1-x)^{15/2}} \, dx\)

Optimal. Leaf size=101 \[ \frac{8 (x+1)^{5/2}}{15015 (1-x)^{5/2}}+\frac{8 (x+1)^{5/2}}{3003 (1-x)^{7/2}}+\frac{4 (x+1)^{5/2}}{429 (1-x)^{9/2}}+\frac{4 (x+1)^{5/2}}{143 (1-x)^{11/2}}+\frac{(x+1)^{5/2}}{13 (1-x)^{13/2}} \]

[Out]

(1 + x)^(5/2)/(13*(1 - x)^(13/2)) + (4*(1 + x)^(5/2))/(143*(1 - x)^(11/2)) + (4*(1 + x)^(5/2))/(429*(1 - x)^(9
/2)) + (8*(1 + x)^(5/2))/(3003*(1 - x)^(7/2)) + (8*(1 + x)^(5/2))/(15015*(1 - x)^(5/2))

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Rubi [A]  time = 0.0192798, 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} \[ \frac{8 (x+1)^{5/2}}{15015 (1-x)^{5/2}}+\frac{8 (x+1)^{5/2}}{3003 (1-x)^{7/2}}+\frac{4 (x+1)^{5/2}}{429 (1-x)^{9/2}}+\frac{4 (x+1)^{5/2}}{143 (1-x)^{11/2}}+\frac{(x+1)^{5/2}}{13 (1-x)^{13/2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(3/2)/(1 - x)^(15/2),x]

[Out]

(1 + x)^(5/2)/(13*(1 - x)^(13/2)) + (4*(1 + x)^(5/2))/(143*(1 - x)^(11/2)) + (4*(1 + x)^(5/2))/(429*(1 - x)^(9
/2)) + (8*(1 + x)^(5/2))/(3003*(1 - x)^(7/2)) + (8*(1 + x)^(5/2))/(15015*(1 - x)^(5/2))

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)^{3/2}}{(1-x)^{15/2}} \, dx &=\frac{(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac{4}{13} \int \frac{(1+x)^{3/2}}{(1-x)^{13/2}} \, dx\\ &=\frac{(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac{4 (1+x)^{5/2}}{143 (1-x)^{11/2}}+\frac{12}{143} \int \frac{(1+x)^{3/2}}{(1-x)^{11/2}} \, dx\\ &=\frac{(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac{4 (1+x)^{5/2}}{143 (1-x)^{11/2}}+\frac{4 (1+x)^{5/2}}{429 (1-x)^{9/2}}+\frac{8}{429} \int \frac{(1+x)^{3/2}}{(1-x)^{9/2}} \, dx\\ &=\frac{(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac{4 (1+x)^{5/2}}{143 (1-x)^{11/2}}+\frac{4 (1+x)^{5/2}}{429 (1-x)^{9/2}}+\frac{8 (1+x)^{5/2}}{3003 (1-x)^{7/2}}+\frac{8 \int \frac{(1+x)^{3/2}}{(1-x)^{7/2}} \, dx}{3003}\\ &=\frac{(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac{4 (1+x)^{5/2}}{143 (1-x)^{11/2}}+\frac{4 (1+x)^{5/2}}{429 (1-x)^{9/2}}+\frac{8 (1+x)^{5/2}}{3003 (1-x)^{7/2}}+\frac{8 (1+x)^{5/2}}{15015 (1-x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0186784, size = 40, normalized size = 0.4 \[ \frac{(x+1)^{5/2} \left (8 x^4-72 x^3+308 x^2-852 x+1763\right )}{15015 (1-x)^{13/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(3/2)/(1 - x)^(15/2),x]

[Out]

((1 + x)^(5/2)*(1763 - 852*x + 308*x^2 - 72*x^3 + 8*x^4))/(15015*(1 - x)^(13/2))

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Maple [A]  time = 0.002, size = 35, normalized size = 0.4 \begin{align*}{\frac{8\,{x}^{4}-72\,{x}^{3}+308\,{x}^{2}-852\,x+1763}{15015} \left ( 1+x \right ) ^{{\frac{5}{2}}} \left ( 1-x \right ) ^{-{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(3/2)/(1-x)^(15/2),x)

[Out]

1/15015*(1+x)^(5/2)*(8*x^4-72*x^3+308*x^2-852*x+1763)/(1-x)^(13/2)

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Maxima [B]  time = 1.03483, size = 363, normalized size = 3.59 \begin{align*} \frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{5 \,{\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{6 \, \sqrt{-x^{2} + 1}}{65 \,{\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} + \frac{3 \, \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}}{429 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac{4 \, \sqrt{-x^{2} + 1}}{3003 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac{4 \, \sqrt{-x^{2} + 1}}{5005 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac{8 \, \sqrt{-x^{2} + 1}}{15015 \,{\left (x^{2} - 2 \, x + 1\right )}} - \frac{8 \, \sqrt{-x^{2} + 1}}{15015 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(1-x)^(15/2),x, algorithm="maxima")

[Out]

1/5*(-x^2 + 1)^(3/2)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) + 6/65*sqrt(-x^2 + 1
)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x - 1) + 3/715*sqrt(-x^2 + 1)/(x^6 - 6*x^5 + 15*x^4 - 2
0*x^3 + 15*x^2 - 6*x + 1) - 1/429*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) + 4/3003*sqrt(-x^2
+ 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) - 4/5005*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) + 8/15015*sqrt(-x^2 + 1)/
(x^2 - 2*x + 1) - 8/15015*sqrt(-x^2 + 1)/(x - 1)

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Fricas [A]  time = 1.60946, size = 333, normalized size = 3.3 \begin{align*} \frac{1763 \, x^{7} - 12341 \, x^{6} + 37023 \, x^{5} - 61705 \, x^{4} + 61705 \, x^{3} - 37023 \, x^{2} -{\left (8 \, x^{6} - 56 \, x^{5} + 172 \, x^{4} - 308 \, x^{3} + 367 \, x^{2} + 2674 \, x + 1763\right )} \sqrt{x + 1} \sqrt{-x + 1} + 12341 \, x - 1763}{15015 \,{\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(1-x)^(15/2),x, algorithm="fricas")

[Out]

1/15015*(1763*x^7 - 12341*x^6 + 37023*x^5 - 61705*x^4 + 61705*x^3 - 37023*x^2 - (8*x^6 - 56*x^5 + 172*x^4 - 30
8*x^3 + 367*x^2 + 2674*x + 1763)*sqrt(x + 1)*sqrt(-x + 1) + 12341*x - 1763)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 3
5*x^3 - 21*x^2 + 7*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)**(3/2)/(1-x)**(15/2),x)

[Out]

Timed out

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Giac [A]  time = 1.12134, size = 57, normalized size = 0.56 \begin{align*} -\frac{{\left (4 \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 12\right )} + 143\right )}{\left (x + 1\right )} - 429\right )}{\left (x + 1\right )} + 3003\right )}{\left (x + 1\right )}^{\frac{5}{2}} \sqrt{-x + 1}}{15015 \,{\left (x - 1\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(1-x)^(15/2),x, algorithm="giac")

[Out]

-1/15015*(4*((2*(x + 1)*(x - 12) + 143)*(x + 1) - 429)*(x + 1) + 3003)*(x + 1)^(5/2)*sqrt(-x + 1)/(x - 1)^7