3.1086 \(\int \frac{(1+x)^{3/2}}{(1-x)^{13/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac{2 (x+1)^{5/2}}{1155 (1-x)^{5/2}}+\frac{2 (x+1)^{5/2}}{231 (1-x)^{7/2}}+\frac{(x+1)^{5/2}}{33 (1-x)^{9/2}}+\frac{(x+1)^{5/2}}{11 (1-x)^{11/2}} \]

[Out]

(1 + x)^(5/2)/(11*(1 - x)^(11/2)) + (1 + x)^(5/2)/(33*(1 - x)^(9/2)) + (2*(1 + x)^(5/2))/(231*(1 - x)^(7/2)) +
(2*(1 + x)^(5/2))/(1155*(1 - x)^(5/2))

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Rubi [A]  time = 0.0131675, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {45, 37} \[ \frac{2 (x+1)^{5/2}}{1155 (1-x)^{5/2}}+\frac{2 (x+1)^{5/2}}{231 (1-x)^{7/2}}+\frac{(x+1)^{5/2}}{33 (1-x)^{9/2}}+\frac{(x+1)^{5/2}}{11 (1-x)^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(3/2)/(1 - x)^(13/2),x]

[Out]

(1 + x)^(5/2)/(11*(1 - x)^(11/2)) + (1 + x)^(5/2)/(33*(1 - x)^(9/2)) + (2*(1 + x)^(5/2))/(231*(1 - x)^(7/2)) +
(2*(1 + x)^(5/2))/(1155*(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)^{13/2}} \, dx &=\frac{(1+x)^{5/2}}{11 (1-x)^{11/2}}+\frac{3}{11} \int \frac{(1+x)^{3/2}}{(1-x)^{11/2}} \, dx\\ &=\frac{(1+x)^{5/2}}{11 (1-x)^{11/2}}+\frac{(1+x)^{5/2}}{33 (1-x)^{9/2}}+\frac{2}{33} \int \frac{(1+x)^{3/2}}{(1-x)^{9/2}} \, dx\\ &=\frac{(1+x)^{5/2}}{11 (1-x)^{11/2}}+\frac{(1+x)^{5/2}}{33 (1-x)^{9/2}}+\frac{2 (1+x)^{5/2}}{231 (1-x)^{7/2}}+\frac{2}{231} \int \frac{(1+x)^{3/2}}{(1-x)^{7/2}} \, dx\\ &=\frac{(1+x)^{5/2}}{11 (1-x)^{11/2}}+\frac{(1+x)^{5/2}}{33 (1-x)^{9/2}}+\frac{2 (1+x)^{5/2}}{231 (1-x)^{7/2}}+\frac{2 (1+x)^{5/2}}{1155 (1-x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0229021, size = 35, normalized size = 0.43 \[ \frac{(x+1)^{5/2} \left (-2 x^3+16 x^2-61 x+152\right )}{1155 (1-x)^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(3/2)/(1 - x)^(13/2),x]

[Out]

((1 + x)^(5/2)*(152 - 61*x + 16*x^2 - 2*x^3))/(1155*(1 - x)^(11/2))

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Maple [A]  time = 0.003, size = 30, normalized size = 0.4 \begin{align*} -{\frac{2\,{x}^{3}-16\,{x}^{2}+61\,x-152}{1155} \left ( 1+x \right ) ^{{\frac{5}{2}}} \left ( 1-x \right ) ^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(3/2)/(1-x)^(13/2),x)

[Out]

-1/1155*(1+x)^(5/2)*(2*x^3-16*x^2+61*x-152)/(1-x)^(11/2)

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Maxima [B]  time = 1.04473, size = 294, normalized size = 3.63 \begin{align*} -\frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{4 \,{\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}}{22 \,{\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{132 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{231 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{385 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac{2 \, \sqrt{-x^{2} + 1}}{1155 \,{\left (x^{2} - 2 \, x + 1\right )}} - \frac{2 \, \sqrt{-x^{2} + 1}}{1155 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(1-x)^(13/2),x, algorithm="maxima")

[Out]

-1/4*(-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) - 3/22*sqrt(-x^2 + 1)/(x^6 -
6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1) - 1/132*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1)
+ 1/231*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) - 1/385*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) + 2/1155
*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) - 2/1155*sqrt(-x^2 + 1)/(x - 1)

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Fricas [A]  time = 1.54324, size = 273, normalized size = 3.37 \begin{align*} \frac{152 \, x^{6} - 912 \, x^{5} + 2280 \, x^{4} - 3040 \, x^{3} + 2280 \, x^{2} -{\left (2 \, x^{5} - 12 \, x^{4} + 31 \, x^{3} - 46 \, x^{2} - 243 \, x - 152\right )} \sqrt{x + 1} \sqrt{-x + 1} - 912 \, x + 152}{1155 \,{\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(1-x)^(13/2),x, algorithm="fricas")

[Out]

1/1155*(152*x^6 - 912*x^5 + 2280*x^4 - 3040*x^3 + 2280*x^2 - (2*x^5 - 12*x^4 + 31*x^3 - 46*x^2 - 243*x - 152)*
sqrt(x + 1)*sqrt(-x + 1) - 912*x + 152)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*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)**(13/2),x)

[Out]

Timed out

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Giac [A]  time = 1.09859, size = 47, normalized size = 0.58 \begin{align*} -\frac{{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 10\right )} + 99\right )}{\left (x + 1\right )} - 231\right )}{\left (x + 1\right )}^{\frac{5}{2}} \sqrt{-x + 1}}{1155 \,{\left (x - 1\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(1-x)^(13/2),x, algorithm="giac")

[Out]

-1/1155*((2*(x + 1)*(x - 10) + 99)*(x + 1) - 231)*(x + 1)^(5/2)*sqrt(-x + 1)/(x - 1)^6