3.1101 \(\int \frac{(1+x)^{5/2}}{(1-x)^{15/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac{2 (x+1)^{7/2}}{3003 (1-x)^{7/2}}+\frac{2 (x+1)^{7/2}}{429 (1-x)^{9/2}}+\frac{3 (x+1)^{7/2}}{143 (1-x)^{11/2}}+\frac{(x+1)^{7/2}}{13 (1-x)^{13/2}} \]

[Out]

(1 + x)^(7/2)/(13*(1 - x)^(13/2)) + (3*(1 + x)^(7/2))/(143*(1 - x)^(11/2)) + (2*(1 + x)^(7/2))/(429*(1 - x)^(9
/2)) + (2*(1 + x)^(7/2))/(3003*(1 - x)^(7/2))

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Rubi [A]  time = 0.0127514, 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} \[ \frac{2 (x+1)^{7/2}}{3003 (1-x)^{7/2}}+\frac{2 (x+1)^{7/2}}{429 (1-x)^{9/2}}+\frac{3 (x+1)^{7/2}}{143 (1-x)^{11/2}}+\frac{(x+1)^{7/2}}{13 (1-x)^{13/2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(5/2)/(1 - x)^(15/2),x]

[Out]

(1 + x)^(7/2)/(13*(1 - x)^(13/2)) + (3*(1 + x)^(7/2))/(143*(1 - x)^(11/2)) + (2*(1 + x)^(7/2))/(429*(1 - x)^(9
/2)) + (2*(1 + x)^(7/2))/(3003*(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)^{15/2}} \, dx &=\frac{(1+x)^{7/2}}{13 (1-x)^{13/2}}+\frac{3}{13} \int \frac{(1+x)^{5/2}}{(1-x)^{13/2}} \, dx\\ &=\frac{(1+x)^{7/2}}{13 (1-x)^{13/2}}+\frac{3 (1+x)^{7/2}}{143 (1-x)^{11/2}}+\frac{6}{143} \int \frac{(1+x)^{5/2}}{(1-x)^{11/2}} \, dx\\ &=\frac{(1+x)^{7/2}}{13 (1-x)^{13/2}}+\frac{3 (1+x)^{7/2}}{143 (1-x)^{11/2}}+\frac{2 (1+x)^{7/2}}{429 (1-x)^{9/2}}+\frac{2}{429} \int \frac{(1+x)^{5/2}}{(1-x)^{9/2}} \, dx\\ &=\frac{(1+x)^{7/2}}{13 (1-x)^{13/2}}+\frac{3 (1+x)^{7/2}}{143 (1-x)^{11/2}}+\frac{2 (1+x)^{7/2}}{429 (1-x)^{9/2}}+\frac{2 (1+x)^{7/2}}{3003 (1-x)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0173916, size = 35, normalized size = 0.43 \[ \frac{(x+1)^{7/2} \left (-2 x^3+20 x^2-97 x+310\right )}{3003 (1-x)^{13/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(5/2)/(1 - x)^(15/2),x]

[Out]

((1 + x)^(7/2)*(310 - 97*x + 20*x^2 - 2*x^3))/(3003*(1 - x)^(13/2))

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Maple [A]  time = 0.003, size = 30, normalized size = 0.4 \begin{align*} -{\frac{2\,{x}^{3}-20\,{x}^{2}+97\,x-310}{3003} \left ( 1+x \right ) ^{{\frac{7}{2}}} \left ( 1-x \right ) ^{-{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(5/2)/(1-x)^(15/2),x)

[Out]

-1/3003*(1+x)^(7/2)*(2*x^3-20*x^2+97*x-310)/(1-x)^(13/2)

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Maxima [B]  time = 1.00994, size = 439, normalized size = 5.42 \begin{align*} -\frac{{\left (-x^{2} + 1\right )}^{\frac{5}{2}}}{4 \,{\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{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{4 \,{\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{3 \, \sqrt{-x^{2} + 1}}{26 \,{\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}}{572 \,{\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}}{1716 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac{5 \, \sqrt{-x^{2} + 1}}{3003 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{1001 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{2 \, \sqrt{-x^{2} + 1}}{3003 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{2 \, \sqrt{-x^{2} + 1}}{3003 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(15/2),x, algorithm="maxima")

[Out]

-1/4*(-x^2 + 1)^(5/2)/(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) - 1/4*(-
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) - 3/26*sqrt(-x^2 + 1)/(x^7
- 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x - 1) - 3/572*sqrt(-x^2 + 1)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3
+ 15*x^2 - 6*x + 1) + 5/1716*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) - 5/3003*sqrt(-x^2 + 1)/
(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 1/1001*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 2/3003*sqrt(-x^2 + 1)/(x^2 -
2*x + 1) + 2/3003*sqrt(-x^2 + 1)/(x - 1)

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Fricas [B]  time = 1.49529, size = 319, normalized size = 3.94 \begin{align*} \frac{310 \, x^{7} - 2170 \, x^{6} + 6510 \, x^{5} - 10850 \, x^{4} + 10850 \, x^{3} - 6510 \, x^{2} +{\left (2 \, x^{6} - 14 \, x^{5} + 43 \, x^{4} - 77 \, x^{3} - 659 \, x^{2} - 833 \, x - 310\right )} \sqrt{x + 1} \sqrt{-x + 1} + 2170 \, x - 310}{3003 \,{\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)^(5/2)/(1-x)^(15/2),x, algorithm="fricas")

[Out]

1/3003*(310*x^7 - 2170*x^6 + 6510*x^5 - 10850*x^4 + 10850*x^3 - 6510*x^2 + (2*x^6 - 14*x^5 + 43*x^4 - 77*x^3 -
659*x^2 - 833*x - 310)*sqrt(x + 1)*sqrt(-x + 1) + 2170*x - 310)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*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)**(5/2)/(1-x)**(15/2),x)

[Out]

Timed out

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Giac [A]  time = 1.12125, size = 47, normalized size = 0.58 \begin{align*} \frac{{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 12\right )} + 143\right )}{\left (x + 1\right )} - 429\right )}{\left (x + 1\right )}^{\frac{7}{2}} \sqrt{-x + 1}}{3003 \,{\left (x - 1\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(15/2),x, algorithm="giac")

[Out]

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