3.1103 \(\int \frac{(1+x)^{5/2}}{(1-x)^{19/2}} \, dx\)

Optimal. Leaf size=121 \[ \frac{8 (x+1)^{7/2}}{153153 (1-x)^{7/2}}+\frac{8 (x+1)^{7/2}}{21879 (1-x)^{9/2}}+\frac{4 (x+1)^{7/2}}{2431 (1-x)^{11/2}}+\frac{4 (x+1)^{7/2}}{663 (1-x)^{13/2}}+\frac{(x+1)^{7/2}}{51 (1-x)^{15/2}}+\frac{(x+1)^{7/2}}{17 (1-x)^{17/2}} \]

[Out]

(1 + x)^(7/2)/(17*(1 - x)^(17/2)) + (1 + x)^(7/2)/(51*(1 - x)^(15/2)) + (4*(1 + x)^(7/2))/(663*(1 - x)^(13/2))
+ (4*(1 + x)^(7/2))/(2431*(1 - x)^(11/2)) + (8*(1 + x)^(7/2))/(21879*(1 - x)^(9/2)) + (8*(1 + x)^(7/2))/(1531
53*(1 - x)^(7/2))

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Rubi [A]  time = 0.0246597, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {45, 37} \[ \frac{8 (x+1)^{7/2}}{153153 (1-x)^{7/2}}+\frac{8 (x+1)^{7/2}}{21879 (1-x)^{9/2}}+\frac{4 (x+1)^{7/2}}{2431 (1-x)^{11/2}}+\frac{4 (x+1)^{7/2}}{663 (1-x)^{13/2}}+\frac{(x+1)^{7/2}}{51 (1-x)^{15/2}}+\frac{(x+1)^{7/2}}{17 (1-x)^{17/2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(5/2)/(1 - x)^(19/2),x]

[Out]

(1 + x)^(7/2)/(17*(1 - x)^(17/2)) + (1 + x)^(7/2)/(51*(1 - x)^(15/2)) + (4*(1 + x)^(7/2))/(663*(1 - x)^(13/2))
+ (4*(1 + x)^(7/2))/(2431*(1 - x)^(11/2)) + (8*(1 + x)^(7/2))/(21879*(1 - x)^(9/2)) + (8*(1 + x)^(7/2))/(1531
53*(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)^{19/2}} \, dx &=\frac{(1+x)^{7/2}}{17 (1-x)^{17/2}}+\frac{5}{17} \int \frac{(1+x)^{5/2}}{(1-x)^{17/2}} \, dx\\ &=\frac{(1+x)^{7/2}}{17 (1-x)^{17/2}}+\frac{(1+x)^{7/2}}{51 (1-x)^{15/2}}+\frac{4}{51} \int \frac{(1+x)^{5/2}}{(1-x)^{15/2}} \, dx\\ &=\frac{(1+x)^{7/2}}{17 (1-x)^{17/2}}+\frac{(1+x)^{7/2}}{51 (1-x)^{15/2}}+\frac{4 (1+x)^{7/2}}{663 (1-x)^{13/2}}+\frac{4}{221} \int \frac{(1+x)^{5/2}}{(1-x)^{13/2}} \, dx\\ &=\frac{(1+x)^{7/2}}{17 (1-x)^{17/2}}+\frac{(1+x)^{7/2}}{51 (1-x)^{15/2}}+\frac{4 (1+x)^{7/2}}{663 (1-x)^{13/2}}+\frac{4 (1+x)^{7/2}}{2431 (1-x)^{11/2}}+\frac{8 \int \frac{(1+x)^{5/2}}{(1-x)^{11/2}} \, dx}{2431}\\ &=\frac{(1+x)^{7/2}}{17 (1-x)^{17/2}}+\frac{(1+x)^{7/2}}{51 (1-x)^{15/2}}+\frac{4 (1+x)^{7/2}}{663 (1-x)^{13/2}}+\frac{4 (1+x)^{7/2}}{2431 (1-x)^{11/2}}+\frac{8 (1+x)^{7/2}}{21879 (1-x)^{9/2}}+\frac{8 \int \frac{(1+x)^{5/2}}{(1-x)^{9/2}} \, dx}{21879}\\ &=\frac{(1+x)^{7/2}}{17 (1-x)^{17/2}}+\frac{(1+x)^{7/2}}{51 (1-x)^{15/2}}+\frac{4 (1+x)^{7/2}}{663 (1-x)^{13/2}}+\frac{4 (1+x)^{7/2}}{2431 (1-x)^{11/2}}+\frac{8 (1+x)^{7/2}}{21879 (1-x)^{9/2}}+\frac{8 (1+x)^{7/2}}{153153 (1-x)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0214536, size = 45, normalized size = 0.37 \[ \frac{(x+1)^{7/2} \left (-8 x^5+96 x^4-556 x^3+2096 x^2-5871 x+13252\right )}{153153 (1-x)^{17/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(5/2)/(1 - x)^(19/2),x]

[Out]

((1 + x)^(7/2)*(13252 - 5871*x + 2096*x^2 - 556*x^3 + 96*x^4 - 8*x^5))/(153153*(1 - x)^(17/2))

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Maple [A]  time = 0.003, size = 40, normalized size = 0.3 \begin{align*} -{\frac{8\,{x}^{5}-96\,{x}^{4}+556\,{x}^{3}-2096\,{x}^{2}+5871\,x-13252}{153153} \left ( 1+x \right ) ^{{\frac{7}{2}}} \left ( 1-x \right ) ^{-{\frac{17}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(5/2)/(1-x)^(19/2),x)

[Out]

-1/153153*(1+x)^(7/2)*(8*x^5-96*x^4+556*x^3-2096*x^2+5871*x-13252)/(1-x)^(17/2)

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Maxima [B]  time = 1.03146, size = 610, normalized size = 5.04 \begin{align*} -\frac{{\left (-x^{2} + 1\right )}^{\frac{5}{2}}}{6 \,{\left (x^{11} - 11 \, x^{10} + 55 \, x^{9} - 165 \, x^{8} + 330 \, x^{7} - 462 \, x^{6} + 462 \, x^{5} - 330 \, x^{4} + 165 \, x^{3} - 55 \, x^{2} + 11 \, x - 1\right )}} - \frac{5 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{42 \,{\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{5 \, \sqrt{-x^{2} + 1}}{119 \,{\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}}{714 \,{\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}}{1326 \,{\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}}{2431 \,{\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}}{21879 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac{20 \, \sqrt{-x^{2} + 1}}{153153 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac{4 \, \sqrt{-x^{2} + 1}}{51051 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{8 \, \sqrt{-x^{2} + 1}}{153153 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{8 \, \sqrt{-x^{2} + 1}}{153153 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(19/2),x, algorithm="maxima")

[Out]

-1/6*(-x^2 + 1)^(5/2)/(x^11 - 11*x^10 + 55*x^9 - 165*x^8 + 330*x^7 - 462*x^6 + 462*x^5 - 330*x^4 + 165*x^3 - 5
5*x^2 + 11*x - 1) - 5/42*(-x^2 + 1)^(3/2)/(x^10 - 10*x^9 + 45*x^8 - 120*x^7 + 210*x^6 - 252*x^5 + 210*x^4 - 12
0*x^3 + 45*x^2 - 10*x + 1) - 5/119*sqrt(-x^2 + 1)/(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) - 1/714*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
) + 1/1326*sqrt(-x^2 + 1)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x - 1) - 1/2431*sqrt(-x^2 + 1)/
(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1) + 5/21879*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5
*x - 1) - 20/153153*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 4/51051*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x
- 1) - 8/153153*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 8/153153*sqrt(-x^2 + 1)/(x - 1)

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Fricas [A]  time = 1.56385, size = 448, normalized size = 3.7 \begin{align*} \frac{13252 \, x^{9} - 119268 \, x^{8} + 477072 \, x^{7} - 1113168 \, x^{6} + 1669752 \, x^{5} - 1669752 \, x^{4} + 1113168 \, x^{3} - 477072 \, x^{2} +{\left (8 \, x^{8} - 72 \, x^{7} + 292 \, x^{6} - 708 \, x^{5} + 1155 \, x^{4} - 1371 \, x^{3} - 24239 \, x^{2} - 33885 \, x - 13252\right )} \sqrt{x + 1} \sqrt{-x + 1} + 119268 \, x - 13252}{153153 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(19/2),x, algorithm="fricas")

[Out]

1/153153*(13252*x^9 - 119268*x^8 + 477072*x^7 - 1113168*x^6 + 1669752*x^5 - 1669752*x^4 + 1113168*x^3 - 477072
*x^2 + (8*x^8 - 72*x^7 + 292*x^6 - 708*x^5 + 1155*x^4 - 1371*x^3 - 24239*x^2 - 33885*x - 13252)*sqrt(x + 1)*sq
rt(-x + 1) + 119268*x - 13252)/(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)

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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)**(19/2),x)

[Out]

Timed out

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Giac [A]  time = 1.16272, size = 65, normalized size = 0.54 \begin{align*} \frac{{\left ({\left (4 \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 16\right )} + 255\right )}{\left (x + 1\right )} - 1105\right )}{\left (x + 1\right )} + 12155\right )}{\left (x + 1\right )} - 21879\right )}{\left (x + 1\right )}^{\frac{7}{2}} \sqrt{-x + 1}}{153153 \,{\left (x - 1\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(5/2)/(1-x)^(19/2),x, algorithm="giac")

[Out]

1/153153*((4*((2*(x + 1)*(x - 16) + 255)*(x + 1) - 1105)*(x + 1) + 12155)*(x + 1) - 21879)*(x + 1)^(7/2)*sqrt(
-x + 1)/(x - 1)^9