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

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

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Rubi [A]  time = 0.02, antiderivative size = 101, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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]

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])

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*}

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Mathematica [A]  time = 0.02, size = 40, normalized size = 0.40 \begin {gather*} \frac {(x+1)^{5/2} \left (8 x^4-72 x^3+308 x^2-852 x+1763\right )}{15015 (1-x)^{13/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.09, size = 76, normalized size = 0.75 \begin {gather*} \frac {(x+1)^{13/2} \left (\frac {3003 (1-x)^4}{(x+1)^4}+\frac {8580 (1-x)^3}{(x+1)^3}+\frac {10010 (1-x)^2}{(x+1)^2}+\frac {5460 (1-x)}{x+1}+1155\right )}{240240 (1-x)^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x)^(13/2)*(1155 + (3003*(1 - x)^4)/(1 + x)^4 + (8580*(1 - x)^3)/(1 + x)^3 + (10010*(1 - x)^2)/(1 + x)^2
+ (5460*(1 - x))/(1 + x)))/(240240*(1 - x)^(13/2))

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fricas [A]  time = 1.23, size = 116, normalized size = 1.15 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.23, size = 42, normalized size = 0.42 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.00, size = 35, normalized size = 0.35 \begin {gather*} \frac {\left (x +1\right )^{\frac {5}{2}} \left (8 x^{4}-72 x^{3}+308 x^{2}-852 x +1763\right )}{15015 \left (-x +1\right )^{\frac {13}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.38, size = 269, normalized size = 2.66 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.33, size = 110, normalized size = 1.09 \begin {gather*} -\frac {\sqrt {1-x}\,\left (\frac {382\,x\,\sqrt {x+1}}{2145}+\frac {1763\,\sqrt {x+1}}{15015}+\frac {367\,x^2\,\sqrt {x+1}}{15015}-\frac {4\,x^3\,\sqrt {x+1}}{195}+\frac {172\,x^4\,\sqrt {x+1}}{15015}-\frac {8\,x^5\,\sqrt {x+1}}{2145}+\frac {8\,x^6\,\sqrt {x+1}}{15015}\right )}{x^7-7\,x^6+21\,x^5-35\,x^4+35\,x^3-21\,x^2+7\,x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(3/2)/(1 - x)^(15/2),x)

[Out]

-((1 - x)^(1/2)*((382*x*(x + 1)^(1/2))/2145 + (1763*(x + 1)^(1/2))/15015 + (367*x^2*(x + 1)^(1/2))/15015 - (4*
x^3*(x + 1)^(1/2))/195 + (172*x^4*(x + 1)^(1/2))/15015 - (8*x^5*(x + 1)^(1/2))/2145 + (8*x^6*(x + 1)^(1/2))/15
015))/(7*x - 21*x^2 + 35*x^3 - 35*x^4 + 21*x^5 - 7*x^6 + x^7 - 1)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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