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

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

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Rubi [A]  time = 0.01, antiderivative size = 81, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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)^{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*}

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Mathematica [A]  time = 0.02, size = 35, normalized size = 0.43 \begin {gather*} \frac {(x+1)^{5/2} \left (-2 x^3+16 x^2-61 x+152\right )}{1155 (1-x)^{11/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.08, size = 62, normalized size = 0.77 \begin {gather*} \frac {(x+1)^{11/2} \left (\frac {231 (1-x)^3}{(x+1)^3}+\frac {495 (1-x)^2}{(x+1)^2}+\frac {385 (1-x)}{x+1}+105\right )}{9240 (1-x)^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x)^(11/2)*(105 + (231*(1 - x)^3)/(1 + x)^3 + (495*(1 - x)^2)/(1 + x)^2 + (385*(1 - x))/(1 + x)))/(9240*(
1 - x)^(11/2))

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fricas [A]  time = 1.53, size = 101, normalized size = 1.25 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.17, size = 35, normalized size = 0.43 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.00, size = 30, normalized size = 0.37 \begin {gather*} -\frac {\left (x +1\right )^{\frac {5}{2}} \left (2 x^{3}-16 x^{2}+61 x -152\right )}{1155 \left (-x +1\right )^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.39, size = 218, normalized size = 2.69 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.31, size = 94, normalized size = 1.16 \begin {gather*} \frac {\sqrt {1-x}\,\left (\frac {81\,x\,\sqrt {x+1}}{385}+\frac {152\,\sqrt {x+1}}{1155}+\frac {46\,x^2\,\sqrt {x+1}}{1155}-\frac {31\,x^3\,\sqrt {x+1}}{1155}+\frac {4\,x^4\,\sqrt {x+1}}{385}-\frac {2\,x^5\,\sqrt {x+1}}{1155}\right )}{x^6-6\,x^5+15\,x^4-20\,x^3+15\,x^2-6\,x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(3/2)/(1 - x)^(13/2),x)

[Out]

((1 - x)^(1/2)*((81*x*(x + 1)^(1/2))/385 + (152*(x + 1)^(1/2))/1155 + (46*x^2*(x + 1)^(1/2))/1155 - (31*x^3*(x
+ 1)^(1/2))/1155 + (4*x^4*(x + 1)^(1/2))/385 - (2*x^5*(x + 1)^(1/2))/1155))/(15*x^2 - 6*x - 20*x^3 + 15*x^4 -
6*x^5 + x^6 + 1)

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sympy [B]  time = 132.96, size = 1753, normalized size = 21.64

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(3/2)/(1-x)**(13/2),x)

[Out]

Piecewise((-2*I*(x + 1)**(17/2)/(1155*sqrt(x - 1)*(x + 1)**8 - 18480*sqrt(x - 1)*(x + 1)**7 + 129360*sqrt(x -
1)*(x + 1)**6 - 517440*sqrt(x - 1)*(x + 1)**5 + 1293600*sqrt(x - 1)*(x + 1)**4 - 2069760*sqrt(x - 1)*(x + 1)**
3 + 2069760*sqrt(x - 1)*(x + 1)**2 - 1182720*sqrt(x - 1)*(x + 1) + 295680*sqrt(x - 1)) + 34*I*(x + 1)**(15/2)/
(1155*sqrt(x - 1)*(x + 1)**8 - 18480*sqrt(x - 1)*(x + 1)**7 + 129360*sqrt(x - 1)*(x + 1)**6 - 517440*sqrt(x -
1)*(x + 1)**5 + 1293600*sqrt(x - 1)*(x + 1)**4 - 2069760*sqrt(x - 1)*(x + 1)**3 + 2069760*sqrt(x - 1)*(x + 1)*
*2 - 1182720*sqrt(x - 1)*(x + 1) + 295680*sqrt(x - 1)) - 255*I*(x + 1)**(13/2)/(1155*sqrt(x - 1)*(x + 1)**8 -
18480*sqrt(x - 1)*(x + 1)**7 + 129360*sqrt(x - 1)*(x + 1)**6 - 517440*sqrt(x - 1)*(x + 1)**5 + 1293600*sqrt(x
- 1)*(x + 1)**4 - 2069760*sqrt(x - 1)*(x + 1)**3 + 2069760*sqrt(x - 1)*(x + 1)**2 - 1182720*sqrt(x - 1)*(x + 1
) + 295680*sqrt(x - 1)) + 1105*I*(x + 1)**(11/2)/(1155*sqrt(x - 1)*(x + 1)**8 - 18480*sqrt(x - 1)*(x + 1)**7 +
129360*sqrt(x - 1)*(x + 1)**6 - 517440*sqrt(x - 1)*(x + 1)**5 + 1293600*sqrt(x - 1)*(x + 1)**4 - 2069760*sqrt
(x - 1)*(x + 1)**3 + 2069760*sqrt(x - 1)*(x + 1)**2 - 1182720*sqrt(x - 1)*(x + 1) + 295680*sqrt(x - 1)) - 2750
*I*(x + 1)**(9/2)/(1155*sqrt(x - 1)*(x + 1)**8 - 18480*sqrt(x - 1)*(x + 1)**7 + 129360*sqrt(x - 1)*(x + 1)**6
- 517440*sqrt(x - 1)*(x + 1)**5 + 1293600*sqrt(x - 1)*(x + 1)**4 - 2069760*sqrt(x - 1)*(x + 1)**3 + 2069760*sq
rt(x - 1)*(x + 1)**2 - 1182720*sqrt(x - 1)*(x + 1) + 295680*sqrt(x - 1)) + 3564*I*(x + 1)**(7/2)/(1155*sqrt(x
- 1)*(x + 1)**8 - 18480*sqrt(x - 1)*(x + 1)**7 + 129360*sqrt(x - 1)*(x + 1)**6 - 517440*sqrt(x - 1)*(x + 1)**5
+ 1293600*sqrt(x - 1)*(x + 1)**4 - 2069760*sqrt(x - 1)*(x + 1)**3 + 2069760*sqrt(x - 1)*(x + 1)**2 - 1182720*
sqrt(x - 1)*(x + 1) + 295680*sqrt(x - 1)) - 1848*I*(x + 1)**(5/2)/(1155*sqrt(x - 1)*(x + 1)**8 - 18480*sqrt(x
- 1)*(x + 1)**7 + 129360*sqrt(x - 1)*(x + 1)**6 - 517440*sqrt(x - 1)*(x + 1)**5 + 1293600*sqrt(x - 1)*(x + 1)*
*4 - 2069760*sqrt(x - 1)*(x + 1)**3 + 2069760*sqrt(x - 1)*(x + 1)**2 - 1182720*sqrt(x - 1)*(x + 1) + 295680*sq
rt(x - 1)), Abs(x + 1)/2 > 1), (2*(x + 1)**(17/2)/(1155*sqrt(1 - x)*(x + 1)**8 - 18480*sqrt(1 - x)*(x + 1)**7
+ 129360*sqrt(1 - x)*(x + 1)**6 - 517440*sqrt(1 - x)*(x + 1)**5 + 1293600*sqrt(1 - x)*(x + 1)**4 - 2069760*sqr
t(1 - x)*(x + 1)**3 + 2069760*sqrt(1 - x)*(x + 1)**2 - 1182720*sqrt(1 - x)*(x + 1) + 295680*sqrt(1 - x)) - 34*
(x + 1)**(15/2)/(1155*sqrt(1 - x)*(x + 1)**8 - 18480*sqrt(1 - x)*(x + 1)**7 + 129360*sqrt(1 - x)*(x + 1)**6 -
517440*sqrt(1 - x)*(x + 1)**5 + 1293600*sqrt(1 - x)*(x + 1)**4 - 2069760*sqrt(1 - x)*(x + 1)**3 + 2069760*sqrt
(1 - x)*(x + 1)**2 - 1182720*sqrt(1 - x)*(x + 1) + 295680*sqrt(1 - x)) + 255*(x + 1)**(13/2)/(1155*sqrt(1 - x)
*(x + 1)**8 - 18480*sqrt(1 - x)*(x + 1)**7 + 129360*sqrt(1 - x)*(x + 1)**6 - 517440*sqrt(1 - x)*(x + 1)**5 + 1
293600*sqrt(1 - x)*(x + 1)**4 - 2069760*sqrt(1 - x)*(x + 1)**3 + 2069760*sqrt(1 - x)*(x + 1)**2 - 1182720*sqrt
(1 - x)*(x + 1) + 295680*sqrt(1 - x)) - 1105*(x + 1)**(11/2)/(1155*sqrt(1 - x)*(x + 1)**8 - 18480*sqrt(1 - x)*
(x + 1)**7 + 129360*sqrt(1 - x)*(x + 1)**6 - 517440*sqrt(1 - x)*(x + 1)**5 + 1293600*sqrt(1 - x)*(x + 1)**4 -
2069760*sqrt(1 - x)*(x + 1)**3 + 2069760*sqrt(1 - x)*(x + 1)**2 - 1182720*sqrt(1 - x)*(x + 1) + 295680*sqrt(1
- x)) + 2750*(x + 1)**(9/2)/(1155*sqrt(1 - x)*(x + 1)**8 - 18480*sqrt(1 - x)*(x + 1)**7 + 129360*sqrt(1 - x)*(
x + 1)**6 - 517440*sqrt(1 - x)*(x + 1)**5 + 1293600*sqrt(1 - x)*(x + 1)**4 - 2069760*sqrt(1 - x)*(x + 1)**3 +
2069760*sqrt(1 - x)*(x + 1)**2 - 1182720*sqrt(1 - x)*(x + 1) + 295680*sqrt(1 - x)) - 3564*(x + 1)**(7/2)/(1155
*sqrt(1 - x)*(x + 1)**8 - 18480*sqrt(1 - x)*(x + 1)**7 + 129360*sqrt(1 - x)*(x + 1)**6 - 517440*sqrt(1 - x)*(x
+ 1)**5 + 1293600*sqrt(1 - x)*(x + 1)**4 - 2069760*sqrt(1 - x)*(x + 1)**3 + 2069760*sqrt(1 - x)*(x + 1)**2 -
1182720*sqrt(1 - x)*(x + 1) + 295680*sqrt(1 - x)) + 1848*(x + 1)**(5/2)/(1155*sqrt(1 - x)*(x + 1)**8 - 18480*s
qrt(1 - x)*(x + 1)**7 + 129360*sqrt(1 - x)*(x + 1)**6 - 517440*sqrt(1 - x)*(x + 1)**5 + 1293600*sqrt(1 - x)*(x
+ 1)**4 - 2069760*sqrt(1 - x)*(x + 1)**3 + 2069760*sqrt(1 - x)*(x + 1)**2 - 1182720*sqrt(1 - x)*(x + 1) + 295
680*sqrt(1 - x)), True))

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