3.11.74 \(\int \frac {\sqrt {1+x}}{(1-x)^{13/2}} \, dx\) [1074]
Optimal. Leaf size=101 \[ \frac {(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac {4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac {4 (1+x)^{3/2}}{231 (1-x)^{7/2}}+\frac {8 (1+x)^{3/2}}{1155 (1-x)^{5/2}}+\frac {8 (1+x)^{3/2}}{3465 (1-x)^{3/2}} \]
[Out]
1/11*(1+x)^(3/2)/(1-x)^(11/2)+4/99*(1+x)^(3/2)/(1-x)^(9/2)+4/231*(1+x)^(3/2)/(1-x)^(7/2)+8/1155*(1+x)^(3/2)/(1
-x)^(5/2)+8/3465*(1+x)^(3/2)/(1-x)^(3/2)
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Rubi [A]
time = 0.01, 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 = {47, 37}
\begin {gather*} \frac {8 (x+1)^{3/2}}{3465 (1-x)^{3/2}}+\frac {8 (x+1)^{3/2}}{1155 (1-x)^{5/2}}+\frac {4 (x+1)^{3/2}}{231 (1-x)^{7/2}}+\frac {4 (x+1)^{3/2}}{99 (1-x)^{9/2}}+\frac {(x+1)^{3/2}}{11 (1-x)^{11/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[1 + x]/(1 - x)^(13/2),x]
[Out]
(1 + x)^(3/2)/(11*(1 - x)^(11/2)) + (4*(1 + x)^(3/2))/(99*(1 - x)^(9/2)) + (4*(1 + x)^(3/2))/(231*(1 - x)^(7/2
)) + (8*(1 + x)^(3/2))/(1155*(1 - x)^(5/2)) + (8*(1 + x)^(3/2))/(3465*(1 - x)^(3/2))
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 47
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 {\sqrt {1+x}}{(1-x)^{13/2}} \, dx &=\frac {(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac {4}{11} \int \frac {\sqrt {1+x}}{(1-x)^{11/2}} \, dx\\ &=\frac {(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac {4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac {4}{33} \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx\\ &=\frac {(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac {4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac {4 (1+x)^{3/2}}{231 (1-x)^{7/2}}+\frac {8}{231} \int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx\\ &=\frac {(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac {4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac {4 (1+x)^{3/2}}{231 (1-x)^{7/2}}+\frac {8 (1+x)^{3/2}}{1155 (1-x)^{5/2}}+\frac {8 \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx}{1155}\\ &=\frac {(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac {4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac {4 (1+x)^{3/2}}{231 (1-x)^{7/2}}+\frac {8 (1+x)^{3/2}}{1155 (1-x)^{5/2}}+\frac {8 (1+x)^{3/2}}{3465 (1-x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 40, normalized size = 0.40 \begin {gather*} \frac {(1+x)^{3/2} \left (547-364 x+180 x^2-56 x^3+8 x^4\right )}{3465 (1-x)^{11/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[1 + x]/(1 - x)^(13/2),x]
[Out]
((1 + x)^(3/2)*(547 - 364*x + 180*x^2 - 56*x^3 + 8*x^4))/(3465*(1 - x)^(11/2))
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Maple [A]
time = 0.16, size = 86, normalized size = 0.85
| | |
| method |
result |
size |
| | |
| gosper |
\(\frac {\left (1+x \right )^{\frac {3}{2}} \left (8 x^{4}-56 x^{3}+180 x^{2}-364 x +547\right )}{3465 \left (1-x \right )^{\frac {11}{2}}}\) |
\(35\) |
| risch |
\(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{6}-40 x^{5}+76 x^{4}-60 x^{3}-x^{2}+730 x +547\right )}{3465 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{5} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) |
\(71\) |
| default |
\(\frac {2 \sqrt {1+x}}{11 \left (1-x \right )^{\frac {11}{2}}}-\frac {\sqrt {1+x}}{99 \left (1-x \right )^{\frac {9}{2}}}-\frac {4 \sqrt {1+x}}{693 \left (1-x \right )^{\frac {7}{2}}}-\frac {4 \sqrt {1+x}}{1155 \left (1-x \right )^{\frac {5}{2}}}-\frac {8 \sqrt {1+x}}{3465 \left (1-x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1+x}}{3465 \sqrt {1-x}}\) |
\(86\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((1+x)^(1/2)/(1-x)^(13/2),x,method=_RETURNVERBOSE)
[Out]
2/11*(1+x)^(1/2)/(1-x)^(11/2)-1/99*(1+x)^(1/2)/(1-x)^(9/2)-4/693*(1+x)^(1/2)/(1-x)^(7/2)-4/1155*(1+x)^(1/2)/(1
-x)^(5/2)-8/3465*(1+x)^(1/2)/(1-x)^(3/2)-8/3465*(1+x)^(1/2)/(1-x)^(1/2)
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 172 vs.
\(2 (71) = 142\).
time = 0.33, size = 172, normalized size = 1.70 \begin {gather*} \frac {2 \, \sqrt {-x^{2} + 1}}{11 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{99 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {4 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {4 \, \sqrt {-x^{2} + 1}}{1155 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {8 \, \sqrt {-x^{2} + 1}}{3465 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {8 \, \sqrt {-x^{2} + 1}}{3465 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+x)^(1/2)/(1-x)^(13/2),x, algorithm="maxima")
[Out]
2/11*sqrt(-x^2 + 1)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1) + 1/99*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10
*x^3 - 10*x^2 + 5*x - 1) - 4/693*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 4/1155*sqrt(-x^2 + 1)/(x^3 -
3*x^2 + 3*x - 1) - 8/3465*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 8/3465*sqrt(-x^2 + 1)/(x - 1)
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Fricas [A]
time = 0.57, size = 100, normalized size = 0.99 \begin {gather*} \frac {547 \, x^{6} - 3282 \, x^{5} + 8205 \, x^{4} - 10940 \, x^{3} + 8205 \, x^{2} + {\left (8 \, x^{5} - 48 \, x^{4} + 124 \, x^{3} - 184 \, x^{2} + 183 \, x + 547\right )} \sqrt {x + 1} \sqrt {-x + 1} - 3282 \, x + 547}{3465 \, {\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)^(1/2)/(1-x)^(13/2),x, algorithm="fricas")
[Out]
1/3465*(547*x^6 - 3282*x^5 + 8205*x^4 - 10940*x^3 + 8205*x^2 + (8*x^5 - 48*x^4 + 124*x^3 - 184*x^2 + 183*x + 5
47)*sqrt(x + 1)*sqrt(-x + 1) - 3282*x + 547)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1)
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Sympy [C] Result contains complex when optimal does not.
time = 134.79, size = 3648, normalized size = 36.12 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+x)**(1/2)/(1-x)**(13/2),x)
[Out]
Piecewise((8*I*(x + 1)**(23/2)/(3465*sqrt(x - 1)*(x + 1)**11 - 76230*sqrt(x - 1)*(x + 1)**10 + 762300*sqrt(x -
1)*(x + 1)**9 - 4573800*sqrt(x - 1)*(x + 1)**8 + 18295200*sqrt(x - 1)*(x + 1)**7 - 51226560*sqrt(x - 1)*(x +
1)**6 + 102453120*sqrt(x - 1)*(x + 1)**5 - 146361600*sqrt(x - 1)*(x + 1)**4 + 146361600*sqrt(x - 1)*(x + 1)**3
- 97574400*sqrt(x - 1)*(x + 1)**2 + 39029760*sqrt(x - 1)*(x + 1) - 7096320*sqrt(x - 1)) - 184*I*(x + 1)**(21/
2)/(3465*sqrt(x - 1)*(x + 1)**11 - 76230*sqrt(x - 1)*(x + 1)**10 + 762300*sqrt(x - 1)*(x + 1)**9 - 4573800*sqr
t(x - 1)*(x + 1)**8 + 18295200*sqrt(x - 1)*(x + 1)**7 - 51226560*sqrt(x - 1)*(x + 1)**6 + 102453120*sqrt(x - 1
)*(x + 1)**5 - 146361600*sqrt(x - 1)*(x + 1)**4 + 146361600*sqrt(x - 1)*(x + 1)**3 - 97574400*sqrt(x - 1)*(x +
1)**2 + 39029760*sqrt(x - 1)*(x + 1) - 7096320*sqrt(x - 1)) + 1932*I*(x + 1)**(19/2)/(3465*sqrt(x - 1)*(x + 1
)**11 - 76230*sqrt(x - 1)*(x + 1)**10 + 762300*sqrt(x - 1)*(x + 1)**9 - 4573800*sqrt(x - 1)*(x + 1)**8 + 18295
200*sqrt(x - 1)*(x + 1)**7 - 51226560*sqrt(x - 1)*(x + 1)**6 + 102453120*sqrt(x - 1)*(x + 1)**5 - 146361600*sq
rt(x - 1)*(x + 1)**4 + 146361600*sqrt(x - 1)*(x + 1)**3 - 97574400*sqrt(x - 1)*(x + 1)**2 + 39029760*sqrt(x -
1)*(x + 1) - 7096320*sqrt(x - 1)) - 12236*I*(x + 1)**(17/2)/(3465*sqrt(x - 1)*(x + 1)**11 - 76230*sqrt(x - 1)*
(x + 1)**10 + 762300*sqrt(x - 1)*(x + 1)**9 - 4573800*sqrt(x - 1)*(x + 1)**8 + 18295200*sqrt(x - 1)*(x + 1)**7
- 51226560*sqrt(x - 1)*(x + 1)**6 + 102453120*sqrt(x - 1)*(x + 1)**5 - 146361600*sqrt(x - 1)*(x + 1)**4 + 146
361600*sqrt(x - 1)*(x + 1)**3 - 97574400*sqrt(x - 1)*(x + 1)**2 + 39029760*sqrt(x - 1)*(x + 1) - 7096320*sqrt(
x - 1)) + 52003*I*(x + 1)**(15/2)/(3465*sqrt(x - 1)*(x + 1)**11 - 76230*sqrt(x - 1)*(x + 1)**10 + 762300*sqrt(
x - 1)*(x + 1)**9 - 4573800*sqrt(x - 1)*(x + 1)**8 + 18295200*sqrt(x - 1)*(x + 1)**7 - 51226560*sqrt(x - 1)*(x
+ 1)**6 + 102453120*sqrt(x - 1)*(x + 1)**5 - 146361600*sqrt(x - 1)*(x + 1)**4 + 146361600*sqrt(x - 1)*(x + 1)
**3 - 97574400*sqrt(x - 1)*(x + 1)**2 + 39029760*sqrt(x - 1)*(x + 1) - 7096320*sqrt(x - 1)) - 155316*I*(x + 1)
**(13/2)/(3465*sqrt(x - 1)*(x + 1)**11 - 76230*sqrt(x - 1)*(x + 1)**10 + 762300*sqrt(x - 1)*(x + 1)**9 - 45738
00*sqrt(x - 1)*(x + 1)**8 + 18295200*sqrt(x - 1)*(x + 1)**7 - 51226560*sqrt(x - 1)*(x + 1)**6 + 102453120*sqrt
(x - 1)*(x + 1)**5 - 146361600*sqrt(x - 1)*(x + 1)**4 + 146361600*sqrt(x - 1)*(x + 1)**3 - 97574400*sqrt(x - 1
)*(x + 1)**2 + 39029760*sqrt(x - 1)*(x + 1) - 7096320*sqrt(x - 1)) + 329588*I*(x + 1)**(11/2)/(3465*sqrt(x - 1
)*(x + 1)**11 - 76230*sqrt(x - 1)*(x + 1)**10 + 762300*sqrt(x - 1)*(x + 1)**9 - 4573800*sqrt(x - 1)*(x + 1)**8
+ 18295200*sqrt(x - 1)*(x + 1)**7 - 51226560*sqrt(x - 1)*(x + 1)**6 + 102453120*sqrt(x - 1)*(x + 1)**5 - 1463
61600*sqrt(x - 1)*(x + 1)**4 + 146361600*sqrt(x - 1)*(x + 1)**3 - 97574400*sqrt(x - 1)*(x + 1)**2 + 39029760*s
qrt(x - 1)*(x + 1) - 7096320*sqrt(x - 1)) - 488224*I*(x + 1)**(9/2)/(3465*sqrt(x - 1)*(x + 1)**11 - 76230*sqrt
(x - 1)*(x + 1)**10 + 762300*sqrt(x - 1)*(x + 1)**9 - 4573800*sqrt(x - 1)*(x + 1)**8 + 18295200*sqrt(x - 1)*(x
+ 1)**7 - 51226560*sqrt(x - 1)*(x + 1)**6 + 102453120*sqrt(x - 1)*(x + 1)**5 - 146361600*sqrt(x - 1)*(x + 1)*
*4 + 146361600*sqrt(x - 1)*(x + 1)**3 - 97574400*sqrt(x - 1)*(x + 1)**2 + 39029760*sqrt(x - 1)*(x + 1) - 70963
20*sqrt(x - 1)) + 479952*I*(x + 1)**(7/2)/(3465*sqrt(x - 1)*(x + 1)**11 - 76230*sqrt(x - 1)*(x + 1)**10 + 7623
00*sqrt(x - 1)*(x + 1)**9 - 4573800*sqrt(x - 1)*(x + 1)**8 + 18295200*sqrt(x - 1)*(x + 1)**7 - 51226560*sqrt(x
- 1)*(x + 1)**6 + 102453120*sqrt(x - 1)*(x + 1)**5 - 146361600*sqrt(x - 1)*(x + 1)**4 + 146361600*sqrt(x - 1)
*(x + 1)**3 - 97574400*sqrt(x - 1)*(x + 1)**2 + 39029760*sqrt(x - 1)*(x + 1) - 7096320*sqrt(x - 1)) - 280896*I
*(x + 1)**(5/2)/(3465*sqrt(x - 1)*(x + 1)**11 - 76230*sqrt(x - 1)*(x + 1)**10 + 762300*sqrt(x - 1)*(x + 1)**9
- 4573800*sqrt(x - 1)*(x + 1)**8 + 18295200*sqrt(x - 1)*(x + 1)**7 - 51226560*sqrt(x - 1)*(x + 1)**6 + 1024531
20*sqrt(x - 1)*(x + 1)**5 - 146361600*sqrt(x - 1)*(x + 1)**4 + 146361600*sqrt(x - 1)*(x + 1)**3 - 97574400*sqr
t(x - 1)*(x + 1)**2 + 39029760*sqrt(x - 1)*(x + 1) - 7096320*sqrt(x - 1)) + 73920*I*(x + 1)**(3/2)/(3465*sqrt(
x - 1)*(x + 1)**11 - 76230*sqrt(x - 1)*(x + 1)**10 + 762300*sqrt(x - 1)*(x + 1)**9 - 4573800*sqrt(x - 1)*(x +
1)**8 + 18295200*sqrt(x - 1)*(x + 1)**7 - 51226560*sqrt(x - 1)*(x + 1)**6 + 102453120*sqrt(x - 1)*(x + 1)**5 -
146361600*sqrt(x - 1)*(x + 1)**4 + 146361600*sqrt(x - 1)*(x + 1)**3 - 97574400*sqrt(x - 1)*(x + 1)**2 + 39029
760*sqrt(x - 1)*(x + 1) - 7096320*sqrt(x - 1)), Abs(x + 1) > 2), (-8*(x + 1)**(23/2)/(3465*sqrt(1 - x)*(x + 1)
**11 - 76230*sqrt(1 - x)*(x + 1)**10 + 762300*sqrt(1 - x)*(x + 1)**9 - 4573800*sqrt(1 - x)*(x + 1)**8 + 182952
00*sqrt(1 - x)*(x + 1)**7 - 51226560*sqrt(1 - x)*(x + 1)**6 + 102453120*sqrt(1 - x)*(x + 1)**5 - 146361600*sqr
t(1 - x)*(x + 1)**4 + 146361600*sqrt(1 - x)*(x + 1)**3 - 97574400*sqrt(1 - x)*(x + 1)**2 + 39029760*sqrt(1 - x
)*(x + 1) - 7096320*sqrt(1 - x)) + 184*(x + 1)*...
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Giac [A]
time = 0.92, size = 42, normalized size = 0.42 \begin {gather*} \frac {{\left (4 \, {\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 10\right )} + 99\right )} {\left (x + 1\right )} - 231\right )} {\left (x + 1\right )} + 1155\right )} {\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1}}{3465 \, {\left (x - 1\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+x)^(1/2)/(1-x)^(13/2),x, algorithm="giac")
[Out]
1/3465*(4*((2*(x + 1)*(x - 10) + 99)*(x + 1) - 231)*(x + 1) + 1155)*(x + 1)^(3/2)*sqrt(-x + 1)/(x - 1)^6
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Mupad [B]
time = 0.29, size = 94, normalized size = 0.93 \begin {gather*} \frac {\sqrt {1-x}\,\left (\frac {61\,x\,\sqrt {x+1}}{1155}+\frac {547\,\sqrt {x+1}}{3465}-\frac {184\,x^2\,\sqrt {x+1}}{3465}+\frac {124\,x^3\,\sqrt {x+1}}{3465}-\frac {16\,x^4\,\sqrt {x+1}}{1155}+\frac {8\,x^5\,\sqrt {x+1}}{3465}\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)^(1/2)/(1 - x)^(13/2),x)
[Out]
((1 - x)^(1/2)*((61*x*(x + 1)^(1/2))/1155 + (547*(x + 1)^(1/2))/3465 - (184*x^2*(x + 1)^(1/2))/3465 + (124*x^3
*(x + 1)^(1/2))/3465 - (16*x^4*(x + 1)^(1/2))/1155 + (8*x^5*(x + 1)^(1/2))/3465))/(15*x^2 - 6*x - 20*x^3 + 15*
x^4 - 6*x^5 + x^6 + 1)
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