3.12.15 \(\int \frac {1}{(1-x)^{11/2} \sqrt {1+x}} \, dx\) [1115]

Optimal. Leaf size=101 \[ \frac {\sqrt {1+x}}{9 (1-x)^{9/2}}+\frac {4 \sqrt {1+x}}{63 (1-x)^{7/2}}+\frac {4 \sqrt {1+x}}{105 (1-x)^{5/2}}+\frac {8 \sqrt {1+x}}{315 (1-x)^{3/2}}+\frac {8 \sqrt {1+x}}{315 \sqrt {1-x}} \]

[Out]

1/9*(1+x)^(1/2)/(1-x)^(9/2)+4/63*(1+x)^(1/2)/(1-x)^(7/2)+4/105*(1+x)^(1/2)/(1-x)^(5/2)+8/315*(1+x)^(1/2)/(1-x)
^(3/2)+8/315*(1+x)^(1/2)/(1-x)^(1/2)

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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 \sqrt {x+1}}{315 \sqrt {1-x}}+\frac {8 \sqrt {x+1}}{315 (1-x)^{3/2}}+\frac {4 \sqrt {x+1}}{105 (1-x)^{5/2}}+\frac {4 \sqrt {x+1}}{63 (1-x)^{7/2}}+\frac {\sqrt {x+1}}{9 (1-x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(11/2)*Sqrt[1 + x]),x]

[Out]

Sqrt[1 + x]/(9*(1 - x)^(9/2)) + (4*Sqrt[1 + x])/(63*(1 - x)^(7/2)) + (4*Sqrt[1 + x])/(105*(1 - x)^(5/2)) + (8*
Sqrt[1 + x])/(315*(1 - x)^(3/2)) + (8*Sqrt[1 + x])/(315*Sqrt[1 - x])

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 {1}{(1-x)^{11/2} \sqrt {1+x}} \, dx &=\frac {\sqrt {1+x}}{9 (1-x)^{9/2}}+\frac {4}{9} \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{9 (1-x)^{9/2}}+\frac {4 \sqrt {1+x}}{63 (1-x)^{7/2}}+\frac {4}{21} \int \frac {1}{(1-x)^{7/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{9 (1-x)^{9/2}}+\frac {4 \sqrt {1+x}}{63 (1-x)^{7/2}}+\frac {4 \sqrt {1+x}}{105 (1-x)^{5/2}}+\frac {8}{105} \int \frac {1}{(1-x)^{5/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{9 (1-x)^{9/2}}+\frac {4 \sqrt {1+x}}{63 (1-x)^{7/2}}+\frac {4 \sqrt {1+x}}{105 (1-x)^{5/2}}+\frac {8 \sqrt {1+x}}{315 (1-x)^{3/2}}+\frac {8}{315} \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{9 (1-x)^{9/2}}+\frac {4 \sqrt {1+x}}{63 (1-x)^{7/2}}+\frac {4 \sqrt {1+x}}{105 (1-x)^{5/2}}+\frac {8 \sqrt {1+x}}{315 (1-x)^{3/2}}+\frac {8 \sqrt {1+x}}{315 \sqrt {1-x}}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 40, normalized size = 0.40 \begin {gather*} \frac {\sqrt {1+x} \left (83-100 x+84 x^2-40 x^3+8 x^4\right )}{315 (1-x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(11/2)*Sqrt[1 + x]),x]

[Out]

(Sqrt[1 + x]*(83 - 100*x + 84*x^2 - 40*x^3 + 8*x^4))/(315*(1 - x)^(9/2))

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 49.64, size = 525, normalized size = 5.20 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {83-100 x+84 x^2-40 x^3+8 x^4}{315 \left (1-4 x+6 x^2-4 x^3+x^4\right ) \sqrt {\frac {1-x}{1+x}}},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},\frac {-315 I}{-10080 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}-2520 \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}+315 \left (1+x\right )^4 \sqrt {1-\frac {2}{1+x}}+5040 \sqrt {1-\frac {2}{1+x}}+7560 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}}-\frac {252 I \left (1+x\right )^2}{-10080 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}-2520 \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}+315 \left (1+x\right )^4 \sqrt {1-\frac {2}{1+x}}+5040 \sqrt {1-\frac {2}{1+x}}+7560 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}}-\frac {8 I \left (1+x\right )^4}{-10080 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}-2520 \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}+315 \left (1+x\right )^4 \sqrt {1-\frac {2}{1+x}}+5040 \sqrt {1-\frac {2}{1+x}}+7560 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}}+\frac {I 72 \left (1+x\right )^3}{-10080 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}-2520 \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}+315 \left (1+x\right )^4 \sqrt {1-\frac {2}{1+x}}+5040 \sqrt {1-\frac {2}{1+x}}+7560 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}}+\frac {I 420 \left (1+x\right )}{-10080 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}-2520 \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}+315 \left (1+x\right )^4 \sqrt {1-\frac {2}{1+x}}+5040 \sqrt {1-\frac {2}{1+x}}+7560 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/((1 - x)^(11/2)*(1 + x)^(1/2)),x]')

[Out]

Piecewise[{{(83 - 100 x + 84 x ^ 2 - 40 x ^ 3 + 8 x ^ 4) / (315 (1 - 4 x + 6 x ^ 2 - 4 x ^ 3 + x ^ 4) Sqrt[(1
- x) / (1 + x)]), 1 / Abs[1 + x] > 1 / 2}}, -315 I / (-10080 (1 + x) Sqrt[1 - 2 / (1 + x)] - 2520 (1 + x) ^ 3
Sqrt[1 - 2 / (1 + x)] + 315 (1 + x) ^ 4 Sqrt[1 - 2 / (1 + x)] + 5040 Sqrt[1 - 2 / (1 + x)] + 7560 (1 + x) ^ 2
Sqrt[1 - 2 / (1 + x)]) - 252 I (1 + x) ^ 2 / (-10080 (1 + x) Sqrt[1 - 2 / (1 + x)] - 2520 (1 + x) ^ 3 Sqrt[1 -
2 / (1 + x)] + 315 (1 + x) ^ 4 Sqrt[1 - 2 / (1 + x)] + 5040 Sqrt[1 - 2 / (1 + x)] + 7560 (1 + x) ^ 2 Sqrt[1 -
2 / (1 + x)]) - 8 I (1 + x) ^ 4 / (-10080 (1 + x) Sqrt[1 - 2 / (1 + x)] - 2520 (1 + x) ^ 3 Sqrt[1 - 2 / (1 +
x)] + 315 (1 + x) ^ 4 Sqrt[1 - 2 / (1 + x)] + 5040 Sqrt[1 - 2 / (1 + x)] + 7560 (1 + x) ^ 2 Sqrt[1 - 2 / (1 +
x)]) + I 72 (1 + x) ^ 3 / (-10080 (1 + x) Sqrt[1 - 2 / (1 + x)] - 2520 (1 + x) ^ 3 Sqrt[1 - 2 / (1 + x)] + 315
(1 + x) ^ 4 Sqrt[1 - 2 / (1 + x)] + 5040 Sqrt[1 - 2 / (1 + x)] + 7560 (1 + x) ^ 2 Sqrt[1 - 2 / (1 + x)]) + I
420 (1 + x) / (-10080 (1 + x) Sqrt[1 - 2 / (1 + x)] - 2520 (1 + x) ^ 3 Sqrt[1 - 2 / (1 + x)] + 315 (1 + x) ^ 4
Sqrt[1 - 2 / (1 + x)] + 5040 Sqrt[1 - 2 / (1 + x)] + 7560 (1 + x) ^ 2 Sqrt[1 - 2 / (1 + x)])]

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Maple [A]
time = 0.16, size = 72, normalized size = 0.71

method	result	size
gosper	\(\frac {\sqrt {1+x}\, \left (8 x^{4}-40 x^{3}+84 x^{2}-100 x +83\right )}{315 \left (1-x \right )^{\frac {9}{2}}}\)	\(35\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{5}-32 x^{4}+44 x^{3}-16 x^{2}-17 x +83\right )}{315 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{4} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(66\)
default	\(\frac {\sqrt {1+x}}{9 \left (1-x \right )^{\frac {9}{2}}}+\frac {4 \sqrt {1+x}}{63 \left (1-x \right )^{\frac {7}{2}}}+\frac {4 \sqrt {1+x}}{105 \left (1-x \right )^{\frac {5}{2}}}+\frac {8 \sqrt {1+x}}{315 \left (1-x \right )^{\frac {3}{2}}}+\frac {8 \sqrt {1+x}}{315 \sqrt {1-x}}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(11/2)/(1+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/9*(1+x)^(1/2)/(1-x)^(9/2)+4/63*(1+x)^(1/2)/(1-x)^(7/2)+4/105*(1+x)^(1/2)/(1-x)^(5/2)+8/315*(1+x)^(1/2)/(1-x)
^(3/2)+8/315*(1+x)^(1/2)/(1-x)^(1/2)

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Maxima [A]
time = 0.35, size = 131, normalized size = 1.30 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{9 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac {4 \, \sqrt {-x^{2} + 1}}{63 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac {4 \, \sqrt {-x^{2} + 1}}{105 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {8 \, \sqrt {-x^{2} + 1}}{315 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {8 \, \sqrt {-x^{2} + 1}}{315 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(11/2)/(1+x)^(1/2),x, algorithm="maxima")

[Out]

-1/9*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) + 4/63*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x
+ 1) - 4/105*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) + 8/315*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) - 8/315*sqrt(-x^2
+ 1)/(x - 1)

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Fricas [A]
time = 0.30, size = 86, normalized size = 0.85 \begin {gather*} \frac {83 \, x^{5} - 415 \, x^{4} + 830 \, x^{3} - 830 \, x^{2} - {\left (8 \, x^{4} - 40 \, x^{3} + 84 \, x^{2} - 100 \, x + 83\right )} \sqrt {x + 1} \sqrt {-x + 1} + 415 \, x - 83}{315 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(11/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

1/315*(83*x^5 - 415*x^4 + 830*x^3 - 830*x^2 - (8*x^4 - 40*x^3 + 84*x^2 - 100*x + 83)*sqrt(x + 1)*sqrt(-x + 1)
+ 415*x - 83)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1)

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Sympy [C] Result contains complex when optimal does not.
time = 57.54, size = 850, normalized size = 8.42

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)**(11/2)/(1+x)**(1/2),x)

[Out]

Piecewise((8*(x + 1)**4/(315*sqrt(-1 + 2/(x + 1))*(x + 1)**4 - 2520*sqrt(-1 + 2/(x + 1))*(x + 1)**3 + 7560*sqr
t(-1 + 2/(x + 1))*(x + 1)**2 - 10080*sqrt(-1 + 2/(x + 1))*(x + 1) + 5040*sqrt(-1 + 2/(x + 1))) - 72*(x + 1)**3
/(315*sqrt(-1 + 2/(x + 1))*(x + 1)**4 - 2520*sqrt(-1 + 2/(x + 1))*(x + 1)**3 + 7560*sqrt(-1 + 2/(x + 1))*(x +
1)**2 - 10080*sqrt(-1 + 2/(x + 1))*(x + 1) + 5040*sqrt(-1 + 2/(x + 1))) + 252*(x + 1)**2/(315*sqrt(-1 + 2/(x +
1))*(x + 1)**4 - 2520*sqrt(-1 + 2/(x + 1))*(x + 1)**3 + 7560*sqrt(-1 + 2/(x + 1))*(x + 1)**2 - 10080*sqrt(-1
+ 2/(x + 1))*(x + 1) + 5040*sqrt(-1 + 2/(x + 1))) - 420*(x + 1)/(315*sqrt(-1 + 2/(x + 1))*(x + 1)**4 - 2520*sq
rt(-1 + 2/(x + 1))*(x + 1)**3 + 7560*sqrt(-1 + 2/(x + 1))*(x + 1)**2 - 10080*sqrt(-1 + 2/(x + 1))*(x + 1) + 50
40*sqrt(-1 + 2/(x + 1))) + 315/(315*sqrt(-1 + 2/(x + 1))*(x + 1)**4 - 2520*sqrt(-1 + 2/(x + 1))*(x + 1)**3 + 7
560*sqrt(-1 + 2/(x + 1))*(x + 1)**2 - 10080*sqrt(-1 + 2/(x + 1))*(x + 1) + 5040*sqrt(-1 + 2/(x + 1))), 1/Abs(x
+ 1) > 1/2), (-8*I*(x + 1)**4/(315*sqrt(1 - 2/(x + 1))*(x + 1)**4 - 2520*sqrt(1 - 2/(x + 1))*(x + 1)**3 + 756
0*sqrt(1 - 2/(x + 1))*(x + 1)**2 - 10080*sqrt(1 - 2/(x + 1))*(x + 1) + 5040*sqrt(1 - 2/(x + 1))) + 72*I*(x + 1
)**3/(315*sqrt(1 - 2/(x + 1))*(x + 1)**4 - 2520*sqrt(1 - 2/(x + 1))*(x + 1)**3 + 7560*sqrt(1 - 2/(x + 1))*(x +
1)**2 - 10080*sqrt(1 - 2/(x + 1))*(x + 1) + 5040*sqrt(1 - 2/(x + 1))) - 252*I*(x + 1)**2/(315*sqrt(1 - 2/(x +
1))*(x + 1)**4 - 2520*sqrt(1 - 2/(x + 1))*(x + 1)**3 + 7560*sqrt(1 - 2/(x + 1))*(x + 1)**2 - 10080*sqrt(1 - 2
/(x + 1))*(x + 1) + 5040*sqrt(1 - 2/(x + 1))) + 420*I*(x + 1)/(315*sqrt(1 - 2/(x + 1))*(x + 1)**4 - 2520*sqrt(
1 - 2/(x + 1))*(x + 1)**3 + 7560*sqrt(1 - 2/(x + 1))*(x + 1)**2 - 10080*sqrt(1 - 2/(x + 1))*(x + 1) + 5040*sqr
t(1 - 2/(x + 1))) - 315*I/(315*sqrt(1 - 2/(x + 1))*(x + 1)**4 - 2520*sqrt(1 - 2/(x + 1))*(x + 1)**3 + 7560*sqr
t(1 - 2/(x + 1))*(x + 1)**2 - 10080*sqrt(1 - 2/(x + 1))*(x + 1) + 5040*sqrt(1 - 2/(x + 1))), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (71) = 142\).
time = 0.02, size = 366, normalized size = 3.62 \begin {gather*} 2 \left (\frac {\frac {1}{9}\cdot 5192296858534827628530496329220096 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{9}+\frac {1}{7}\cdot 46730671726813448656774466962980864 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{7}+\frac {1}{5}\cdot 186922686907253794627097867851923456 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{5}+145384312038975173598853897218162688 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{3}-\frac {327114702087694140597421268740866048 \left (-2 \sqrt {x+1}+2 \sqrt {2}\right )}{\sqrt {-x+1}}}{85070591730234615865843651857942052864}+\frac {-39690 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{8}-8820 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{6}-2268 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{4}-405 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{2}-35}{5160960 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{9}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(11/2)/(1+x)^(1/2),x)

[Out]

-1/73728*(sqrt(2) - sqrt(x + 1))^9/(-x + 1)^(9/2) - 9/57344*(sqrt(2) - sqrt(x + 1))^7/(-x + 1)^(7/2) - 9/10240
*(sqrt(2) - sqrt(x + 1))^5/(-x + 1)^(5/2) - 7/2048*(sqrt(2) - sqrt(x + 1))^3/(-x + 1)^(3/2) - 63/4096*(sqrt(2)
- sqrt(x + 1))/sqrt(-x + 1) + 1/2580480*(39690*(sqrt(2) - sqrt(x + 1))^8/(x - 1)^4 - 8820*(sqrt(2) - sqrt(x +
1))^6/(x - 1)^3 + 2268*(sqrt(2) - sqrt(x + 1))^4/(x - 1)^2 - 405*(sqrt(2) - sqrt(x + 1))^2/(x - 1) + 35)*(-x
+ 1)^(9/2)/(sqrt(2) - sqrt(x + 1))^9

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Mupad [B]
time = 0.36, size = 80, normalized size = 0.79 \begin {gather*} \frac {17\,x\,\sqrt {1-x}-83\,\sqrt {1-x}+16\,x^2\,\sqrt {1-x}-44\,x^3\,\sqrt {1-x}+32\,x^4\,\sqrt {1-x}-8\,x^5\,\sqrt {1-x}}{315\,{\left (x-1\right )}^5\,\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x)^(11/2)*(x + 1)^(1/2)),x)

[Out]

(17*x*(1 - x)^(1/2) - 83*(1 - x)^(1/2) + 16*x^2*(1 - x)^(1/2) - 44*x^3*(1 - x)^(1/2) + 32*x^4*(1 - x)^(1/2) -
8*x^5*(1 - x)^(1/2))/(315*(x - 1)^5*(x + 1)^(1/2))

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