∫ √ ___ 1 + x (1−x)11∕2 dx [1073]

3.11.73 \(\int \frac {\sqrt {1+x}}{(1-x)^{11/2}} \, dx\) [1073]

Optimal. Leaf size=81 \[ \frac {(1+x)^{3/2}}{9 (1-x)^{9/2}}+\frac {(1+x)^{3/2}}{21 (1-x)^{7/2}}+\frac {2 (1+x)^{3/2}}{105 (1-x)^{5/2}}+\frac {2 (1+x)^{3/2}}{315 (1-x)^{3/2}} \]

[Out]

1/9*(1+x)^(3/2)/(1-x)^(9/2)+1/21*(1+x)^(3/2)/(1-x)^(7/2)+2/105*(1+x)^(3/2)/(1-x)^(5/2)+2/315*(1+x)^(3/2)/(1-x)

^(3/2)

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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 = {47, 37} \begin {gather*} \frac {2 (x+1)^{3/2}}{315 (1-x)^{3/2}}+\frac {2 (x+1)^{3/2}}{105 (1-x)^{5/2}}+\frac {(x+1)^{3/2}}{21 (1-x)^{7/2}}+\frac {(x+1)^{3/2}}{9 (1-x)^{9/2}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(1 + x)^(3/2)/(9*(1 - x)^(9/2)) + (1 + x)^(3/2)/(21*(1 - x)^(7/2)) + (2*(1 + x)^(3/2))/(105*(1 - x)^(5/2)) + (

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

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Mathematica [A]
time = 0.05, size = 35, normalized size = 0.43 \begin {gather*} \frac {(1+x)^{3/2} \left (58-33 x+12 x^2-2 x^3\right )}{315 (1-x)^{9/2}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((1 + x)^(3/2)*(58 - 33*x + 12*x^2 - 2*x^3))/(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 = 43.37, size = 867, normalized size = 10.70 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-58-25 x+21 x^2-10 x^3+2 x^4\right ) \sqrt {1+x}}{315 \sqrt {-1+x} \left (1-4 x+6 x^2-4 x^3+x^4\right )},\text {Abs}\left [1+x\right ]>2\right \}\right \},\frac {-1530 \left (1+x\right )^{\frac {7}{2}}}{-211680 \left (1+x\right )^2 \sqrt {1-x}-88200 \left (1+x\right )^4 \sqrt {1-x}-40320 \sqrt {1-x}-4410 \left (1+x\right )^6 \sqrt {1-x}+315 \left (1+x\right )^7 \sqrt {1-x}+26460 \left (1+x\right )^5 \sqrt {1-x}+141120 \left (1+x\right ) \sqrt {1-x}+176400 \left (1+x\right )^3 \sqrt {1-x}}-\frac {840 \left (1+x\right )^{\frac {3}{2}}}{-211680 \left (1+x\right )^2 \sqrt {1-x}-88200 \left (1+x\right )^4 \sqrt {1-x}-40320 \sqrt {1-x}-4410 \left (1+x\right )^6 \sqrt {1-x}+315 \left (1+x\right )^7 \sqrt {1-x}+26460 \left (1+x\right )^5 \sqrt {1-x}+141120 \left (1+x\right ) \sqrt {1-x}+176400 \left (1+x\right )^3 \sqrt {1-x}}-\frac {195 \left (1+x\right )^{\frac {11}{2}}}{-211680 \left (1+x\right )^2 \sqrt {1-x}-88200 \left (1+x\right )^4 \sqrt {1-x}-40320 \sqrt {1-x}-4410 \left (1+x\right )^6 \sqrt {1-x}+315 \left (1+x\right )^7 \sqrt {1-x}+26460 \left (1+x\right )^5 \sqrt {1-x}+141120 \left (1+x\right ) \sqrt {1-x}+176400 \left (1+x\right )^3 \sqrt {1-x}}-\frac {2 \left (1+x\right )^{\frac {15}{2}}}{-211680 \left (1+x\right )^2 \sqrt {1-x}-88200 \left (1+x\right )^4 \sqrt {1-x}-40320 \sqrt {1-x}-4410 \left (1+x\right )^6 \sqrt {1-x}+315 \left (1+x\right )^7 \sqrt {1-x}+26460 \left (1+x\right )^5 \sqrt {1-x}+141120 \left (1+x\right ) \sqrt {1-x}+176400 \left (1+x\right )^3 \sqrt {1-x}}+\frac {30 \left (1+x\right )^{\frac {13}{2}}}{-211680 \left (1+x\right )^2 \sqrt {1-x}-88200 \left (1+x\right )^4 \sqrt {1-x}-40320 \sqrt {1-x}-4410 \left (1+x\right )^6 \sqrt {1-x}+315 \left (1+x\right )^7 \sqrt {1-x}+26460 \left (1+x\right )^5 \sqrt {1-x}+141120 \left (1+x\right ) \sqrt {1-x}+176400 \left (1+x\right )^3 \sqrt {1-x}}+\frac {715 \left (1+x\right )^{\frac {9}{2}}}{-211680 \left (1+x\right )^2 \sqrt {1-x}-88200 \left (1+x\right )^4 \sqrt {1-x}-40320 \sqrt {1-x}-4410 \left (1+x\right )^6 \sqrt {1-x}+315 \left (1+x\right )^7 \sqrt {1-x}+26460 \left (1+x\right )^5 \sqrt {1-x}+141120 \left (1+x\right ) \sqrt {1-x}+176400 \left (1+x\right )^3 \sqrt {1-x}}+\frac {1764 \left (1+x\right )^{\frac {5}{2}}}{-211680 \left (1+x\right )^2 \sqrt {1-x}-88200 \left (1+x\right )^4 \sqrt {1-x}-40320 \sqrt {1-x}-4410 \left (1+x\right )^6 \sqrt {1-x}+315 \left (1+x\right )^7 \sqrt {1-x}+26460 \left (1+x\right )^5 \sqrt {1-x}+141120 \left (1+x\right ) \sqrt {1-x}+176400 \left (1+x\right )^3 \sqrt {1-x}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{I / 315 (-58 - 25 x + 21 x ^ 2 - 10 x ^ 3 + 2 x ^ 4) Sqrt[1 + x] / (Sqrt[-1 + x] (1 - 4 x + 6 x ^

2 - 4 x ^ 3 + x ^ 4)), Abs[1 + x] > 2}}, -1530 (1 + x) ^ (7 / 2) / (-211680 (1 + x) ^ 2 Sqrt[1 - x] - 88200 (1

+ x) ^ 4 Sqrt[1 - x] - 40320 Sqrt[1 - x] - 4410 (1 + x) ^ 6 Sqrt[1 - x] + 315 (1 + x) ^ 7 Sqrt[1 - x] + 26460

(1 + x) ^ 5 Sqrt[1 - x] + 141120 (1 + x) Sqrt[1 - x] + 176400 (1 + x) ^ 3 Sqrt[1 - x]) - 840 (1 + x) ^ (3 / 2

) / (-211680 (1 + x) ^ 2 Sqrt[1 - x] - 88200 (1 + x) ^ 4 Sqrt[1 - x] - 40320 Sqrt[1 - x] - 4410 (1 + x) ^ 6 Sq

rt[1 - x] + 315 (1 + x) ^ 7 Sqrt[1 - x] + 26460 (1 + x) ^ 5 Sqrt[1 - x] + 141120 (1 + x) Sqrt[1 - x] + 176400

(1 + x) ^ 3 Sqrt[1 - x]) - 195 (1 + x) ^ (11 / 2) / (-211680 (1 + x) ^ 2 Sqrt[1 - x] - 88200 (1 + x) ^ 4 Sqrt[

1 - x] - 40320 Sqrt[1 - x] - 4410 (1 + x) ^ 6 Sqrt[1 - x] + 315 (1 + x) ^ 7 Sqrt[1 - x] + 26460 (1 + x) ^ 5 Sq

rt[1 - x] + 141120 (1 + x) Sqrt[1 - x] + 176400 (1 + x) ^ 3 Sqrt[1 - x]) - 2 (1 + x) ^ (15 / 2) / (-211680 (1

+ x) ^ 2 Sqrt[1 - x] - 88200 (1 + x) ^ 4 Sqrt[1 - x] - 40320 Sqrt[1 - x] - 4410 (1 + x) ^ 6 Sqrt[1 - x] + 315

(1 + x) ^ 7 Sqrt[1 - x] + 26460 (1 + x) ^ 5 Sqrt[1 - x] + 141120 (1 + x) Sqrt[1 - x] + 176400 (1 + x) ^ 3 Sqrt

[1 - x]) + 30 (1 + x) ^ (13 / 2) / (-211680 (1 + x) ^ 2 Sqrt[1 - x] - 88200 (1 + x) ^ 4 Sqrt[1 - x] - 40320 Sq

rt[1 - x] - 4410 (1 + x) ^ 6 Sqrt[1 - x] + 315 (1 + x) ^ 7 Sqrt[1 - x] + 26460 (1 + x) ^ 5 Sqrt[1 - x] + 14112

0 (1 + x) Sqrt[1 - x] + 176400 (1 + x) ^ 3 Sqrt[1 - x]) + 715 (1 + x) ^ (9 / 2) / (-211680 (1 + x) ^ 2 Sqrt[1

- x] - 88200 (1 + x) ^ 4 Sqrt[1 - x] - 40320 Sqrt[1 - x] - 4410 (1 + x) ^ 6 Sqrt[1 - x] + 315 (1 + x) ^ 7 Sqrt

[1 - x] + 26460 (1 + x) ^ 5 Sqrt[1 - x] + 141120 (1 + x) Sqrt[1 - x] + 176400 (1 + x) ^ 3 Sqrt[1 - x]) + 1764

(1 + x) ^ (5 / 2) / (-211680 (1 + x) ^ 2 Sqrt[1 - x] - 88200 (1 + x) ^ 4 Sqrt[1 - x] - 40320 Sqrt[1 - x] - 441

0 (1 + x) ^ 6 Sqrt[1 - x] + 315 (1 + x) ^ 7 Sqrt[1 - x] + 26460 (1 + x) ^ 5 Sqrt[1 - x] + 141120 (1 + x) Sqrt[

1 - x] + 176400 (1 + x) ^ 3 Sqrt[1 - x])]

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Maple [A]
time = 0.17, size = 72, normalized size = 0.89


method	result	size

gosper	\(-\frac {\left (1+x \right )^{\frac {3}{2}} \left (2 x^{3}-12 x^{2}+33 x -58\right )}{315 \left (1-x \right )^{\frac {9}{2}}}\)	\(30\)
risch	\(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{5}-8 x^{4}+11 x^{3}-4 x^{2}-83 x -58\right )}{315 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{4} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(66\)
default	\(\frac {2 \sqrt {1+x}}{9 \left (1-x \right )^{\frac {9}{2}}}-\frac {\sqrt {1+x}}{63 \left (1-x \right )^{\frac {7}{2}}}-\frac {\sqrt {1+x}}{105 \left (1-x \right )^{\frac {5}{2}}}-\frac {2 \sqrt {1+x}}{315 \left (1-x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1+x}}{315 \sqrt {1-x}}\)	\(72\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

^(3/2)-2/315*(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. 131 vs. \(2 (57) = 114\).
time = 0.26, size = 131, normalized size = 1.62 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} + 1}}{9 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{63 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{105 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{315 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{315 \, {\left (x - 1\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

+ 1) + 1/105*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 2/315*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 2/315*sqrt(-x^2

+ 1)/(x - 1)

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Fricas [A]
time = 0.29, size = 85, normalized size = 1.05 \begin {gather*} \frac {58 \, x^{5} - 290 \, x^{4} + 580 \, x^{3} - 580 \, x^{2} + {\left (2 \, x^{4} - 10 \, x^{3} + 21 \, x^{2} - 25 \, x - 58\right )} \sqrt {x + 1} \sqrt {-x + 1} + 290 \, x - 58}{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+x)^(1/2)/(1-x)^(11/2),x, algorithm="fricas")

[Out]

1/315*(58*x^5 - 290*x^4 + 580*x^3 - 580*x^2 + (2*x^4 - 10*x^3 + 21*x^2 - 25*x - 58)*sqrt(x + 1)*sqrt(-x + 1) +

290*x - 58)/(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 = 61.44, size = 1561, normalized size = 19.27

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*I*(x + 1)**(15/2)/(315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x - 1)*(x + 1)**6 + 26460*sqrt(x - 1)*(

x + 1)**5 - 88200*sqrt(x - 1)*(x + 1)**4 + 176400*sqrt(x - 1)*(x + 1)**3 - 211680*sqrt(x - 1)*(x + 1)**2 + 141

120*sqrt(x - 1)*(x + 1) - 40320*sqrt(x - 1)) - 30*I*(x + 1)**(13/2)/(315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x

- 1)*(x + 1)**6 + 26460*sqrt(x - 1)*(x + 1)**5 - 88200*sqrt(x - 1)*(x + 1)**4 + 176400*sqrt(x - 1)*(x + 1)**3

- 211680*sqrt(x - 1)*(x + 1)**2 + 141120*sqrt(x - 1)*(x + 1) - 40320*sqrt(x - 1)) + 195*I*(x + 1)**(11/2)/(315

*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x - 1)*(x + 1)**6 + 26460*sqrt(x - 1)*(x + 1)**5 - 88200*sqrt(x - 1)*(x +

1)**4 + 176400*sqrt(x - 1)*(x + 1)**3 - 211680*sqrt(x - 1)*(x + 1)**2 + 141120*sqrt(x - 1)*(x + 1) - 40320*sqr

t(x - 1)) - 715*I*(x + 1)**(9/2)/(315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x - 1)*(x + 1)**6 + 26460*sqrt(x - 1)

*(x + 1)**5 - 88200*sqrt(x - 1)*(x + 1)**4 + 176400*sqrt(x - 1)*(x + 1)**3 - 211680*sqrt(x - 1)*(x + 1)**2 + 1

41120*sqrt(x - 1)*(x + 1) - 40320*sqrt(x - 1)) + 1530*I*(x + 1)**(7/2)/(315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt

(x - 1)*(x + 1)**6 + 26460*sqrt(x - 1)*(x + 1)**5 - 88200*sqrt(x - 1)*(x + 1)**4 + 176400*sqrt(x - 1)*(x + 1)*

*3 - 211680*sqrt(x - 1)*(x + 1)**2 + 141120*sqrt(x - 1)*(x + 1) - 40320*sqrt(x - 1)) - 1764*I*(x + 1)**(5/2)/(

315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x - 1)*(x + 1)**6 + 26460*sqrt(x - 1)*(x + 1)**5 - 88200*sqrt(x - 1)*(x

+ 1)**4 + 176400*sqrt(x - 1)*(x + 1)**3 - 211680*sqrt(x - 1)*(x + 1)**2 + 141120*sqrt(x - 1)*(x + 1) - 40320*

sqrt(x - 1)) + 840*I*(x + 1)**(3/2)/(315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x - 1)*(x + 1)**6 + 26460*sqrt(x -

1)*(x + 1)**5 - 88200*sqrt(x - 1)*(x + 1)**4 + 176400*sqrt(x - 1)*(x + 1)**3 - 211680*sqrt(x - 1)*(x + 1)**2

+ 141120*sqrt(x - 1)*(x + 1) - 40320*sqrt(x - 1)), Abs(x + 1) > 2), (-2*(x + 1)**(15/2)/(315*sqrt(1 - x)*(x +

1)**7 - 4410*sqrt(1 - x)*(x + 1)**6 + 26460*sqrt(1 - x)*(x + 1)**5 - 88200*sqrt(1 - x)*(x + 1)**4 + 176400*sqr

t(1 - x)*(x + 1)**3 - 211680*sqrt(1 - x)*(x + 1)**2 + 141120*sqrt(1 - x)*(x + 1) - 40320*sqrt(1 - x)) + 30*(x

+ 1)**(13/2)/(315*sqrt(1 - x)*(x + 1)**7 - 4410*sqrt(1 - x)*(x + 1)**6 + 26460*sqrt(1 - x)*(x + 1)**5 - 88200*

sqrt(1 - x)*(x + 1)**4 + 176400*sqrt(1 - x)*(x + 1)**3 - 211680*sqrt(1 - x)*(x + 1)**2 + 141120*sqrt(1 - x)*(x

+ 1) - 40320*sqrt(1 - x)) - 195*(x + 1)**(11/2)/(315*sqrt(1 - x)*(x + 1)**7 - 4410*sqrt(1 - x)*(x + 1)**6 + 2

6460*sqrt(1 - x)*(x + 1)**5 - 88200*sqrt(1 - x)*(x + 1)**4 + 176400*sqrt(1 - x)*(x + 1)**3 - 211680*sqrt(1 - x

)*(x + 1)**2 + 141120*sqrt(1 - x)*(x + 1) - 40320*sqrt(1 - x)) + 715*(x + 1)**(9/2)/(315*sqrt(1 - x)*(x + 1)**

7 - 4410*sqrt(1 - x)*(x + 1)**6 + 26460*sqrt(1 - x)*(x + 1)**5 - 88200*sqrt(1 - x)*(x + 1)**4 + 176400*sqrt(1

- x)*(x + 1)**3 - 211680*sqrt(1 - x)*(x + 1)**2 + 141120*sqrt(1 - x)*(x + 1) - 40320*sqrt(1 - x)) - 1530*(x +

1)**(7/2)/(315*sqrt(1 - x)*(x + 1)**7 - 4410*sqrt(1 - x)*(x + 1)**6 + 26460*sqrt(1 - x)*(x + 1)**5 - 88200*sqr

t(1 - x)*(x + 1)**4 + 176400*sqrt(1 - x)*(x + 1)**3 - 211680*sqrt(1 - x)*(x + 1)**2 + 141120*sqrt(1 - x)*(x +

1) - 40320*sqrt(1 - x)) + 1764*(x + 1)**(5/2)/(315*sqrt(1 - x)*(x + 1)**7 - 4410*sqrt(1 - x)*(x + 1)**6 + 2646

0*sqrt(1 - x)*(x + 1)**5 - 88200*sqrt(1 - x)*(x + 1)**4 + 176400*sqrt(1 - x)*(x + 1)**3 - 211680*sqrt(1 - x)*(

x + 1)**2 + 141120*sqrt(1 - x)*(x + 1) - 40320*sqrt(1 - x)) - 840*(x + 1)**(3/2)/(315*sqrt(1 - x)*(x + 1)**7 -

4410*sqrt(1 - x)*(x + 1)**6 + 26460*sqrt(1 - x)*(x + 1)**5 - 88200*sqrt(1 - x)*(x + 1)**4 + 176400*sqrt(1 - x

)*(x + 1)**3 - 211680*sqrt(1 - x)*(x + 1)**2 + 141120*sqrt(1 - x)*(x + 1) - 40320*sqrt(1 - x)), True))

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Giac [A]
time = 0.02, size = 96, normalized size = 1.19 \begin {gather*} \frac {2 \left (\left (\left (\frac 1{35}-\frac {1}{315} \sqrt {x+1} \sqrt {x+1}\right ) \sqrt {x+1} \sqrt {x+1}-\frac 1{10}\right ) \sqrt {x+1} \sqrt {x+1}+\frac 1{6}\right ) \sqrt {x+1} \sqrt {x+1} \sqrt {x+1} \sqrt {-x+1}}{\left (-x+1\right )^{5}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.28, size = 80, normalized size = 0.99 \begin {gather*} -\frac {\sqrt {1-x}\,\left (\frac {5\,x\,\sqrt {x+1}}{63}+\frac {58\,\sqrt {x+1}}{315}-\frac {x^2\,\sqrt {x+1}}{15}+\frac {2\,x^3\,\sqrt {x+1}}{63}-\frac {2\,x^4\,\sqrt {x+1}}{315}\right )}{x^5-5\,x^4+10\,x^3-10\,x^2+5\,x-1} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((1 - x)^(1/2)*((5*x*(x + 1)^(1/2))/63 + (58*(x + 1)^(1/2))/315 - (x^2*(x + 1)^(1/2))/15 + (2*x^3*(x + 1)^(1/

2))/63 - (2*x^4*(x + 1)^(1/2))/315))/(5*x - 10*x^2 + 10*x^3 - 5*x^4 + x^5 - 1)

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