3.12.12 \(\int \frac {1}{(1-x)^{5/2} \sqrt {1+x}} \, dx\) [1112]

Optimal. Leaf size=41 \[ \frac {\sqrt {1+x}}{3 (1-x)^{3/2}}+\frac {\sqrt {1+x}}{3 \sqrt {1-x}} \]

[Out]

1/3*(1+x)^(1/2)/(1-x)^(3/2)+1/3*(1+x)^(1/2)/(1-x)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, 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 {\sqrt {x+1}}{3 \sqrt {1-x}}+\frac {\sqrt {x+1}}{3 (1-x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(5/2)*Sqrt[1 + x]),x]

[Out]

Sqrt[1 + x]/(3*(1 - x)^(3/2)) + Sqrt[1 + x]/(3*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)^{5/2} \sqrt {1+x}} \, dx &=\frac {\sqrt {1+x}}{3 (1-x)^{3/2}}+\frac {1}{3} \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{3 (1-x)^{3/2}}+\frac {\sqrt {1+x}}{3 \sqrt {1-x}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 23, normalized size = 0.56 \begin {gather*} -\frac {(-2+x) \sqrt {1+x}}{3 (1-x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(5/2)*Sqrt[1 + x]),x]

[Out]

-1/3*((-2 + x)*Sqrt[1 + x])/(1 - x)^(3/2)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 3.87, size = 106, normalized size = 2.59 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {-2+x}{3 \left (-1+x\right ) \sqrt {\frac {1-x}{1+x}}},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},-\frac {I \left (1+x\right )}{-6 \sqrt {1-\frac {2}{1+x}}+3 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}+\frac {I 3}{-6 \sqrt {1-\frac {2}{1+x}}+3 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/((1 - x)^(5/2)*(1 + x)^(1/2)),x]')

[Out]

Piecewise[{{(-2 + x) / (3 (-1 + x) Sqrt[(1 - x) / (1 + x)]), 1 / Abs[1 + x] > 1 / 2}}, -I (1 + x) / (-6 Sqrt[1
- 2 / (1 + x)] + 3 (1 + x) Sqrt[1 - 2 / (1 + x)]) + I 3 / (-6 Sqrt[1 - 2 / (1 + x)] + 3 (1 + x) Sqrt[1 - 2 /
(1 + x)])]

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Maple [A]
time = 0.13, size = 30, normalized size = 0.73

method	result	size
gosper	\(-\frac {\sqrt {1+x}\, \left (-2+x \right )}{3 \left (1-x \right )^{\frac {3}{2}}}\)	\(18\)
default	\(\frac {\sqrt {1+x}}{3 \left (1-x \right )^{\frac {3}{2}}}+\frac {\sqrt {1+x}}{3 \sqrt {1-x}}\)	\(30\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{2}-x -2\right )}{3 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right ) \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(5/2)/(1+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(1+x)^(1/2)/(1-x)^(3/2)+1/3*(1+x)^(1/2)/(1-x)^(1/2)

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Maxima [A]
time = 0.35, size = 38, normalized size = 0.93 \begin {gather*} \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(5/2)/(1+x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.30, size = 39, normalized size = 0.95 \begin {gather*} \frac {2 \, x^{2} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} - 4 \, x + 2}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(5/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

1/3*(2*x^2 - sqrt(x + 1)*(x - 2)*sqrt(-x + 1) - 4*x + 2)/(x^2 - 2*x + 1)

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Sympy [A]
time = 1.33, size = 128, normalized size = 3.12 \begin {gather*} \begin {cases} \frac {x + 1}{3 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 6 \sqrt {-1 + \frac {2}{x + 1}}} - \frac {3}{3 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 6 \sqrt {-1 + \frac {2}{x + 1}}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {i \left (x + 1\right )}{3 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 6 \sqrt {1 - \frac {2}{x + 1}}} + \frac {3 i}{3 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 6 \sqrt {1 - \frac {2}{x + 1}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)**(5/2)/(1+x)**(1/2),x)

[Out]

Piecewise(((x + 1)/(3*sqrt(-1 + 2/(x + 1))*(x + 1) - 6*sqrt(-1 + 2/(x + 1))) - 3/(3*sqrt(-1 + 2/(x + 1))*(x +
1) - 6*sqrt(-1 + 2/(x + 1))), 1/Abs(x + 1) > 1/2), (-I*(x + 1)/(3*sqrt(1 - 2/(x + 1))*(x + 1) - 6*sqrt(1 - 2/(
x + 1))) + 3*I/(3*sqrt(1 - 2/(x + 1))*(x + 1) - 6*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. 87 vs. \(2 (29) = 58\).
time = 0.00, size = 149, normalized size = 3.63 \begin {gather*} -2 \left (\frac {-\frac {1024}{3} \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{3}+\frac {1536 \left (-2 \sqrt {x+1}+2 \sqrt {2}\right )}{\sqrt {-x+1}}}{32768}+\frac {9 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{2}+1}{96 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(5/2)/(1+x)^(1/2),x)

[Out]

-1/48*(sqrt(2) - sqrt(x + 1))^3/(-x + 1)^(3/2) - 3/16*(sqrt(2) - sqrt(x + 1))/sqrt(-x + 1) - 1/48*(-x + 1)^(3/
2)*(9*(sqrt(2) - sqrt(x + 1))^2/(x - 1) - 1)/(sqrt(2) - sqrt(x + 1))^3

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Mupad [B]
time = 0.31, size = 43, normalized size = 1.05 \begin {gather*} \frac {x\,\sqrt {1-x}+2\,\sqrt {1-x}-x^2\,\sqrt {1-x}}{3\,{\left (x-1\right )}^2\,\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x)^(5/2)*(x + 1)^(1/2)),x)

[Out]

(x*(1 - x)^(1/2) + 2*(1 - x)^(1/2) - x^2*(1 - x)^(1/2))/(3*(x - 1)^2*(x + 1)^(1/2))

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