3.9.32 \(\int \frac {\sqrt {1+x}}{(-1+x)^{5/2}} \, dx\) [832]

Optimal. Leaf size=18 \[ -\frac {(1+x)^{3/2}}{3 (-1+x)^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \begin {gather*} -\frac {(x+1)^{3/2}}{3 (x-1)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x]/(-1 + x)^(5/2),x]

[Out]

-1/3*(1 + x)^(3/2)/(-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]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} -\frac {(1+x)^{3/2}}{3 (-1+x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(1 + x)^(3/2)/(-1 + x)^(3/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(25\) vs. \(2(12)=24\).
time = 0.09, size = 26, normalized size = 1.44

method	result	size
gosper	\(-\frac {\left (1+x \right )^{\frac {3}{2}}}{3 \left (-1+x \right )^{\frac {3}{2}}}\)	\(13\)
risch	\(-\frac {x^{2}+2 x +1}{3 \left (-1+x \right )^{\frac {3}{2}} \sqrt {1+x}}\)	\(21\)
default	\(-\frac {2 \sqrt {1+x}}{3 \left (-1+x \right )^{\frac {3}{2}}}-\frac {\sqrt {1+x}}{3 \sqrt {-1+x}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/(-1+x)^(3/2)*(1+x)^(1/2)-1/3/(-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. 34 vs. \(2 (12) = 24\).
time = 0.29, size = 34, normalized size = 1.89 \begin {gather*} -\frac {2 \, \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+x)^(1/2)/(-1+x)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).
time = 0.70, size = 31, normalized size = 1.72 \begin {gather*} -\frac {{\left (x + 1\right )}^{\frac {3}{2}} \sqrt {x - 1} + x^{2} - 2 \, x + 1}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.96, size = 60, normalized size = 3.33 \begin {gather*} \begin {cases} - \frac {\left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {x - 1} \left (x + 1\right ) - 6 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {i \left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {1 - x} \left (x + 1\right ) - 6 \sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(x + 1)**(3/2)/(3*sqrt(x - 1)*(x + 1) - 6*sqrt(x - 1)), Abs(x + 1) > 2), (I*(x + 1)**(3/2)/(3*sqrt
(1 - x)*(x + 1) - 6*sqrt(1 - x)), True))

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Giac [A]
time = 1.50, size = 12, normalized size = 0.67 \begin {gather*} -\frac {{\left (x + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x - 1\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(x + 1)^(3/2)/(x - 1)^(3/2)

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Mupad [B]
time = 1.08, size = 27, normalized size = 1.50 \begin {gather*} -\frac {x\,\sqrt {x+1}+\sqrt {x+1}}{\left (3\,x-3\right )\,\sqrt {x-1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(1/2)/(x - 1)^(5/2),x)

[Out]

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

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