3.7.33 \(\int \frac {1}{x^{3/2} \sqrt {2-b x}} \, dx\) [633]

Optimal. Leaf size=17 \[ -\frac {\sqrt {2-b x}}{\sqrt {x}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {37} \begin {gather*} -\frac {\sqrt {2-b x}}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*Sqrt[2 - b*x]),x]

[Out]

-(Sqrt[2 - b*x]/Sqrt[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]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 17, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2-b x}}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*Sqrt[2 - b*x]),x]

[Out]

-(Sqrt[2 - b*x]/Sqrt[x])

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Maple [A]
time = 0.11, size = 14, normalized size = 0.82

method	result	size
gosper	\(-\frac {\sqrt {-b x +2}}{\sqrt {x}}\)	\(14\)
default	\(-\frac {\sqrt {-b x +2}}{\sqrt {x}}\)	\(14\)
meijerg	\(-\frac {\sqrt {2}\, \sqrt {-\frac {b x}{2}+1}}{\sqrt {x}}\)	\(17\)
risch	\(\frac {\left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{\sqrt {-x \left (b x -2\right )}\, \sqrt {x}\, \sqrt {-b x +2}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(3/2)/(-b*x+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 13, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {-b x + 2}}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(-b*x+2)^(1/2),x, algorithm="maxima")

[Out]

-sqrt(-b*x + 2)/sqrt(x)

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Fricas [A]
time = 0.99, size = 13, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {-b x + 2}}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(-b*x+2)^(1/2),x, algorithm="fricas")

[Out]

-sqrt(-b*x + 2)/sqrt(x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.46, size = 41, normalized size = 2.41 \begin {gather*} \begin {cases} - \sqrt {b} \sqrt {-1 + \frac {2}{b x}} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\- i \sqrt {b} \sqrt {1 - \frac {2}{b x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(3/2)/(-b*x+2)**(1/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (13) = 26\).
time = 2.33, size = 30, normalized size = 1.76 \begin {gather*} -\frac {\sqrt {-b x + 2} b^{2}}{\sqrt {{\left (b x - 2\right )} b + 2 \, b} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(-b*x+2)^(1/2),x, algorithm="giac")

[Out]

-sqrt(-b*x + 2)*b^2/(sqrt((b*x - 2)*b + 2*b)*abs(b))

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Mupad [B]
time = 0.31, size = 13, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {2-b\,x}}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2)*(2 - b*x)^(1/2)),x)

[Out]

-(2 - b*x)^(1/2)/x^(1/2)

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