3.7.38 \(\int \frac {1}{\sqrt {x} (2-b x)^{3/2}} \, dx\)

Optimal. Leaf size=16 \[ \frac {\sqrt {x}}{\sqrt {2-b x}} \]

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Rubi [A]  time = 0.00, antiderivative size = 16, 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 {x}}{\sqrt {2-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(2 - b*x)^(3/2)),x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x}}{\sqrt {2-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[x]/Sqrt[2 - b*x]

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IntegrateAlgebraic [A]  time = 0.03, size = 16, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x}}{\sqrt {2-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[x]/Sqrt[2 - b*x]

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fricas [A]  time = 1.16, size = 20, normalized size = 1.25 \begin {gather*} -\frac {\sqrt {-b x + 2} \sqrt {x}}{b x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-b*x + 2)*sqrt(x)/(b*x - 2)

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giac [B]  time = 1.03, size = 50, normalized size = 3.12 \begin {gather*} -\frac {4 \, \sqrt {-b} b}{{\left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*sqrt(-b)*b/(((sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2 - 2*b)*abs(b))

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maple [A]  time = 0.00, size = 13, normalized size = 0.81 \begin {gather*} \frac {\sqrt {x}}{\sqrt {-b x +2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.27, size = 12, normalized size = 0.75 \begin {gather*} \frac {\sqrt {x}}{\sqrt {-b x + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x)/sqrt(-b*x + 2)

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mupad [B]  time = 0.30, size = 12, normalized size = 0.75 \begin {gather*} \frac {\sqrt {x}}{\sqrt {2-b\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(2 - b*x)^(3/2)),x)

[Out]

x^(1/2)/(2 - b*x)^(1/2)

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sympy [A]  time = 0.93, size = 39, normalized size = 2.44 \begin {gather*} \begin {cases} \frac {1}{\sqrt {b} \sqrt {-1 + \frac {2}{b x}}} & \text {for}\: \frac {2}{\left |{b x}\right |} > 1 \\- \frac {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/(-b*x+2)**(3/2)/x**(1/2),x)

[Out]

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

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