3.7.19 \(\int \frac {1}{\sqrt {x} (2+b x)^{3/2}} \, dx\) [619]

Optimal. Leaf size=15 \[ \frac {\sqrt {x}}{\sqrt {2+b x}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 15, 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 {\sqrt {x}}{\sqrt {b x+2}} \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 = 0.01, size = 15, 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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Mathics [A]
time = 1.96, size = 16, normalized size = 1.07 \begin {gather*} \frac {1}{\sqrt {b} \sqrt {1+\frac {2}{b x}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/(Sqrt[x]*(2 + b*x)^(3/2)),x]')

[Out]

1 / (Sqrt[b] Sqrt[1 + 2 / (b x)])

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Maple [A]
time = 0.13, size = 12, normalized size = 0.80

method	result	size
gosper	\(\frac {\sqrt {x}}{\sqrt {b x +2}}\)	\(12\)
default	\(\frac {\sqrt {x}}{\sqrt {b x +2}}\)	\(12\)
meijerg	\(\frac {\sqrt {x}\, \sqrt {2}}{2 \sqrt {\frac {b x}{2}+1}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 11, normalized size = 0.73 \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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Fricas [A]
time = 0.30, size = 11, normalized size = 0.73 \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="fricas")

[Out]

sqrt(x)/sqrt(b*x + 2)

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Sympy [A]
time = 0.47, size = 15, normalized size = 1.00 \begin {gather*} \frac {1}{\sqrt {b} \sqrt {1 + \frac {2}{b x}}} \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]

1/(sqrt(b)*sqrt(1 + 2/(b*x)))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (11) = 22\).
time = 0.00, size = 53, normalized size = 3.53 \begin {gather*} \frac {8 b \sqrt {b}}{2 \left |b\right | \left (\left (\sqrt {b \left (b x+2\right )-2 b}-\sqrt {b} \sqrt {b x+2}\right )^{2}+2 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)

[Out]

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

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Mupad [B]
time = 0.31, size = 11, normalized size = 0.73 \begin {gather*} \frac {\sqrt {x}}{\sqrt {b\,x+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(b*x + 2)^(3/2)),x)

[Out]

x^(1/2)/(b*x + 2)^(1/2)

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