3.7.12 \(\int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx\) [612]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, 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 {b x+2}}{\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 = 0.01, size = 16, 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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Mathics [A]
time = 1.97, size = 17, normalized size = 1.06 \begin {gather*} -\sqrt {b} \sqrt {1+\frac {2}{b x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-Sqrt[b] Sqrt[1 + 2 / (b x)]

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Maple [A]
time = 0.10, size = 13, normalized size = 0.81

method	result	size
gosper	\(-\frac {\sqrt {b x +2}}{\sqrt {x}}\)	\(13\)
default	\(-\frac {\sqrt {b x +2}}{\sqrt {x}}\)	\(13\)
risch	\(-\frac {\sqrt {b x +2}}{\sqrt {x}}\)	\(13\)
meijerg	\(-\frac {\sqrt {2}\, \sqrt {\frac {b x}{2}+1}}{\sqrt {x}}\)	\(17\)

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 = 0.25, size = 12, normalized size = 0.75 \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 = 0.31, size = 12, normalized size = 0.75 \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 [A]
time = 0.45, size = 15, normalized size = 0.94 \begin {gather*} - \sqrt {b} \sqrt {1 + \frac {2}{b 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)

[Out]

-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. 28 vs. \(2 (12) = 24\).
time = 0.00, size = 36, normalized size = 2.25 \begin {gather*} \frac {8 \sqrt {b}}{2 \left (\left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )^{2}-2\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)

[Out]

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

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Mupad [B]
time = 0.33, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {b\,x+2}}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2)*(b*x + 2)^(1/2)),x)

[Out]

-(b*x + 2)^(1/2)/x^(1/2)

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